From 6fff2967dfd8d779d599a444163283f5ec8befe4 Mon Sep 17 00:00:00 2001 From: Tony Arcieri Date: Wed, 12 Feb 2025 11:53:23 -0700 Subject: [PATCH] Add proptests with round trip check - Ensures we can verify signatures we've produced - Ensures we can decode signatures we've encoded --- Cargo.lock | 1 + ml-dsa/Cargo.toml | 1 + ml-dsa/src/lib.rs | 25 +++++++-- ml-dsa/tests/key-gen.rs | 1 - ml-dsa/tests/pkcs8.rs | 3 +- ml-dsa/tests/proptests.proptest-regressions | 8 +++ ml-dsa/tests/proptests.rs | 56 +++++++++++++++++++++ 7 files changed, 89 insertions(+), 6 deletions(-) create mode 100644 ml-dsa/tests/proptests.proptest-regressions create mode 100644 ml-dsa/tests/proptests.rs diff --git a/Cargo.lock b/Cargo.lock index c4f2d006..6667fa03 100644 --- a/Cargo.lock +++ b/Cargo.lock @@ -690,6 +690,7 @@ dependencies = [ "hybrid-array", "num-traits", "pkcs8", + "proptest", "rand", "rand_core", "serde", diff --git a/ml-dsa/Cargo.toml b/ml-dsa/Cargo.toml index ebecf487..4ab3515f 100644 --- a/ml-dsa/Cargo.toml +++ b/ml-dsa/Cargo.toml @@ -37,6 +37,7 @@ criterion = "0.5.1" hex = { version = "0.4.3", features = ["serde"] } hex-literal = "0.4.1" pkcs8 = { version = "=0.11.0-rc.1", features = ["pem"] } +proptest = "1" rand = "0.8.5" serde = { version = "1.0.215", features = ["derive"] } serde_json = "1.0.132" diff --git a/ml-dsa/src/lib.rs b/ml-dsa/src/lib.rs index 96d856bf..e68d56a4 100644 --- a/ml-dsa/src/lib.rs +++ b/ml-dsa/src/lib.rs @@ -17,8 +17,7 @@ //! # Quickstart //! //! ``` -//! use ml_dsa::{MlDsa65, KeyGen}; -//! use signature::{Keypair, Signer, Verifier}; +//! use ml_dsa::{MlDsa65, KeyGen, signature::{Keypair, Signer, Verifier}}; //! //! let mut rng = rand::thread_rng(); //! let kp = MlDsa65::key_gen(&mut rng); @@ -83,10 +82,11 @@ use crate::ntt::{Ntt, NttInverse}; use crate::param::{ParameterSet, QMinus1, SamplingSize, SpecQ}; use crate::sampling::{expand_a, expand_mask, expand_s, sample_in_ball}; use crate::util::B64; +use core::fmt; pub use crate::param::{EncodedSignature, EncodedSigningKey, EncodedVerifyingKey, MlDsaParams}; pub use crate::util::B32; -pub use signature::Error; +pub use signature::{self, Error}; /// An ML-DSA signature #[derive(Clone, PartialEq, Debug)] @@ -192,6 +192,11 @@ impl KeyPair

{ pub fn signing_key(&self) -> &SigningKey

{ &self.signing_key } + + /// The verifying key of the key pair + pub fn verifying_key(&self) -> &VerifyingKey

{ + &self.verifying_key + } } impl AsRef> for KeyPair

{ @@ -200,6 +205,14 @@ impl AsRef> for KeyPair

{ } } +impl fmt::Debug for KeyPair

{ + fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result { + f.debug_struct("KeyPair") + .field("verifying_key", &self.verifying_key) + .finish_non_exhaustive() + } +} + impl signature::KeypairRef for KeyPair

{ type VerifyingKey = VerifyingKey

; } @@ -276,6 +289,12 @@ pub struct SigningKey { A_hat: NttMatrix, } +impl fmt::Debug for SigningKey

{ + fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result { + f.debug_struct("SigningKey").finish_non_exhaustive() + } +} + #[cfg(feature = "zeroize")] impl Drop for SigningKey

{ fn drop(&mut self) { diff --git a/ml-dsa/tests/key-gen.rs b/ml-dsa/tests/key-gen.rs index 5ac5d600..5cdc478a 100644 --- a/ml-dsa/tests/key-gen.rs +++ b/ml-dsa/tests/key-gen.rs @@ -1,7 +1,6 @@ use ml_dsa::*; use hybrid_array::Array; -use signature::Keypair; use std::{fs::read_to_string, path::PathBuf}; #[test] diff --git a/ml-dsa/tests/pkcs8.rs b/ml-dsa/tests/pkcs8.rs index ffc8474e..d6210e03 100644 --- a/ml-dsa/tests/pkcs8.rs +++ b/ml-dsa/tests/pkcs8.rs @@ -7,7 +7,6 @@ use pkcs8::{ spki::AssociatedAlgorithmIdentifier, DecodePrivateKey, DecodePublicKey, EncodePrivateKey, EncodePublicKey, }; -use signature::Keypair; #[test] fn private_key_serialization() { @@ -34,7 +33,7 @@ fn private_key_serialization() { public_bytes ); - assert_eq!(kp.verifying_key(), pk); + assert_eq!(kp.verifying_key(), &pk); } test_roundtrip::( diff --git a/ml-dsa/tests/proptests.proptest-regressions b/ml-dsa/tests/proptests.proptest-regressions new file mode 100644 index 00000000..9aaee664 --- /dev/null +++ b/ml-dsa/tests/proptests.proptest-regressions @@ -0,0 +1,8 @@ +# Seeds for failure cases proptest has generated in the past. It is +# automatically read and these particular cases re-run before any +# novel cases are generated. +# +# It is recommended to check this file in to source control so that +# everyone who runs the test benefits from these saved cases. +cc f6919c464ada8c5022a1cf1817b50e0bd6fd1c02aed7474bd50fc21b03ea4ae1 # shrinks to keypair = KeyPair { verifying_key: VerifyingKey { rho: Array([63, 82, 240, 28, 70, 19, 224, 10, 174, 246, 14, 214, 195, 159, 45, 88, 250, 187, 227, 223, 246, 243, 64, 182, 99, 210, 167, 115, 201, 132, 170, 243]), t1: Vector(Array([Polynomial(Array([Elem(418), Elem(532), Elem(466), Elem(162), Elem(1016), Elem(497), Elem(487), Elem(1014), Elem(605), Elem(413), Elem(197), Elem(470), Elem(209), Elem(126), Elem(304), Elem(1006), Elem(435), Elem(717), Elem(227), Elem(834), Elem(426), Elem(126), Elem(137), Elem(916), Elem(1022), Elem(46), Elem(155), Elem(135), Elem(244), Elem(337), Elem(103), Elem(667), Elem(164), Elem(890), Elem(481), Elem(774), Elem(236), Elem(412), Elem(979), 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Elem(946), Elem(922), Elem(103)])), Polynomial(Array([Elem(67), Elem(134), Elem(767), Elem(187), Elem(306), Elem(359), Elem(971), Elem(928), Elem(677), Elem(517), Elem(391), Elem(424), Elem(392), Elem(788), Elem(905), Elem(960), Elem(441), Elem(906), Elem(917), Elem(381), Elem(599), Elem(11), Elem(660), Elem(332), Elem(210), Elem(252), Elem(317), Elem(88), Elem(878), Elem(690), Elem(622), Elem(101), Elem(491), Elem(1007), Elem(537), Elem(307), Elem(379), Elem(718), Elem(902), Elem(238), Elem(267), Elem(13), Elem(501), Elem(512), Elem(904), Elem(162), Elem(44), Elem(268), Elem(176), Elem(504), Elem(367), Elem(122), Elem(759), Elem(498), Elem(583), Elem(518), Elem(469), Elem(93), Elem(978), Elem(340), Elem(1001), Elem(267), Elem(639), Elem(499), Elem(388), Elem(663), Elem(962), Elem(187), Elem(46), Elem(805), Elem(814), Elem(822), Elem(376), Elem(530), Elem(913), Elem(729), Elem(0), Elem(308), Elem(478), Elem(843), Elem(816), Elem(942), Elem(80), Elem(988), Elem(871), Elem(642), 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Elem(346), Elem(28), Elem(206), Elem(654), Elem(235), Elem(952), Elem(288), Elem(1022), Elem(43), Elem(756), Elem(295), Elem(841), Elem(396), Elem(188), Elem(43), Elem(598), Elem(162), Elem(51), Elem(176), Elem(245), Elem(573), Elem(761), Elem(696), Elem(612), Elem(293), Elem(188), Elem(99), Elem(74), Elem(579), Elem(465), Elem(1016), Elem(61), Elem(624), Elem(404), Elem(102), Elem(417), Elem(449), Elem(295), Elem(416), Elem(866), Elem(197), Elem(366), Elem(901), Elem(564), Elem(271), Elem(761), Elem(986), Elem(800), Elem(399), Elem(691), Elem(272), Elem(9), Elem(334), Elem(959), Elem(83), Elem(550), Elem(62), Elem(137), Elem(365), Elem(498), Elem(441), Elem(183), Elem(649), Elem(486), Elem(291), Elem(569), Elem(375), Elem(473), Elem(327), Elem(136), Elem(934), Elem(994), Elem(398), Elem(92), Elem(306), Elem(844), Elem(700), Elem(979), Elem(809), Elem(543), Elem(909), Elem(926), Elem(50), Elem(771), Elem(1006), Elem(337), Elem(360), Elem(254), Elem(502), Elem(331), Elem(22), Elem(885), Elem(457), Elem(416), Elem(815), Elem(891), Elem(511), Elem(731), Elem(979), Elem(920), Elem(890), Elem(861), Elem(460), Elem(801), Elem(850), Elem(431), Elem(400), Elem(984), Elem(715), Elem(356), Elem(164), Elem(988), Elem(245), Elem(423), Elem(892), Elem(469), Elem(433), Elem(667), Elem(636), Elem(990), Elem(407), Elem(367), Elem(506), Elem(546), Elem(982), Elem(348), Elem(588), Elem(390), Elem(86), Elem(203), Elem(169), Elem(831), Elem(652), Elem(85), Elem(245), Elem(263), Elem(96), Elem(84), Elem(208), Elem(789), Elem(111), Elem(357), Elem(146), Elem(732), Elem(66), Elem(582), Elem(670), Elem(25), Elem(666), Elem(677), Elem(636), Elem(203), Elem(206), Elem(410), Elem(217), Elem(211), Elem(630), Elem(191), Elem(265), Elem(895), Elem(910), Elem(1003), Elem(195), Elem(649), Elem(199), Elem(619), Elem(545), Elem(1022), Elem(941), Elem(631), Elem(762), Elem(765), Elem(744), Elem(259), Elem(83), Elem(432), Elem(979), Elem(972), Elem(512), Elem(47), Elem(301), Elem(360), Elem(991), Elem(10), Elem(632), Elem(878), Elem(284), Elem(555), Elem(674), Elem(908), Elem(684), Elem(636), Elem(504), Elem(623), Elem(549), Elem(536), Elem(688), Elem(543), Elem(883), Elem(73), Elem(149), Elem(947), Elem(885), Elem(981), Elem(386), Elem(177), Elem(606), Elem(824), Elem(560), Elem(620), Elem(335), Elem(99), Elem(684), Elem(437), Elem(384), Elem(517), Elem(695), Elem(219), Elem(169), Elem(847), Elem(310), Elem(184), Elem(756), Elem(580), Elem(625), Elem(838), Elem(622), Elem(325), Elem(880), Elem(659), Elem(307), Elem(956), Elem(533), Elem(179), Elem(357), Elem(609), Elem(353), Elem(713), Elem(388), Elem(36), Elem(320), Elem(43), Elem(999), Elem(679), Elem(434), Elem(286)])), Polynomial(Array([Elem(683), Elem(501), Elem(221), Elem(20), Elem(879), Elem(500), Elem(357), Elem(790), Elem(676), Elem(905), Elem(406), Elem(986), Elem(823), Elem(204), Elem(670), Elem(144), Elem(316), Elem(529), Elem(817), Elem(175), Elem(599), Elem(929), Elem(613), Elem(17), Elem(608), Elem(42), Elem(849), Elem(842), Elem(211), Elem(948), Elem(644), Elem(950), Elem(977), Elem(783), Elem(715), Elem(648), Elem(783), Elem(703), Elem(144), Elem(663), Elem(939), Elem(267), Elem(89), Elem(265), Elem(1018), Elem(258), Elem(34), Elem(917), Elem(433), Elem(31), Elem(96), Elem(260), Elem(628), Elem(191), Elem(423), Elem(158), Elem(840), Elem(151), Elem(391), Elem(967), Elem(38), Elem(54), Elem(81), Elem(769), Elem(770), Elem(823), Elem(158), Elem(362), Elem(735), Elem(309), Elem(400), Elem(772), Elem(729), Elem(679), Elem(83), Elem(329), Elem(324), Elem(296), Elem(436), Elem(997), Elem(388), Elem(956), Elem(199), Elem(623), Elem(491), Elem(590), Elem(215), Elem(9), Elem(102), Elem(82), Elem(733), Elem(358), Elem(159), Elem(555), Elem(593), Elem(1002), Elem(465), Elem(14), Elem(434), Elem(636), Elem(705), Elem(808), Elem(444), Elem(865), Elem(319), Elem(767), Elem(549), Elem(539), Elem(310), Elem(877), Elem(242), Elem(998), Elem(477), Elem(919), Elem(906), Elem(102), Elem(662), Elem(912), Elem(1008), Elem(11), Elem(633), Elem(198), Elem(261), Elem(1003), Elem(740), Elem(637), Elem(160), Elem(650), Elem(521), Elem(914), Elem(715), Elem(595), Elem(316), Elem(181), Elem(170), Elem(223), Elem(811), Elem(659), Elem(398), Elem(144), Elem(970), Elem(915), Elem(237), Elem(199), Elem(323), Elem(618), Elem(653), Elem(929), Elem(847), Elem(120), Elem(538), Elem(768), Elem(324), Elem(709), Elem(843), Elem(380), Elem(536), Elem(686), Elem(142), Elem(161), Elem(776), Elem(1000), Elem(568), Elem(610), Elem(382), Elem(494), Elem(357), Elem(785), Elem(315), Elem(565), Elem(783), Elem(378), Elem(900), Elem(750), Elem(760), Elem(463), Elem(979), Elem(575), Elem(414), Elem(203), Elem(514), Elem(715), Elem(696), Elem(614), Elem(748), Elem(647), Elem(337), Elem(116), Elem(28), Elem(658), Elem(420), Elem(116), Elem(106), Elem(1022), Elem(284), Elem(51), Elem(818), Elem(786), Elem(724), Elem(13), Elem(167), Elem(436), Elem(704), Elem(129), Elem(765), Elem(878), Elem(110), Elem(497), Elem(297), Elem(335), Elem(894), Elem(232), Elem(683), Elem(981), Elem(608), Elem(565), Elem(942), Elem(818), Elem(798), Elem(843), Elem(901), Elem(809), Elem(474), Elem(774), Elem(747), Elem(257), Elem(388), Elem(748), Elem(89), Elem(752), Elem(273), Elem(277), Elem(397), Elem(605), Elem(12), Elem(190), Elem(439), Elem(113), Elem(605), Elem(285), Elem(456), Elem(544), Elem(1010), Elem(388), Elem(609), Elem(444), Elem(24), Elem(619), Elem(391), Elem(559), Elem(416), Elem(517), Elem(133), Elem(205), Elem(136), Elem(618)])), Polynomial(Array([Elem(963), Elem(250), Elem(51), Elem(814), Elem(180), Elem(563), Elem(902), Elem(954), Elem(400), Elem(124), Elem(874), Elem(914), Elem(208), Elem(503), Elem(661), Elem(977), Elem(857), Elem(778), Elem(957), Elem(677), Elem(229), Elem(561), Elem(84), Elem(794), Elem(321), Elem(595), Elem(849), Elem(777), Elem(253), Elem(836), Elem(333), Elem(15), Elem(216), Elem(698), Elem(63), Elem(438), Elem(189), Elem(528), Elem(34), Elem(204), Elem(469), Elem(888), Elem(636), Elem(870), Elem(381), Elem(473), Elem(696), Elem(177), Elem(238), Elem(940), Elem(842), Elem(524), Elem(137), Elem(813), Elem(477), Elem(813), Elem(753), Elem(616), Elem(404), Elem(622), Elem(887), Elem(302), Elem(111), Elem(462), Elem(621), Elem(596), Elem(230), Elem(60), Elem(229), Elem(379), Elem(32), Elem(867), Elem(323), Elem(731), Elem(929), Elem(875), Elem(815), Elem(953), Elem(758), Elem(495), Elem(835), Elem(15), Elem(117), Elem(148), Elem(976), Elem(835), Elem(91), Elem(921), Elem(892), Elem(319), Elem(287), Elem(851), Elem(985), Elem(767), Elem(860), Elem(941), Elem(442), Elem(59), Elem(117), Elem(588), Elem(60), Elem(942), Elem(233), Elem(366), Elem(494), Elem(362), Elem(493), Elem(1007), Elem(370), Elem(390), Elem(475), Elem(21), Elem(230), Elem(777), Elem(50), Elem(326), Elem(907), Elem(768), Elem(803), Elem(474), Elem(318), Elem(942), Elem(265), Elem(695), Elem(365), Elem(442), Elem(969), Elem(526), Elem(455), Elem(497), Elem(750), Elem(70), Elem(917), Elem(689), Elem(820), Elem(364), Elem(430), Elem(482), Elem(897), Elem(629), Elem(92), Elem(229), Elem(719), Elem(967), Elem(949), Elem(425), Elem(426), Elem(813), Elem(880), Elem(808), Elem(627), Elem(872), Elem(655), Elem(485), Elem(719), Elem(908), Elem(177), Elem(532), Elem(927), Elem(333), Elem(800), Elem(191), Elem(169), Elem(857), Elem(517), Elem(993), Elem(981), Elem(954), Elem(719), Elem(193), Elem(477), Elem(100), Elem(59), Elem(460), Elem(338), Elem(140), Elem(65), Elem(522), Elem(939), Elem(792), Elem(771), Elem(642), Elem(772), Elem(507), Elem(982), Elem(589), Elem(371), Elem(843), Elem(310), Elem(481), Elem(285), Elem(666), Elem(610), Elem(378), Elem(769), Elem(153), Elem(447), Elem(875), Elem(740), Elem(875), Elem(410), Elem(795), Elem(226), Elem(184), Elem(531), Elem(652), Elem(619), Elem(249), Elem(924), Elem(670), Elem(564), Elem(526), Elem(96), Elem(745), Elem(958), Elem(443), Elem(218), Elem(881), Elem(982), Elem(254), Elem(553), Elem(540), Elem(359), Elem(981), Elem(445), Elem(452), Elem(27), Elem(601), Elem(558), Elem(889), Elem(62), Elem(582), Elem(858), Elem(599), Elem(238), Elem(542), Elem(174), Elem(892), Elem(637), Elem(114), Elem(178), Elem(457), Elem(673), Elem(775), Elem(708), Elem(778), Elem(754), Elem(754), Elem(600), Elem(1002), Elem(158), Elem(37), Elem(840), Elem(464), Elem(337), Elem(26)])), Polynomial(Array([Elem(82), Elem(926), Elem(438), Elem(994), Elem(286), Elem(795), Elem(421), Elem(612), Elem(347), Elem(44), Elem(554), Elem(622), Elem(652), Elem(243), Elem(597), Elem(206), Elem(658), Elem(972), Elem(194), Elem(365), Elem(515), Elem(1002), Elem(286), Elem(97), Elem(724), Elem(120), Elem(610), Elem(788), Elem(150), Elem(680), Elem(333), Elem(205), Elem(453), Elem(260), Elem(269), Elem(59), Elem(31), Elem(519), Elem(114), Elem(285), Elem(715), Elem(403), Elem(787), Elem(220), Elem(485), Elem(425), Elem(429), Elem(841), Elem(419), Elem(484), Elem(614), Elem(1002), Elem(958), Elem(192), Elem(363), Elem(1012), Elem(514), Elem(862), Elem(826), Elem(68), Elem(67), Elem(174), Elem(766), Elem(496), Elem(535), Elem(989), Elem(151), Elem(614), Elem(719), Elem(806), Elem(993), Elem(603), Elem(53), Elem(685), Elem(366), Elem(828), Elem(223), Elem(387), Elem(158), Elem(68), Elem(892), Elem(11), Elem(1015), Elem(481), Elem(735), Elem(98), Elem(478), Elem(58), Elem(139), Elem(1007), Elem(736), Elem(857), Elem(19), Elem(349), Elem(698), Elem(979), Elem(136), Elem(804), Elem(213), Elem(656), Elem(168), Elem(233), Elem(232), Elem(127), Elem(954), Elem(76), Elem(58), Elem(575), Elem(172), Elem(801), Elem(461), Elem(126), Elem(712), Elem(294), Elem(199), Elem(850), Elem(922), Elem(161), Elem(983), Elem(386), Elem(809), Elem(889), Elem(544), Elem(334), Elem(651), Elem(67), Elem(118), Elem(749), Elem(479), Elem(555), Elem(425), Elem(434), Elem(53), Elem(437), Elem(31), Elem(765), Elem(773), Elem(817), Elem(326), Elem(237), Elem(919), Elem(768), Elem(196), Elem(199), Elem(714), Elem(975), Elem(947), Elem(303), Elem(644), Elem(451), Elem(748), Elem(409), Elem(739), Elem(111), Elem(265), Elem(694), Elem(177), Elem(149), Elem(406), Elem(648), Elem(209), Elem(298), Elem(616), Elem(460), Elem(400), Elem(993), Elem(517), Elem(660), Elem(15), Elem(843), Elem(637), Elem(375), Elem(135), Elem(837), Elem(700), Elem(414), Elem(93), Elem(109), Elem(920), Elem(156), Elem(132), Elem(484), Elem(537), Elem(158), Elem(845), Elem(501), Elem(907), Elem(119), Elem(549), Elem(214), Elem(970), Elem(339), Elem(462), Elem(903), Elem(772), Elem(641), Elem(91), Elem(1022), Elem(343), Elem(910), Elem(495), Elem(885), Elem(310), Elem(1019), Elem(163), Elem(663), Elem(118), Elem(249), Elem(154), Elem(832), Elem(743), Elem(809), Elem(906), Elem(151), Elem(787), Elem(185), Elem(292), Elem(865), Elem(24), Elem(154), Elem(300), Elem(611), Elem(111), Elem(18), Elem(559), Elem(936), Elem(169), Elem(401), Elem(271), Elem(991), Elem(249), Elem(440), Elem(674), Elem(313), Elem(235), Elem(969), Elem(893), Elem(554), Elem(681), Elem(845), Elem(876), Elem(802), Elem(1018), Elem(909), Elem(981), Elem(30), Elem(710), Elem(126), Elem(716), Elem(236), Elem(627), Elem(23), Elem(894), Elem(752), Elem(766), Elem(803)]))])), A_hat: NttMatrix(Array([NttVector(Array([NttPolynomial(Array([Elem(5573637), Elem(5093), Elem(5160886), Elem(1498998), Elem(6919930), Elem(962790), Elem(2035880), Elem(55268), Elem(3956320), Elem(3862222), Elem(4706551), Elem(5894282), Elem(4727079), Elem(2368784), Elem(4796784), Elem(6304401), Elem(7493165), Elem(7795297), Elem(1193661), Elem(3090242), Elem(1524738), Elem(4972964), Elem(833542), Elem(506970), Elem(2038231), Elem(1722538), Elem(5462173), Elem(902267), Elem(5452112), Elem(8376754), Elem(5097971), Elem(5392841), Elem(1363559), Elem(4024324), Elem(4300719), Elem(6723998), Elem(1665334), Elem(5709373), Elem(950124), Elem(2263827), Elem(7138631), Elem(258853), Elem(3395892), Elem(7209074), Elem(1202965), Elem(7077458), Elem(2348344), Elem(7130642), Elem(5406104), Elem(1330193), Elem(1750653), Elem(71028), Elem(7450587), Elem(5860247), Elem(7545812), Elem(7578536), Elem(209742), Elem(497816), Elem(6211329), Elem(2616148), Elem(2375170), Elem(2506674), Elem(1369691), Elem(7469868), Elem(6199340), Elem(7919730), Elem(1319402), Elem(1464166), Elem(5057891), Elem(7252548), Elem(5848030), Elem(21422), Elem(6208760), Elem(3776919), Elem(16810), Elem(3701727), Elem(746657), Elem(7358618), Elem(1441312), Elem(4731986), Elem(3936474), Elem(1088074), Elem(3680113), Elem(5109929), Elem(1957352), Elem(2502364), Elem(1391079), Elem(2452977), Elem(7177179), Elem(4868127), Elem(6263462), Elem(2015779), Elem(8095820), Elem(6274699), Elem(5995581), Elem(5665571), Elem(2172824), Elem(3665240), Elem(4496325), Elem(5473183), Elem(5212668), Elem(7112794), Elem(7451029), Elem(1294362), Elem(379346), Elem(337233), Elem(6988768), Elem(2039354), Elem(6933682), Elem(2082728), Elem(7259726), Elem(3434707), Elem(7064704), Elem(2955339), Elem(7792300), Elem(8143620), Elem(4961747), Elem(3710748), Elem(7700435), Elem(7845229), Elem(6504830), Elem(7271408), Elem(5175662), Elem(5957549), Elem(5103428), Elem(746412), Elem(910395), Elem(4168393), Elem(1708347), Elem(6551184), Elem(3376677), Elem(3479703), Elem(4350509), Elem(883629), Elem(2504319), Elem(7277004), Elem(4673726), Elem(2513676), Elem(2415763), Elem(7392345), Elem(2555870), Elem(756410), Elem(5167052), Elem(8173735), Elem(7283296), Elem(7021075), Elem(5195146), Elem(2772805), Elem(3708871), Elem(216338), Elem(6851150), Elem(1078834), Elem(3465883), Elem(859451), Elem(409624), Elem(808036), Elem(3145030), Elem(7829385), Elem(3454604), Elem(5639000), Elem(7382570), Elem(3590546), Elem(4446804), Elem(2491337), Elem(7644554), Elem(2630681), Elem(3079708), Elem(2223210), Elem(1844529), Elem(4016716), Elem(1478982), Elem(2776826), Elem(4233015), Elem(48758), Elem(6659430), Elem(4116201), Elem(5103120), Elem(2216204), Elem(3634360), Elem(4560646), Elem(4984459), Elem(7239783), Elem(7437491), Elem(7105431), Elem(628861), Elem(5822642), Elem(5573209), Elem(1310541), Elem(2609137), Elem(7053648), Elem(1278669), Elem(2129862), Elem(5660429), Elem(5273735), Elem(4819795), Elem(4054741), Elem(3527870), Elem(1010491), Elem(1669648), Elem(4068688), Elem(1902184), Elem(1167961), Elem(6340271), Elem(1710100), Elem(8054795), Elem(6760367), Elem(3245397), Elem(3782482), Elem(6125252), Elem(719555), Elem(8329421), Elem(2085083), Elem(431030), Elem(6454875), Elem(1919029), Elem(276490), Elem(5048539), Elem(3847046), Elem(7844002), Elem(1351491), Elem(7876310), Elem(2439419), Elem(7559623), Elem(5785005), Elem(7440635), Elem(3945432), Elem(1371814), Elem(4216565), Elem(7150184), Elem(984630), Elem(506573), Elem(4746736), Elem(267999), Elem(5547556), Elem(5382955), Elem(613212), Elem(6230776), Elem(1756887), Elem(7617335), Elem(2103017), Elem(4646142), Elem(6824837), Elem(1670794), Elem(2323428), Elem(13360), Elem(5543891), Elem(7455806), Elem(1332943), Elem(4508595), Elem(7431111), Elem(4439433), Elem(1670876), Elem(4477599), Elem(7707772), Elem(878368), Elem(73041)])), NttPolynomial(Array([Elem(5850310), Elem(5900276), Elem(5855802), Elem(2378171), Elem(7644332), Elem(3791218), Elem(1859379), Elem(3881050), Elem(4425176), Elem(6842684), Elem(2464038), Elem(4212355), Elem(7797078), Elem(5793365), Elem(1366314), Elem(2366711), Elem(2556807), Elem(2847035), Elem(5734755), Elem(4851579), Elem(5332423), Elem(8067860), Elem(2344083), Elem(6895839), Elem(471699), Elem(6034771), Elem(8051568), Elem(8009438), Elem(2331350), Elem(2839382), Elem(6581925), Elem(6751738), Elem(7563854), Elem(901170), Elem(6410716), Elem(2706647), Elem(2164895), Elem(4956767), Elem(7631462), Elem(5609238), Elem(3494728), Elem(1565306), Elem(4821239), Elem(5970632), Elem(474811), Elem(846762), Elem(6853446), Elem(6652103), Elem(5920352), Elem(5151234), Elem(3775928), Elem(4602470), Elem(6237292), Elem(1915877), Elem(4938872), Elem(4983887), Elem(7280655), Elem(5514941), Elem(2563668), Elem(2216323), Elem(1431143), Elem(1560838), Elem(7778622), Elem(3130463), Elem(5734666), Elem(5291168), Elem(1764390), Elem(7817235), Elem(2944073), Elem(6867482), Elem(1060566), Elem(4680285), Elem(8100016), Elem(467144), Elem(762569), Elem(4803799), Elem(1246736), Elem(2550799), Elem(7287379), Elem(7562637), Elem(6807347), Elem(7885425), Elem(1331639), Elem(6224339), Elem(1675395), Elem(347810), Elem(8033616), Elem(6232031), Elem(4433101), Elem(4139277), Elem(2309584), Elem(991286), Elem(6071178), Elem(6236978), Elem(1127495), Elem(3926134), Elem(5748167), Elem(2203807), Elem(3332681), Elem(5398383), Elem(1668982), Elem(39254), Elem(4228406), Elem(3236815), Elem(5252740), Elem(6650731), Elem(4779111), Elem(6682639), Elem(6081978), Elem(114742), Elem(300036), Elem(3912414), Elem(3380802), Elem(616745), Elem(7702038), Elem(7378042), Elem(4417296), Elem(6437538), Elem(7191926), Elem(2769904), Elem(5078993), Elem(6810124), Elem(2106416), Elem(107123), Elem(7231746), Elem(1579620), Elem(6444969), Elem(3522991), Elem(1636495), Elem(3267574), Elem(7012747), Elem(4864816), Elem(5056833), Elem(4672024), Elem(3165066), Elem(7466876), Elem(7751134), Elem(1169289), Elem(7078555), Elem(1701423), Elem(83743), Elem(8285345), Elem(2666914), Elem(6908910), Elem(5710008), Elem(929843), Elem(5687040), Elem(4024190), Elem(6135142), Elem(5927555), Elem(4076234), Elem(5940516), Elem(4884622), Elem(4042950), Elem(684462), Elem(6076374), Elem(7756893), Elem(3097683), Elem(202404), Elem(5226039), Elem(5385749), Elem(3398985), Elem(1077302), Elem(3149968), Elem(1223839), Elem(6947016), Elem(7646444), Elem(7116100), Elem(5610670), Elem(3757985), Elem(2813391), Elem(4234696), Elem(6682231), Elem(8324967), Elem(5802841), Elem(586821), Elem(1395218), Elem(4498230), Elem(365976), Elem(6634445), Elem(1320149), Elem(7985896), Elem(1609833), Elem(5515022), Elem(4022736), Elem(1946906), Elem(7221714), Elem(855851), Elem(8280315), Elem(7362310), Elem(566046), Elem(5728099), Elem(4695196), Elem(6524792), Elem(3877065), Elem(7099296), Elem(6416576), Elem(1977844), Elem(6328055), Elem(950037), Elem(7969148), Elem(8287359), Elem(3159593), Elem(7587345), Elem(2740851), Elem(6555930), Elem(7564992), Elem(2779691), Elem(5783448), Elem(3544518), Elem(6629286), Elem(2243262), Elem(1088434), Elem(6892568), Elem(3679789), Elem(860023), Elem(2108337), Elem(7138677), Elem(755773), Elem(7443412), Elem(5068405), Elem(7261208), Elem(4458934), Elem(4804962), Elem(7427217), Elem(7127902), Elem(2585285), Elem(7154994), Elem(2152994), Elem(2506377), Elem(8038379), Elem(7011666), Elem(7071506), Elem(6094684), Elem(7480623), Elem(2623091), Elem(7538650), Elem(2291267), Elem(6207399), Elem(268913), Elem(268563), Elem(2678671), Elem(1551607), Elem(6669889), Elem(2469188), Elem(4129351), Elem(1754466), Elem(6238867), Elem(56404), Elem(7368809), Elem(6993116), Elem(7479656), Elem(7342866), Elem(5656033), Elem(6135688), Elem(1332565)])), NttPolynomial(Array([Elem(4521361), Elem(137175), Elem(699265), Elem(3341517), Elem(2283787), Elem(5668194), Elem(3281069), Elem(2561783), Elem(5905741), Elem(3991720), Elem(6941077), Elem(5722183), Elem(485431), Elem(3150320), Elem(5667757), Elem(4168822), Elem(6665208), Elem(3572990), Elem(6062509), Elem(7281372), Elem(5976783), Elem(7549297), Elem(6538171), Elem(8004580), Elem(2848586), Elem(2048829), Elem(2427754), Elem(5609520), Elem(1720582), Elem(7626805), Elem(7184691), Elem(6759732), Elem(4459296), Elem(2554289), Elem(4483413), Elem(5197134), Elem(3539309), Elem(8316304), Elem(4025744), Elem(211696), Elem(7510910), Elem(6355509), Elem(2007944), Elem(4436260), Elem(450058), Elem(292627), Elem(847029), Elem(4192183), Elem(584007), Elem(5979926), Elem(6536827), Elem(4137606), Elem(3021610), Elem(5532255), Elem(234818), Elem(4501960), Elem(1197227), Elem(1557425), Elem(5442879), Elem(6374771), Elem(3265767), Elem(4445617), Elem(1484421), Elem(6002665), Elem(5199420), Elem(7899066), Elem(3977767), Elem(2175313), Elem(4827291), Elem(7557540), Elem(8200079), Elem(664987), Elem(6488037), Elem(2504034), Elem(2853079), Elem(7118957), Elem(1915871), Elem(4365549), Elem(3355639), Elem(6832263), Elem(2309765), Elem(5788611), Elem(5390749), Elem(2584364), Elem(8066942), Elem(4300618), Elem(136849), Elem(5555817), Elem(598227), Elem(5749273), Elem(780581), Elem(3303269), Elem(2348040), Elem(2321008), Elem(8331481), Elem(6010421), Elem(6098951), Elem(6787113), Elem(3611403), Elem(55975), Elem(211945), Elem(7205872), Elem(2230239), Elem(136710), Elem(7720245), Elem(6862663), Elem(5404695), Elem(2862296), Elem(5041466), Elem(4698088), Elem(6672051), Elem(6121328), Elem(6468048), Elem(1733936), Elem(4759539), Elem(4560243), Elem(729121), Elem(1279864), Elem(1086054), Elem(1671818), Elem(1630786), Elem(5272444), Elem(4374599), Elem(1556485), Elem(2564914), Elem(7441411), Elem(3152164), Elem(1720467), Elem(2887478), Elem(870813), Elem(6549344), Elem(6125981), Elem(8010759), Elem(7752488), Elem(3239580), Elem(169377), Elem(3052337), Elem(8019899), Elem(4291830), Elem(2939502), Elem(3866376), Elem(5021976), Elem(5129626), Elem(1062043), Elem(5708799), Elem(7076026), Elem(5232087), Elem(7707696), Elem(2515391), Elem(7973802), Elem(5483639), Elem(7511555), Elem(7786186), Elem(3278871), Elem(2865010), Elem(3375247), Elem(3522055), Elem(5785402), Elem(7786509), Elem(5574944), Elem(4648710), Elem(3897749), Elem(5060282), Elem(691838), Elem(8289665), Elem(3539732), Elem(7024886), Elem(6329096), Elem(6896632), Elem(3861583), Elem(168616), Elem(5274144), Elem(489296), Elem(6502623), Elem(1210085), Elem(1835599), Elem(370267), Elem(16920), Elem(6423279), Elem(711978), Elem(3905122), Elem(1436813), Elem(8344431), Elem(175476), Elem(6856342), Elem(3103686), Elem(6931923), Elem(6810099), Elem(1535865), Elem(5706368), Elem(2723444), Elem(6852016), Elem(7325724), Elem(683490), Elem(4054157), Elem(6103804), Elem(6758686), Elem(6454570), Elem(4309973), Elem(5413041), Elem(5864385), Elem(3551649), Elem(7372285), Elem(3520225), Elem(7894259), Elem(6149264), Elem(7561543), Elem(33659), Elem(6741240), Elem(539565), Elem(5609894), Elem(1541111), Elem(7106473), Elem(6071782), Elem(649321), Elem(713658), Elem(1353487), Elem(2380771), Elem(1515418), Elem(1511399), Elem(7443502), Elem(1680505), Elem(4311530), Elem(2458212), Elem(1897245), Elem(5159872), Elem(1884505), Elem(1397245), Elem(5410310), Elem(5905259), Elem(4583568), Elem(3344094), Elem(8116644), Elem(5480079), Elem(391132), Elem(2086721), Elem(7351113), Elem(2852637), Elem(1260570), Elem(2093424), Elem(5113319), Elem(903045), Elem(4651330), Elem(6021876), Elem(6005605), Elem(301895), Elem(8186735), Elem(2447739), Elem(5762469), Elem(3810984), Elem(4003443), Elem(2119821), Elem(1468388), Elem(1521009), Elem(1186457), Elem(8159644)])), NttPolynomial(Array([Elem(5574657), Elem(7316144), Elem(6607876), Elem(6680215), Elem(1167601), Elem(2829068), Elem(3934362), Elem(6829069), Elem(3444007), Elem(6757830), Elem(7253769), Elem(6286652), Elem(1854766), Elem(6202433), Elem(7075300), Elem(2427491), Elem(620570), Elem(5119562), Elem(3245350), Elem(6164114), Elem(3927215), Elem(1535817), Elem(2316327), Elem(1740448), Elem(7479865), Elem(7968485), Elem(4868147), Elem(5754199), Elem(4626090), Elem(5608384), Elem(165024), Elem(3273543), Elem(2207691), Elem(3522638), Elem(1540019), Elem(4290603), Elem(117091), Elem(2571659), Elem(1157887), Elem(6769530), Elem(880124), Elem(2959648), Elem(6147617), Elem(172265), Elem(7859175), Elem(4544413), Elem(5599796), Elem(5123823), Elem(5037192), Elem(1072802), Elem(4883532), Elem(4303961), Elem(3208660), Elem(2794273), Elem(990583), Elem(7702790), Elem(1671445), Elem(486659), Elem(8342665), Elem(2302448), Elem(1998495), Elem(4134028), Elem(6721330), Elem(2393517), Elem(4891886), Elem(6748789), Elem(5772763), Elem(2027962), Elem(8263194), Elem(7540830), Elem(5285287), Elem(3715611), Elem(423904), Elem(5083507), Elem(3857018), Elem(5875290), Elem(3247356), Elem(7595725), Elem(6231590), Elem(7281613), 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Elem(3922986), Elem(2613168), Elem(1648244), Elem(4491318), Elem(6239890), Elem(6185969), Elem(979612), Elem(7511017), Elem(339125), Elem(1997936), Elem(2331474), Elem(959192), Elem(3220923), Elem(7329025), Elem(1976424), Elem(4241779), Elem(1804401), Elem(2557201), Elem(738355), Elem(6082962), Elem(6923969), Elem(2339785), Elem(4068351), Elem(1540540), Elem(1238755), Elem(6513486), Elem(4367320), Elem(2554018), Elem(7761788), Elem(1369001), Elem(262937), Elem(1645786), Elem(5984312), Elem(3029271), Elem(4688477), Elem(1396006), Elem(7284355), Elem(3532188), Elem(7519636), Elem(421353), Elem(1194596), Elem(5736850), Elem(5843232), Elem(5729811), Elem(400415), Elem(7451815), Elem(6883346), Elem(1232893), Elem(1405632), Elem(4188288), Elem(6063984), Elem(3912108), Elem(6906179), Elem(4355994), Elem(6382449), Elem(3918565), Elem(675567), Elem(1955437), Elem(4363990), Elem(5338441), Elem(2675618), Elem(4602803), Elem(3995476), Elem(5937651), Elem(6681543), Elem(7544351), Elem(1170365), 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Elem(2594306), Elem(6914287), Elem(2737643), Elem(5452187), Elem(5646006), Elem(204959), Elem(7924051), Elem(8029679), Elem(6041650), Elem(2723726), Elem(626726), Elem(7863306), Elem(3859921), Elem(4670327), Elem(3041661), Elem(1073062), Elem(6180715), Elem(4528726), Elem(3807435), Elem(5572395), Elem(6681828), Elem(5882648), Elem(3013420), Elem(4100767), Elem(1168516), Elem(4218590), Elem(5086616), Elem(7411906), Elem(3763169), Elem(3602020), Elem(2483133), Elem(2629795), Elem(4182984), Elem(863864), Elem(2966090), Elem(3903297), Elem(1385664), Elem(6037621), Elem(8321687), Elem(2553808), Elem(1840945), Elem(801134), Elem(6460163), Elem(6565206), Elem(850160), Elem(7035501), Elem(897005), Elem(7618463), Elem(8230516), Elem(4181496), Elem(50578), Elem(1112091), Elem(3239353), Elem(191700), Elem(6835571), Elem(657365), Elem(4244817), Elem(3412748), Elem(778548), Elem(7482589), Elem(3251097), Elem(4677274), Elem(2581308), Elem(7615891), Elem(2794033), Elem(954795), Elem(4139270), 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Elem(6712404), Elem(7840283), Elem(6736435), Elem(6280409), Elem(5150476), Elem(5272127), Elem(1028856), Elem(2394486), Elem(1390556), Elem(2550311), Elem(5260480), Elem(4341832), Elem(6021876), Elem(4843028), Elem(8314853), Elem(1664056), Elem(6154609), Elem(4186420), Elem(5086357), Elem(8204858), Elem(7141611), Elem(1777955), Elem(5855797), Elem(6405511), Elem(4143553), Elem(5727447), Elem(5345585), Elem(3006837), Elem(1799943), Elem(5438327), Elem(4628111), Elem(6486212), Elem(3622072), Elem(1773669), Elem(7948426), Elem(2299609), Elem(4677932), Elem(7963530), Elem(5352940), Elem(4957447), Elem(7762701), Elem(138515), Elem(7474993), Elem(3707890), Elem(5709183), Elem(913488), Elem(6887574), Elem(7122781), Elem(4255634), Elem(5119132), Elem(2587578), Elem(1491988), Elem(2145509), Elem(6930146), Elem(5426258), Elem(6859944), Elem(5008052), Elem(3529860), Elem(835524), Elem(2529672), Elem(1833146), Elem(682990), Elem(7904441), Elem(3267230), Elem(2046586), Elem(600277), Elem(3743859), 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Elem(1387869), Elem(838977), Elem(646997), Elem(8114061), Elem(5615362), Elem(217598), Elem(8021566), Elem(6810561), Elem(4391953), Elem(5049169), Elem(6636643), Elem(6090391), Elem(4613528), Elem(4278462), Elem(3611796), Elem(6933099), Elem(3044058), Elem(2436630), Elem(957073), Elem(2524196), Elem(81505), Elem(3408567)])), NttPolynomial(Array([Elem(1982747), Elem(7507778), Elem(526969), Elem(2577586), Elem(2226744), Elem(1797227), Elem(2242659), Elem(5873244), Elem(1637108), Elem(7293454), Elem(5772748), Elem(7892302), Elem(2447043), Elem(3390042), Elem(1886535), Elem(5284001), Elem(900170), Elem(1458886), Elem(2777894), Elem(7279316), Elem(832788), Elem(5757933), Elem(4634389), Elem(2742823), Elem(6915056), Elem(414203), Elem(5035540), Elem(1611340), Elem(4649082), Elem(6248094), Elem(7368387), Elem(6269924), Elem(6521937), Elem(6486741), Elem(2749870), Elem(798415), Elem(3148311), Elem(3473811), Elem(5982405), Elem(7142938), Elem(2704816), Elem(378037), Elem(5628942), 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Elem(606241), Elem(191346), Elem(2134338), Elem(1147259), Elem(1878816), Elem(7670782), Elem(8039693), Elem(8288134), Elem(885092), Elem(4002858), Elem(7496113), Elem(4942094), Elem(6656372), Elem(7568191), Elem(3606785), Elem(1641023), Elem(6568414), Elem(2914158), Elem(4822142), Elem(4235307), Elem(6093954), Elem(2016402), Elem(3982748), Elem(4879867), Elem(8338986), Elem(3090753), Elem(8236620), Elem(7320160), Elem(4871580), Elem(3589301), Elem(4178844), Elem(2641781), Elem(381461), Elem(735202), Elem(4638004), Elem(5329764), Elem(1714611), Elem(2182747), Elem(983398), Elem(1242475), Elem(2301934), Elem(4318945), Elem(4388257), Elem(4803783), Elem(4527915), Elem(8337889), Elem(1248176), Elem(1988114), Elem(6139949), Elem(6317182), Elem(3614815), Elem(2293095), Elem(4081729), Elem(8186063), Elem(5398530), Elem(2885656), Elem(6970950), Elem(2635873), Elem(5661543), Elem(1780221), Elem(3715230), Elem(1546257), Elem(5057762), Elem(3478707), Elem(5771936), Elem(781791), Elem(6539750), 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Elem(2965397), Elem(8079748), Elem(6422122), Elem(4248778), Elem(5698721), Elem(5185796), Elem(8141808), Elem(1619323), Elem(1132492), Elem(3967590), Elem(913481), Elem(6893767), Elem(2538410), Elem(6206267), Elem(1378893), Elem(4637522), Elem(4562950), Elem(122850), Elem(2128832), Elem(1131098), Elem(4175589), Elem(4689063), Elem(6458433), Elem(7104446), Elem(7675810), Elem(2211593), Elem(3356006), Elem(3596694), Elem(3951867), Elem(7028514), Elem(1885480), Elem(1813769), Elem(8013513), Elem(4921767), Elem(3158742), Elem(3167769), Elem(7509215), Elem(2409796), Elem(4417009), Elem(2121025), Elem(494326), Elem(3827429), Elem(3971740), Elem(7963588), Elem(1654235), Elem(4370544), Elem(6620428), Elem(2565810), Elem(6738696), Elem(6824492), Elem(6710325), Elem(5165570), Elem(5160764), Elem(5856212), Elem(4057692), Elem(4734483), Elem(6851944), Elem(2399294), Elem(1738451), Elem(1685701), Elem(5821046), Elem(5701982), Elem(6191393), Elem(5182930), Elem(122904), Elem(7405643), Elem(7350453), Elem(3320732), Elem(3583019), Elem(4049992), Elem(3636563), Elem(4373982), Elem(7848596), Elem(485567), Elem(1318369), Elem(4821614), Elem(7969934), Elem(2297936), Elem(2077860), Elem(2625667), Elem(58004), Elem(4679051), Elem(2274232), Elem(1309864), Elem(4024634), Elem(3398), Elem(6224449), Elem(1739178), Elem(1033826), Elem(5638332), Elem(6746000), Elem(4969065), Elem(2521394), Elem(7033146), Elem(6571053), Elem(8285576), Elem(1105864), Elem(6470791), Elem(4926142), Elem(7005179), Elem(4687286), Elem(8350046), Elem(5023005), Elem(6614807), Elem(6909733), Elem(1600844), Elem(817885), Elem(2487174), Elem(2734790), Elem(7431353), Elem(4513227), Elem(6000073), Elem(5610728), Elem(2032222), Elem(2730436), Elem(2753662), Elem(5515843), Elem(1365526), Elem(3769797), Elem(7001861), Elem(849996), Elem(3684167), Elem(7561178), Elem(3493793), Elem(8281100), Elem(161095), Elem(8340715), Elem(5070470), Elem(4654355), Elem(6472503), Elem(2890136), Elem(7795993), Elem(7124658), Elem(1457139), Elem(3528181), Elem(103221), Elem(2677386), Elem(177497), Elem(5555740), Elem(1143123), Elem(383582), Elem(7102615), Elem(7683310), Elem(3882441), Elem(807101), Elem(6086353), Elem(1653452), Elem(4000910), Elem(8316558), Elem(7076230), Elem(1725670), Elem(6550599), Elem(887199), Elem(6189958), Elem(1853025), Elem(4538272), Elem(4527984), Elem(2279304), Elem(4107788), Elem(158874), Elem(4067555), Elem(1014406), Elem(6193327), Elem(2072908), Elem(4924205), Elem(2976589), Elem(1789011), Elem(7803803), Elem(202371), Elem(217171), Elem(5306334), Elem(4841703), Elem(5602658), Elem(5428978), Elem(6758270), Elem(3874463), Elem(1081527), Elem(6310129), Elem(4016139), Elem(5987612), Elem(3915810), Elem(2201509), Elem(4794395), Elem(7500825)])), NttPolynomial(Array([Elem(1350694), Elem(3907975), Elem(7519117), Elem(630791), Elem(8247633), Elem(8355823), Elem(5266230), Elem(1226890), Elem(1976068), Elem(6749288), Elem(6471861), Elem(6580605), Elem(1523754), Elem(1584985), Elem(6201005), Elem(6601139), Elem(4367525), Elem(2083607), Elem(3113023), Elem(2419954), Elem(6084079), Elem(2729891), Elem(7216678), Elem(6625537), Elem(2357637), Elem(5063245), Elem(4232250), Elem(8076551), Elem(8136874), Elem(6985181), Elem(537310), Elem(1569074), Elem(4533989), Elem(3332935), Elem(2577749), Elem(3948090), Elem(689071), Elem(3680145), Elem(4741948), Elem(6461999), Elem(8004959), Elem(8302286), Elem(7303486), Elem(3585518), Elem(7813250), Elem(6808905), Elem(6424662), Elem(386144), Elem(638158), Elem(2777487), Elem(7540279), Elem(3108098), Elem(3505515), Elem(1924205), Elem(5468438), Elem(2171495), Elem(306600), Elem(3962327), Elem(7655901), Elem(695003), Elem(6240818), Elem(6083753), Elem(3789945), Elem(1076211), Elem(869442), Elem(4295628), Elem(7080289), Elem(7002721), Elem(582548), Elem(8206888), Elem(922835), Elem(5753828), Elem(7537482), Elem(5140316), Elem(2154887), Elem(2529301), Elem(3891764), Elem(1843674), Elem(1450620), Elem(5381617), Elem(197441), Elem(1754592), Elem(3741561), Elem(2089424), Elem(6192918), Elem(5191945), Elem(2048127), Elem(3567984), Elem(1852566), Elem(6264868), Elem(5504615), Elem(7596766), Elem(5467195), Elem(7835369), Elem(7018826), Elem(4955079), Elem(4223558), Elem(2917234), Elem(1490088), Elem(637902), Elem(7854788), Elem(4471022), Elem(1778737), Elem(5917378), Elem(7285941), Elem(4764741), Elem(6091769), Elem(514615), Elem(2357208), Elem(3559592), Elem(543084), Elem(6216763), Elem(7125351), Elem(3403550), Elem(4573410), Elem(6078843), Elem(2668212), Elem(4160382), Elem(5375059), Elem(5496111), Elem(4043186), Elem(4726584), Elem(7790359), Elem(5375327), Elem(903360), Elem(6684090), Elem(4465212), Elem(2899243), Elem(4894776), Elem(3982443), Elem(8188342), Elem(4217597), Elem(7574794), Elem(6996130), Elem(3990868), Elem(7872728), Elem(651015), Elem(2966329), Elem(1993524), Elem(2739925), Elem(5168328), Elem(4013558), Elem(3885871), Elem(81944), Elem(2658330), Elem(5396242), Elem(1270633), Elem(166875), Elem(3512358), 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Elem(1304352), Elem(606773), Elem(4797034), Elem(5818805), Elem(5198222), Elem(3993388), Elem(5011168), Elem(3465014), Elem(3819316), Elem(134039), Elem(6999702), Elem(7488514), Elem(6622593), Elem(3678633), Elem(5719092), Elem(594490), Elem(3978099), Elem(7990955), Elem(2630242), Elem(4552017), Elem(7968408), Elem(1946323), Elem(1848327), Elem(7852533), Elem(5424515), Elem(4848696), Elem(7305859), Elem(5816679), Elem(2266378), Elem(2384053), Elem(1950925), Elem(4523786), Elem(7295897), Elem(263801), Elem(4936482), Elem(6342059), Elem(3128884), Elem(1388994), Elem(7623154), Elem(8014965)])), NttPolynomial(Array([Elem(6294073), Elem(4115816), Elem(4871090), Elem(6102622), Elem(587239), Elem(7750413), Elem(452413), Elem(6815701), Elem(7557322), Elem(4598268), Elem(8307214), Elem(3462378), Elem(3172877), Elem(7329244), Elem(1601445), Elem(1375212), Elem(5626233), Elem(3237511), Elem(6202333), Elem(1052082), Elem(2235639), Elem(7352662), Elem(6314717), Elem(3936990), Elem(283004), Elem(3805285), Elem(6765538), Elem(2834858), Elem(8119880), Elem(369000), Elem(3097758), Elem(6684568), Elem(4086183), Elem(6290851), Elem(4489328), Elem(4060307), Elem(5683784), Elem(4165306), Elem(1196701), Elem(6990396), Elem(1315001), Elem(8079404), Elem(1866668), Elem(5769878), Elem(6677302), Elem(2268146), Elem(7409084), Elem(3343393), Elem(2618528), Elem(4289589), Elem(8167639), Elem(7347779), Elem(6505883), Elem(342753), Elem(2087352), Elem(7949042), Elem(2988398), Elem(6533938), Elem(2931305), Elem(2150884), Elem(2910178), Elem(5720253), Elem(2348737), Elem(699535), Elem(4094427), Elem(7382038), Elem(1748370), Elem(4265566), Elem(2673533), Elem(6241665), Elem(880113), Elem(3441674), Elem(6490745), Elem(6543585), Elem(5469183), Elem(3088828), Elem(3940065), Elem(2159569), Elem(4617340), Elem(5865201), Elem(7366471), Elem(3639417), Elem(5300694), Elem(1012195), Elem(5481609), Elem(7483861), Elem(486486), Elem(7850811), Elem(2520630), Elem(1144750), Elem(754709), Elem(6826792), 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Elem(5215989), Elem(8134478), Elem(7745304), Elem(550163), Elem(7940184), Elem(8066177), Elem(6005819), Elem(7609291), Elem(3444024), Elem(4049099), Elem(6043367), Elem(7087810), Elem(2856217), Elem(7943909), Elem(1143433), Elem(7545275), Elem(3123729), Elem(1995904), Elem(1535139), Elem(2622101), Elem(3118583), Elem(4780090), Elem(2371202), Elem(8266652), Elem(7340183), Elem(1077308), Elem(77201), Elem(4119333), Elem(6435572), Elem(7080862), Elem(6182285), Elem(8141416), Elem(3150856), Elem(3052393), Elem(7695374), Elem(8064839), Elem(2216000), Elem(3669146), Elem(7506654), Elem(81065), Elem(3771398), Elem(4796548), Elem(2644238), Elem(6261227), Elem(7049942), Elem(1930463), Elem(105576), Elem(180242), Elem(2330259), Elem(1420066), Elem(5428042), Elem(6634854), Elem(3143582), Elem(3523934), Elem(1149108), Elem(1151351), Elem(1235401), Elem(768623), Elem(4381543), Elem(3083579), Elem(1373340), Elem(2541308), Elem(906363), Elem(670571), Elem(4809809), Elem(462425), Elem(2556932), 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Elem(8020575), Elem(407816), Elem(4792079), Elem(7679214), Elem(459132), Elem(439151), Elem(4392276), Elem(765026), Elem(726231), Elem(7930158), Elem(3073656), Elem(4207439), Elem(4692301), Elem(5525304), Elem(269816), Elem(3240485), Elem(2400372), Elem(6968461), Elem(8365972), Elem(6251582), Elem(3508980), Elem(6002830), Elem(4293923), Elem(839659), Elem(546817), Elem(6960722), Elem(816028), Elem(6824290), Elem(1263490), Elem(3825088), Elem(579315), Elem(8215740), Elem(6546219), Elem(4628107), Elem(1568762), Elem(4531086), Elem(2369568), Elem(8002887), Elem(6030438), Elem(1116505), Elem(7165805), Elem(4147664), Elem(6203460), Elem(3090109), Elem(2322233), Elem(3126062), Elem(4682346), Elem(3896450), Elem(441658), Elem(4539509), Elem(2437861), Elem(1581399), Elem(1647986), Elem(5073589), Elem(3051163), Elem(2565947), Elem(7685089), Elem(980738), Elem(4291542), Elem(2779061), Elem(4313082), Elem(2055944), Elem(582345), Elem(4109360), Elem(1412210), Elem(5783927), Elem(3047194), 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Elem(2037425), Elem(7364491), Elem(1724745), Elem(2967609), Elem(5594205), Elem(524440), Elem(4919686), Elem(229657), Elem(3700929), Elem(1104279), Elem(3914676), Elem(1229009), Elem(6234759), Elem(5455703), Elem(1960859), Elem(5239003), Elem(3539377), Elem(7809121), Elem(1738527), Elem(8162598), Elem(4909766), Elem(8177724), Elem(4373130), Elem(5404706), Elem(7999454), Elem(7990427), Elem(5020757), Elem(6574034), Elem(821749), Elem(686471), Elem(1679917), Elem(3728910), Elem(4570476), Elem(4607447), Elem(5543435), Elem(4606682), Elem(6085828), Elem(380396), Elem(5719491), Elem(1872106), Elem(802487), Elem(2822410), Elem(2661688), Elem(3512906), Elem(119818), Elem(5573374), Elem(2298484), Elem(2649550), Elem(37199), Elem(6727522), Elem(376348), Elem(4366995), Elem(5653994), Elem(1848075), Elem(8133053), Elem(6014107), Elem(2211715), Elem(7265632), Elem(1847834), Elem(1106018), Elem(1533187), Elem(8094876), Elem(2454312), Elem(4893998), Elem(5685373), Elem(4955913), Elem(4339377), Elem(1324245), Elem(5109603), Elem(5246147), Elem(4990575), Elem(2275845), Elem(5098719), Elem(5080750), Elem(4453749), Elem(5007706), Elem(2792125), Elem(5536360), Elem(1089448), Elem(3774581), Elem(2222267), Elem(2792652), Elem(4722292), Elem(1713373), Elem(1861982), Elem(8206748), Elem(4947629), Elem(6787238), Elem(2508580), Elem(6454712), Elem(728254), Elem(7594424), Elem(5590937), Elem(8367775), Elem(2628596), Elem(6347714), Elem(4562685), Elem(8024082), Elem(2506483), Elem(1051064), Elem(3969798), Elem(3589208), Elem(265815), Elem(6222305), Elem(5768118), Elem(5215951), Elem(1631589), Elem(5162443), Elem(5668856), Elem(5508981), Elem(6799141), Elem(865954), Elem(3324982), Elem(4268261), Elem(6886627), Elem(3121906), Elem(880253), Elem(5102017), Elem(2822881), Elem(5808705), Elem(2787185), Elem(847529), Elem(6312598), Elem(4889594), Elem(5623992), Elem(548181), Elem(4454799), Elem(3247645), Elem(3594558), Elem(3207540), Elem(3625468), Elem(4041303), Elem(4882138), Elem(4361146), Elem(4483177), Elem(1305898), Elem(5821019), Elem(7005996), Elem(19546), Elem(425491), Elem(1875472), Elem(1742033), Elem(2853695), Elem(2378387), Elem(8016292), Elem(3051439), Elem(2343312), Elem(7145362), Elem(1823877), Elem(4984857), Elem(5770197), Elem(2484774), Elem(5957846), Elem(217956), Elem(3627435), Elem(7861577), Elem(4950364), Elem(6466115), Elem(3362431), Elem(2457327), Elem(5608514), Elem(2318038), Elem(5691337), Elem(6658445), Elem(6176236), Elem(7452476), Elem(1502746), Elem(1543123), Elem(279846), Elem(8162526), Elem(8136741), Elem(4841561), Elem(8004059), Elem(6384084), Elem(8196915), Elem(6948860), Elem(1612821), Elem(5300536), Elem(2079911), Elem(3409894), Elem(4530214), Elem(5154167), Elem(5606353), Elem(5692990), Elem(935607), Elem(6851707), Elem(2413631), Elem(2131837), Elem(2474689), Elem(7334579), Elem(6007983), Elem(2111166), Elem(2371858), Elem(4821045), Elem(4082948), Elem(5596207), Elem(8056370), Elem(58860), Elem(4261786), Elem(4536001), Elem(2160599), Elem(782098), Elem(1121195), Elem(6778172), Elem(1390860), Elem(710088), Elem(1348825), Elem(5153216), Elem(5443099), Elem(2820383), Elem(899624), Elem(1284300), Elem(5514809), Elem(5730686), Elem(1801859), Elem(5720093), Elem(7116059), Elem(1196418), Elem(5495639), Elem(2409554), Elem(932962), Elem(903307), Elem(6030293), Elem(6681110), Elem(4289076), Elem(8320446), Elem(5367871), Elem(2745669), Elem(5545183), Elem(6667283), Elem(6622099), Elem(7358737), Elem(5994982), Elem(1639012), Elem(7968645), Elem(4009514), Elem(4619465), Elem(8211803), Elem(3729766), Elem(7542081), Elem(1430376)])), NttPolynomial(Array([Elem(3797975), Elem(6669192), Elem(93146), Elem(2422382), Elem(2976344), Elem(4500050), Elem(7763920), Elem(8029035), Elem(547911), Elem(7075194), Elem(607079), Elem(7905793), Elem(3517067), Elem(3161198), Elem(1063653), Elem(7541229), Elem(826335), Elem(1578387), Elem(5637505), Elem(1867614), Elem(6887637), Elem(7048840), Elem(5876794), Elem(3057635), Elem(388923), 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Elem(3456408), Elem(3503278), Elem(7562800), Elem(8029066), Elem(2868686), Elem(7328352), Elem(477591), Elem(5098229), Elem(3527654), Elem(2489313), Elem(3783115), Elem(6932737), Elem(3511860), Elem(5963557), Elem(481109), Elem(228422), Elem(6876736), Elem(1142438), Elem(3901113), Elem(547234), Elem(2849000), Elem(2438111), Elem(424000), Elem(6098560), Elem(6072628), Elem(7057952), Elem(2789058), Elem(2015030), Elem(718883), Elem(8051057), Elem(881020), Elem(3043755), Elem(7566557), Elem(4217293), Elem(3866709), Elem(6378193), Elem(1638663), Elem(5182813), Elem(3585619), Elem(4557192), Elem(311672), Elem(3454425), Elem(550594), Elem(5381645), Elem(1181451), Elem(3050496), Elem(4511849), Elem(337424), Elem(4232382), Elem(8292339), Elem(5331359), Elem(6149482), Elem(7592792), Elem(4979100), Elem(6436449), Elem(6076034), Elem(1480116), Elem(7199367), Elem(2986939), Elem(8269679), Elem(7651577), Elem(974410), Elem(599790), Elem(6389436), Elem(3679789), Elem(7377166), Elem(3031635), Elem(3132387), Elem(2167661), Elem(6407623), Elem(30596), Elem(7525758), Elem(4232690), Elem(7366367), Elem(8085854), Elem(1818383), Elem(4474601), Elem(3307063), Elem(6760253), Elem(2836110), Elem(7231279), Elem(3198774), Elem(8311608), Elem(1420383), Elem(6677121), Elem(6254890), Elem(6860824), Elem(6355288), Elem(2488086), Elem(5928447), Elem(925928), Elem(6388242), Elem(2179469), Elem(4570371), Elem(3533806), Elem(6618932), Elem(1973510), Elem(798918), Elem(2694314), Elem(3729531), Elem(3652209), Elem(2777756), Elem(2108394), Elem(583127), Elem(2704829), Elem(7729590), Elem(6121197), Elem(92514), Elem(1013577), Elem(3301084), Elem(7463678), Elem(1771201), Elem(6001152), Elem(5716013), Elem(1597469), Elem(1684977), Elem(1996563), Elem(2478405), Elem(1794338), Elem(1302901), Elem(2300574), Elem(3115168), Elem(6019732), Elem(3316082), Elem(118142), Elem(2681633), Elem(3520723), Elem(1014086), Elem(8174295), Elem(1095836), Elem(4292827), Elem(2926471), Elem(5581735), Elem(1019416), Elem(6743228), Elem(3752307), Elem(4758878), Elem(728249), Elem(1755495), Elem(851184), Elem(412135), Elem(897227), Elem(418391), Elem(7757022), Elem(685666), Elem(7369140), Elem(3307289), Elem(2629982), Elem(6443730), Elem(4261841), Elem(7622369), Elem(2199324), Elem(3720721), Elem(1389850), Elem(62518), Elem(7323536), Elem(876617), Elem(6850353), Elem(8311852), Elem(92441), Elem(5397396), Elem(6976171), Elem(6247444), Elem(1844726)])), NttPolynomial(Array([Elem(5868009), Elem(2163464), Elem(473897), Elem(8030826), Elem(6357445), Elem(7382613), Elem(7797426), Elem(1581759), Elem(4077709), Elem(992850), Elem(1562085), Elem(1350912), Elem(2266888), Elem(781674), Elem(4548561), Elem(1118759), Elem(7394435), Elem(4914754), Elem(5180760), Elem(2262460), Elem(3349740), Elem(602188), Elem(1627253), Elem(2164715), Elem(5859189), Elem(1080450), Elem(3274558), Elem(2327764), Elem(1519006), Elem(7355479), Elem(4286874), Elem(1927539), Elem(7193145), Elem(1836043), Elem(6830917), Elem(5854866), Elem(2565027), Elem(4077609), Elem(2197034), Elem(3772137), Elem(660075), Elem(7971457), Elem(5048271), Elem(694813), Elem(8269574), Elem(8182713), Elem(2898980), Elem(4728005), Elem(5809522), Elem(4155640), Elem(3520829), Elem(3365581), Elem(8304095), Elem(5284797), Elem(1662088), Elem(3546244), Elem(2315372), Elem(7079872), Elem(614696), Elem(2729923), Elem(1197907), Elem(7372252), Elem(4284708), Elem(3218453), Elem(83376), Elem(7293815), Elem(1921065), Elem(2479178), Elem(4645379), Elem(5965691), Elem(7493667), Elem(4288448), Elem(175931), Elem(2443078), Elem(1288581), Elem(8122048), Elem(6702456), Elem(8231939), Elem(3519967), Elem(1854317), Elem(1447261), Elem(7439328), Elem(4674263), Elem(8375068), Elem(2871470), Elem(6139567), Elem(1946893), Elem(1254782), Elem(6232900), Elem(4366793), Elem(7774835), Elem(4685295), Elem(3126533), Elem(2579630), Elem(6925592), Elem(7988372), Elem(5028165), Elem(4971975), Elem(2165657), Elem(3544969), Elem(2491160), Elem(1042883), Elem(8237510), 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Elem(5055556), Elem(5610335), Elem(3130070), Elem(6286697), Elem(6816407), Elem(7178071), Elem(772106), Elem(1540004), Elem(1576638), Elem(2358802), Elem(5617339), Elem(1850277), Elem(1970399), Elem(7205351), Elem(941183), Elem(3293340), Elem(6355127), Elem(4903784), Elem(5242758), Elem(7864447), Elem(2053702), Elem(360219), Elem(2187306), Elem(3710459), Elem(2533858), Elem(2204649), Elem(5824233), Elem(4124375), Elem(8259854), Elem(1099525), Elem(5835439), Elem(7336821), Elem(36624), Elem(3472971), Elem(5659519), Elem(4570005), Elem(7768263), Elem(5994874), Elem(3683492), Elem(7092280), Elem(7973678), Elem(3220600), Elem(146676), Elem(2964908), Elem(7649261), Elem(7759724), Elem(7716997), Elem(7228240), Elem(6459883), Elem(5135883), Elem(7888345), Elem(3427496), Elem(2140850), Elem(6797247), Elem(6625548), Elem(6094305), Elem(5586369), Elem(5545258), Elem(2278475), Elem(1600874), Elem(6857050), Elem(1875336), Elem(8213350), Elem(1379067), Elem(6195853), Elem(1847096), Elem(3430960), Elem(7879575), Elem(5941741), Elem(3956522), Elem(1352199), Elem(7509852), Elem(7510154), Elem(6930157), Elem(405782), Elem(3797278), Elem(7708658), Elem(3607296), Elem(6960679), Elem(7693020), Elem(3706878), Elem(4003284), Elem(2263817), Elem(6023721), Elem(637648), Elem(6448053)])), NttPolynomial(Array([Elem(3816033), Elem(2695499), Elem(7684200), Elem(7481611), Elem(3812957), Elem(714045), Elem(8350999), Elem(1859077), Elem(3044660), Elem(1309241), Elem(6616588), Elem(2494021), Elem(4222968), Elem(6118162), Elem(7954660), Elem(4122003), Elem(2891415), Elem(6821425), Elem(546197), Elem(6871305), Elem(6946357), Elem(1696751), Elem(2067246), Elem(8214155), Elem(1860726), Elem(4035865), Elem(5574134), Elem(7021593), Elem(6626049), Elem(3175378), Elem(1672783), Elem(3105862), Elem(7073656), Elem(5587158), Elem(4704184), Elem(5002713), Elem(5825632), Elem(2529272), Elem(6965317), Elem(4501175), Elem(3027972), Elem(3353942), Elem(5838464), Elem(6891626), Elem(6123331), Elem(1380779), 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Elem(8198814), Elem(5824317), Elem(5888425), Elem(5076119), Elem(3144715), Elem(4709351), Elem(3968920), Elem(2812853), Elem(7581590), Elem(3721490), Elem(2659020), Elem(2238406), Elem(8202305), Elem(264667), Elem(4724258), Elem(8113238), Elem(756217), Elem(2780148), Elem(3622587), Elem(7312216), Elem(996372), Elem(5420108), Elem(8123741), Elem(6510953), Elem(3513714), Elem(8218209), Elem(7266331), Elem(2322362), Elem(5245703), Elem(4541375), Elem(4140760), Elem(3984192), Elem(6727892), Elem(2889686), Elem(5424639), Elem(179310), Elem(3940214), Elem(5429947), Elem(4825475), Elem(4548191), Elem(1048051), Elem(4127937), Elem(6877072), Elem(3556403), Elem(5748085), Elem(4573163), Elem(1241769), Elem(729066), Elem(8206832), Elem(1589690), Elem(1066717), Elem(1991758), Elem(2087911), Elem(1237938), Elem(6938277), Elem(609740), Elem(4707738), Elem(3387380), Elem(2300645), Elem(7006744), Elem(6819728), Elem(863076), Elem(3432450), Elem(6690973), Elem(3637341), Elem(6077646)])), NttPolynomial(Array([Elem(4937330), Elem(599434), Elem(182675), Elem(503301), Elem(6976184), Elem(7894445), Elem(7672523), Elem(7677812), Elem(6312541), Elem(4626451), Elem(1755168), Elem(509655), Elem(4783521), Elem(7631106), Elem(1794025), Elem(2166249), Elem(2372841), Elem(5859822), Elem(4285643), Elem(1906742), Elem(2463016), Elem(7271365), Elem(4310048), Elem(837107), Elem(5363406), Elem(3869872), Elem(3313143), Elem(5229174), Elem(8003915), Elem(3346886), Elem(296912), Elem(3613179), Elem(2330880), Elem(3201127), Elem(7023913), Elem(6523936), Elem(7661066), Elem(2520039), Elem(2072355), Elem(6085299), Elem(6224759), Elem(3901759), Elem(5535157), Elem(5931555), Elem(6718171), Elem(3602450), Elem(5778663), Elem(7421670), Elem(8230426), Elem(8335369), Elem(4308909), Elem(7848782), Elem(215539), Elem(8301487), Elem(4823184), Elem(2114316), Elem(7063517), Elem(477631), Elem(7605136), Elem(6029160), Elem(7595865), Elem(8027746), Elem(7632281), Elem(1485811), Elem(4158856), Elem(1941236), Elem(980776), Elem(3169303), Elem(7617704), Elem(6343456), Elem(7828751), Elem(5604999), Elem(3288589), Elem(6115812), Elem(5627380), Elem(7422782), Elem(259427), Elem(3800188), Elem(3515840), Elem(8007673), Elem(6916846), Elem(2584292), Elem(8303795), Elem(4143943), Elem(2285823), Elem(6815811), Elem(5511411), Elem(7413982), Elem(5103764), Elem(4583682), Elem(4650754), Elem(2883333), Elem(4010195), Elem(1449411), Elem(4522684), Elem(6761228), Elem(2480822), Elem(1531886), Elem(2965737), Elem(4059799), Elem(464262), Elem(7292619), Elem(5840702), Elem(3603327), Elem(5705618), Elem(87020), Elem(3361746), Elem(3942053), Elem(2272514), Elem(3611823), Elem(6209797), Elem(6865590), Elem(7301799), Elem(4198657), Elem(1155904), Elem(2374570), Elem(8292763), Elem(7913305), Elem(8241384), Elem(1683746), Elem(2516511), Elem(5234924), Elem(6967754), Elem(6154448), Elem(8358450), Elem(411124), Elem(1105995), Elem(5664453), Elem(1416273), Elem(3911222), Elem(4311605), Elem(6868038), Elem(5379504), Elem(7400082), Elem(5774862), Elem(3081559), Elem(4584230), Elem(4970362), Elem(1719096), Elem(3204102), Elem(4394790), Elem(742215), Elem(8152534), Elem(7975478), Elem(7619736), Elem(6322445), Elem(661843), Elem(149766), Elem(2229288), Elem(5799814), Elem(7560487), Elem(7005174), Elem(2946157), Elem(3777833), Elem(6436627), Elem(6352413), Elem(3422595), Elem(2846591), Elem(1491583), Elem(8218869), Elem(5443075), Elem(3371498), Elem(422324), Elem(8247374), Elem(314339), Elem(6123344), Elem(5181419), Elem(1701724), Elem(349722), Elem(6139327), Elem(5873640), Elem(3813767), Elem(990705), Elem(437068), Elem(7855504), Elem(3229603), Elem(5225320), Elem(1214449), Elem(6839977), Elem(243390), Elem(7320640), Elem(2972119), Elem(7046719), Elem(2071994), Elem(1602832), Elem(1205636), Elem(5061835), Elem(3481680), Elem(1801010), Elem(1156219), Elem(3632599), Elem(3076756), Elem(5416458), Elem(7681978), Elem(5828694), Elem(4018146), Elem(5349008), Elem(925041), Elem(947531), Elem(6782681), Elem(1002025), Elem(3854194), Elem(2621147), Elem(7839685), Elem(5802464), Elem(7461704), Elem(2694139), Elem(7661926), Elem(6130705), Elem(3063061), Elem(4558851), Elem(8312537), Elem(2279448), Elem(1352192), Elem(7001478), Elem(5260758), Elem(1195610), Elem(7590013), Elem(3339732), Elem(3388422), Elem(3677283), Elem(3645956), Elem(685437), Elem(3923886), Elem(3324606), Elem(3430881), Elem(4032147), Elem(526574), Elem(2236312), Elem(5791927), Elem(3144736), Elem(2092529), Elem(109667), Elem(6402733), Elem(2842582), Elem(1932928), Elem(3292828), Elem(2255053), Elem(3594400), Elem(5655767), Elem(4353137), Elem(8343434), Elem(7138618), Elem(6677945), Elem(1141658), Elem(3101792), Elem(5561571), Elem(7217755), Elem(6543487), Elem(560532), Elem(2314081), Elem(8175896), Elem(1567749), Elem(4745166), Elem(7473853), Elem(5699644)])), NttPolynomial(Array([Elem(628061), Elem(7634672), Elem(7213734), Elem(2629411), Elem(6239525), Elem(6004088), Elem(4414375), Elem(1547173), Elem(8258366), Elem(5520328), Elem(7447188), Elem(1606067), Elem(7118854), Elem(658804), Elem(7886514), Elem(5679246), Elem(6774626), Elem(6151370), Elem(7350110), Elem(4827037), Elem(6167821), Elem(3108332), Elem(793086), Elem(6807775), Elem(279178), Elem(1733287), Elem(7739218), Elem(6738942), Elem(6992665), Elem(7624533), Elem(1599003), Elem(2067779), Elem(3861018), Elem(3467363), Elem(3992859), Elem(5325073), Elem(3454603), Elem(6358806), Elem(3647951), Elem(5478861), Elem(1996559), Elem(975781), Elem(4839637), Elem(3487925), Elem(305313), Elem(7681135), Elem(6226493), Elem(166677), Elem(4247430), Elem(6410506), Elem(7889961), Elem(85392), Elem(1639639), Elem(541652), Elem(2107729), Elem(7229635), Elem(5811673), Elem(5409987), Elem(3934882), Elem(710506), Elem(7376355), Elem(415450), Elem(2226088), Elem(6117501), Elem(4013784), Elem(4942957), Elem(5625106), Elem(7439204), Elem(1838060), Elem(3380367), Elem(6130635), Elem(4791175), Elem(929331), Elem(4185141), Elem(2605251), Elem(6564031), Elem(2013956), Elem(701606), Elem(552905), Elem(6175762), Elem(8207009), Elem(690864), Elem(4793236), Elem(4125172), Elem(6170799), Elem(5549604), Elem(5349991), Elem(2371729), Elem(3786892), Elem(3858973), Elem(5543728), Elem(7184383), Elem(577018), Elem(6916928), Elem(7248996), Elem(995950), Elem(694553), Elem(4532302), Elem(6889758), Elem(6721926), Elem(5272741), Elem(2286703), Elem(7295734), Elem(3986577), Elem(1154201), Elem(2161194), Elem(6485694), Elem(5792463), Elem(7211125), Elem(7276232), Elem(2230106), Elem(2332517), Elem(4625929), Elem(4553695), Elem(7287309), Elem(6396546), Elem(6281592), Elem(5524140), Elem(2367339), Elem(4930948), Elem(1063096), Elem(1828076), Elem(3055381), Elem(4788618), Elem(5497829), Elem(6343994), Elem(2285129), Elem(1560020), Elem(5567448), Elem(776161), Elem(2745202), Elem(3005694), Elem(5483176), Elem(5227951), Elem(2704130), Elem(2722921), Elem(1366314), Elem(2789681), Elem(1449194), Elem(4548995), Elem(2815578), Elem(8277022), Elem(5225035), Elem(1533260), Elem(830322), Elem(3961551), Elem(914622), Elem(5345461), Elem(4214913), Elem(409160), Elem(3457904), Elem(790536), Elem(6611182), Elem(6700443), Elem(6933285), Elem(1629976), Elem(4705213), Elem(829914), Elem(725753), Elem(4998807), Elem(8128583), Elem(3707607), Elem(5668724), Elem(5923752), Elem(180289), Elem(225173), Elem(4145066), Elem(5142867), Elem(7945450), Elem(7583171), Elem(4736731), Elem(7770535), Elem(1471254), Elem(3763241), Elem(4985875), Elem(7877425), Elem(324766), Elem(3368587), Elem(5012758), Elem(2984952), Elem(4277840), Elem(1581827), Elem(1861115), Elem(1714294), Elem(6398821), Elem(7084889), Elem(813945), Elem(38641), Elem(3123671), Elem(2162908), Elem(4767201), Elem(6947733), Elem(323569), Elem(6408179), Elem(6819998), Elem(3309426), Elem(7444246), Elem(4345529), Elem(5740327), Elem(314864), Elem(6025341), Elem(1801548), Elem(2836017), Elem(4889598), Elem(659187), Elem(865578), Elem(2967462), Elem(7297316), Elem(186244), Elem(7285381), Elem(5344646), Elem(6870977), Elem(6241298), Elem(6416606), Elem(3602753), Elem(7170287), Elem(1480383), Elem(6001213), Elem(2418737), Elem(1412648), Elem(1616546), Elem(7282859), Elem(4700805), Elem(704591), Elem(1658841), Elem(7079292), Elem(4002873), Elem(1544567), Elem(3743323), Elem(3881644), Elem(2729169), Elem(2055790), Elem(2710990), Elem(1105209), Elem(5831962), Elem(5955134), Elem(5986976), Elem(3116870), Elem(3808670), Elem(189295), Elem(7888046), Elem(2466667), Elem(6836090), Elem(170739), Elem(7085532), Elem(4197279), Elem(6521950), Elem(1470547), Elem(6543765), Elem(3572251), Elem(2269314), Elem(5997322), Elem(4980871), Elem(3132706), Elem(7380920), Elem(2581742)])), NttPolynomial(Array([Elem(6927279), Elem(890785), Elem(1721029), Elem(2454991), Elem(2111442), Elem(7372403), Elem(2883457), Elem(6265505), Elem(5610032), Elem(5745879), Elem(7131232), Elem(1498452), Elem(5947408), Elem(7637830), Elem(6878980), Elem(73829), Elem(7153032), Elem(4287414), Elem(4298538), Elem(2362409), Elem(6942651), Elem(3851343), Elem(5139836), Elem(7254830), Elem(6158406), Elem(3806544), Elem(395213), Elem(4507980), Elem(2349726), Elem(3205387), Elem(4835980), Elem(3200019), Elem(5462012), Elem(4997718), Elem(3282511), Elem(2496653), Elem(6442232), Elem(4253609), Elem(3819692), Elem(4402337), Elem(4697234), Elem(3831701), Elem(1944678), Elem(7576665), Elem(3350266), Elem(8362524), Elem(6858037), Elem(7529938), Elem(1640150), Elem(5083546), Elem(1692135), Elem(2437082), Elem(122298), Elem(5454364), Elem(3309399), Elem(1808809), Elem(8079373), Elem(2075829), Elem(6641814), Elem(5354556), Elem(7439153), Elem(4596714), Elem(5659450), Elem(4594451), Elem(5562066), Elem(1708201), Elem(7367481), Elem(597288), Elem(4125988), Elem(7466272), Elem(4688079), Elem(7978514), Elem(201493), Elem(1859368), Elem(5822360), Elem(5739295), Elem(4957699), Elem(221146), Elem(6066374), Elem(2195943), Elem(5487206), Elem(8214369), Elem(5678095), Elem(4688335), Elem(2118566), Elem(1562599), Elem(1776066), Elem(2354406), Elem(6297033), Elem(1046690), Elem(5248956), Elem(6124505), Elem(6591752), Elem(5296833), Elem(706161), Elem(5399493), Elem(6609767), Elem(3511590), Elem(2992496), Elem(2807059), Elem(5066939), Elem(764684), Elem(2723889), Elem(2010230), Elem(3785039), Elem(7266522), Elem(2070030), Elem(5988514), Elem(524509), Elem(589499), Elem(4194129), Elem(2866925), Elem(536235), Elem(3977716), Elem(5117974), Elem(719674), Elem(854822), Elem(842572), Elem(2405460), Elem(2640028), Elem(6373608), Elem(4811855), Elem(5204159), Elem(1963820), Elem(4980025), Elem(4473953), Elem(7436759), Elem(2500181), Elem(4158136), Elem(3673031), Elem(911158), Elem(2299321), Elem(5652682), Elem(7614019), Elem(6134522), Elem(7217263), Elem(2748227), Elem(2273364), Elem(1422), Elem(1175599), Elem(8311881), Elem(4355510), Elem(961073), Elem(7750267), Elem(6098047), Elem(1685809), Elem(5573922), Elem(14583), Elem(2775901), Elem(3720268), Elem(719571), Elem(458329), Elem(983407), Elem(8333418), Elem(4162405), Elem(8099030), Elem(6650077), Elem(7053064), Elem(3781889), Elem(3470882), Elem(2750838), Elem(1881531), Elem(1822104), Elem(6899429), Elem(4103774), Elem(4630488), Elem(6912768), Elem(1919807), Elem(8012670), Elem(3842630), Elem(2809691), Elem(7072467), Elem(6736124), Elem(7941444), Elem(7931762), Elem(1664206), Elem(2935083), Elem(2105305), Elem(6674909), Elem(7291452), Elem(2814809), Elem(5698203), Elem(8206072), Elem(1909258), Elem(4523641), Elem(6135501), Elem(1756586), Elem(5163251), Elem(4184700), Elem(3380883), Elem(5048505), Elem(3279096), Elem(7576993), Elem(4457232), Elem(593565), Elem(7801404), Elem(6207804), Elem(7174438), Elem(6209883), Elem(3040477), Elem(3110324), Elem(3970799), Elem(8148397), Elem(3967197), Elem(2454623), Elem(4112399), Elem(882083), Elem(5175872), Elem(2524769), Elem(2101511), Elem(6377806), Elem(4699590), Elem(920730), Elem(6795042), Elem(6659259), Elem(4008510), Elem(3645138), Elem(8232564), Elem(5714067), Elem(8044719), Elem(3243158), Elem(4371430), Elem(4577939), Elem(4722768), Elem(7072874), Elem(164498), Elem(886645), Elem(5720071), Elem(723678), Elem(4852844), Elem(4980780), Elem(103971), Elem(4256432), Elem(2728565), Elem(6995347), Elem(583897), Elem(7628708), Elem(388373), Elem(2897251), Elem(8078034), Elem(7699306), Elem(6524067), Elem(3042230), Elem(1511802), Elem(3476006), Elem(2156894), Elem(6146418), Elem(296719), Elem(4572478), Elem(8144871), Elem(5661956), Elem(4895198), Elem(4480288), Elem(2924151), Elem(1713997), Elem(6219682)])), NttPolynomial(Array([Elem(7883870), Elem(6891823), Elem(113175), Elem(1780254), Elem(7299748), Elem(7886120), Elem(2243094), Elem(1705104), Elem(3796735), Elem(7438363), Elem(560317), Elem(5187261), Elem(6630872), Elem(7234911), Elem(7318974), Elem(4951667), Elem(6431400), Elem(7980540), Elem(1769232), Elem(5094095), Elem(2395430), Elem(2065752), Elem(5449083), Elem(2632694), Elem(7819659), Elem(3399882), Elem(3902207), Elem(2573385), 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Elem(1653681), Elem(7504161), Elem(6530666), Elem(2724331), Elem(4858238), Elem(2326132), Elem(2355841), Elem(4153666), Elem(387827), Elem(5408297), Elem(21812), Elem(900722), Elem(3015463), Elem(137335), Elem(2308359), Elem(5163452), Elem(7770775), Elem(7203723), Elem(1954938), Elem(4220118), Elem(7182321), Elem(2892935), Elem(6339704), Elem(8031227), Elem(6810432), Elem(5542938), Elem(1338387), Elem(2420016), Elem(8304737), Elem(2545574), Elem(223571), Elem(7822301), Elem(285486), Elem(1457593), Elem(2458702), Elem(220905), Elem(8091046), Elem(3374164), Elem(5521587), Elem(7634828), Elem(2054180), Elem(1112804), Elem(4114603), Elem(107868), Elem(5718687), Elem(7820961), Elem(7099818), Elem(716011), Elem(7653865), Elem(3048817), Elem(4384263), Elem(4561925), Elem(8250625), Elem(4344282), Elem(3980961), Elem(8092521), Elem(2950488), Elem(6952470), Elem(4216822), Elem(8197894), Elem(2842694), Elem(2152019), Elem(6893052), Elem(4651337), Elem(5395840), Elem(3822542), Elem(4080674), Elem(6928623), Elem(3686778), Elem(3254214), Elem(5013714), Elem(1814223), Elem(6883085), Elem(6336980), Elem(2612179), Elem(2139560), Elem(1624935), Elem(6753103), Elem(8274501), Elem(3382601), Elem(2305932), Elem(2606859), Elem(3861762), Elem(5527265), Elem(4535486), Elem(1547083), Elem(181793), Elem(1785865), Elem(4748603), Elem(7844289), Elem(3528779), Elem(663143), Elem(6810750), Elem(4804814)])), NttPolynomial(Array([Elem(271037), Elem(7507918), Elem(1904152), Elem(5593956), Elem(3882023), Elem(3164002), Elem(1525941), Elem(6409484), Elem(6698970), Elem(6115948), Elem(3446447), Elem(5400864), Elem(204410), Elem(5811804), Elem(3781075), Elem(6660751), Elem(665039), Elem(4534124), Elem(2299905), Elem(3131015), Elem(1050664), Elem(7449005), Elem(4006740), Elem(8111521), Elem(6479477), Elem(2592387), Elem(2282325), Elem(2610142), Elem(7668705), Elem(7879652), Elem(5210690), Elem(3911854), Elem(542307), Elem(1470172), Elem(1623631), Elem(7429199), Elem(1136416), Elem(6269739), Elem(92782), Elem(6091328), Elem(5983309), Elem(2416984), Elem(6308622), Elem(8137913), Elem(6586022), Elem(8118521), Elem(3274635), Elem(4195108), Elem(40494), Elem(4824983), Elem(2186392), Elem(3218840), Elem(6158776), Elem(6594534), Elem(7569409), Elem(2173898), Elem(4194906), Elem(3849934), Elem(3848465), Elem(138306), Elem(4133337), Elem(3272598), Elem(724606), Elem(147436), Elem(2847098), Elem(1115308), Elem(2701505), Elem(5054708), Elem(4752420), Elem(7035706), Elem(4782613), Elem(4910534), Elem(6887052), Elem(6421718), Elem(8320262), Elem(532784), Elem(5369286), Elem(3318443), Elem(1353502), Elem(1341095), Elem(8268059), Elem(759006), Elem(484689), Elem(5238762), Elem(1665269), Elem(5857938), Elem(2161758), Elem(3429018), Elem(2324417), Elem(900457), Elem(4376673), Elem(6712890), Elem(6106059), Elem(4826694), Elem(958491), Elem(4348523), Elem(594295), Elem(5302355), Elem(3391467), Elem(1854128), Elem(7041865), Elem(2431688), Elem(2253285), Elem(656007), Elem(5176933), Elem(6857900), Elem(7296640), Elem(7290578), Elem(3428821), Elem(2216524), Elem(1824271), Elem(1230211), Elem(3670249), Elem(4032081), Elem(5990630), Elem(4856255), Elem(721898), Elem(4721709), Elem(6109712), Elem(3365999), Elem(1550188), Elem(7297989), Elem(6942012), Elem(6707203), Elem(1674359), Elem(253674), Elem(2153994), Elem(5091345), Elem(2853212), Elem(2444301), Elem(658192), Elem(4598975), Elem(871308), Elem(4840600), Elem(5224657), Elem(2265444), Elem(5884913), Elem(4109160), Elem(7516700), Elem(603905), Elem(1670562), Elem(4225442), Elem(7736307), Elem(4001073), Elem(7052654), Elem(1131101), Elem(526558), Elem(6411079), Elem(6951824), Elem(1125725), Elem(5305402), Elem(6271327), Elem(4858261), Elem(719588), Elem(7662907), Elem(1267314), Elem(7547741), Elem(6266759), Elem(3403723), Elem(222198), Elem(2583356), Elem(2583036), Elem(6347555), Elem(5030537), Elem(355004), Elem(7492881), Elem(6632498), Elem(5184059), Elem(1354205), Elem(6961055), Elem(5687961), Elem(4335573), Elem(7787496), Elem(774871), Elem(2929279), Elem(4869161), Elem(2616674), Elem(6938768), Elem(1501597), Elem(5206596), Elem(1486409), Elem(2163105), Elem(4934328), Elem(1573115), Elem(4613355), Elem(8079032), Elem(3155040), Elem(7558966), Elem(52465), Elem(5502484), Elem(4490004), Elem(1242868), Elem(5504733), Elem(6901277), Elem(2940116), Elem(7817163), Elem(5457158), Elem(188421), Elem(5764480), Elem(7352656), Elem(4430695), Elem(7034159), Elem(340566), Elem(3307064), Elem(900095), Elem(3221621), Elem(1766906), Elem(933906), Elem(6578373), Elem(5038359), Elem(7662015), Elem(2452404), Elem(4326698), Elem(7936344), Elem(5334476), Elem(1873208), Elem(5823600), Elem(7618366), Elem(4981595), Elem(7755668), Elem(4630512), Elem(3091600), Elem(5792337), Elem(4013431), Elem(7961688), Elem(6348861), Elem(6845185), Elem(608397), Elem(7652749), Elem(359735), Elem(3627652), Elem(8352926), Elem(4463632), Elem(3173791), Elem(888843), Elem(1608657), Elem(3717361), Elem(5229018), Elem(7312858), Elem(1158752), Elem(1368958), Elem(3789696), Elem(3502678), Elem(8174192), Elem(6556556), Elem(6615724), Elem(4959805), Elem(4608195), Elem(4209042), Elem(7409975), Elem(18647), Elem(4913304), Elem(5602935), Elem(5947885), Elem(5342460), Elem(7010193)])), NttPolynomial(Array([Elem(5987586), Elem(6466200), Elem(6077231), Elem(4732956), Elem(187686), Elem(4216527), Elem(4865119), Elem(2195818), Elem(2356640), Elem(6314669), Elem(909549), Elem(5444054), Elem(4044112), Elem(3671418), Elem(30568), Elem(3443723), Elem(7807172), Elem(4868653), Elem(6781845), Elem(4750386), Elem(4805873), Elem(1955779), Elem(7580945), Elem(5117012), Elem(975311), Elem(3738650), Elem(6015173), Elem(4039870), Elem(2096012), Elem(1269937), Elem(4877229), Elem(265083), Elem(3005612), Elem(8351854), Elem(6855870), Elem(3747361), Elem(4018619), Elem(3820952), Elem(7449583), Elem(4478706), Elem(2221098), Elem(44166), Elem(5236398), Elem(1920935), Elem(5419405), Elem(6092919), Elem(4252105), Elem(8235555), Elem(7675345), Elem(7622719), Elem(5806103), Elem(2160542), Elem(1901167), Elem(5512102), Elem(8299953), Elem(5846384), Elem(3713362), Elem(7917689), Elem(449124), Elem(6129409), Elem(6669385), Elem(3928781), Elem(724674), Elem(7058307), Elem(2070709), Elem(2206491), Elem(5858475), Elem(2009622), Elem(1052394), Elem(1944796), Elem(264286), Elem(1331007), Elem(458002), Elem(4707990), Elem(1853625), Elem(5433137), Elem(1422903), Elem(5382824), Elem(5440692), Elem(1029288), Elem(203515), Elem(94497), Elem(6048485), Elem(7366950), Elem(5589905), Elem(6591192), Elem(5967763), Elem(5418864), Elem(7563294), Elem(7208663), Elem(5678606), Elem(5121549), Elem(7246170), Elem(1618979), Elem(6129874), Elem(4779112), Elem(5182830), Elem(1778892), Elem(2448555), Elem(1175202), Elem(4136400), Elem(3371652), Elem(250474), Elem(4348219), Elem(581501), Elem(5390219), Elem(444319), Elem(1497445), Elem(1390111), Elem(6469906), Elem(1392758), Elem(2563202), Elem(7625845), Elem(5756965), Elem(4069812), Elem(1660287), Elem(6800373), Elem(3074985), Elem(3953883), Elem(2740396), Elem(5302814), Elem(1373550), Elem(1589954), Elem(4663975), Elem(4989511), Elem(5463179), Elem(5148901), Elem(7606577), Elem(1908801), Elem(3063005), Elem(6835260), Elem(601995), Elem(3222912), Elem(7735920), Elem(3321758), Elem(4786121), Elem(5460430), Elem(2175824), Elem(2921982), Elem(6815618), Elem(1378641), Elem(4082906), Elem(2083155), Elem(5163940), Elem(8165193), Elem(58160), Elem(2690447), Elem(8251606), Elem(73215), Elem(7410506), Elem(6480711), Elem(1376278), Elem(3340233), Elem(4293828), Elem(8158754), Elem(6852511), Elem(7620188), Elem(7503370), Elem(5411109), Elem(7419852), Elem(7246951), Elem(220425), Elem(5551837), Elem(6498149), Elem(5092441), Elem(4165370), Elem(5929038), Elem(5450081), Elem(5732170), Elem(8053756), Elem(8098095), Elem(432189), Elem(1097037), Elem(3021435), Elem(2489146), Elem(5816705), Elem(1019673), Elem(307053), Elem(3014762), Elem(1242498), Elem(1850131), Elem(73238), Elem(3087567), Elem(4026647), Elem(7011863), Elem(284897), Elem(3432023), Elem(8148534), Elem(4531204), Elem(1367260), Elem(3133101), Elem(5627055), Elem(2338843), Elem(4983187), Elem(1618844), Elem(4480458), Elem(204379), Elem(5318100), Elem(6074256), Elem(746998), Elem(7589743), Elem(1137291), Elem(1399557), Elem(6073596), Elem(2540287), Elem(6686771), Elem(6531649), Elem(4227560), Elem(4383132), Elem(8341054), Elem(2532803), Elem(2495257), Elem(7647876), Elem(4664599), Elem(3838779), Elem(2021098), Elem(583421), Elem(6436655), Elem(7273918), Elem(2370689), Elem(3003734), Elem(784500), Elem(2916660), Elem(5705400), Elem(7979419), Elem(3105626), Elem(3555385), Elem(3661582), Elem(922631), Elem(6062579), Elem(4081752), Elem(5740441), Elem(5652557), Elem(6645807), Elem(7408189), Elem(6448036), Elem(3882825), Elem(3204212), Elem(777393), Elem(6325612), Elem(2664471), Elem(5386204), Elem(3643357), Elem(3864286), Elem(1775035), Elem(2115317), Elem(4935388), Elem(7315958), Elem(399087), Elem(6258163), Elem(200398), Elem(5012830), Elem(7206386), Elem(3917277), Elem(7608816), Elem(619821)]))])), NttVector(Array([NttPolynomial(Array([Elem(3872740), Elem(866695), Elem(2803642), Elem(5153509), Elem(6066646), Elem(7828486), Elem(8301585), Elem(6434674), Elem(4543535), Elem(4934218), Elem(792445), Elem(6457533), Elem(3086710), Elem(2864078), Elem(1020878), Elem(1066611), Elem(8323343), Elem(2331961), Elem(3345088), Elem(5416411), Elem(7848005), Elem(2452596), Elem(3232174), Elem(6794414), Elem(1963728), Elem(3397095), Elem(6255540), Elem(7958813), Elem(6187169), Elem(5117515), Elem(5954491), Elem(4560748), Elem(3899763), Elem(942529), Elem(882960), Elem(5617470), Elem(6557743), Elem(5296784), Elem(2912737), Elem(57954), Elem(1663367), Elem(4092395), Elem(7095500), Elem(2035180), Elem(7330994), Elem(4529094), Elem(5884323), Elem(807682), Elem(3853874), Elem(2638067), Elem(3866560), Elem(8119253), Elem(6141000), Elem(338791), Elem(2953919), Elem(523739), Elem(193697), 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Elem(2205789), Elem(2500910), Elem(6570873), Elem(4534916), Elem(583071), Elem(4483335), Elem(969002), Elem(7127873), Elem(2818012), Elem(3508312), Elem(6865537), Elem(6615489), Elem(875094), Elem(7542713), Elem(1224827), Elem(249627), Elem(7771725), Elem(6205087), Elem(8058452), Elem(4356683), Elem(5478378), Elem(8276910), Elem(4598503), Elem(4225024), Elem(7248295), Elem(246679), Elem(7243322), Elem(1744700), Elem(2852646), Elem(2576174), Elem(2554314), Elem(1832673), Elem(3979169), Elem(3827966), Elem(3788923), Elem(3110344), Elem(759892), Elem(809549), Elem(1973395), Elem(1122129), Elem(7719801), Elem(7271964), Elem(5241840), Elem(8084463), Elem(6626879), Elem(5401006), Elem(7758656), Elem(2033317), Elem(4916383), Elem(1016248), Elem(2429467), Elem(3490940), Elem(7735630), Elem(6325811), Elem(5287925), Elem(6676104), Elem(2210093), Elem(2432854), Elem(3505158), Elem(7405406), Elem(8145913), Elem(7230561), Elem(2113071), Elem(6038500), Elem(1434529), Elem(3490295), Elem(2992453), 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NttPolynomial(Array([Elem(204097), Elem(6033627), Elem(663896), Elem(2126366), Elem(3154505), Elem(7860462), Elem(1466610), Elem(2885591), Elem(1845863), Elem(31383), Elem(7586250), Elem(1175129), Elem(3762547), Elem(7818735), Elem(5274621), Elem(91445), Elem(7120023), Elem(6581161), Elem(1703599), Elem(3891626), Elem(1288117), Elem(2358320), Elem(1473864), Elem(7512134), Elem(7514990), Elem(7933438), Elem(1353895), Elem(3545056), Elem(5770861), Elem(7613006), Elem(1395633), Elem(7763881), Elem(3163102), Elem(7260121), Elem(3937876), Elem(1983341), Elem(5880492), Elem(7060312), Elem(1260425), Elem(4962908), Elem(4355026), Elem(1870440), Elem(4982562), Elem(7789864), Elem(7577540), Elem(6544595), Elem(274758), Elem(3418560), Elem(4419792), Elem(2174451), Elem(1091730), Elem(1435542), Elem(3408550), Elem(5418240), Elem(6862635), Elem(6580436), Elem(1433047), Elem(3432087), Elem(5281597), Elem(4086483), Elem(2742570), Elem(4354519), Elem(1148570), Elem(5520835), Elem(3524865), 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Elem(7957223), Elem(448622), Elem(3273359), Elem(8311994), Elem(2149664), Elem(4833061), Elem(4463171), Elem(2879344), Elem(6498435), Elem(6837896), Elem(6076913), Elem(6904422), Elem(4027515), Elem(971053), Elem(3069566), Elem(7401574), Elem(5617053), Elem(740695), Elem(4015358), Elem(1124074), Elem(2688697), Elem(2758897), Elem(3852464), Elem(4274053), Elem(4558117), Elem(4971719), Elem(4424267), Elem(2798362), Elem(1294725), Elem(4328943), Elem(3447653), Elem(5714268), Elem(5491273), Elem(2619558), Elem(7195626), Elem(969026), Elem(5066424), Elem(3100217), Elem(3190707), Elem(4291414), Elem(6751886), Elem(773348), Elem(6265212), Elem(377132), Elem(7901934), Elem(7984337), Elem(7999334), Elem(7694395), Elem(7001342), Elem(7857643), Elem(8097644), Elem(2346302), Elem(6799279), Elem(3647039), Elem(4094629), Elem(6169719), Elem(2895601), Elem(5538413), Elem(4330940), Elem(2286573), Elem(1749797), Elem(2080811), Elem(320211), Elem(5604620), Elem(6660688), Elem(2496716), Elem(1533029), Elem(1255687), Elem(3333468), Elem(6025976), Elem(7580829), Elem(5368335), Elem(6241685), Elem(1472397), Elem(6462468), Elem(1637002), Elem(6586947), Elem(5833381), Elem(1431568), Elem(908910), Elem(6010360), Elem(3036382), Elem(5056181), Elem(3797739), Elem(2006595), Elem(6850146), Elem(7092934), Elem(6968696), Elem(3221269), Elem(7705210), Elem(6967449), Elem(6171165), Elem(8049487), Elem(1702869), Elem(1126644), Elem(2947206), Elem(6097748), Elem(4440631), Elem(6053605), Elem(1560565), Elem(2256564), Elem(2265022), Elem(30860), Elem(148254), Elem(2873252), Elem(2065531), Elem(3232130), Elem(1730508), Elem(3675219), Elem(1103296), Elem(4645043), Elem(7769191), Elem(552731), Elem(4820319), Elem(686881), Elem(1758482), Elem(3815639), Elem(999226), Elem(5596148), Elem(4982829), Elem(4380672), Elem(925160), Elem(2722654), Elem(509334)])), NttPolynomial(Array([Elem(7006821), Elem(2400531), Elem(7623405), Elem(6301247), Elem(6952552), Elem(4826434), Elem(6232909), Elem(5599864), 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Elem(1649867), Elem(716406), Elem(7425192), Elem(649214), Elem(4190579), Elem(7091208), Elem(5196989), Elem(1640722), Elem(7794330), Elem(3222007), Elem(1039886), Elem(1771833), Elem(935287), Elem(1867976), Elem(2128686), Elem(3310006), Elem(7675390), Elem(1168731), Elem(678331), Elem(4547141), Elem(5934076), Elem(2707070), Elem(994828), Elem(8208705), Elem(2796873), Elem(3180707), Elem(718622), Elem(5329843), Elem(6689320), Elem(6663213), Elem(6445401), Elem(2297155), Elem(1200919), Elem(6780575), Elem(4707637), Elem(794478), Elem(6683592), Elem(3673993), Elem(7718937), Elem(1258518), Elem(692725), Elem(1078059), Elem(5938272), Elem(6269801), Elem(6537508), Elem(1211428), Elem(5348702), Elem(1245522), Elem(3553196), Elem(692535), Elem(2859831), Elem(4081060), Elem(4363666), Elem(662813), Elem(5997820), Elem(3701569), Elem(5782802), Elem(1925867), Elem(4076539), Elem(778562), Elem(4826506), Elem(2291964), Elem(682753), Elem(732285), Elem(3138711), Elem(970682), Elem(6407764), Elem(4486311), Elem(5351879), Elem(4500809), Elem(5187109), Elem(1922747), Elem(1989153), Elem(5080548), Elem(2805840), Elem(4639128), Elem(2819316), Elem(2312765), Elem(6060859), Elem(7831222), Elem(4674594), Elem(5606604), Elem(456140), Elem(5954361), Elem(5770348), Elem(6494148), Elem(753719), Elem(7100073), Elem(3981338), Elem(8286809), Elem(1270155), Elem(1112847), Elem(5140188), Elem(5040877), Elem(1331768), Elem(4168617), Elem(249117), Elem(6227778), Elem(1545427), Elem(5854318), Elem(1461433), Elem(1298491), Elem(888276), Elem(2271040), Elem(1012604), Elem(7750376), Elem(953183), Elem(4314553), Elem(5725830), Elem(3337730), Elem(4994284), Elem(1726687), Elem(369651), Elem(5572671), Elem(2512205), Elem(5825238), Elem(2636966), Elem(340535), Elem(8057540), Elem(5064388), Elem(5517341), Elem(1983827), Elem(4117010), Elem(3501918), Elem(975285), Elem(6870115), Elem(1757886), Elem(3811321), Elem(6756209), Elem(2318210), Elem(7442031), Elem(4201983), Elem(5299691), Elem(4793343), 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Elem(2554191), Elem(7211196), Elem(2946191), Elem(5928177), Elem(6914650), Elem(6801031), Elem(4358799), Elem(5130130), Elem(4843386), Elem(6180690), Elem(413162), Elem(6123514), Elem(1417009), Elem(3261448), Elem(924135), Elem(6463648), Elem(3968495), Elem(6628613), Elem(789681), Elem(8186851), Elem(3212610), Elem(6085229), Elem(4202597), Elem(2486168), Elem(1501740), Elem(7085683), Elem(5343618), Elem(6257091), Elem(3897737), Elem(5598439), Elem(6326842), Elem(2987588), Elem(1216323), Elem(3738424), Elem(2417095), Elem(997172), Elem(173289), Elem(6163046), Elem(3969637), Elem(7808060), Elem(5359341), Elem(4893871), Elem(5113971), Elem(1810664), Elem(3094210), Elem(3802231), Elem(7486734), Elem(3193897), Elem(1114017), Elem(2036532), Elem(7975866), Elem(6494359), Elem(2454130), Elem(3014240), Elem(8117788), Elem(6795509), Elem(4880853), Elem(3036138), Elem(7213156), Elem(7555661), Elem(2422554), Elem(7263160), Elem(6123976), Elem(6672142), Elem(5122673), Elem(5748088), Elem(3064330), 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Elem(662758), Elem(2674226), Elem(5909129), Elem(1979515), Elem(8032131), Elem(5069850), Elem(5768118), Elem(7712777), Elem(2803822), Elem(1198056), Elem(3552961), Elem(7698687), Elem(4959022), Elem(2682304), Elem(4506563), Elem(5648331), Elem(2634979), Elem(5668116), Elem(2044315), Elem(3754690), Elem(6964615), Elem(2049279), Elem(3963035), Elem(2088377), Elem(2807275), Elem(4725552), Elem(809732), Elem(2763488), Elem(826123), Elem(207750), Elem(7469753), Elem(4213438), Elem(7937343), Elem(6437761), Elem(5340751), Elem(8085765), Elem(6807220), Elem(2988331), Elem(4409098), Elem(4419500), Elem(6582702), Elem(1001886), Elem(6727020), Elem(625118), Elem(1041591), Elem(6135532), Elem(4165064), Elem(1073794), Elem(1166168), Elem(4669917), Elem(7608230), Elem(3044992), Elem(5413731), Elem(7846713), Elem(3793204), Elem(2430828), Elem(1596386), Elem(8113389), Elem(227745), Elem(1542683), Elem(3391035), Elem(2380514), Elem(4196353), Elem(5095104), Elem(1052272), Elem(6110952), Elem(690107), 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Elem(7134853), Elem(5739503), Elem(3386077), Elem(2603427), Elem(2451946), Elem(5856366), Elem(2116796), Elem(1548643), Elem(7758918), Elem(2024617), Elem(1243390), Elem(5807044), Elem(3829307), Elem(143836), Elem(4129313), Elem(2396092), Elem(1476331), Elem(4106132), Elem(8363864), Elem(3420000), Elem(565298), Elem(5818370), Elem(8173578), Elem(5027821), Elem(3830418), Elem(7629795), Elem(7168624), Elem(5664926), Elem(4433471), Elem(2244006), Elem(1893773), Elem(2165483), Elem(82681), Elem(6128099), Elem(7125079), Elem(5860725), Elem(189739), Elem(4056067), Elem(5107593), Elem(2111418), Elem(8132880), Elem(1149002), Elem(5075866), Elem(8181833), Elem(3318060), Elem(1149167), Elem(8178626), Elem(2549066), Elem(1600963), Elem(2905970), Elem(5809252), Elem(343840), Elem(1225926), Elem(5221514), Elem(7546081), Elem(4519677), Elem(1617176), Elem(7899579), Elem(2143170), Elem(6205590), Elem(6474823), Elem(2592836), Elem(5629141), Elem(2088786), Elem(606944), Elem(4398931), Elem(7037777), 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Elem(4962473), Elem(1555761), Elem(1512704), Elem(2652650), Elem(432802), Elem(6295248), Elem(2082381), Elem(5286324), Elem(3439641), Elem(773074), Elem(876018), Elem(1672760), Elem(8041220), Elem(6373819), Elem(1923939), Elem(920924), Elem(7447343), Elem(1511238), Elem(3259678), Elem(5895733), Elem(5550552), Elem(4585166), Elem(2422126), Elem(3391814), Elem(7322534), Elem(6000499), Elem(4709926), Elem(8248482), Elem(7804954), Elem(5156399), Elem(6620655), Elem(7807833), Elem(6842987), Elem(6323351), Elem(2728114), Elem(5617401), Elem(84365), Elem(456396), Elem(490023), Elem(5623678), Elem(176552), Elem(165385), Elem(2107593), Elem(6879172), Elem(6853495), Elem(4599517), Elem(2052536), Elem(4561769), Elem(1479115), Elem(5059903), Elem(8199618), Elem(350538), Elem(4777827), Elem(1586531), Elem(4216269), Elem(3914008), Elem(4855066), Elem(3667957), Elem(773322), Elem(2706689), Elem(2803919), Elem(2514311), Elem(2861257), Elem(2148309), Elem(6245899), Elem(8058799), Elem(235254), 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Elem(8305972), Elem(7799638), Elem(299183), Elem(470160), Elem(8344330), Elem(2639410), Elem(819707), Elem(4326094), Elem(2122008), Elem(5892618), Elem(4802475), Elem(202443), Elem(2549449), Elem(6779336), Elem(1789565), Elem(6845817), Elem(169826)])), NttPolynomial(Array([Elem(3264505), Elem(5414951), Elem(6459440), Elem(3603319), Elem(3856817), Elem(2160068), Elem(3660985), Elem(5369949), Elem(8242073), Elem(5792550), Elem(573805), Elem(7316689), Elem(3651048), Elem(7935801), Elem(5916177), Elem(6664930), Elem(3099356), Elem(315768), Elem(3404495), Elem(1276191), Elem(698824), Elem(3869883), Elem(2632988), Elem(3884766), Elem(2036256), Elem(746564), Elem(6156846), Elem(4672396), Elem(6651954), Elem(8380346), Elem(6665822), Elem(2987888), Elem(1691099), Elem(413791), Elem(5194557), Elem(6738553), Elem(6199894), Elem(4485748), Elem(3017407), Elem(5409622), Elem(3994118), Elem(5139347), Elem(8105690), Elem(3962193), Elem(1800539), Elem(469798), Elem(3614718), Elem(4136241), Elem(7962819), Elem(7384192), Elem(5548806), Elem(905486), Elem(8076909), Elem(3451516), Elem(2740336), Elem(5178665), Elem(2504309), Elem(6586412), Elem(4723270), Elem(2296567), Elem(1326538), Elem(2400228), Elem(7072155), Elem(5039887), Elem(7487853), Elem(5836259), Elem(1972231), Elem(8244534), Elem(8138678), Elem(5953796), Elem(2595232), Elem(3193399), Elem(886502), Elem(2393716), Elem(5463831), Elem(2647910), Elem(2052506), Elem(7333769), Elem(5264466), Elem(5356447), Elem(3462311), Elem(3101891), Elem(4785195), Elem(6838913), Elem(2113924), Elem(5512212), Elem(2360473), Elem(4740751), Elem(733259), Elem(2525811), Elem(6913730), Elem(2484820), Elem(232732), Elem(7228004), Elem(8007238), Elem(1207545), Elem(843404), Elem(4129620), Elem(8346138), Elem(957842), Elem(7054196), Elem(1534114), Elem(947564), Elem(6221322), Elem(645105), Elem(5454049), Elem(232649), Elem(2596753), Elem(2408510), Elem(5613853), Elem(8270163), Elem(3733107), Elem(5506771), Elem(7994955), Elem(1048307), Elem(4528528), Elem(7800623), Elem(4277288), Elem(7846277), Elem(2217796), Elem(2134407), Elem(5589194), Elem(6006794), Elem(6692555), Elem(1766844), Elem(2115637), Elem(4372283), Elem(3685681), Elem(824592), Elem(402949), Elem(2119362), Elem(3609986), Elem(395090), Elem(2486001), Elem(4075720), Elem(1216439), Elem(4497237), Elem(2920707), Elem(8194000), Elem(1908456), Elem(2350001), Elem(5599278), Elem(2586600), Elem(7839577), Elem(2383484), Elem(7859613), Elem(2521632), Elem(2826877), Elem(2456976), Elem(6449040), Elem(4424334), Elem(2855778), Elem(5283098), Elem(6484390), Elem(6778319), Elem(5297986), Elem(6315182), Elem(7824008), Elem(6188205), Elem(6799456), Elem(6772696), Elem(2481682), Elem(6858863), Elem(1435058), Elem(7937333), Elem(7640134), Elem(2800876), Elem(1178038), Elem(6055740), Elem(309196), Elem(6916116), Elem(5589373), Elem(3393438), Elem(1520915), Elem(6555476), Elem(1733740), Elem(7686620), Elem(3452879), Elem(776262), Elem(3427974), Elem(880146), Elem(6188007), Elem(7711669), Elem(2790102), Elem(1761076), Elem(1728412), Elem(3844161), Elem(6563849), Elem(6494967), Elem(2752732), Elem(4129124), Elem(1821132), Elem(438618), Elem(5195560), Elem(8311558), Elem(2847324), Elem(2476287), Elem(6525550), Elem(4244124), Elem(7594664), Elem(3184592), Elem(7237795), Elem(8349325), Elem(6478645), Elem(3657036), Elem(2777282), Elem(466324), Elem(7837307), Elem(2224453), Elem(2741029), Elem(5188172), Elem(8055800), Elem(5128471), Elem(6114564), Elem(3116481), Elem(5035144), Elem(1926386), Elem(5452162), Elem(6282507), Elem(3908974), Elem(3591476), Elem(1310438), Elem(2278624), Elem(5861389), Elem(5379015), Elem(3797550), Elem(170485), Elem(3395085), Elem(2456593), Elem(7761962), Elem(4251010), Elem(5686244), Elem(7799548), Elem(7226421), Elem(7509455), Elem(13412), Elem(3947664), Elem(4531931), Elem(2191030), Elem(2324171), Elem(1317721), Elem(2840319), Elem(7777370), Elem(7852335), Elem(1652115), Elem(7927878), Elem(4620921), Elem(5000191), Elem(7783137), Elem(5412620), Elem(3497597), Elem(1329433), Elem(6410551), Elem(6373198), Elem(2065270), Elem(4001606)]))])), NttVector(Array([NttPolynomial(Array([Elem(321782), Elem(5819979), Elem(2478034), Elem(5124621), Elem(7659924), Elem(379972), Elem(3880306), Elem(6248199), Elem(5564521), Elem(2300798), Elem(1935610), Elem(1194754), Elem(4245699), Elem(7943770), Elem(6443404), Elem(6364350), Elem(4704718), Elem(2114601), Elem(8344537), Elem(1769170), Elem(3769690), Elem(7371870), Elem(6661843), Elem(3854168), Elem(5069286), Elem(2873615), Elem(1493493), Elem(6676493), Elem(5510507), Elem(2997154), Elem(3071520), Elem(2229305), Elem(6608877), Elem(3155753), Elem(183130), Elem(6635563), Elem(7002109), Elem(3991412), Elem(7441259), Elem(7028821), Elem(598204), Elem(4744655), Elem(6858481), Elem(6197429), Elem(7699753), Elem(769819), Elem(5269859), Elem(499576), Elem(4780946), Elem(5193520), Elem(7147668), Elem(6024503), Elem(515354), Elem(2930979), Elem(4040722), Elem(1898307), Elem(980901), Elem(7429573), Elem(7086748), Elem(4429329), Elem(2142769), Elem(2970608), Elem(7370287), Elem(2303655), Elem(7818504), Elem(4441196), Elem(1802108), Elem(1227396), Elem(4462005), Elem(805722), Elem(4479492), Elem(2524259), Elem(7102527), Elem(1799997), Elem(5778843), Elem(5064340), Elem(6076683), Elem(798625), Elem(5905852), Elem(4303531), Elem(6736187), Elem(6248291), Elem(3340149), Elem(5560624), Elem(8162171), Elem(6761145), Elem(144359), Elem(4226676), Elem(4581440), Elem(5907854), Elem(5656040), Elem(1667842), Elem(1839468), Elem(6884935), Elem(5726213), Elem(2105992), Elem(1488062), Elem(2930884), Elem(1370338), Elem(2012528), Elem(4951637), Elem(5312474), Elem(5259041), Elem(7155654), Elem(1531431), Elem(7704325), Elem(6035395), Elem(7461122), Elem(4462553), Elem(2667011), Elem(1142091), Elem(1966975), Elem(496764), Elem(5458079), Elem(1202701), Elem(4413665), Elem(6523714), Elem(7300477), Elem(3623495), Elem(6565740), Elem(753388), Elem(7132617), Elem(2653059), Elem(1857828), Elem(3503029), Elem(1346838), Elem(2841812), Elem(2537191), Elem(4645332), Elem(7788314), Elem(2685003), Elem(4497603), Elem(6784240), Elem(1455581), Elem(1741721), Elem(5966142), Elem(3308696), Elem(473483), Elem(473159), Elem(6292743), Elem(4117294), Elem(4347464), Elem(6713746), Elem(3468934), Elem(4663887), Elem(8080284), Elem(6969473), Elem(3835269), Elem(60274), Elem(3817107), Elem(6790465), Elem(1389502), Elem(3680230), Elem(3896974), Elem(4214544), Elem(966722), Elem(5097659), Elem(1717119), Elem(4560464), Elem(5837516), Elem(4403930), Elem(5146287), Elem(865261), Elem(2718877), Elem(4797775), Elem(2149940), Elem(1179965), Elem(2054751), Elem(5900461), Elem(1883537), Elem(2741160), Elem(3557054), Elem(5972113), Elem(6492040), Elem(1042960), Elem(715968), Elem(3235671), Elem(6206898), Elem(4267909), Elem(5296291), Elem(3451700), Elem(6931280), Elem(3171327), Elem(7046970), Elem(1172080), Elem(6836788), Elem(342005), Elem(2440688), Elem(5448960), Elem(7790438), Elem(3994805), 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NttPolynomial(Array([Elem(6633328), Elem(606762), Elem(4013738), Elem(6914101), Elem(1281899), Elem(2495521), Elem(6736324), Elem(7901480), Elem(4352557), Elem(7780275), Elem(3977684), Elem(6860109), Elem(5323741), Elem(3687174), Elem(1723130), Elem(2626151), Elem(3386395), Elem(5010594), Elem(5372686), Elem(7583447), Elem(365799), Elem(5107560), Elem(3940752), Elem(2400830), Elem(6355810), Elem(6022401), Elem(263284), Elem(2341654), Elem(1179629), Elem(6412855), Elem(8339568), Elem(4170093), Elem(8366878), Elem(7727330), Elem(6023351), Elem(2750812), Elem(1366872), Elem(8223842), Elem(3226210), Elem(5071270), Elem(4472740), Elem(1423715), Elem(6211764), Elem(7792364), Elem(917178), Elem(2895652), Elem(7644666), Elem(4801487), Elem(5934617), Elem(7182804), Elem(4817469), Elem(7473665), Elem(7400545), Elem(3676216), Elem(3367318), Elem(1066467), Elem(7966870), Elem(466627), Elem(6207050), Elem(5033807), Elem(1414869), Elem(2828189), Elem(799142), Elem(7122828), Elem(4009206), Elem(1085932), Elem(1317076), Elem(1835096), Elem(6684186), Elem(3867140), Elem(5043855), Elem(1692364), Elem(6152928), Elem(7269432), Elem(5954109), Elem(1529762), Elem(2022069), Elem(623769), Elem(5624817), Elem(3583777), Elem(1264888), Elem(1870118), Elem(874034), Elem(1935043), Elem(345974), Elem(5972547), Elem(7823880), Elem(2985318), Elem(1018175), Elem(2399750), Elem(4247375), Elem(861886), Elem(6187891), Elem(189438), Elem(1560561), Elem(7369804), Elem(6977325), Elem(4884775), Elem(2142709), Elem(774881), Elem(4956657), Elem(6719222), Elem(4903175), Elem(6540415), Elem(1119748), Elem(2609257), Elem(1152499), Elem(2807791), Elem(4036715), Elem(4443331), Elem(5658591), Elem(1801134), Elem(757587), Elem(724071), Elem(6324508), Elem(508897), Elem(2913511), Elem(836618), Elem(6009830), Elem(4770930), Elem(2985866), Elem(353560), Elem(3004128), Elem(8363531), Elem(383057), Elem(6464119), Elem(2272197), Elem(8025994), Elem(3559586), Elem(3441231), Elem(4593544), Elem(7438641), 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Elem(1367505), Elem(6771978), Elem(6377402), Elem(5282538), Elem(333140), Elem(1266241), Elem(506104), Elem(797942), Elem(2590675), Elem(2180503), Elem(4116593), Elem(4368763), Elem(7459779), Elem(1692440), Elem(5509245), Elem(6979511), Elem(6906425), Elem(6250587), Elem(3870300), Elem(7661405), Elem(1075411), Elem(4333726), Elem(4987270), Elem(2302223), Elem(6943299), Elem(5671836), Elem(4909697), Elem(8288086), Elem(181603), Elem(5931755), Elem(1883554), Elem(5817626), Elem(1105522), Elem(321642), Elem(2251560), Elem(722523), Elem(550371), Elem(6579354), Elem(6824484), Elem(4084998), Elem(3702130), Elem(6278713), Elem(2421370), Elem(4689930), Elem(1696863), Elem(1692997), Elem(4680714), Elem(5908398), Elem(1760107), Elem(840937), Elem(226373), Elem(2961490), Elem(8153969), Elem(6626680), Elem(565905), Elem(6905173), Elem(2458665)])), NttPolynomial(Array([Elem(7792217), Elem(4477063), Elem(7622221), Elem(8220331), Elem(404793), Elem(2919472), Elem(5081046), Elem(6146054), 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Elem(7574989), Elem(4213797), Elem(6981821), Elem(8000329), Elem(5468831), Elem(925289), Elem(4191951), Elem(6034455), Elem(6779345), Elem(2436812), Elem(8063883), Elem(2337115), Elem(3219483), Elem(6792835), Elem(2316946), Elem(7745250), Elem(1990241), Elem(5265429), Elem(5026082), Elem(4473872), Elem(5554740), Elem(4117314), Elem(6886093), Elem(8192052), Elem(1925874), Elem(3363310), Elem(7687096), Elem(890051), Elem(2499500), Elem(7038202), Elem(4315965), Elem(3038138), Elem(1790115), Elem(6687647), Elem(5525754), Elem(4260173), Elem(4689520), Elem(5976395), Elem(7412955), Elem(4082431), Elem(6832154), Elem(1319186), Elem(3925619), Elem(2835724), Elem(5083584), Elem(3385986), Elem(1444624), Elem(2182125), Elem(5583898), Elem(5068205), Elem(4128902), Elem(3497185), Elem(5662882), Elem(5832457), Elem(5349160), Elem(10468), Elem(2388186), Elem(4306380), Elem(2083437), Elem(2039629), Elem(3002188), Elem(3735875), Elem(6971479), Elem(4201677), Elem(107869), Elem(172377), Elem(7339955), Elem(4307253), Elem(5186339), Elem(5435012), Elem(1493597), Elem(3284512), Elem(427651), Elem(2069571), Elem(4956811), Elem(3353267), Elem(2946131), Elem(6658683), Elem(6260901), Elem(1336280), Elem(5658549), Elem(826321), Elem(6853324), Elem(1765568), Elem(1176646), Elem(5790857), Elem(6086039), Elem(4030565), Elem(7549011), Elem(7829573), Elem(3020769), Elem(2932367), Elem(4753823), Elem(2066119), Elem(1102163), Elem(7796563), Elem(7114765), Elem(2731981), Elem(6193947), Elem(2607550), Elem(6900980), Elem(1488514), Elem(6329645), Elem(6760936), Elem(3730995), Elem(72616), Elem(4146584), Elem(6104519), Elem(2791079), Elem(1027643), Elem(1669927), Elem(6316491), Elem(7652943), Elem(369065), Elem(3259768), Elem(5573238), Elem(1904009), Elem(2258904), Elem(7092396), Elem(5613928), Elem(5549080), Elem(1320125), Elem(5740709), Elem(5708492), Elem(1294783), Elem(2838757), Elem(7938299), Elem(6277488), Elem(809782), Elem(6393265), Elem(4810372), Elem(2387262), Elem(1984348), Elem(1651098), Elem(4729800), Elem(329576), Elem(2055548), Elem(4167264), Elem(4411946), Elem(2720326), Elem(1012122), Elem(6988033), Elem(4033266), Elem(891285), Elem(8165656), Elem(2513990), Elem(4981226), Elem(1798912), Elem(5806040), Elem(1627691), Elem(795724), Elem(8338096), Elem(6500309), Elem(3125054), Elem(5928103), Elem(1085993), Elem(5384422), Elem(3225119), Elem(8288371), Elem(2116759), Elem(5080214), Elem(4273763), Elem(7153997), Elem(1569714), Elem(2698108), Elem(6101859), Elem(1050868), Elem(6926116), Elem(489688), Elem(7992024), Elem(2685158), Elem(3657030), Elem(5654827), Elem(745733), Elem(6038390), Elem(1103665), Elem(7475206), Elem(318966), Elem(666825), Elem(7528321), Elem(605126)])), NttPolynomial(Array([Elem(7244441), Elem(723940), Elem(7641773), Elem(3329555), Elem(68598), Elem(1300412), Elem(5107360), Elem(3836338), Elem(5390800), Elem(5529108), Elem(5576852), Elem(2277955), Elem(2365419), Elem(3129557), Elem(5955633), Elem(2951033), Elem(4127812), Elem(7065857), Elem(8359171), Elem(3889067), Elem(4659182), Elem(1699425), Elem(7004677), Elem(4510233), Elem(7548549), Elem(1457221), Elem(8112104), Elem(1216682), Elem(3416353), Elem(553809), Elem(7972878), Elem(7392645), Elem(3433911), Elem(5718029), Elem(6658227), Elem(4851513), Elem(3992948), Elem(7665630), Elem(3857765), Elem(1323122), Elem(4887834), Elem(103101), Elem(7185586), Elem(4871298), Elem(2180344), Elem(4442363), Elem(7103851), Elem(5988147), Elem(6379329), Elem(7113470), Elem(3883681), Elem(3514598), Elem(2882384), Elem(3755112), Elem(1429260), Elem(3176441), Elem(5128283), Elem(409435), Elem(4336952), Elem(6912853), Elem(4334046), Elem(2475577), Elem(3399477), Elem(6143532), Elem(2631164), Elem(2140476), Elem(5042020), Elem(4620734), Elem(2153600), Elem(3256727), Elem(6575282), Elem(1657245), Elem(3366609), Elem(2647445), Elem(6162453), Elem(2136825), Elem(73801), Elem(6078570), Elem(620506), Elem(8104654), Elem(3016192), Elem(4716064), Elem(2493601), Elem(2327301), Elem(183138), Elem(8328392), Elem(7401031), Elem(4763030), Elem(7740934), Elem(1304861), Elem(3974573), Elem(3395508), Elem(4064306), Elem(5593230), Elem(4402781), Elem(5048074), Elem(6158412), Elem(5133123), Elem(5938305), Elem(7734181), Elem(2359356), Elem(8178467), Elem(6838997), Elem(5087378), Elem(4570301), Elem(3356320), Elem(1239917), Elem(4279330), Elem(6312887), Elem(5527582), Elem(2243397), Elem(3810843), Elem(6821067), Elem(4438899), Elem(2748616), Elem(7036642), Elem(3592090), Elem(7786058), Elem(7060141), Elem(4795221), Elem(6655185), Elem(2083898), Elem(4171788), Elem(7907935), Elem(4703596), Elem(345693), Elem(4678394), Elem(6294335), Elem(3539152), Elem(937498), Elem(697817), Elem(2683323), Elem(2601988), Elem(13540), Elem(8329857), Elem(4380009), Elem(2920453), Elem(5049137), Elem(7745360), Elem(1705091), Elem(225065), Elem(7670458), Elem(2200196), Elem(3003782), Elem(2017993), Elem(5385236), Elem(5953713), Elem(7954164), Elem(2089667), Elem(3699024), Elem(6191490), Elem(6780201), Elem(4090852), Elem(5511631), Elem(5685875), Elem(2849063), Elem(1189790), Elem(1970571), Elem(8365847), Elem(2317392), Elem(5051067), Elem(8138659), Elem(6660293), Elem(1970126), Elem(5963061), Elem(3554152), Elem(7578412), Elem(3331007), Elem(2035801), Elem(1059660), Elem(7223475), Elem(6757330), Elem(2696909), Elem(6196435), Elem(5869513), Elem(3945851), Elem(3866498), Elem(7722685), Elem(6298395), Elem(31189), Elem(138916), Elem(2470800), Elem(4489907), Elem(2196019), Elem(6310884), Elem(7780797), Elem(1070355), Elem(5859856), Elem(3702421), Elem(5566500), Elem(7808581), Elem(7525184), Elem(3192673), Elem(2884120), Elem(3783626), Elem(7217142), Elem(7849693), Elem(790087), Elem(8290904), Elem(7031540), Elem(6258070), Elem(6255491), Elem(1710368), Elem(1391380), Elem(6782460), Elem(3662460), Elem(6037944), Elem(7782851), Elem(899658), Elem(4209587), Elem(5932203), Elem(6765033), Elem(4101406), Elem(2836105), Elem(6218169), Elem(3666389), Elem(5843056), Elem(5817880), Elem(1197308), Elem(231906), Elem(2161259), Elem(1947788), Elem(75294), Elem(6219964), Elem(8224485), Elem(966267), Elem(3632378), Elem(152030), Elem(7035145), Elem(4342756), Elem(2934366), Elem(5635056), Elem(7253418), Elem(2802774), Elem(7875945), Elem(557792), Elem(1410128), Elem(2381306), Elem(4683842), Elem(8009111), Elem(2879077), Elem(7192809), Elem(3864375), Elem(5812012), Elem(8201987), Elem(2983679), Elem(7727410), Elem(3663415), Elem(8235119), Elem(6858109), Elem(5546970), Elem(2892650), Elem(4444257), Elem(5166996), Elem(866504), Elem(459846)])), NttPolynomial(Array([Elem(8073907), Elem(7096190), Elem(5728837), Elem(1255417), Elem(5994693), Elem(1959970), Elem(3492254), Elem(90250), Elem(6299717), Elem(385571), Elem(1771080), Elem(4566459), Elem(1744292), Elem(2424253), Elem(1968041), Elem(3875572), Elem(8067990), Elem(3377913), Elem(3137466), Elem(2211941), Elem(648244), Elem(5702229), Elem(4934774), Elem(2611774), Elem(2092697), Elem(4840684), Elem(5727740), Elem(6787711), Elem(1163348), Elem(2003241), Elem(7945957), Elem(7687974), Elem(6533186), Elem(5845973), Elem(7188465), Elem(2264279), Elem(610459), Elem(893038), Elem(1383790), Elem(5627086), Elem(4980405), Elem(1035731), Elem(3149866), Elem(7844147), Elem(5955987), Elem(3510468), Elem(5559403), Elem(3532890), Elem(4078005), Elem(3265406), Elem(463297), Elem(7446552), Elem(5584558), Elem(2903732), Elem(1885326), Elem(3417283), Elem(2987310), Elem(6720266), Elem(4239919), Elem(3752828), Elem(7861220), Elem(1896989), Elem(1352464), Elem(3091309), Elem(1160109), Elem(6055008), Elem(4371771), Elem(7962113), Elem(2755507), Elem(710318), Elem(7705808), Elem(3296362), Elem(3119998), Elem(8144027), Elem(4248347), Elem(4337536), Elem(4670330), Elem(6779566), Elem(6351262), Elem(6159306), Elem(2832151), Elem(2333469), Elem(5615187), Elem(2257807), Elem(4967959), Elem(2703547), Elem(2819330), Elem(1618143), Elem(4832005), Elem(804624), Elem(1211422), Elem(5516722), Elem(6156870), Elem(4080805), Elem(934166), 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Elem(4024654), Elem(3108465), Elem(6350445), Elem(4727710), Elem(2458221), Elem(491681), Elem(3081927), Elem(4665359), Elem(2371240), Elem(6308200), Elem(4525633), Elem(909072), Elem(7623675), Elem(5922406), Elem(4626302), Elem(2640418), Elem(744206), Elem(6783549), Elem(4480021), Elem(835875), Elem(4851742), Elem(5410983), Elem(2842764), Elem(6230504), Elem(2342409), Elem(1188412), Elem(6941525), Elem(4061016), Elem(7279010), Elem(439587), Elem(4004349), Elem(4961258), Elem(5928671), Elem(7078144), Elem(2236093), Elem(8072345), Elem(3352485), Elem(1074530), Elem(72119), Elem(4525154), Elem(2350522), Elem(6797425), Elem(3455119), Elem(6441459), Elem(1164945), Elem(7045258), Elem(5094531), Elem(5625474), Elem(2056567), Elem(7413626), Elem(4098093), Elem(4066223), Elem(7947660), Elem(6019325), Elem(5666968), Elem(3332369), Elem(519340), Elem(444474), Elem(4806597), Elem(303103), Elem(6293608), Elem(5058734), Elem(1554505), Elem(1288359), Elem(6056366), Elem(8322002), Elem(6711915), 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Elem(3915653), Elem(2799958), Elem(2723882), Elem(5884076), Elem(5229096), Elem(4873093), Elem(6151305), Elem(1485025), Elem(4155658), Elem(5670318), Elem(8297442), Elem(3987849), Elem(6130842), Elem(6754613), Elem(6655235), Elem(6756332), Elem(2076804), Elem(858440), Elem(850730), Elem(3256680), Elem(2344461), Elem(1365384), Elem(8279137), Elem(2201565), Elem(2841876), Elem(3485429), Elem(824415), Elem(3049861), Elem(7902883), Elem(60279), Elem(1165862), Elem(8128359), Elem(7154128), Elem(973742), Elem(851414), Elem(3553395), Elem(599355), Elem(5750856), Elem(2538716), Elem(3978412), Elem(1957345), Elem(7794251), Elem(7949751), Elem(5966471), Elem(3878458), Elem(5560596), Elem(4369875), Elem(2787381), Elem(6456560), Elem(5625668), Elem(7293400), Elem(3828119), Elem(2253687), Elem(6697151), Elem(5739536), Elem(2520844), Elem(905801), Elem(3145285), Elem(7462377), Elem(5910453), Elem(4302512), Elem(540018), Elem(3636655), Elem(6454532), Elem(3005089), Elem(1353630), Elem(6503507), Elem(2628233), Elem(201476), Elem(954304), Elem(6556430), Elem(1800181), Elem(7789822), Elem(6361204), Elem(8133484), Elem(5390897), Elem(2535917), Elem(1786452), Elem(5180306), Elem(6076468), Elem(1845260), Elem(7615300), Elem(387883), Elem(4353103), Elem(5646079), Elem(8250381), Elem(2922352), Elem(2896316), Elem(225822), Elem(54322), Elem(3953577), Elem(8278726), Elem(8218570), Elem(1626378), Elem(966573), Elem(4796921), Elem(4795430), Elem(5832034), Elem(1609968), Elem(6389438), Elem(2750609), Elem(7743562), Elem(3936988), Elem(7743831), Elem(1927958), Elem(315812), Elem(7339084), Elem(4055655), Elem(3256743), Elem(2890960), Elem(702609), Elem(5736641), Elem(6355847), Elem(3676992), Elem(2890768), Elem(1749930), Elem(1187970), Elem(401529), Elem(34588), Elem(2967562), Elem(5730666), Elem(5759412), Elem(2590360), Elem(2058416), Elem(5937952), Elem(5556105), Elem(1404805), Elem(2738211), Elem(307343), Elem(2913189), Elem(6802305), Elem(3673005), Elem(6615454), Elem(2624902), 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Elem(7639171), Elem(675024), Elem(5190349), Elem(246038), Elem(4281962), Elem(8093437), Elem(96116), Elem(7020197), Elem(7261053), Elem(3605608), Elem(4474339), Elem(2904435), Elem(119761), Elem(3130539), Elem(1030500), Elem(6485428), Elem(6709687), Elem(2283650), Elem(8865), Elem(3708323), Elem(5130169), Elem(1224273), Elem(5789485), Elem(2490380), Elem(7155558), Elem(3511417), Elem(2102625), Elem(7652190), Elem(7180554), Elem(6841744), Elem(2993789), Elem(2801369), Elem(1315967), Elem(2452806), Elem(5971227), Elem(8241847), Elem(6564284), Elem(5346334), Elem(1156014), Elem(2366327), Elem(7691435), Elem(3596867), Elem(4395462), Elem(7665709), Elem(4310664), Elem(2091045), Elem(7637257), Elem(1830557), Elem(134627), Elem(1238119), Elem(4959714), Elem(6689429), Elem(7761592), Elem(4626262), Elem(6309297), Elem(3533071), Elem(1929873), Elem(1366435), Elem(1956647), Elem(6775015), Elem(5060175), Elem(2004516), Elem(5596422), Elem(7544094), Elem(3154131), Elem(2866935), Elem(7900863), 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Elem(5254916), Elem(7888613), Elem(1656276), Elem(5265091), Elem(2691655), Elem(494861), Elem(6369941), Elem(4419563), Elem(8048609), Elem(3935743), Elem(645147), Elem(2741089), Elem(431334), Elem(865019), Elem(6704245), Elem(6854601), Elem(301310), Elem(6986546), Elem(454320), Elem(4253242), Elem(952644), Elem(3769158), Elem(1999497), Elem(5959247), Elem(2465274), Elem(713238), Elem(8013422), Elem(2606658), Elem(15143), Elem(325054), Elem(928158), Elem(1532538), Elem(2131712), Elem(2482309), Elem(5098174), Elem(2608836), Elem(2516367), Elem(5282061), Elem(6642495), Elem(8371433), Elem(3416639), Elem(989050), Elem(6807952), Elem(636571), Elem(1013874), Elem(2327331), Elem(3418860), Elem(2319387), Elem(6524782), Elem(1831000), Elem(5043228), Elem(8328973), Elem(736278), Elem(4133802), Elem(5357852), Elem(3123919), Elem(5510505), Elem(450666), Elem(6431884), Elem(3650017), Elem(2592770), Elem(2677667), Elem(5999703), Elem(2868342), Elem(7682830), Elem(2787150), Elem(6602567), Elem(112403), Elem(3807560), Elem(4357354), Elem(7574744), Elem(6770086), Elem(4928126), Elem(7090594), Elem(4720156), Elem(1650457), Elem(2588405), Elem(2008789), Elem(5038674), Elem(7511851), Elem(3063645), Elem(3418034), Elem(1060263), Elem(5771974), Elem(4558571), Elem(1549520), Elem(2139), Elem(5954910), Elem(6252917), Elem(2398554), Elem(5719602), Elem(6978416), Elem(7728038), Elem(1726803), Elem(7247099), Elem(6687591), Elem(5216794), Elem(3782795), Elem(5587762), Elem(5559243), Elem(3784270), Elem(4154388), Elem(1765525), Elem(4452502), Elem(803052), Elem(986678), Elem(4536900), Elem(6638789), Elem(4354981), Elem(524995), Elem(16654), Elem(2172303), Elem(2343898), Elem(5127469), Elem(2301470), Elem(6793509), Elem(5403519), Elem(1316843), Elem(4247844), Elem(2241104), Elem(6766254), Elem(6063959), Elem(2605865), Elem(3919543), Elem(8099441), Elem(354136), Elem(303798), Elem(3939362), Elem(6623582), Elem(716255), Elem(4213859), Elem(7018147), Elem(7371043), Elem(2387695), Elem(4664628), Elem(3697907), Elem(4830625), Elem(613754), Elem(1019737), Elem(1043143), Elem(7723239), Elem(977149), Elem(3327194), Elem(4443716), Elem(2757533), Elem(1098552), Elem(5230470), Elem(4642766), Elem(7150610), Elem(602989), Elem(6262450), Elem(7850817), Elem(6544366), Elem(4660969), Elem(3212884), Elem(151497), Elem(6568113), Elem(6808907), Elem(3581403), Elem(1998291), Elem(563369), Elem(6506341), Elem(1065516), Elem(4867198), Elem(5693240), Elem(1316206), Elem(425326), Elem(1209645), Elem(2467877), Elem(6157878), Elem(5130282), Elem(1914322), Elem(4943785), Elem(3919092), Elem(2335736), Elem(7720091), Elem(5070239), Elem(6964044), Elem(7135355), Elem(5923911), Elem(3899146), Elem(2344048), Elem(1260535), Elem(3548007), Elem(6578690), Elem(4383700), Elem(5206513), Elem(1226479), Elem(6194976), Elem(2372892), Elem(1321761), Elem(815568), Elem(3872683), Elem(189734), Elem(7150329), Elem(2903806), Elem(1008705), Elem(2689984), Elem(4441297), Elem(4215583), Elem(2857011), Elem(4419215), Elem(2207233), Elem(4839322), Elem(2666513), Elem(2773757), Elem(6360002), Elem(5541526), Elem(2342038), Elem(4055233), Elem(1590023), Elem(2401243), Elem(5980257), Elem(6594632), Elem(6009073), Elem(5392732), Elem(6757195), Elem(4114467), Elem(1772663), Elem(688288), Elem(793067), Elem(3930085), Elem(6530958), Elem(3451203), Elem(6078160), Elem(1631757), Elem(4286373), Elem(4785223), Elem(1010426), Elem(7208177), Elem(3010126), Elem(2444311), Elem(2763424), Elem(3459757)])), NttPolynomial(Array([Elem(2667407), Elem(7119102), Elem(3993369), Elem(818940), Elem(2772438), Elem(6449387), Elem(3667599), Elem(2596829), Elem(5649438), Elem(4946709), Elem(4217539), Elem(6150139), Elem(8352087), Elem(71297), Elem(3607323), Elem(2573801), Elem(8272799), Elem(7251854), Elem(2579312), Elem(7357203), Elem(3917760), Elem(7903072), Elem(5214091), Elem(5647247), Elem(2165222), Elem(3246074), Elem(3538072), Elem(6658916), Elem(3466834), Elem(8374673), Elem(2854476), Elem(627126), 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Elem(3500803), Elem(6982726), Elem(4301489), Elem(3406452), Elem(6681782), Elem(4433123), Elem(2527679), Elem(3640664), Elem(1712598), Elem(3663834), Elem(7511573), Elem(5968901), Elem(6291979), Elem(2928798), Elem(7927415), Elem(2029837), Elem(4284165), Elem(6899902), Elem(855187), Elem(3598986), Elem(4769415), Elem(1612568), Elem(563340), Elem(6861083), Elem(1826627), Elem(3771708), Elem(3215074), Elem(1589498), Elem(4196362), Elem(5215976), Elem(5635148), Elem(5643423), Elem(2037407), Elem(3154425), Elem(3137297), Elem(5999889), Elem(5128590), Elem(51538), Elem(4983107), Elem(6339649), Elem(1308726), Elem(2021184), Elem(7243000), Elem(7849867), Elem(8187016), Elem(4618666), Elem(3895587), Elem(8341227), Elem(2564279), Elem(2929891), Elem(2910954), Elem(3758468), Elem(7890428), Elem(1412468), Elem(2737766), Elem(8113372), Elem(6375188), Elem(7927596), Elem(2966282), Elem(3927154), Elem(6056712), Elem(3485212), Elem(7205890), Elem(3537678), Elem(8349060), Elem(678263), Elem(4449), Elem(7035320), Elem(7670827), Elem(7766), Elem(288944), Elem(6057610), Elem(1182077), Elem(5527906), Elem(4186169), Elem(2967499), Elem(2130900), Elem(4074080), Elem(5637051), Elem(4516348), Elem(7106414), Elem(6407272), Elem(7447146), Elem(7923104), Elem(1219740), Elem(1450534), Elem(764701), Elem(6396253), Elem(1726570), Elem(7386463), Elem(2974082), Elem(1306725), Elem(6064721), Elem(2925687), Elem(4424323), Elem(5700057), Elem(7987520), Elem(216148), Elem(1231747), Elem(6783205), Elem(1654450), Elem(5409454), Elem(1645180), Elem(493361), Elem(6648278), Elem(4412829), Elem(6010190), Elem(6125785), Elem(4457535), Elem(315062), Elem(6593986), Elem(4059938), Elem(5421407), Elem(6266888), Elem(99653), Elem(3954506), Elem(6058217), Elem(4818352), Elem(3918405), Elem(2235838), Elem(566336), Elem(6109200), Elem(5090514), Elem(7814050), Elem(1325040), Elem(1132504), Elem(1963656), Elem(3078115), Elem(1423442), Elem(3312972), Elem(906126), Elem(1593894), Elem(2890810), Elem(6959884), Elem(8017062), Elem(5063639), Elem(4999807), Elem(6205830), Elem(3337654), Elem(3166203), Elem(3574158), Elem(5883692), Elem(7441032), Elem(5952252), Elem(7246337), Elem(7866396), Elem(3089527), Elem(5989873), Elem(5273248), Elem(7670300), Elem(5604202), Elem(1969654), Elem(6848921), Elem(2804705), Elem(2505290), Elem(4901919), Elem(1210808)])), NttPolynomial(Array([Elem(2695741), Elem(7415370), Elem(7291665), Elem(6358465), Elem(5163921), Elem(1024230), Elem(4001417), Elem(7398648), Elem(4676776), Elem(2590426), Elem(3626900), Elem(1695622), Elem(6835271), Elem(3276224), Elem(7874978), Elem(4354689), Elem(7532716), Elem(7842040), Elem(612631), Elem(3676152), Elem(6346585), Elem(2379478), Elem(6912496), Elem(2352512), Elem(480844), Elem(7686963), Elem(2085354), Elem(1062359), Elem(3331893), Elem(6795130), Elem(4081149), Elem(6765614), Elem(3888001), Elem(7618386), Elem(8074889), Elem(2871634), Elem(6833449), Elem(2337892), Elem(4101298), Elem(889857), Elem(2061965), Elem(1512535), Elem(2717382), Elem(3639324), Elem(7654620), Elem(7127797), Elem(5516044), Elem(2424293), Elem(5862551), Elem(8351981), Elem(964140), Elem(4551224), Elem(41570), Elem(6133067), Elem(5232599), Elem(5332389), Elem(6299840), Elem(109344), Elem(1043750), Elem(5572637), Elem(1063091), Elem(1984803), Elem(465885), Elem(3658462), Elem(8277131), Elem(6438232), Elem(3031452), Elem(1673259), Elem(6698942), Elem(7820816), Elem(7764219), Elem(5244993), Elem(5194210), Elem(6918313), Elem(2623026), Elem(4550630), Elem(2669842), Elem(4329877), Elem(8027841), Elem(7183252), Elem(995814), Elem(834799), Elem(6519292), Elem(8302151), Elem(2450195), Elem(138632), Elem(7977895), Elem(4650143), Elem(5734889), Elem(3101217), Elem(6454955), Elem(7521580), Elem(1934226), Elem(6960047), Elem(3177197), Elem(3225661), Elem(1794090), Elem(8187900), Elem(6839236), Elem(4820017), Elem(4740818), Elem(4488921), Elem(1361665), Elem(4224635), Elem(1009855), Elem(5802520), Elem(1976460), Elem(1587015), Elem(4456793), Elem(1763285), Elem(6593173), Elem(5977356), Elem(1007904), Elem(6074188), Elem(6224546), Elem(6232463), Elem(912002), Elem(4542096), Elem(2581254), Elem(3205561), Elem(7998275), Elem(3110721), Elem(8122949), Elem(5026532), Elem(1364350), Elem(2428910), Elem(1289139), Elem(2641648), Elem(4297359), Elem(7807000), Elem(8302091), Elem(485257), Elem(1090050), Elem(3147471), Elem(7808293), Elem(2179868), Elem(3719219), Elem(8065577), Elem(7009330), Elem(1615382), Elem(4273876), Elem(6381892), Elem(761142), Elem(6278751), Elem(3278915), Elem(5103604), Elem(6636710), Elem(6492014), Elem(2227066), Elem(22818), Elem(3987054), Elem(6127946), Elem(1677438), Elem(6899498), Elem(3340000), Elem(7863542), Elem(1843410), Elem(1408222), Elem(5231692), Elem(1793472), Elem(2823104), Elem(6269166), Elem(4548655), Elem(6314669), Elem(8281683), Elem(2372273), Elem(4375948), Elem(4714977), Elem(2497413), Elem(953850), Elem(2795566), Elem(5287233), Elem(2557098), Elem(1000308), Elem(6454380), Elem(4065056), Elem(7828551), Elem(4667924), Elem(5699098), Elem(244928), Elem(4885872), Elem(604964), Elem(6515925), Elem(6638610), Elem(4753299), Elem(5491635), Elem(987971), Elem(8096443), Elem(3039646), Elem(390871), Elem(790393), Elem(3190084), Elem(5549068), Elem(3513343), Elem(7031296), Elem(759874), Elem(3941688), Elem(5008682), Elem(6308905), Elem(3123953), Elem(3556774), Elem(5486943), Elem(7760024), Elem(5341768), Elem(4275500), Elem(3109609), Elem(4966222), Elem(3787337), Elem(3579233), Elem(4506011), Elem(7600110), Elem(4401084), Elem(5268392), Elem(7030893), Elem(7647597), Elem(2514041), Elem(1639903), Elem(6310293), Elem(5423092), Elem(3670005), Elem(402309), Elem(2153004), Elem(5691566), Elem(7747417), Elem(630060), Elem(1709326), Elem(6220403), Elem(3289221), Elem(6404500), Elem(1037142), Elem(7313982), Elem(5775929), Elem(3493839), Elem(3356871), Elem(6689811), Elem(3186543), Elem(3803317), Elem(5214410), Elem(4219582), Elem(5362682), Elem(7336326), Elem(7733308), Elem(6851787), 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Elem(6624568), Elem(7612132), Elem(3730283), Elem(5336905), Elem(7084481), Elem(8189994), Elem(4012422), Elem(7409192), Elem(5885401), Elem(4808740), Elem(8338243), Elem(1875582), Elem(4146984), Elem(8170041), Elem(6370238), Elem(8243670), Elem(7421770), Elem(1198960), Elem(7071178), Elem(712892), Elem(8243506), Elem(2073260), Elem(6236155), Elem(7491490), Elem(6187033), Elem(4818973), Elem(4513245), Elem(7410620), Elem(7277951), Elem(1310070), Elem(7752410), Elem(734312), Elem(6940772), Elem(5085402), Elem(4979307), Elem(4734183), Elem(723653), Elem(3674524), Elem(4740118), Elem(7470191), Elem(3239057), Elem(529368), Elem(2060936), Elem(685246), Elem(1342112), Elem(736445), Elem(8268), Elem(4060721), Elem(7455187), Elem(1057139), Elem(4030575), Elem(573354), Elem(1420753), Elem(4825565), Elem(1904021), Elem(4071881), Elem(7505107), Elem(1146194), Elem(3789485), Elem(6942405), Elem(5122055), Elem(4549360), Elem(4981609), Elem(2900052), Elem(5077949), Elem(6719075), Elem(2303185), Elem(7410988), Elem(929490), Elem(5812202), Elem(3299151), Elem(5909312), Elem(6617083), Elem(3379452), Elem(2158291), Elem(7250130), Elem(539281), Elem(8309566), Elem(4219932), Elem(7039778), Elem(7098872), Elem(6349554), Elem(5647785), Elem(4352550), Elem(1060625), Elem(3898577), Elem(812189), Elem(5226474), Elem(1221152), Elem(7595287), Elem(4502), Elem(3296586), Elem(7734816), Elem(1500672), Elem(751644), Elem(6728758), Elem(2838982), Elem(6037969), Elem(3534237), Elem(495427), Elem(233313), Elem(4117573), Elem(2562392), Elem(272054), Elem(131796), Elem(6856476), Elem(4630691), Elem(5576058), Elem(4471709), Elem(28062), Elem(5532021), Elem(4097338), Elem(4473347), Elem(930767), Elem(5335942), Elem(3237521), Elem(2582621), Elem(4782650), Elem(1695155), Elem(1463169), Elem(5992847), Elem(2953517), Elem(4592698), Elem(5417339), Elem(1010760), Elem(3869428), Elem(6471516), Elem(726248), Elem(4863763), Elem(646122), Elem(6485074), Elem(6614284), Elem(5028210), Elem(2380345), Elem(2571413), Elem(6712473), Elem(6883688)]))]))])), t1_2d_hat: NttVector(Array([NttPolynomial(Array([Elem(4226766), Elem(6719417), Elem(3788079), Elem(708627), Elem(1922144), Elem(159342), Elem(4011593), Elem(4446021), Elem(1589639), Elem(3437419), Elem(3257549), Elem(3346628), Elem(1008296), Elem(3518015), Elem(398301), Elem(2185082), Elem(5722796), Elem(2295379), Elem(6368051), Elem(419593), Elem(3203411), Elem(4910639), Elem(80077), Elem(4456974), Elem(3311378), Elem(2176071), Elem(657213), Elem(6652487), Elem(5573563), Elem(3738439), Elem(5725543), Elem(2465780), Elem(484384), Elem(3110314), Elem(6530373), Elem(1858413), Elem(2285865), Elem(7727799), Elem(7169733), Elem(6481866), Elem(2918550), Elem(1608403), Elem(6107179), Elem(3988790), Elem(2370310), Elem(6689515), Elem(3857229), Elem(6612780), Elem(7211057), Elem(966304), Elem(3063413), Elem(1883903), Elem(3040539), Elem(3229456), Elem(179088), Elem(457187), Elem(7510568), Elem(6759758), Elem(7565261), Elem(3858521), Elem(8143822), Elem(8336767), Elem(2584363), Elem(4011872), Elem(1206459), Elem(6388117), Elem(7728130), Elem(7643367), Elem(1856049), Elem(7287442), Elem(1253288), Elem(6281900), Elem(3380925), Elem(1193539), Elem(6476573), Elem(5958736), Elem(5764524), Elem(5421158), Elem(1501734), Elem(6806110), Elem(3933850), Elem(2894458), Elem(7717950), Elem(6537397), Elem(7424453), Elem(3433317), Elem(8354097), Elem(947504), Elem(2685648), Elem(741229), Elem(1115173), Elem(4846800), Elem(5041000), Elem(8030216), Elem(7081138), Elem(7394820), Elem(1788358), Elem(3919767), Elem(2010159), Elem(2863471), Elem(1538500), Elem(1235219), Elem(268383), Elem(7726078), Elem(8109706), Elem(6047045), Elem(8150147), Elem(6459646), Elem(1653173), Elem(416261), Elem(74539), Elem(1826633), Elem(6413381), Elem(3788943), Elem(5264480), Elem(6938830), Elem(4366491), Elem(1121991), Elem(1608013), Elem(3290522), Elem(2044481), Elem(6294969), Elem(6973643), Elem(3617982), Elem(1252393), Elem(1644253), Elem(3503110), Elem(6485188), Elem(5861459), Elem(4452040), Elem(1477039), Elem(2051114), Elem(4069624), Elem(4601547), Elem(4105784), Elem(6903398), Elem(3623566), Elem(8072628), Elem(4751418), Elem(3382794), Elem(8226232), Elem(612675), Elem(2922777), Elem(1214846), Elem(5323303), Elem(3923562), Elem(5775057), Elem(5460334), Elem(7841970), Elem(7561894), Elem(8308047), Elem(6268740), Elem(6995666), Elem(2916622), Elem(5549685), Elem(3613578), Elem(7624241), Elem(7309053), Elem(4605899), Elem(3180596), Elem(3153800), Elem(166284), Elem(5837926), Elem(5926594), Elem(3313646), Elem(5260347), Elem(6318128), Elem(4730497), Elem(4312964), Elem(7293867), Elem(5127658), Elem(1797968), Elem(6043245), Elem(4263235), Elem(7214696), Elem(2244963), Elem(7619556), Elem(1295072), Elem(4662471), Elem(6422558), Elem(6492845), Elem(3053663), Elem(6646365), Elem(5689949), Elem(651055), Elem(4194653), Elem(8301453), Elem(6939993), Elem(2710618), Elem(5970189), Elem(5300161), Elem(3950235), Elem(2586387), Elem(7794038), Elem(500718), Elem(4944902), Elem(1200873), Elem(4310537), Elem(8022327), Elem(2227715), Elem(2176775), Elem(3810375), Elem(3944353), Elem(7502470), Elem(7248040), Elem(4884443), Elem(2195481), Elem(2149906), Elem(5336248), Elem(3906146), Elem(5662287), Elem(8104905), Elem(5770968), Elem(2167120), Elem(3586579), Elem(4059807), Elem(2732565), Elem(7332143), Elem(3719116), Elem(4143575), Elem(4589228), Elem(5846518), Elem(4292994), Elem(1372710), Elem(7048515), Elem(7476232), Elem(3974411), Elem(3878230), Elem(1231018), Elem(518113), Elem(1342108), Elem(6642463), Elem(6498409), Elem(1529651), Elem(3505970), Elem(6059709), Elem(4427987), Elem(7261211), Elem(5808102), Elem(4020329), Elem(5296771), Elem(6883703), Elem(679998), Elem(8038762), Elem(6807792), Elem(2010435), Elem(2416690), Elem(3959740), Elem(5121643), Elem(2442982), Elem(252133), Elem(7643449), Elem(3479475), Elem(381767), Elem(4887276), Elem(7633080)])), NttPolynomial(Array([Elem(216562), Elem(4396750), Elem(547732), Elem(4469336), Elem(7060952), Elem(6498752), Elem(4025793), Elem(7526929), Elem(2088521), Elem(493961), Elem(1725413), Elem(4690681), Elem(1957856), Elem(5674165), Elem(3193759), Elem(3081588), Elem(2781464), Elem(7309557), Elem(4313763), Elem(2535263), Elem(6299931), Elem(6324674), Elem(2683682), Elem(7617995), Elem(3378672), Elem(7240879), Elem(7464152), Elem(7629352), Elem(4667695), Elem(4247795), Elem(3957762), Elem(8033527), Elem(3728273), Elem(6847270), Elem(1960293), Elem(5061032), Elem(3225425), Elem(3544210), Elem(7257566), Elem(1670527), Elem(6520568), Elem(3580706), Elem(1717369), Elem(8267738), Elem(5172374), Elem(2305247), Elem(2323887), Elem(3327294), Elem(1596738), Elem(7689386), Elem(7737384), Elem(5483050), Elem(982560), Elem(6028974), Elem(8219695), Elem(6797177), Elem(4345941), Elem(2288177), Elem(2752812), Elem(4795842), Elem(5696479), Elem(1147416), Elem(919030), Elem(3913212), Elem(1350802), Elem(2874749), Elem(4228163), Elem(7224430), Elem(3774563), Elem(6788290), Elem(4320867), Elem(1963428), Elem(5918990), Elem(2091027), Elem(524975), Elem(6691292), Elem(1315251), Elem(5238493), Elem(979149), Elem(6079040), Elem(1215220), Elem(1937673), Elem(1358480), Elem(460407), Elem(4532066), Elem(6798266), Elem(2469840), Elem(6689778), Elem(804991), Elem(4550144), Elem(5662787), Elem(6831775), Elem(6889937), Elem(4903688), Elem(4737882), Elem(4798304), Elem(4614315), Elem(5591377), Elem(6307348), Elem(3513434), Elem(7161625), Elem(833460), Elem(3438156), Elem(5912281), Elem(346835), Elem(7889053), Elem(3678755), Elem(6343547), Elem(1920867), Elem(3792398), Elem(3123170), Elem(5961072), Elem(4470747), Elem(7207122), Elem(3516713), Elem(5905637), Elem(5964622), Elem(3297906), Elem(5578520), Elem(8171232), Elem(4925469), Elem(6437187), Elem(6524295), Elem(7964036), Elem(3980893), Elem(1894197), Elem(1598012), Elem(1099860), Elem(4620424), Elem(2616945), Elem(3522711), Elem(6405097), Elem(4442543), Elem(2684800), Elem(1424972), Elem(4151467), Elem(8061346), Elem(499481), Elem(6892533), Elem(3029572), Elem(724729), Elem(5889707), Elem(1892250), Elem(5185573), Elem(8110097), Elem(6145570), Elem(6995757), Elem(213659), Elem(4897311), Elem(6423732), Elem(67019), Elem(214631), Elem(3994232), Elem(776382), Elem(6166937), Elem(4856531), Elem(5113647), Elem(6587216), Elem(7045162), Elem(7563422), Elem(1397478), Elem(6681553), Elem(405060), Elem(7061120), Elem(3930840), Elem(818982), Elem(233414), Elem(4485795), Elem(5132611), Elem(5062876), Elem(2908401), Elem(613107), Elem(2796492), Elem(3496621), Elem(648041), Elem(3839646), Elem(6211002), Elem(1514834), Elem(2749088), Elem(4672799), Elem(3348197), Elem(6561103), Elem(6571991), Elem(4964541), Elem(8174596), Elem(2022686), Elem(1417410), Elem(3189828), Elem(3388862), Elem(6470509), Elem(948809), Elem(1363043), Elem(6673220), Elem(3798558), Elem(5057346), Elem(5942412), Elem(3056976), Elem(433531), Elem(6989775), Elem(1695266), Elem(2978026), Elem(4388824), Elem(1759319), Elem(6620624), Elem(4426477), Elem(2419784), Elem(1163442), Elem(55386), Elem(3790374), Elem(950219), Elem(8072727), Elem(1996784), Elem(6347291), Elem(8285587), Elem(304958), Elem(6047756), Elem(1610588), Elem(2629519), Elem(4422035), Elem(2128906), Elem(4809645), Elem(7588492), Elem(3650333), Elem(5746989), Elem(4131507), Elem(3042026), Elem(4523189), Elem(5985554), Elem(6468626), Elem(2975529), Elem(5885752), Elem(2494546), Elem(7614090), Elem(5917313), Elem(8029672), Elem(7406715), Elem(4930353), Elem(315392), Elem(3411077), Elem(212369), Elem(2239774), Elem(6885214), Elem(7281054), Elem(1349684), Elem(1604372), Elem(1766750), Elem(8323906), Elem(545926), Elem(4951050), Elem(5421259), Elem(601286), Elem(3487523), Elem(2150197), Elem(2268392), Elem(6201688), Elem(4892502)])), NttPolynomial(Array([Elem(5357725), Elem(5265157), Elem(3015550), Elem(7020983), Elem(3299949), Elem(8263812), Elem(4415410), Elem(1933827), Elem(125297), Elem(1292144), Elem(5467326), Elem(4714400), Elem(4342376), Elem(7496137), Elem(5200270), Elem(2886077), Elem(4736024), Elem(8205633), Elem(2624977), Elem(3948754), Elem(6284987), Elem(5580967), Elem(3336999), Elem(1354144), Elem(4122173), Elem(8093478), Elem(2500544), Elem(3790017), Elem(5989120), Elem(4471217), Elem(553105), Elem(6003187), Elem(1868722), Elem(4356693), Elem(3529492), Elem(6718914), Elem(3292048), Elem(6424063), Elem(6568803), Elem(1433807), Elem(6385286), Elem(834167), Elem(6394217), Elem(4835705), Elem(2902881), Elem(632454), Elem(6724819), Elem(4982908), Elem(1148482), Elem(8130260), Elem(6288228), Elem(6383847), Elem(646368), Elem(3913310), Elem(5322041), Elem(1291240), Elem(5336258), Elem(3413270), Elem(2910834), Elem(1399907), Elem(4314635), Elem(3259943), Elem(6205168), Elem(4497396), Elem(1789108), Elem(3623331), Elem(2575961), Elem(3843161), Elem(6538520), Elem(84334), Elem(1233989), Elem(1033835), Elem(7729706), Elem(1651531), Elem(3297188), Elem(467512), Elem(5937789), Elem(7504057), Elem(4926031), Elem(1412274), Elem(3606047), Elem(1820405), Elem(6723799), Elem(5318667), Elem(7120758), Elem(6243784), Elem(3884260), Elem(3210852), Elem(6292941), Elem(7520456), Elem(3357259), Elem(6545090), Elem(2416544), Elem(4926161), Elem(1544460), Elem(2916076), Elem(4962423), Elem(3439622), Elem(1796476), Elem(3557496), Elem(1415075), Elem(7443132), Elem(4853057), Elem(1007), Elem(1720053), Elem(1222961), Elem(5736778), Elem(46494), Elem(3651080), Elem(1658761), Elem(2472604), Elem(7501262), Elem(3070782), Elem(1795121), Elem(3705545), Elem(1627405), Elem(4813561), Elem(4218701), Elem(1678194), Elem(113255), Elem(7264852), Elem(1213681), Elem(2210721), Elem(7334980), Elem(5882460), Elem(5639138), Elem(665500), Elem(2180332), Elem(4980339), Elem(2163444), Elem(5038494), Elem(1053022), Elem(574720), Elem(4700019), Elem(2672302), Elem(8250282), Elem(1423572), Elem(3720736), Elem(5741301), Elem(6342768), Elem(4977411), Elem(490275), Elem(3583332), Elem(760312), Elem(364297), Elem(7314301), Elem(6333198), Elem(3184784), Elem(6275458), Elem(4629772), Elem(2978358), Elem(4197140), Elem(841551), Elem(7839443), Elem(543169), Elem(6748110), Elem(3033113), Elem(4317022), Elem(4732580), Elem(3003265), Elem(1816594), Elem(6366703), Elem(3451720), Elem(5162522), Elem(6412814), Elem(7643369), Elem(1901469), Elem(2527319), Elem(3187772), Elem(5387686), Elem(2888963), Elem(6647857), Elem(778225), Elem(5187535), Elem(3755794), Elem(7557617), Elem(2739370), Elem(1614056), Elem(7471515), Elem(1181844), Elem(5705460), Elem(2371657), Elem(7247456), Elem(7245942), Elem(8082386), Elem(7751093), Elem(522110), Elem(3854394), Elem(4897823), Elem(4738202), Elem(641886), Elem(4293069), Elem(5687346), Elem(5036541), Elem(6514290), Elem(7297711), Elem(4207046), Elem(7075296), Elem(3614670), Elem(5321084), Elem(1754383), Elem(6252202), Elem(2301748), Elem(3690817), Elem(7461827), Elem(5231892), Elem(6952945), Elem(470646), Elem(171143), Elem(2890891), Elem(2631345), Elem(2312920), Elem(6364031), Elem(4406944), Elem(1960197), Elem(4779987), Elem(3156821), Elem(3575303), Elem(6821539), Elem(4332816), Elem(2006166), Elem(3379139), Elem(6257551), Elem(1053997), Elem(761032), Elem(2898153), Elem(4988501), Elem(242790), Elem(2876827), Elem(6390234), Elem(6826832), Elem(6495347), Elem(1903748), Elem(7539181), Elem(5734286), Elem(1996592), Elem(594269), Elem(229472), Elem(7942790), Elem(2821050), Elem(7099971), Elem(1895375), Elem(6915271), Elem(7848367), Elem(4680315), Elem(7680296), Elem(5420166), Elem(4992856), Elem(2281475), Elem(5295261), Elem(827204), Elem(1372564), Elem(2348180), Elem(3767002), Elem(2279613), Elem(25422)])), NttPolynomial(Array([Elem(2714502), Elem(783906), Elem(1566989), Elem(4885965), Elem(5894027), Elem(3012280), Elem(7360148), Elem(2153289), Elem(6050024), Elem(401620), Elem(5854645), Elem(8345192), Elem(7913361), Elem(2861837), Elem(3713997), Elem(5046391), Elem(7577270), Elem(412083), Elem(6283177), Elem(821740), Elem(603327), Elem(5792807), Elem(1408323), Elem(4640238), Elem(774201), Elem(2839339), Elem(1602883), Elem(4107792), Elem(224545), Elem(4850373), Elem(3406444), Elem(1367004), Elem(845219), Elem(6712004), Elem(5382303), Elem(5511435), Elem(2690887), Elem(5595908), Elem(5591236), Elem(2284861), Elem(2931594), Elem(6154130), Elem(4940503), Elem(475854), Elem(6844140), Elem(5532672), Elem(4329925), Elem(7986306), Elem(7951855), Elem(6312875), Elem(3836583), Elem(3021731), Elem(1409854), Elem(3718381), Elem(1711938), Elem(3208450), Elem(3156890), Elem(6178524), Elem(4946771), Elem(5699962), Elem(6768642), Elem(3706876), Elem(2580896), Elem(6342450), Elem(1125237), Elem(6816262), Elem(6695354), Elem(3563680), Elem(2613932), Elem(3441324), Elem(3558401), Elem(1767026), Elem(5042376), Elem(3833363), Elem(7750766), Elem(7959873), Elem(4864030), Elem(757233), Elem(7787562), Elem(2108723), Elem(6581754), Elem(965362), Elem(7744214), Elem(5925943), Elem(1979628), Elem(7974044), Elem(3564418), Elem(1877005), Elem(4922710), Elem(3306027), Elem(6086722), Elem(2423991), Elem(1620468), Elem(521192), Elem(7414127), Elem(4979342), Elem(5274907), Elem(4229588), Elem(1829010), Elem(6885322), Elem(4155521), Elem(5565368), Elem(6934052), Elem(226351), Elem(6060967), Elem(7795952), Elem(7228917), Elem(6899414), Elem(6536583), Elem(778235), Elem(3055753), Elem(7476545), Elem(5454776), Elem(5242769), Elem(528943), Elem(2804090), Elem(3101696), Elem(7575500), Elem(7302640), Elem(1298472), Elem(6044941), Elem(7951703), Elem(5692211), Elem(4840051), Elem(4983053), Elem(6549831), Elem(6738585), Elem(8020134), Elem(5462417), Elem(3722010), Elem(3040541), Elem(3331986), Elem(2034441), Elem(2002898), Elem(2531099), Elem(7849608), Elem(2964492), Elem(5479750), Elem(3128274), Elem(5900156), Elem(4193574), Elem(5707481), Elem(6223910), Elem(2588038), Elem(7909502), Elem(283451), Elem(4214698), Elem(3816994), Elem(6957878), Elem(2294363), Elem(5489221), Elem(6596106), Elem(4360519), Elem(6558488), Elem(4976969), Elem(4014564), Elem(1752401), Elem(1714652), Elem(3192539), Elem(2615085), Elem(635994), Elem(2638093), Elem(6365089), Elem(7653407), Elem(1792321), Elem(7042007), Elem(7394859), Elem(6322571), Elem(1530136), Elem(5849370), Elem(8297461), Elem(7551522), Elem(4563783), Elem(7231895), Elem(4166720), Elem(6344084), Elem(7093728), Elem(4309025), Elem(4020063), Elem(5851667), Elem(7059536), Elem(7597889), Elem(3595809), Elem(2317683), Elem(7505284), Elem(7038792), Elem(5082429), Elem(2070007), Elem(71323), Elem(7108345), Elem(2840807), Elem(1382444), Elem(1028780), Elem(8259300), Elem(1522284), Elem(2772413), Elem(5623385), Elem(978594), Elem(2099815), Elem(4264038), Elem(3689695), Elem(7638740), Elem(522147), Elem(786500), Elem(1112487), Elem(7486405), Elem(3946201), Elem(4315768), Elem(6799320), Elem(2471216), Elem(3068625), Elem(174280), Elem(5365482), Elem(4032122), Elem(5557130), Elem(3244440), Elem(8095631), Elem(7907156), Elem(2256737), Elem(479290), Elem(1739551), Elem(4570206), Elem(2459231), Elem(115391), Elem(411688), Elem(2682659), Elem(4581331), Elem(7606610), Elem(3601553), Elem(6239541), Elem(1773501), Elem(6479185), Elem(4459760), Elem(8352301), Elem(339759), Elem(5367418), Elem(2210740), Elem(6750527), Elem(826854), Elem(5962298), Elem(2607415), Elem(2382568), Elem(3774182), Elem(7477163), Elem(7494473), Elem(6435504), Elem(1304895), Elem(284456), Elem(7081790), Elem(6462892), Elem(3281223), Elem(1800228), Elem(7074379), Elem(4833661), Elem(2780968), Elem(1655666)])), NttPolynomial(Array([Elem(4478518), Elem(6831094), Elem(3818158), Elem(4416410), Elem(3499622), Elem(5007144), Elem(1521314), Elem(4572968), Elem(5912614), Elem(3830938), Elem(3862520), Elem(4359251), Elem(6761261), Elem(2095982), Elem(7020579), Elem(226515), Elem(121953), Elem(7596073), Elem(6603779), Elem(6618711), Elem(4677110), Elem(7413745), Elem(4543210), Elem(552555), Elem(2004808), Elem(3304285), Elem(7720231), Elem(7785668), Elem(6476059), Elem(5815093), Elem(2309963), Elem(4697385), Elem(2104280), Elem(307001), Elem(3429252), Elem(8040544), Elem(6591738), Elem(1822478), Elem(1332979), Elem(5555039), Elem(3221400), Elem(1771290), Elem(446791), Elem(2983949), Elem(2855271), Elem(7986559), Elem(2454261), Elem(6527856), Elem(676721), Elem(7075878), Elem(6357920), Elem(6326576), Elem(5711779), Elem(5158702), Elem(7008376), Elem(55751), Elem(553165), Elem(5685908), Elem(2995387), Elem(3696463), Elem(8245508), Elem(4320100), Elem(869371), Elem(5160818), Elem(5002227), Elem(648327), Elem(1674071), Elem(5318069), Elem(7721896), Elem(3688853), Elem(5136905), Elem(5229309), Elem(3793127), Elem(5730182), Elem(4017393), Elem(7332351), Elem(3610446), Elem(4551562), Elem(7921363), Elem(6942665), Elem(3155148), Elem(4423450), Elem(3705163), Elem(7197579), Elem(7094401), Elem(8131884), Elem(917373), Elem(1267615), Elem(4829071), Elem(419951), Elem(6151965), Elem(593201), Elem(1248928), Elem(4157762), Elem(3328096), Elem(244392), Elem(3429576), Elem(3722130), Elem(6067843), Elem(3916937), Elem(2708826), Elem(3481950), Elem(1963379), Elem(2229444), Elem(4683970), Elem(7117458), Elem(1583955), Elem(3511367), Elem(7196862), Elem(45291), Elem(3979112), Elem(2399723), Elem(3187693), Elem(5550336), Elem(3643836), Elem(6907390), Elem(5429270), Elem(4735861), Elem(4451050), Elem(3319004), Elem(3168609), Elem(3375847), Elem(6508535), Elem(12198), Elem(2548323), Elem(2486292), Elem(994582), Elem(4655257), Elem(9250), Elem(5394466), Elem(8215893), Elem(4947098), Elem(282039), Elem(5798542), Elem(6431561), Elem(5106582), Elem(1594177), Elem(4659314), Elem(5597712), Elem(1318926), Elem(4794827), Elem(1524935), Elem(2976516), Elem(5773007), Elem(2704515), Elem(2015408), Elem(6012996), Elem(222020), Elem(7868087), Elem(6437365), Elem(6516739), Elem(3272641), Elem(7217666), Elem(7006718), Elem(7515649), Elem(6918888), Elem(4165355), Elem(609343), Elem(3565621), Elem(5347080), Elem(5626437), Elem(5395631), Elem(1897024), Elem(1988005), Elem(2467440), Elem(3481865), Elem(7284817), Elem(570759), Elem(2809709), Elem(1514309), Elem(3631158), Elem(7087457), Elem(7096525), Elem(5552007), Elem(614296), Elem(7821859), Elem(8271775), Elem(7847520), Elem(8370885), Elem(8099368), Elem(7189954), Elem(5427168), Elem(6141767), Elem(1097514), Elem(8136155), Elem(5652090), Elem(5507754), Elem(6163351), Elem(1051931), Elem(7192924), Elem(1455485), Elem(4117436), Elem(8265391), Elem(3591710), Elem(5867839), Elem(4133442), Elem(2948870), Elem(7457795), Elem(3829104), Elem(1814243), Elem(7219188), Elem(6202595), Elem(4249803), Elem(2090957), Elem(6975199), Elem(6885106), Elem(6059746), Elem(2481719), Elem(7469307), Elem(3796295), Elem(6630572), Elem(680631), Elem(3554919), Elem(1735712), Elem(3926761), Elem(3508123), Elem(2427544), Elem(7111954), Elem(2233961), Elem(6431541), Elem(2527408), Elem(464881), Elem(762687), Elem(5208745), Elem(2481510), Elem(4177507), Elem(3898612), Elem(5260680), Elem(2274603), Elem(7579788), Elem(865771), Elem(5037896), Elem(8310529), Elem(5264258), Elem(7151132), Elem(3310592), Elem(8039931), Elem(616596), Elem(5173118), Elem(6230203), Elem(5336956), Elem(6950263), Elem(471670), Elem(7992745), Elem(180480), Elem(2737466), Elem(5073694), Elem(6522440), Elem(5883671), Elem(4415221), Elem(7972982), Elem(317320), Elem(2138868), Elem(6558762), Elem(5036920), Elem(3975339)])), NttPolynomial(Array([Elem(3718314), Elem(7930581), Elem(2881280), Elem(130594), Elem(6440124), Elem(4056864), Elem(6198293), Elem(2628692), Elem(6831197), Elem(4887540), Elem(1359681), Elem(7186171), Elem(3439595), Elem(6972984), Elem(3390876), Elem(6024600), Elem(2470686), Elem(4647779), Elem(986432), Elem(6626376), Elem(1122280), Elem(4772083), Elem(2168125), Elem(477624), Elem(1331547), Elem(336999), Elem(6105449), Elem(302749), Elem(4441124), Elem(1222770), Elem(4921191), Elem(5671528), Elem(548282), Elem(4801078), Elem(4924686), Elem(3416577), Elem(1329654), Elem(6651238), Elem(2982568), Elem(8060145), Elem(4610854), Elem(3649045), Elem(6796791), Elem(6303644), Elem(7755133), Elem(6261128), Elem(7251640), Elem(7495752), Elem(1805409), Elem(314575), Elem(2754515), Elem(8082898), Elem(4925159), Elem(3860898), Elem(4906724), Elem(6876918), Elem(6798022), Elem(4461016), Elem(3225537), Elem(7641151), Elem(4318123), Elem(1701687), Elem(3255491), Elem(7220002), Elem(3254255), Elem(1416801), Elem(1251153), Elem(6391935), Elem(732629), Elem(3819382), Elem(6126526), Elem(3052651), Elem(1817194), Elem(3322532), Elem(2519259), Elem(7255399), Elem(3748900), Elem(2909328), Elem(7065068), Elem(153940), Elem(2943811), Elem(3548713), Elem(7806033), Elem(3543134), Elem(2451053), Elem(1429801), Elem(7449934), Elem(1445712), Elem(5694629), Elem(6693339), Elem(2338686), Elem(603144), Elem(6126355), Elem(730300), Elem(3653932), Elem(7239511), Elem(3823280), Elem(27154), Elem(3250023), Elem(6311319), Elem(497409), Elem(5593211), Elem(2575177), Elem(5284832), Elem(8247928), Elem(6334273), Elem(7157671), Elem(5891273), Elem(2770666), Elem(4908148), Elem(6219669), Elem(199106), Elem(2718043), Elem(5373630), Elem(7185814), Elem(4846195), Elem(7795595), Elem(7668517), Elem(7108322), Elem(2362107), Elem(1776613), Elem(6842043), Elem(3775042), Elem(2258974), Elem(3604009), Elem(870335), Elem(7884503), Elem(6612157), Elem(7884195), Elem(56430), Elem(5280131), Elem(164011), Elem(2672408), Elem(7305773), Elem(214068), Elem(4554532), Elem(6508519), Elem(7577538), Elem(8159694), Elem(7807626), Elem(1095394), Elem(3562), Elem(8074145), Elem(2549816), Elem(1592291), Elem(4881805), Elem(950187), Elem(3518490), Elem(3241749), Elem(8047600), Elem(3270648), Elem(2069110), Elem(1288101), Elem(2496504), Elem(8041535), Elem(1981254), Elem(4456668), Elem(7348657), Elem(1337403), Elem(7418841), Elem(56608), Elem(8322605), Elem(929678), Elem(4227281), Elem(2198953), Elem(7157873), Elem(4510099), Elem(6505186), Elem(6834516), Elem(5463089), Elem(7833693), Elem(2412478), Elem(5415845), Elem(5528613), Elem(7745284), Elem(8071219), 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Elem(7046665), Elem(6311968), Elem(6437357), Elem(7027654), Elem(3301167), Elem(2758634), Elem(1782895), Elem(7872531), Elem(1042276), Elem(7773332), Elem(1794207), Elem(3631720), Elem(6132664), Elem(4409828), Elem(3830628), Elem(8244113), Elem(2909431), Elem(6301848), Elem(4739425), Elem(4052392), Elem(4303931), Elem(1710608), Elem(1084839), Elem(5699990), Elem(4254219), Elem(2857259), Elem(3380249), Elem(789800), Elem(780795), Elem(6326929), Elem(3148607), Elem(1742966), Elem(5923695), Elem(2026829), Elem(8315393), Elem(1630327), Elem(1643323), Elem(3396760), Elem(2108901), Elem(6566798), Elem(4719953), Elem(75187), Elem(8307407), Elem(3242927), Elem(262075), Elem(7576325), Elem(728447), Elem(1317765), Elem(4895070), Elem(3735541), Elem(8370925), Elem(2090660), Elem(2250606), Elem(8156192), Elem(8032463), Elem(6277822), Elem(5425644), Elem(5734184), Elem(2393913), Elem(1706710), Elem(3590431), Elem(2136358), Elem(7587599), Elem(2466427), Elem(5704408), Elem(4315646), Elem(7084894), Elem(2086422), Elem(7091939), Elem(7242230)])), NttPolynomial(Array([Elem(7505315), Elem(7104564), Elem(5390841), Elem(3865673), Elem(6482903), Elem(8137740), Elem(5828167), Elem(2836092), Elem(2146513), Elem(3338300), Elem(3521226), Elem(3031951), Elem(176560), Elem(6986912), Elem(82362), Elem(5693132), Elem(85055), Elem(3441066), Elem(2146428), Elem(2368350), Elem(7965917), Elem(3361433), Elem(7478902), Elem(1710081), Elem(2851680), Elem(183825), Elem(3767328), Elem(3729662), Elem(6854209), Elem(8255571), Elem(492806), Elem(7516061), Elem(740373), Elem(1605718), Elem(7433334), Elem(7737670), Elem(2261644), Elem(50710), Elem(3213888), Elem(1806062), Elem(4211058), Elem(2076615), Elem(3880389), Elem(6994345), Elem(7171472), Elem(7526059), Elem(5268285), Elem(7331037), Elem(4945052), Elem(2257270), Elem(8313388), Elem(7371372), Elem(2597649), Elem(7016884), Elem(5011925), Elem(7333189), Elem(7495522), Elem(5563509), Elem(164489), Elem(8172843), Elem(6175714), Elem(790091), Elem(2253779), Elem(6520176), Elem(894338), Elem(3939938), Elem(4115377), Elem(3952932), Elem(577258), Elem(7105171), Elem(3591721), Elem(4979204), Elem(1169044), Elem(8267059), Elem(5186136), Elem(1044807), Elem(7799920), Elem(102420), Elem(7483293), Elem(4013356), Elem(4685323), Elem(5742644), Elem(2975217), Elem(6778294), Elem(1358788), Elem(6906422), Elem(3971096), Elem(7767996), Elem(5690212), Elem(1163108), Elem(7989983), Elem(2545643), Elem(4618376), Elem(5860669), Elem(7531776), Elem(1054831), Elem(3545783), Elem(3561004), Elem(7723109), Elem(5008659), Elem(4133891), Elem(60450), Elem(5152252), Elem(1053417), Elem(3060713), Elem(3012417), Elem(3065212), Elem(1837588), Elem(6531268), Elem(5720618), Elem(1759795), Elem(2438081), Elem(347211), Elem(7797879), Elem(6275003), Elem(5894903), Elem(7938658), Elem(4193398), Elem(6791951), Elem(1781100), Elem(555258), Elem(2817911), Elem(2055908), Elem(385047), Elem(1376838), Elem(2273376), Elem(1944364), Elem(5003993), Elem(3893392), Elem(4268021), Elem(968812), Elem(7471877), Elem(6491077), Elem(5815530), Elem(6929153), Elem(6202286), Elem(3653648), Elem(4615912), Elem(2884453), Elem(2950141), Elem(6858170), Elem(3536779), Elem(1407928), Elem(5370431), Elem(7368060), Elem(6176901), Elem(6477169), Elem(4772375), Elem(7089220), Elem(3917392), Elem(1750872), Elem(5301582), Elem(6838699), Elem(5821552), Elem(8163644), Elem(3035241), Elem(2854135), Elem(1740721), Elem(2161758), Elem(2309131), Elem(4673903), Elem(7208262), Elem(3161926), Elem(5001170), Elem(4927357), Elem(7662824), Elem(4980342), Elem(5146979), Elem(6502144), Elem(404984), Elem(681298), Elem(4614269), Elem(5736905), Elem(6273035), Elem(2418130), Elem(1435066), Elem(2764472), Elem(6388789), Elem(108407), Elem(1205988), Elem(565019), Elem(100562), Elem(1493182), Elem(6986026), Elem(6087574), Elem(5750203), Elem(5006494), Elem(7976544), Elem(7659217), Elem(2811866), Elem(7993332), Elem(7020822), Elem(3075615), Elem(5531363), Elem(6789657), Elem(515164), Elem(5492020), Elem(2962119), Elem(7693115), Elem(4956218), Elem(7060038), Elem(169330), Elem(4725297), Elem(6773601), Elem(2199258), Elem(1251625), Elem(6010631), Elem(5607448), Elem(2494342), Elem(6118597), Elem(417779), Elem(2189847), Elem(3335738), Elem(2303676), Elem(5943379), Elem(5365629), Elem(7376395), Elem(2832732), Elem(152419), Elem(34243), Elem(3168261), Elem(1083504), Elem(6376906), Elem(373068), Elem(553095), Elem(5986021), Elem(458312), Elem(1784136), Elem(6453294), Elem(6567674), Elem(5996348), Elem(4513259), Elem(928720), Elem(7663184), Elem(630931), Elem(2419691), Elem(4221047), Elem(2943721), Elem(7352067), Elem(1854825), Elem(3922546), Elem(2834541), Elem(111248), Elem(4247074), Elem(1787342), Elem(1654001), Elem(2343516), Elem(390875), Elem(1147318), Elem(705368), Elem(5921748), Elem(1850298), Elem(2951091), Elem(6267472), Elem(2134926), Elem(1504850)]))])), tr: Array([0, 44, 109, 167, 173, 52, 69, 121, 119, 128, 18, 108, 111, 237, 188, 126, 183, 210, 137, 174, 197, 75, 182, 63, 127, 217, 64, 16, 37, 75, 86, 188, 20, 131, 65, 185, 160, 11, 97, 119, 249, 22, 137, 211, 238, 164, 71, 196, 87, 119, 121, 28, 171, 91, 68, 51, 242, 138, 33, 158, 239, 86, 79, 203]) }, .. } +cc c7beac23efb5c5022f2fb07d2710c36a6801d885362cc7aabbab922f7c3b2b10 # shrinks to keypair = KeyPair { verifying_key: VerifyingKey { rho: Array([103, 184, 172, 136, 84, 33, 60, 13, 42, 48, 121, 130, 56, 52, 135, 250, 66, 26, 75, 110, 5, 26, 46, 20, 15, 181, 62, 78, 45, 65, 110, 114]), t1: Vector(Array([Polynomial(Array([Elem(481), Elem(131), Elem(1007), Elem(214), Elem(997), Elem(398), Elem(868), Elem(67), Elem(622), Elem(311), Elem(309), Elem(589), Elem(1007), Elem(93), Elem(810), Elem(999), Elem(555), Elem(5), Elem(838), Elem(439), Elem(634), Elem(1005), Elem(779), Elem(94), Elem(734), Elem(108), Elem(357), Elem(661), Elem(796), Elem(667), Elem(567), Elem(778), Elem(212), Elem(98), Elem(798), Elem(675), Elem(724), Elem(334), Elem(434), Elem(583), Elem(87), Elem(170), Elem(115), Elem(905), Elem(304), Elem(255), Elem(458), Elem(615), Elem(390), Elem(911), Elem(120), Elem(666), Elem(127), Elem(67), Elem(663), Elem(291), Elem(61), Elem(754), Elem(479), Elem(936), Elem(331), Elem(587), Elem(369), Elem(836), Elem(481), Elem(892), Elem(467), Elem(801), Elem(245), Elem(283), Elem(406), Elem(389), Elem(198), Elem(375), Elem(956), Elem(530), Elem(19), Elem(845), Elem(483), Elem(744), Elem(232), Elem(762), Elem(789), Elem(84), Elem(891), Elem(912), Elem(940), Elem(750), Elem(433), Elem(626), Elem(562), Elem(609), Elem(544), Elem(182), Elem(370), Elem(773), Elem(308), Elem(613), Elem(995), Elem(308), Elem(605), Elem(416), Elem(667), Elem(251), Elem(740), Elem(950), Elem(404), Elem(306), Elem(169), Elem(309), Elem(688), Elem(303), Elem(142), Elem(593), Elem(188), Elem(32), Elem(677), Elem(402), Elem(336), Elem(169), Elem(967), Elem(221), Elem(223), Elem(834), Elem(634), Elem(538), Elem(120), Elem(132), Elem(669), Elem(89), Elem(540), Elem(568), Elem(95), Elem(1013), Elem(734), Elem(981), Elem(119), Elem(150), Elem(295), Elem(753), Elem(317), Elem(641), Elem(384), Elem(558), Elem(982), Elem(808), Elem(15), Elem(802), Elem(983), Elem(302), Elem(627), Elem(327), Elem(688), Elem(710), Elem(972), Elem(159), Elem(612), Elem(268), Elem(174), Elem(902), Elem(202), Elem(530), Elem(542), Elem(1021), Elem(426), Elem(294), Elem(933), Elem(24), Elem(372), Elem(954), Elem(109), Elem(53), Elem(109), Elem(755), Elem(110), Elem(474), Elem(607), Elem(402), Elem(578), Elem(22), Elem(522), Elem(989), Elem(463), Elem(452), Elem(128), Elem(221), Elem(630), Elem(602), Elem(12), Elem(15), Elem(334), Elem(983), Elem(233), Elem(638), Elem(1015), Elem(957), Elem(180), Elem(800), Elem(655), Elem(879), Elem(381), Elem(777), Elem(816), Elem(676), Elem(706), Elem(331), Elem(292), Elem(752), Elem(942), Elem(800), Elem(344), Elem(536), Elem(234), Elem(570), Elem(650), Elem(21), Elem(540), Elem(849), Elem(53), Elem(154), Elem(213), Elem(693), Elem(193), Elem(614), Elem(707), Elem(203), Elem(604), Elem(509), Elem(598), Elem(567), Elem(675), Elem(776), Elem(151), Elem(762), Elem(293), Elem(420), Elem(848), Elem(160), Elem(461), Elem(309), Elem(193), Elem(423), Elem(935), Elem(268), Elem(895), Elem(62), Elem(658), Elem(166), Elem(317), Elem(24), Elem(27), Elem(695), Elem(697), Elem(980), Elem(525), Elem(821)])), Polynomial(Array([Elem(657), Elem(434), Elem(509), Elem(487), Elem(614), Elem(981), Elem(207), Elem(684), Elem(7), Elem(112), Elem(989), Elem(901), Elem(895), Elem(471), Elem(451), Elem(379), Elem(450), Elem(575), Elem(9), Elem(854), Elem(33), Elem(688), Elem(218), Elem(350), Elem(432), Elem(128), Elem(524), Elem(919), Elem(758), Elem(975), Elem(1010), Elem(177), Elem(515), Elem(266), Elem(197), Elem(495), Elem(442), Elem(297), Elem(486), Elem(766), Elem(288), Elem(943), Elem(800), Elem(905), Elem(454), Elem(163), Elem(470), Elem(203), Elem(271), Elem(331), Elem(409), Elem(980), Elem(26), Elem(742), Elem(45), Elem(68), Elem(292), Elem(627), Elem(939), Elem(185), Elem(411), Elem(527), Elem(100), Elem(871), Elem(220), Elem(788), Elem(833), Elem(590), Elem(833), Elem(862), Elem(455), Elem(188), Elem(638), Elem(574), Elem(229), Elem(618), Elem(174), Elem(725), Elem(128), Elem(47), Elem(61), Elem(612), Elem(832), Elem(591), Elem(189), Elem(653), Elem(448), Elem(524), Elem(3), Elem(302), Elem(190), Elem(497), Elem(151), Elem(115), Elem(31), Elem(496), Elem(161), Elem(362), Elem(30), Elem(809), Elem(660), Elem(630), Elem(46), Elem(512), Elem(599), Elem(229), Elem(283), Elem(177), Elem(76), Elem(258), Elem(424), Elem(6), Elem(591), Elem(350), Elem(426), Elem(841), Elem(319), Elem(879), Elem(500), Elem(495), Elem(56), Elem(954), Elem(781), Elem(829), Elem(266), Elem(327), Elem(926), Elem(878), Elem(498), Elem(452), Elem(180), Elem(154), Elem(16), Elem(1001), Elem(397), Elem(482), Elem(70), Elem(302), Elem(598), Elem(642), Elem(196), Elem(304), Elem(177), Elem(618), Elem(549), Elem(47), Elem(971), Elem(376), Elem(327), Elem(589), Elem(828), Elem(887), Elem(206), Elem(463), Elem(34), Elem(129), Elem(275), Elem(129), Elem(687), Elem(12), Elem(754), Elem(455), Elem(692), Elem(685), Elem(49), Elem(26), Elem(402), Elem(927), Elem(480), Elem(151), Elem(989), Elem(146), Elem(302), Elem(853), Elem(266), Elem(435), Elem(1010), Elem(452), Elem(33), Elem(969), Elem(12), Elem(111), Elem(734), Elem(258), Elem(518), Elem(200), Elem(670), Elem(309), Elem(756), Elem(546), Elem(99), Elem(770), Elem(648), Elem(702), Elem(291), Elem(783), Elem(211), Elem(1012), Elem(950), Elem(101), Elem(898), Elem(14), Elem(874), Elem(598), Elem(315), Elem(493), Elem(917), Elem(705), Elem(231), Elem(19), Elem(67), Elem(1018), Elem(311), Elem(191), Elem(943), Elem(126), Elem(940), Elem(651), Elem(361), Elem(625), Elem(356), Elem(74), Elem(347), Elem(167), Elem(72), Elem(892), Elem(473), Elem(855), Elem(511), Elem(371), Elem(582), Elem(793), Elem(845), Elem(996), Elem(892), Elem(55), Elem(700), Elem(229), Elem(44), Elem(138), Elem(955), Elem(67), Elem(111), Elem(21), Elem(863), Elem(375), Elem(583), Elem(664), Elem(220), Elem(503), Elem(56), Elem(872), Elem(379), Elem(594), Elem(432), Elem(504)])), Polynomial(Array([Elem(118), Elem(322), Elem(1016), Elem(228), Elem(660), Elem(197), Elem(147), Elem(372), Elem(205), Elem(1000), Elem(129), Elem(481), Elem(89), Elem(85), Elem(1021), Elem(43), Elem(634), Elem(868), Elem(1002), Elem(901), Elem(600), Elem(606), Elem(343), Elem(230), Elem(671), Elem(279), Elem(175), Elem(475), Elem(209), Elem(316), Elem(449), Elem(575), Elem(687), Elem(825), Elem(891), Elem(153), Elem(795), Elem(328), Elem(350), Elem(761), Elem(427), Elem(727), Elem(841), Elem(73), Elem(339), Elem(732), Elem(177), Elem(938), Elem(144), Elem(722), Elem(331), Elem(976), Elem(604), Elem(855), Elem(489), Elem(131), Elem(934), Elem(972), Elem(138), Elem(576), Elem(771), Elem(406), Elem(73), Elem(907), Elem(940), Elem(167), Elem(398), Elem(754), Elem(185), Elem(802), Elem(590), Elem(781), Elem(145), Elem(258), Elem(701), Elem(925), Elem(79), Elem(420), Elem(31), Elem(924), Elem(557), Elem(916), Elem(733), Elem(116), Elem(1009), Elem(263), Elem(29), Elem(294), Elem(609), Elem(129), Elem(726), Elem(204), Elem(362), Elem(801), Elem(57), Elem(998), Elem(192), Elem(552), Elem(629), Elem(53), Elem(700), Elem(114), Elem(185), Elem(952), Elem(367), Elem(171), Elem(216), Elem(936), Elem(571), Elem(865), Elem(741), Elem(314), Elem(138), Elem(457), Elem(528), Elem(407), Elem(295), Elem(967), Elem(610), Elem(202), Elem(663), Elem(793), Elem(568), Elem(351), Elem(216), Elem(697), Elem(750), Elem(548), Elem(520), Elem(126), Elem(952), Elem(555), Elem(958), Elem(517), Elem(374), Elem(868), Elem(183), Elem(908), Elem(600), Elem(515), Elem(297), Elem(190), Elem(480), Elem(953), Elem(206), Elem(496), Elem(982), Elem(676), Elem(931), Elem(643), Elem(156), Elem(747), Elem(248), Elem(660), Elem(235), Elem(963), Elem(703), Elem(406), Elem(931), Elem(306), Elem(544), Elem(233), Elem(866), Elem(466), Elem(357), Elem(227), Elem(167), Elem(298), Elem(553), Elem(703), Elem(495), Elem(356), Elem(48), Elem(491), Elem(651), Elem(561), Elem(535), Elem(145), Elem(837), Elem(617), Elem(564), Elem(823), Elem(54), Elem(947), Elem(199), Elem(57), Elem(613), Elem(884), Elem(940), Elem(917), Elem(589), Elem(895), Elem(815), Elem(327), Elem(158), Elem(593), Elem(977), Elem(707), Elem(624), Elem(459), Elem(597), Elem(146), Elem(220), Elem(165), Elem(650), Elem(947), Elem(2), Elem(333), Elem(596), Elem(337), Elem(558), Elem(287), Elem(846), Elem(873), Elem(310), Elem(322), Elem(44), Elem(871), Elem(941), Elem(986), Elem(127), Elem(449), Elem(964), Elem(549), Elem(988), Elem(434), Elem(790), Elem(185), Elem(788), Elem(819), Elem(245), Elem(421), Elem(464), Elem(926), Elem(825), Elem(81), Elem(600), Elem(872), Elem(35), Elem(687), Elem(595), Elem(297), Elem(349), Elem(586), Elem(418), Elem(908), Elem(484), Elem(574), Elem(149), Elem(212), Elem(382), Elem(592), Elem(478), Elem(732), Elem(745), Elem(698)])), Polynomial(Array([Elem(43), Elem(408), Elem(430), Elem(613), Elem(818), Elem(951), Elem(166), Elem(953), Elem(559), Elem(510), Elem(703), Elem(595), Elem(192), Elem(496), Elem(264), Elem(622), Elem(993), Elem(529), Elem(626), Elem(872), Elem(870), Elem(292), Elem(674), Elem(595), Elem(739), Elem(233), Elem(223), Elem(110), Elem(264), Elem(685), Elem(244), Elem(96), Elem(448), Elem(476), Elem(1000), Elem(829), Elem(287), Elem(276), Elem(449), Elem(709), Elem(328), Elem(870), Elem(352), Elem(274), Elem(711), Elem(559), Elem(49), Elem(555), Elem(976), Elem(484), Elem(259), Elem(10), Elem(364), Elem(27), Elem(91), Elem(44), Elem(187), Elem(527), Elem(232), Elem(20), Elem(44), Elem(447), Elem(370), Elem(678), Elem(372), Elem(465), Elem(458), Elem(815), Elem(469), Elem(10), Elem(889), Elem(394), Elem(943), Elem(156), Elem(354), Elem(385), Elem(143), Elem(201), Elem(3), Elem(147), Elem(170), Elem(271), Elem(891), Elem(164), Elem(978), Elem(665), Elem(755), Elem(748), Elem(603), Elem(865), Elem(69), Elem(12), Elem(89), Elem(669), Elem(791), Elem(893), Elem(653), Elem(446), Elem(793), Elem(942), Elem(708), Elem(717), Elem(244), Elem(196), Elem(690), Elem(909), Elem(603), Elem(72), Elem(468), Elem(387), Elem(972), Elem(712), Elem(91), Elem(53), Elem(664), Elem(706), Elem(397), Elem(806), Elem(887), Elem(757), Elem(543), Elem(3), Elem(144), Elem(222), Elem(155), Elem(479), Elem(196), Elem(973), Elem(576), Elem(73), Elem(539), Elem(564), Elem(623), Elem(259), Elem(762), Elem(563), Elem(93), Elem(471), Elem(352), Elem(804), Elem(372), Elem(277), Elem(1009), Elem(120), Elem(27), Elem(545), Elem(639), Elem(991), Elem(959), Elem(522), Elem(360), Elem(442), Elem(699), Elem(640), Elem(550), Elem(924), Elem(231), Elem(875), Elem(3), Elem(696), Elem(464), Elem(1015), Elem(427), Elem(695), Elem(443), Elem(903), Elem(478), Elem(1006), Elem(831), Elem(297), Elem(185), Elem(368), Elem(350), Elem(13), Elem(809), Elem(378), Elem(559), Elem(833), Elem(409), Elem(723), Elem(803), Elem(1018), Elem(444), Elem(235), Elem(0), Elem(474), Elem(56), Elem(770), Elem(771), Elem(161), Elem(124), Elem(580), Elem(196), Elem(451), Elem(158), Elem(515), Elem(352), Elem(965), Elem(631), Elem(628), Elem(69), Elem(653), Elem(211), Elem(657), Elem(30), Elem(568), Elem(491), Elem(943), Elem(225), Elem(875), Elem(129), Elem(899), Elem(67), Elem(521), Elem(480), Elem(692), Elem(345), Elem(228), Elem(426), Elem(368), Elem(56), Elem(848), Elem(411), Elem(522), Elem(419), Elem(800), Elem(169), Elem(653), Elem(21), Elem(342), Elem(83), Elem(901), Elem(162), Elem(196), Elem(978), Elem(417), Elem(634), Elem(825), Elem(240), Elem(798), Elem(316), Elem(830), Elem(132), Elem(783), Elem(877), Elem(680), Elem(327), Elem(854), Elem(408), Elem(20), Elem(923), Elem(475), Elem(332), Elem(801), Elem(129), Elem(75)])), Polynomial(Array([Elem(344), Elem(215), Elem(425), Elem(87), Elem(604), Elem(255), Elem(433), Elem(811), Elem(119), Elem(637), Elem(463), Elem(640), Elem(337), Elem(755), Elem(949), Elem(833), Elem(45), Elem(60), Elem(288), Elem(1002), Elem(952), Elem(31), Elem(836), Elem(731), Elem(346), Elem(181), Elem(583), Elem(964), Elem(1017), Elem(453), Elem(397), Elem(942), Elem(584), Elem(321), Elem(296), Elem(905), Elem(332), Elem(170), Elem(1005), Elem(921), Elem(666), Elem(474), Elem(243), Elem(215), Elem(930), Elem(434), Elem(70), Elem(179), Elem(894), Elem(688), Elem(913), Elem(188), Elem(769), Elem(38), Elem(231), Elem(358), Elem(702), Elem(698), Elem(358), Elem(713), Elem(540), Elem(867), Elem(415), Elem(339), Elem(781), Elem(725), Elem(947), Elem(103), Elem(387), Elem(884), Elem(364), Elem(213), Elem(835), Elem(857), Elem(209), Elem(391), Elem(187), Elem(479), Elem(761), Elem(932), Elem(1008), Elem(225), Elem(277), Elem(357), Elem(1014), Elem(984), Elem(78), Elem(22), Elem(654), Elem(1023), Elem(721), Elem(75), Elem(864), Elem(387), Elem(756), Elem(936), Elem(925), Elem(217), Elem(659), Elem(634), Elem(311), Elem(195), Elem(120), Elem(366), Elem(329), Elem(717), Elem(788), Elem(375), Elem(85), Elem(510), Elem(416), Elem(454), Elem(272), Elem(833), Elem(1008), Elem(421), Elem(528), Elem(656), Elem(89), Elem(493), Elem(788), Elem(701), Elem(48), Elem(341), Elem(251), Elem(271), Elem(96), Elem(204), Elem(996), Elem(488), Elem(466), Elem(125), Elem(317), Elem(103), Elem(869), Elem(992), Elem(331), Elem(243), Elem(728), Elem(261), Elem(460), Elem(675), Elem(650), Elem(886), Elem(668), Elem(580), Elem(98), Elem(209), Elem(58), Elem(34), Elem(133), Elem(454), Elem(262), Elem(29), Elem(179), Elem(807), Elem(800), Elem(96), Elem(147), Elem(840), Elem(963), Elem(360), Elem(531), Elem(260), Elem(61), Elem(618), Elem(277), Elem(885), Elem(683), Elem(592), Elem(147), Elem(22), Elem(503), Elem(630), Elem(986), Elem(624), Elem(195), Elem(501), Elem(719), Elem(266), Elem(437), Elem(124), Elem(974), Elem(699), Elem(308), Elem(874), Elem(965), Elem(501), Elem(376), Elem(211), Elem(874), Elem(926), Elem(201), Elem(1018), Elem(209), Elem(167), Elem(179), Elem(348), Elem(805), Elem(23), Elem(212), Elem(744), Elem(936), Elem(235), Elem(599), Elem(151), Elem(69), Elem(78), Elem(647), Elem(279), Elem(415), Elem(339), Elem(92), Elem(741), Elem(716), Elem(522), Elem(180), Elem(652), Elem(868), Elem(309), Elem(429), Elem(908), Elem(898), Elem(387), Elem(115), Elem(1013), Elem(816), Elem(700), Elem(300), Elem(730), Elem(808), Elem(233), Elem(819), Elem(887), Elem(382), Elem(351), Elem(921), Elem(393), Elem(589), Elem(572), Elem(785), Elem(682), Elem(151), Elem(844), Elem(895), Elem(9), Elem(85), Elem(450), Elem(847), Elem(23), Elem(608), Elem(751), Elem(55), Elem(755), Elem(975), Elem(503)])), Polynomial(Array([Elem(436), Elem(162), Elem(341), Elem(551), Elem(464), Elem(728), Elem(822), Elem(22), Elem(343), Elem(390), Elem(812), Elem(677), Elem(519), Elem(827), Elem(1017), Elem(931), Elem(381), Elem(958), Elem(671), Elem(493), Elem(618), Elem(701), Elem(59), Elem(809), Elem(434), Elem(271), Elem(926), Elem(724), Elem(551), Elem(348), Elem(321), Elem(35), Elem(878), Elem(796), Elem(188), Elem(206), Elem(532), Elem(1015), Elem(584), Elem(920), Elem(482), Elem(677), Elem(326), Elem(733), Elem(852), Elem(334), Elem(867), Elem(653), Elem(950), Elem(159), Elem(577), Elem(166), Elem(649), Elem(278), Elem(552), Elem(269), Elem(717), Elem(503), Elem(5), Elem(537), Elem(553), Elem(230), Elem(98), Elem(880), Elem(390), Elem(783), Elem(520), Elem(673), Elem(798), Elem(704), Elem(996), Elem(906), Elem(855), Elem(525), Elem(27), Elem(308), Elem(601), Elem(718), Elem(229), Elem(324), Elem(108), Elem(48), Elem(39), Elem(33), Elem(17), Elem(527), Elem(816), Elem(905), Elem(3), Elem(191), Elem(262), Elem(994), Elem(424), Elem(8), Elem(92), Elem(245), Elem(807), Elem(302), Elem(118), Elem(890), Elem(333), Elem(944), Elem(624), Elem(773), Elem(361), Elem(755), Elem(734), Elem(346), Elem(407), Elem(936), Elem(795), Elem(478), Elem(781), Elem(658), Elem(141), Elem(237), Elem(82), Elem(577), Elem(432), Elem(82), Elem(84), Elem(189), Elem(434), Elem(379), Elem(441), Elem(66), Elem(256), Elem(917), Elem(984), Elem(23), Elem(754), Elem(87), Elem(381), Elem(68), Elem(183), Elem(790), Elem(550), Elem(263), Elem(659), Elem(1005), Elem(782), Elem(327), Elem(1008), Elem(975), Elem(399), Elem(623), Elem(736), Elem(782), Elem(373), Elem(783), Elem(795), Elem(504), Elem(450), Elem(250), Elem(663), Elem(924), Elem(256), Elem(181), Elem(188), Elem(866), Elem(210), Elem(796), Elem(927), Elem(525), Elem(701), Elem(271), Elem(62), Elem(464), Elem(918), Elem(858), Elem(854), Elem(21), Elem(786), Elem(362), Elem(408), Elem(525), Elem(35), Elem(82), Elem(710), Elem(573), Elem(607), Elem(448), Elem(1008), Elem(602), Elem(475), Elem(27), Elem(745), Elem(816), Elem(471), Elem(756), Elem(778), Elem(54), Elem(647), Elem(230), Elem(447), Elem(235), Elem(622), Elem(96), Elem(728), Elem(719), Elem(525), Elem(566), Elem(852), Elem(569), Elem(936), Elem(635), Elem(487), Elem(677), Elem(934), Elem(118), Elem(720), Elem(927), Elem(708), Elem(612), Elem(302), Elem(802), Elem(633), Elem(130), Elem(599), Elem(955), Elem(317), Elem(524), Elem(863), Elem(96), Elem(122), Elem(122), Elem(522), Elem(698), Elem(298), Elem(194), Elem(608), Elem(766), Elem(758), Elem(836), Elem(384), Elem(10), Elem(810), Elem(668), Elem(359), Elem(303), Elem(910), Elem(831), Elem(871), Elem(833), Elem(522), Elem(299), Elem(301), Elem(751), Elem(250), Elem(254), Elem(484), Elem(409), Elem(821), Elem(500), Elem(285), Elem(558)]))])), A_hat: NttMatrix(Array([NttVector(Array([NttPolynomial(Array([Elem(7016696), Elem(2213964), Elem(6592559), Elem(3984089), Elem(7147248), Elem(7197290), Elem(8122802), Elem(5948796), Elem(8169172), Elem(3563315), Elem(4140015), Elem(6101285), Elem(5236061), Elem(7876643), Elem(2263581), Elem(843194), Elem(4003980), Elem(3336927), Elem(1640061), Elem(6677681), Elem(72723), Elem(2687195), Elem(596146), Elem(723041), Elem(2917244), Elem(7237533), Elem(2868457), Elem(7008793), Elem(8257283), Elem(3265462), Elem(5317987), Elem(4928540), Elem(2010929), Elem(1697705), Elem(5706156), Elem(1937928), Elem(8078683), Elem(6859175), Elem(3586973), Elem(6395885), Elem(6974964), Elem(857405), Elem(2647362), Elem(5818318), Elem(2403111), Elem(2590886), Elem(2654361), Elem(7323180), Elem(4550102), Elem(1699277), Elem(5710818), Elem(5201107), Elem(815928), Elem(8098637), Elem(1883805), Elem(6386104), Elem(1568253), Elem(6896271), Elem(7850477), Elem(7307729), Elem(3592333), Elem(4237131), Elem(5773497), Elem(8373534), Elem(1602605), Elem(7000259), Elem(4543668), Elem(3809662), Elem(6837808), Elem(2092242), Elem(1905373), Elem(6190688), Elem(6220177), Elem(8112174), Elem(2377237), Elem(1950730), Elem(5175870), Elem(964001), Elem(6898241), Elem(5883540), Elem(4802746), Elem(6473479), Elem(7063201), Elem(2875617), Elem(4093195), Elem(5963008), Elem(1508954), Elem(3858738), Elem(8359145), Elem(3652022), Elem(7429514), Elem(3599171), Elem(6287765), Elem(4157805), Elem(7253386), Elem(2065019), Elem(7932593), Elem(2055720), Elem(4576511), Elem(8222770), Elem(1043456), Elem(6189669), Elem(689256), Elem(2077286), Elem(1814027), Elem(7772931), Elem(844891), Elem(7313479), Elem(24377), Elem(4463473), Elem(3652233), Elem(6984577), Elem(4775197), Elem(2776174), Elem(5250236), Elem(7931363), Elem(4882904), Elem(2651127), Elem(2025707), Elem(4684572), Elem(4353278), Elem(2915124), Elem(4853969), Elem(2643455), Elem(3157153), Elem(794751), Elem(8018696), Elem(1022257), Elem(6346151), Elem(2470247), Elem(6702819), Elem(4327995), Elem(5759306), Elem(2082999), Elem(6866713), Elem(2222175), Elem(1554645), Elem(3805221), Elem(7008822), Elem(2275891), Elem(5131690), Elem(3901265), Elem(6255381), Elem(4792781), Elem(1776165), Elem(6790101), Elem(6031523), Elem(2940348), Elem(5184494), Elem(4003648), Elem(6667326), Elem(8074216), Elem(1521304), Elem(7416158), Elem(4989330), Elem(3565470), Elem(6307033), Elem(4234359), Elem(457926), Elem(1017191), Elem(5487659), Elem(2334127), Elem(11930), Elem(6748502), Elem(4215423), Elem(615246), Elem(7969569), Elem(6551579), Elem(8031443), Elem(5218593), Elem(7034960), Elem(6647960), Elem(1514981), Elem(3912990), Elem(7179171), Elem(7241887), Elem(5707573), Elem(8356275), Elem(877012), Elem(1159216), Elem(7539959), Elem(2028990), Elem(1103255), Elem(5636125), Elem(7054253), Elem(8115513), Elem(4545368), Elem(224450), Elem(7035494), Elem(6942405), Elem(6229186), Elem(2495161), Elem(2153969), Elem(1172579), Elem(7322784), Elem(1664733), Elem(3916100), Elem(6933782), Elem(4741891), Elem(1021194), Elem(6724325), Elem(1512613), Elem(7521953), Elem(6864168), Elem(8316109), Elem(5793758), Elem(4795949), Elem(1727577), Elem(1040000), Elem(187865), Elem(3913923), Elem(6181002), Elem(8345786), Elem(6578518), Elem(8370261), Elem(2753891), Elem(4704033), Elem(6202932), Elem(360857), Elem(7878240), Elem(1804094), Elem(8074643), Elem(8214140), Elem(3878683), Elem(8270209), Elem(1551362), Elem(324685), Elem(1591511), Elem(287055), Elem(5291132), Elem(6111974), Elem(3545985), Elem(3113377), Elem(1778736), Elem(6356146), Elem(4132996), Elem(7482205), Elem(8023566), Elem(5036924), Elem(8212579), Elem(90379), Elem(988187), Elem(359142), Elem(6585490), Elem(5661215), Elem(7337792), Elem(4013646), Elem(4943703), Elem(5436194), Elem(6494965), Elem(954988), Elem(512137), Elem(473639), Elem(1824425), Elem(1239677), Elem(6882698)])), NttPolynomial(Array([Elem(603296), Elem(2254346), Elem(4871045), Elem(3299304), Elem(7562713), Elem(2126504), Elem(4102232), Elem(2214043), Elem(2436937), Elem(1485973), Elem(6495335), Elem(4074234), Elem(1928414), Elem(3763193), Elem(8248244), Elem(3659941), Elem(1937922), Elem(5484586), Elem(22216), Elem(7748240), Elem(5427620), Elem(6088302), Elem(1679317), Elem(2673534), Elem(695724), Elem(5930747), Elem(1669776), Elem(3261221), Elem(2812444), Elem(216859), Elem(5676516), Elem(3675401), Elem(7390660), Elem(8152681), Elem(5625421), Elem(4832647), Elem(744407), Elem(652397), Elem(2466403), Elem(2408268), Elem(3309386), Elem(6652665), Elem(5076654), Elem(4584003), Elem(7814176), Elem(2812840), Elem(2923805), Elem(7932716), Elem(5033594), Elem(2430041), Elem(6323430), Elem(8329179), Elem(2268957), Elem(8113590), Elem(930087), Elem(4437349), Elem(1270611), Elem(2487075), Elem(4970137), Elem(756126), Elem(1970306), Elem(3517686), Elem(299321), Elem(7781339), Elem(7793313), Elem(4931559), Elem(7614061), Elem(448564), Elem(2464651), Elem(1177078), Elem(2307234), Elem(3453271), Elem(1934556), Elem(7957789), Elem(3043071), Elem(841993), Elem(1385526), Elem(4576968), Elem(3323242), Elem(4121785), Elem(1094895), Elem(522319), Elem(976561), Elem(279203), Elem(2358828), Elem(3182740), Elem(7202824), Elem(1607495), Elem(2216398), Elem(762798), Elem(2381884), Elem(3836706), Elem(3605608), Elem(4921769), Elem(4389835), Elem(1454023), Elem(6192365), Elem(237246), Elem(3258519), Elem(3083680), Elem(5265594), Elem(2895737), Elem(915837), Elem(2847554), Elem(454281), Elem(6161844), Elem(3504879), Elem(4177120), Elem(2490553), Elem(4729118), Elem(8136202), Elem(3113117), Elem(1639673), Elem(2566466), Elem(7838656), Elem(2333499), Elem(5525829), Elem(2686019), Elem(1435794), Elem(7096227), Elem(4390171), Elem(5826814), Elem(2933726), Elem(1876977), Elem(197155), Elem(1640263), Elem(5790847), Elem(7486726), Elem(3019348), Elem(8078493), Elem(7399419), Elem(7952398), Elem(5990960), Elem(6733823), Elem(5084293), Elem(1693208), Elem(8288184), Elem(4341413), Elem(2614567), Elem(2008492), Elem(4789432), Elem(4655811), Elem(338609), Elem(1360972), Elem(7450891), Elem(5080756), Elem(2412246), Elem(798640), Elem(4464231), Elem(5819273), Elem(3917864), Elem(2557379), Elem(6535924), Elem(6618074), Elem(6310654), Elem(4640708), Elem(7403152), Elem(2605503), Elem(4588721), Elem(5361255), Elem(7009288), Elem(1228348), Elem(4441117), Elem(6800872), Elem(7015899), Elem(1722153), Elem(4278746), Elem(3272359), Elem(7879908), Elem(5478593), Elem(6644714), Elem(122907), Elem(5112332), Elem(7917163), Elem(6589556), Elem(5414525), Elem(6134127), Elem(3940177), Elem(7758857), Elem(7710211), Elem(1867733), Elem(3258422), Elem(7229135), Elem(1853852), Elem(2049828), Elem(7696783), Elem(4735358), Elem(4026798), Elem(2683394), Elem(7950442), Elem(1195911), Elem(786890), Elem(1313091), Elem(1091662), Elem(2186940), Elem(6031203), Elem(6903165), Elem(4265711), Elem(5744682), Elem(6583547), Elem(3464884), Elem(3431071), Elem(906409), Elem(1650038), Elem(4463754), Elem(3775272), Elem(4212164), Elem(857576), Elem(3658514), Elem(4835628), Elem(772355), Elem(3433633), Elem(2108784), Elem(7107788), Elem(5531658), Elem(3088565), Elem(6153731), Elem(656009), Elem(2217527), Elem(4884332), Elem(1144215), Elem(6044966), Elem(5455116), Elem(5641717), Elem(1585440), Elem(976661), Elem(7944466), Elem(3184756), Elem(8048325), Elem(1149788), Elem(2803213), Elem(4999357), Elem(2081946), Elem(6552049), Elem(3489973), Elem(8235182), Elem(6131432), Elem(8365704), Elem(7097467), Elem(1541660), Elem(6924634), Elem(4816592), Elem(180027), Elem(760536), Elem(5733332), Elem(216973), Elem(7165194), Elem(1434910), Elem(2606826), Elem(5293363), Elem(6168078), Elem(7194737), Elem(4489150), Elem(6159135), Elem(5495683), Elem(3859127)])), NttPolynomial(Array([Elem(3528808), Elem(6897131), Elem(1393915), Elem(7934274), Elem(4857674), Elem(3922128), Elem(5657638), Elem(2336301), Elem(1799505), Elem(6744181), Elem(1805565), Elem(629679), Elem(3226086), Elem(96214), Elem(6625089), Elem(8005866), Elem(2935201), Elem(3816395), Elem(6475020), Elem(3605765), Elem(7169738), Elem(6931151), Elem(7127459), Elem(4301176), Elem(2739056), Elem(3816243), Elem(8281968), Elem(6833699), Elem(7675830), Elem(1725435), Elem(5357395), Elem(5878822), Elem(6583522), Elem(3412846), Elem(2611302), Elem(3673143), Elem(5276296), Elem(7609727), Elem(115776), Elem(1385697), Elem(7150848), Elem(718102), Elem(4049807), Elem(7077477), Elem(2358194), Elem(1025558), Elem(7851372), Elem(5124780), Elem(754627), Elem(5002184), Elem(1302790), 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Elem(2987820), Elem(2625163), Elem(4216940), Elem(2088587), Elem(3863089), Elem(3615354), Elem(1641450), Elem(6509826), Elem(7010047), Elem(8187614), Elem(6996195), Elem(2329444), Elem(2566010), Elem(2213706), Elem(6281885), Elem(7342571), Elem(385552), Elem(6525593), Elem(7453268), Elem(4858369), Elem(366776), Elem(6609212), Elem(3548581), Elem(312861), Elem(6579928), Elem(3941643), Elem(4356583), Elem(6965392), Elem(1157510), Elem(7168829), Elem(5269310), Elem(487851), Elem(3210444), Elem(3909697), Elem(6274855), Elem(8276344), Elem(5230052), Elem(2816822), Elem(5662912), Elem(7201884), Elem(3030800), Elem(1562168), Elem(4700256), Elem(4897846), Elem(5836873), Elem(5741078), Elem(7139323), Elem(5333944), Elem(5320133), Elem(807790), Elem(4364049), Elem(7585378), Elem(3166067), Elem(2062647), Elem(2938052), Elem(6027129), Elem(6989494), Elem(6345989), Elem(5251852), Elem(4475744), Elem(3986418), Elem(147800), Elem(2743061), Elem(4771534), Elem(7746308), Elem(2446845), Elem(662344), 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Elem(1663102), Elem(7866150), Elem(7129816), Elem(1239271), Elem(5069597), Elem(4716223), Elem(4518720), Elem(6719227), Elem(1023816), Elem(2296191), Elem(989474), Elem(1702129), Elem(1770154), Elem(1813565), Elem(5205987), Elem(7089985), Elem(2008130), Elem(7566491), Elem(7355431), Elem(4667376), Elem(1447389), Elem(3277018), Elem(735468), Elem(5364195), Elem(6028766), Elem(1181748), Elem(3629389), Elem(411836), Elem(5976489), Elem(2794581), Elem(6659784), Elem(3985711), Elem(3136442), Elem(7195741), Elem(1175985), Elem(1065449), Elem(6927181), Elem(1812881), Elem(833214), Elem(7225081), Elem(1345003), Elem(4240350), Elem(4670351), Elem(7074263), Elem(4965341), Elem(852521), Elem(9780), Elem(5453505), Elem(4080725), Elem(4284021), Elem(2759644), Elem(1940565), Elem(4266699), Elem(5996784), Elem(2425519), Elem(2587707), Elem(6414497), Elem(7080158), Elem(5585000), Elem(6833328), Elem(493795)])), NttPolynomial(Array([Elem(4554471), Elem(4012591), Elem(6447043), Elem(3682378), Elem(7281874), Elem(3893075), Elem(6602040), Elem(7495267), Elem(1643418), Elem(6663319), Elem(7999909), Elem(2592704), Elem(54257), Elem(8320277), Elem(5230964), Elem(7858254), Elem(4900192), Elem(258129), Elem(3788444), Elem(823808), Elem(2414919), Elem(4709637), Elem(2358243), Elem(4020111), Elem(147989), Elem(4652740), Elem(6464076), Elem(3659207), Elem(2915188), Elem(2370229), Elem(840177), Elem(2402658), Elem(8037367), Elem(1074822), Elem(3690271), Elem(2869652), Elem(7688636), Elem(1813110), Elem(2293288), Elem(2957109), Elem(4614462), Elem(8249383), Elem(564909), Elem(3547734), Elem(1324318), Elem(4643028), Elem(6985472), Elem(1516713), Elem(5474147), Elem(359385), Elem(1321841), Elem(328101), Elem(4608079), Elem(7206598), Elem(8083784), Elem(7657256), Elem(2798440), Elem(1604198), Elem(5187753), Elem(3378213), Elem(6258657), Elem(4146742), Elem(8235772), Elem(8354957), Elem(5670759), Elem(2713992), Elem(5320236), Elem(1208049), Elem(5885019), Elem(2630221), Elem(7837794), Elem(2103092), Elem(3557766), Elem(2598646), Elem(1229802), Elem(6836990), Elem(46341), Elem(3749230), Elem(3633959), Elem(8351582), Elem(3638782), Elem(3531933), Elem(2444895), Elem(5481015), Elem(452763), Elem(2434292), Elem(879760), Elem(590631), Elem(718317), Elem(3256839), Elem(3137117), Elem(3817342), Elem(1682990), Elem(4646327), Elem(1346215), Elem(2243853), Elem(5644208), Elem(4099281), Elem(6561693), Elem(3349283), Elem(731244), Elem(1562717), Elem(7717899), Elem(5705828), Elem(5945499), Elem(669071), Elem(5381452), Elem(4033765), Elem(8069376), Elem(3927346), Elem(6578769), Elem(6456942), Elem(1309387), Elem(6859596), Elem(807883), Elem(2606720), Elem(7725617), Elem(1697638), Elem(1966937), Elem(3705129), Elem(59883), Elem(7473410), Elem(7367232), Elem(5291227), Elem(6267744), Elem(1113591), Elem(748882), Elem(2015496), Elem(7056480), Elem(7492608), Elem(5958923), Elem(7440535), Elem(1393202), Elem(8139143), Elem(1197531), Elem(5906436), Elem(1829577), Elem(1449703), Elem(721686), Elem(851778), Elem(4765528), Elem(3278209), Elem(3589362), Elem(6315648), Elem(8205291), Elem(1032310), Elem(4923064), Elem(6423452), Elem(1602456), Elem(7761486), Elem(3568728), Elem(7070835), Elem(4035554), Elem(5901856), Elem(2465432), Elem(1970057), Elem(5817993), Elem(2561666), Elem(7411303), Elem(6643577), Elem(505712), Elem(3587377), Elem(6768938), Elem(737270), Elem(3575208), Elem(1658591), Elem(5001484), Elem(7005360), Elem(4484944), Elem(5674171), Elem(7052541), Elem(5880040), Elem(7453447), Elem(6023748), Elem(1014209), Elem(3971967), Elem(6019438), Elem(186600), Elem(7047844), Elem(500981), Elem(1901900), Elem(5783734), Elem(2616575), Elem(1220461), Elem(5343849), Elem(5308828), Elem(6034156), Elem(6503510), Elem(7599346), Elem(5805705), Elem(7528140), Elem(2242864), Elem(1417338), Elem(420349), Elem(3490015), Elem(6148661), Elem(7431076), Elem(4883922), Elem(6480137), Elem(5743087), Elem(7557517), Elem(3190975), Elem(4916858), Elem(5371492), Elem(2668437), Elem(5027752), Elem(7554842), Elem(337710), Elem(196593), Elem(1148811), Elem(5963087), Elem(5868691), Elem(3685861), Elem(2817079), Elem(5641726), Elem(1243985), Elem(7918947), Elem(3025213), Elem(7568667), Elem(8210646), Elem(805799), Elem(6955092), Elem(5274696), Elem(4618311), Elem(923432), Elem(4044098), Elem(2912434), Elem(7592965), Elem(1779706), Elem(7228247), Elem(4810017), Elem(2442696), Elem(3176474), Elem(3517530), Elem(5856412), Elem(4491422), Elem(2762771), Elem(2283397), Elem(6238955), Elem(2344208), Elem(2353464), Elem(5602326), Elem(424448), Elem(5442720), Elem(6391886), Elem(7807020), Elem(1741025), Elem(3916776), Elem(4962579), Elem(1389518), Elem(748999), Elem(5676216), Elem(1170019), Elem(5121716), Elem(1758779), Elem(2581097)])), NttPolynomial(Array([Elem(449852), Elem(7326321), Elem(5900671), Elem(5797803), Elem(3525668), Elem(5299753), Elem(4457078), Elem(526467), Elem(2263772), Elem(3306514), Elem(7778801), Elem(2625228), Elem(6409186), Elem(212849), Elem(4269783), Elem(4678416), Elem(798116), Elem(4045653), Elem(3200151), Elem(817769), Elem(5931612), Elem(7305235), Elem(1769595), Elem(7192154), Elem(8028705), Elem(3459301), Elem(6476118), Elem(5660515), Elem(4249361), Elem(3401984), Elem(4006142), Elem(2000143), Elem(3504255), Elem(478101), Elem(684219), Elem(729273), Elem(7009321), Elem(7989499), Elem(7121666), Elem(4450468), Elem(7684159), Elem(1942228), Elem(3436088), Elem(7500429), Elem(6843377), Elem(2916626), Elem(7131149), Elem(1559862), Elem(5527779), Elem(1029860), Elem(3493094), Elem(1551663), Elem(4279079), Elem(6306841), Elem(4328743), Elem(5938533), Elem(1443281), Elem(1203556), Elem(4058685), Elem(614111), Elem(4436769), Elem(4318144), Elem(7482685), Elem(8356612), Elem(5253588), Elem(4127644), Elem(5676664), Elem(5810028), Elem(6322957), Elem(5050200), Elem(4978393), Elem(6029445), Elem(6508401), Elem(1557006), Elem(8337869), Elem(7720818), Elem(5183399), Elem(4720328), Elem(6881357), Elem(4022515), Elem(5783808), Elem(4794999), Elem(4328464), Elem(4066103), Elem(7007684), Elem(1923450), Elem(140599), Elem(2421722), Elem(4271729), Elem(4726064), Elem(5031140), Elem(4694511), Elem(1525471), Elem(3469911), Elem(6254696), Elem(2763416), Elem(1215651), Elem(767311), Elem(5546293), Elem(486991), Elem(7739138), Elem(4712202), Elem(6957071), Elem(6780662), Elem(6545863), Elem(7082592), Elem(7654947), Elem(7661516), Elem(5825537), Elem(2902311), Elem(5830060), Elem(7581887), Elem(2135633), Elem(84002), Elem(7237390), Elem(1174056), Elem(6071638), Elem(5952854), Elem(2693304), Elem(8335758), Elem(5534063), Elem(5262466), Elem(578701), Elem(1127666), Elem(4363583), Elem(5080871), Elem(5022453), Elem(370985), Elem(5451930), Elem(2455835), Elem(2150097), Elem(2893788), Elem(1102661), Elem(1420317), Elem(6263691), Elem(1194203), Elem(1707418), Elem(2148450), Elem(2885007), Elem(5281396), Elem(5610646), Elem(7373830), Elem(7701029), Elem(1061431), Elem(5282312), Elem(430953), Elem(5751123), Elem(4249066), Elem(2225719), Elem(8234587), Elem(882770), Elem(3596758), Elem(6493041), Elem(2036881), Elem(5704402), Elem(6301352), Elem(5015481), Elem(5927570), Elem(4157424), Elem(2404398), Elem(1720625), Elem(1603684), Elem(4602232), Elem(5357506), Elem(6098250), Elem(7589344), Elem(2300700), Elem(653087), Elem(8088262), Elem(3418388), Elem(6970514), Elem(7538876), Elem(4394896), Elem(7804515), Elem(5404970), Elem(736072), Elem(7395855), Elem(2160883), Elem(1932679), Elem(4451754), Elem(1944893), Elem(3627139), Elem(4243362), Elem(6593934), Elem(2504336), Elem(2625862), Elem(5492372), Elem(5852812), Elem(1252741), Elem(4604681), Elem(1868873), Elem(3613588), Elem(6727887), Elem(6587155), Elem(4704702), Elem(7758899), Elem(4379675), Elem(6295389), Elem(3841526), Elem(2363112), Elem(3686161), Elem(744960), Elem(6428603), Elem(7258437), Elem(151706), Elem(6724746), Elem(7212632), Elem(3534913), Elem(8046626), Elem(5659502), Elem(4539092), Elem(4915509), Elem(4928654), Elem(7887222), Elem(6456880), Elem(5293025), Elem(3618579), Elem(1884250), Elem(3009944), Elem(5202169), Elem(5159356), Elem(5797679), Elem(7301808), Elem(5580463), Elem(1076668), Elem(5882119), Elem(4059086), Elem(356104), Elem(4431825), Elem(7453698), Elem(2868512), Elem(6912568), Elem(2666615), Elem(6191163), Elem(7579088), Elem(2838270), Elem(2699464), Elem(7783329), Elem(1506845), Elem(2605967), Elem(1631615), Elem(7584784), Elem(2937066), Elem(7288093), Elem(4350853), Elem(5062093), Elem(5879053), Elem(7901971), Elem(981306), Elem(7618396), Elem(689818), Elem(1077043), Elem(3082565), Elem(2365214), Elem(1017177), Elem(774332)])), NttPolynomial(Array([Elem(4197539), Elem(5429365), Elem(3257082), Elem(7359549), Elem(5140832), Elem(5403472), Elem(1635126), Elem(6173052), Elem(5087560), Elem(5914314), Elem(514938), Elem(698305), Elem(5349421), Elem(1785297), Elem(4115278), Elem(5720221), Elem(4505211), Elem(2268765), Elem(3510618), Elem(3408780), Elem(1955800), Elem(2657927), Elem(7389169), Elem(238296), 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Elem(5976300), Elem(4055330), Elem(2595765), Elem(3054193), Elem(5962106), Elem(4083366), Elem(7770769), Elem(4240103), Elem(7726593), Elem(2522697), Elem(6127021), Elem(6095666), Elem(7952739), Elem(6752946), Elem(7506897), Elem(6220882), Elem(917893), Elem(3546928), Elem(921946), Elem(4693532), Elem(4276910), Elem(5519918), Elem(6916852), Elem(3252349), Elem(945533), Elem(8051348), Elem(2650872), Elem(3857523), Elem(4463870), Elem(2987200), Elem(5137506), Elem(7610152), Elem(3553017), Elem(5151577), Elem(2831706), Elem(3555465), Elem(6103041), Elem(3204572), Elem(3269328), Elem(1620320), Elem(7529069), Elem(3353340), Elem(2735352), Elem(7116524), Elem(2365201), Elem(480651), Elem(1282415), Elem(7170220), Elem(6119964), Elem(7538787), Elem(592532), Elem(6439263), Elem(7688429), Elem(7083259), Elem(5084724), Elem(4975967), Elem(2939011), Elem(1575502), Elem(8000121), Elem(2044935), Elem(5998655), Elem(7453542), Elem(6936067), Elem(5732569), Elem(877445), Elem(6681696), Elem(3164510), 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Elem(6427627), Elem(3931054), Elem(8288394), Elem(33284), Elem(5193316), Elem(5614978), Elem(6689778), Elem(194002), Elem(6106387), Elem(6683883), Elem(7706145), Elem(564726), Elem(2873136), Elem(1476012), Elem(7840165), Elem(4017705), Elem(6022767), Elem(2442393), Elem(4515203), Elem(1995418), Elem(1629431), Elem(3388189), Elem(5845371), Elem(6837990), Elem(4637078), Elem(4114168), Elem(4676169), Elem(3974386), Elem(1544293), Elem(1590655), Elem(4908631), Elem(6256719), Elem(6907764), Elem(3421021), Elem(4917244), Elem(6597605), Elem(8068734), Elem(5052987), Elem(5164742), Elem(613841), Elem(5702980), Elem(8250454), Elem(264635), Elem(2951729), Elem(6648757), Elem(7591170), Elem(2282500), Elem(1000025), Elem(7218246), Elem(7023862), Elem(7476182), Elem(5694696), Elem(4780719), Elem(617739), Elem(1680100), Elem(3098216), Elem(3744976), Elem(3756881), Elem(2706088), Elem(7690297), Elem(3629439), Elem(6213571), Elem(286646), Elem(2858708), Elem(8148014), Elem(2065410), Elem(570969), 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Elem(6336053), Elem(4957842), Elem(7274528), Elem(1069513), Elem(4555179), Elem(7694725), Elem(3362588), Elem(5636843), Elem(7647323), Elem(1099401), Elem(169769), Elem(2517886), Elem(1798040), Elem(2103118), Elem(8341281), Elem(2295943), Elem(6993589), Elem(1127263), Elem(2510778), Elem(6062277), Elem(2170358), Elem(364946), Elem(560848), Elem(2352589), Elem(6002010), Elem(5697266), Elem(8308455), Elem(7515868), Elem(57669), Elem(4876769), Elem(7216331), Elem(4601566), Elem(5465993), Elem(2880401), Elem(5391659), Elem(778004), Elem(7867793), Elem(7176643), Elem(2642275), Elem(6261296), Elem(3854393), Elem(4720682), Elem(3197387), Elem(3991607), Elem(5993093), Elem(8016731), Elem(2866207), Elem(175141), Elem(6477063), Elem(2492253), Elem(2228845), Elem(6313791), Elem(7128355), Elem(5334981), Elem(1375738), Elem(7735988), Elem(846265), Elem(8257442), Elem(1887681), Elem(5256655), Elem(1537133), Elem(4424541), Elem(8062915), Elem(7048742), Elem(4563182), Elem(224209), Elem(631492), Elem(2819325), Elem(167381), Elem(1519173), Elem(3331472), Elem(7150510), Elem(524561), Elem(7774541), Elem(1412076), Elem(6597842), Elem(3795135), Elem(1659743), Elem(7277540), Elem(7778949), Elem(8249316), Elem(920199), Elem(7221973), Elem(3599569), Elem(4763365), Elem(6343166), Elem(6994836), Elem(6612323), Elem(8126964), Elem(7291332), Elem(3350830), Elem(2911751), Elem(950492), Elem(7758232), Elem(8122831), Elem(4721630), Elem(343919), Elem(1262476), Elem(3523015), Elem(6762884), Elem(4109187), Elem(6432699), Elem(3565165), Elem(4517760), Elem(6127693), Elem(5461475), Elem(7514778), Elem(3370579), Elem(698811), Elem(4486663), Elem(6484544), Elem(1782320), Elem(5717723), Elem(1906937), Elem(7865408), Elem(7507051), Elem(6874810), Elem(2783012), Elem(3709785), Elem(3143582), Elem(3790098), Elem(3303165), Elem(6152800), Elem(8101203), Elem(7908243), Elem(1106280), Elem(3453376), Elem(2053884), Elem(3365559), Elem(4864156), Elem(1081297), Elem(3937026), Elem(6523880), Elem(6187805), Elem(6730700), Elem(3707434), Elem(8231522), Elem(3464348), Elem(2117161), Elem(6575522), Elem(1500220), Elem(5975394), Elem(6431798), Elem(6638829), Elem(7946318), Elem(4467443), Elem(3051010), Elem(4024848), Elem(1108139), Elem(4308545), Elem(7563213), Elem(1197205), Elem(4919335), Elem(3111199), Elem(581398), Elem(2777022), Elem(7450579), Elem(2187577), Elem(1487066), Elem(7964488), Elem(760595), Elem(5795717), Elem(2654919), Elem(5195092), Elem(1130330), Elem(4392224), Elem(7489320), Elem(5311792), Elem(8055797), Elem(1362109), Elem(2987179)])), NttPolynomial(Array([Elem(4351652), Elem(7055612), Elem(5765018), Elem(2754123), Elem(4447513), Elem(699757), Elem(4866945), Elem(5815310), Elem(3381915), Elem(3323081), Elem(5130290), Elem(2874138), Elem(3127198), Elem(3655641), Elem(1271495), Elem(2385475), Elem(1841995), Elem(7049029), Elem(3148190), Elem(3063111), Elem(777336), Elem(2226307), Elem(7701690), Elem(7572562), Elem(4634063), Elem(5647707), Elem(2367575), Elem(473790), Elem(6822819), Elem(2105530), Elem(8095102), Elem(3703449), Elem(6833181), Elem(530495), Elem(7804564), Elem(2990068), Elem(2683951), Elem(8025191), Elem(5489266), Elem(1601455), Elem(4669764), Elem(8300718), Elem(1352472), Elem(295588), Elem(5513174), Elem(5255998), Elem(399348), Elem(3225278), Elem(4834829), Elem(3412160), Elem(3539539), Elem(2201348), Elem(1814029), Elem(2352189), Elem(3083969), Elem(5995933), Elem(6868160), Elem(62059), Elem(6164801), Elem(3941195), Elem(2125443), Elem(7563400), Elem(764858), Elem(7561474), Elem(1115234), Elem(6759666), Elem(2121525), Elem(6831527), Elem(465710), Elem(4943572), Elem(6323325), Elem(8054880), Elem(7161187), Elem(4047534), Elem(3577876), Elem(4694430), Elem(1617776), Elem(4789709), Elem(3162034), Elem(4830949), Elem(1483907), Elem(4555131), Elem(7372016), Elem(1518077), Elem(4991933), Elem(1916556), Elem(7697898), Elem(1714619), Elem(5669491), Elem(1026561), Elem(4479268), Elem(6991373), Elem(6033387), Elem(5944839), Elem(5430185), Elem(652288), Elem(4491569), Elem(4697037), Elem(5965292), Elem(2715390), Elem(1258448), Elem(6572090), Elem(1287579), Elem(3604293), Elem(7882288), Elem(5311597), Elem(4153521), Elem(7394694), Elem(8336061), Elem(1072904), Elem(7629131), Elem(4006170), Elem(209252), Elem(7824448), Elem(5196153), Elem(8147912), Elem(269713), Elem(4898814), Elem(3216058), Elem(6001301), Elem(6828416), Elem(5861475), Elem(3513542), Elem(1372312), Elem(2157335), Elem(7164559), Elem(5948829), Elem(503419), Elem(3160712), Elem(5609207), Elem(7558693), Elem(7980926), Elem(3105991), Elem(7683610), Elem(4089508), Elem(6622708), Elem(847104), Elem(4419488), Elem(6344826), Elem(975347), Elem(4227038), Elem(2751251), Elem(2158649), Elem(7373899), Elem(7255086), Elem(1776834), Elem(5348712), Elem(3340606), Elem(8037032), Elem(442832), Elem(611233), Elem(2986485), Elem(2956652), Elem(4969151), Elem(4205643), Elem(8330500), Elem(891872), Elem(4287824), Elem(1663607), Elem(2011370), Elem(2303345), Elem(4435684), Elem(1112915), Elem(317004), Elem(166213), Elem(8033418), Elem(372564), Elem(2818682), Elem(5089987), Elem(1587945), Elem(8205580), Elem(5604278), Elem(462710), Elem(8288346), Elem(1093188), Elem(196411), Elem(6559505), Elem(1797387), Elem(1558937), Elem(3188787), Elem(817368), Elem(3503515), Elem(950), Elem(5762428), Elem(1376880), Elem(5390698), Elem(3482071), Elem(8051322), Elem(7434536), Elem(3186960), Elem(6096539), Elem(2995622), Elem(709323), Elem(6568610), Elem(1557656), Elem(3076807), Elem(1721748), Elem(6115181), Elem(7365188), Elem(474793), Elem(1711101), Elem(3903160), Elem(5723056), Elem(5997127), Elem(441347), Elem(7373453), Elem(3180870), Elem(3332992), Elem(7105466), Elem(6142596), Elem(641415), Elem(7044481), Elem(8014192), Elem(5139222), Elem(5743292), Elem(4291595), Elem(3492488), Elem(7662534), Elem(6212372), Elem(1719655), Elem(4960087), Elem(5347028), Elem(1965821), Elem(760436), Elem(2269462), Elem(1615432), Elem(4318225), Elem(7330288), Elem(7674647), Elem(5960347), Elem(5092442), Elem(7401306), Elem(5681619), Elem(8114410), Elem(6740115), Elem(3583298), Elem(658904), Elem(4105791), Elem(7032006), Elem(4198735), Elem(6892131), Elem(2655579), Elem(738422), Elem(3608673), Elem(3980633), Elem(950242), Elem(6944840), Elem(8274249), Elem(6130099), Elem(3273397), Elem(5148061), Elem(2046595), Elem(1574771), Elem(6062840), Elem(2451982), Elem(273925)])), NttPolynomial(Array([Elem(66494), Elem(6650566), Elem(4760432), Elem(1906903), Elem(3852702), Elem(4086526), Elem(1076607), Elem(518713), Elem(7560297), Elem(5143716), Elem(571302), Elem(1973512), Elem(2937928), Elem(3051059), Elem(3706562), Elem(1024793), Elem(3601962), Elem(7377629), Elem(1363898), Elem(1741995), Elem(976986), Elem(3110324), Elem(1155902), Elem(6715973), Elem(1658966), Elem(595220), Elem(8194473), Elem(4431921), Elem(2248108), Elem(364706), Elem(7660028), Elem(6679712), Elem(5069150), Elem(4798302), Elem(1898818), Elem(3463952), Elem(4350877), Elem(821265), Elem(7681328), Elem(188866), Elem(2832113), Elem(3673295), Elem(2922050), Elem(5056579), Elem(7487386), Elem(4452600), Elem(202450), Elem(7462754), Elem(7741119), Elem(3396579), Elem(7086130), Elem(1028070), Elem(8328133), Elem(5180151), Elem(8359643), Elem(6781643), Elem(2889344), Elem(2588046), Elem(4971058), Elem(1175337), Elem(4947756), Elem(7850397), Elem(7854765), Elem(4633038), Elem(4529641), Elem(6662998), Elem(5603240), Elem(1477196), Elem(1258639), Elem(6579962), Elem(3819748), Elem(449410), Elem(4358966), Elem(3429283), Elem(8345088), Elem(7697839), Elem(7459288), Elem(3243720), Elem(1600330), Elem(7068552), Elem(3044786), Elem(5336081), Elem(827994), Elem(2238367), Elem(944194), Elem(5615857), Elem(3419758), Elem(4565615), Elem(4565077), Elem(4617807), Elem(4129240), Elem(5199448), Elem(3508108), Elem(4962345), Elem(8087454), Elem(4028546), Elem(1312060), Elem(3355105), Elem(431612), Elem(6875223), Elem(3988627), Elem(7013157), Elem(6513419), Elem(4175069), Elem(1055841), Elem(7745729), Elem(7915371), Elem(3827097), Elem(3372057), Elem(4670148), Elem(2546456), Elem(7725069), Elem(2869002), Elem(3502590), Elem(903897), Elem(1723273), Elem(781643), Elem(1820090), Elem(1674957), Elem(2110375), Elem(51780), Elem(5917501), Elem(1616581), Elem(4769933), Elem(1484793), Elem(3142659), Elem(7189933), Elem(1394692), Elem(4940332), Elem(7170607), Elem(6112051), Elem(5682542), Elem(4543871), Elem(4034826), Elem(3521102), Elem(6796551), Elem(1211267), Elem(3173148), Elem(6923006), Elem(8281530), Elem(5685275), Elem(3662599), Elem(3566757), Elem(2434220), Elem(2713922), Elem(2456060), Elem(4507766), Elem(2759593), Elem(4661843), Elem(4337449), Elem(1478041), Elem(1895617), Elem(1960317), Elem(6909823), Elem(2996906), Elem(3941344), Elem(971129), Elem(6545692), Elem(5712634), Elem(2591996), Elem(1040120), Elem(542517), Elem(1287039), Elem(5331667), Elem(1090153), Elem(7503201), Elem(6880222), Elem(6328061), Elem(1232407), Elem(2864919), Elem(656199), Elem(3232952), Elem(3334308), Elem(5846076), Elem(8127760), Elem(922284), Elem(7056513), Elem(3588636), Elem(1702073), Elem(5372206), Elem(7952896), Elem(2471770), Elem(8254923), Elem(308214), Elem(4226998), Elem(1816421), Elem(2036830), Elem(1632029), Elem(5823275), Elem(7379257), Elem(5813777), Elem(578388), Elem(3318043), Elem(7651082), Elem(1917680), Elem(7964385), Elem(3047945), Elem(8133567), Elem(4647314), Elem(839336), Elem(4829881), Elem(4623080), Elem(687260), Elem(6420654), Elem(3958134), Elem(799116), Elem(1970966), Elem(3241750), Elem(6737285), Elem(3645457), Elem(2540443), Elem(2180036), Elem(5260391), Elem(6614447), Elem(311197), Elem(7780858), Elem(4082769), Elem(966533), Elem(7675652), Elem(2788338), Elem(1848998), Elem(2513938), Elem(4632178), Elem(3194204), Elem(2694433), Elem(3177440), Elem(6312354), Elem(258115), Elem(2158255), Elem(2154962), Elem(1917354), Elem(2883652), Elem(7148333), Elem(2590155), Elem(5909361), Elem(7809954), Elem(5381954), Elem(4941903), Elem(5695223), Elem(3695773), Elem(3693681), Elem(7452726), Elem(3753656), Elem(5248234), Elem(6616685), Elem(6577297), Elem(1205365), Elem(4135778), Elem(8371003), Elem(8040655), Elem(4304878), Elem(3123240), Elem(6016072), Elem(7449517), Elem(4711053), Elem(3997038)])), NttPolynomial(Array([Elem(5076924), Elem(7416243), Elem(4846474), Elem(6153180), Elem(4675823), Elem(5864350), Elem(6755606), Elem(6134541), Elem(304825), Elem(8101894), Elem(474910), Elem(1485867), Elem(7221455), Elem(4838002), Elem(2058117), Elem(6899809), Elem(81740), Elem(1087763), Elem(7394195), Elem(5014052), Elem(2381967), Elem(5498580), Elem(128668), Elem(5986670), Elem(328697), Elem(761546), Elem(5772899), Elem(5554674), Elem(3569237), Elem(1556056), Elem(4474763), Elem(5451037), Elem(3347341), Elem(5491171), Elem(3162839), Elem(654298), Elem(879122), Elem(5648931), Elem(6816650), Elem(5014238), Elem(6670759), Elem(6013511), Elem(4324988), Elem(4081697), Elem(753121), Elem(7313328), Elem(291472), Elem(1331812), Elem(4235149), Elem(5537759), Elem(937849), Elem(3717950), Elem(7051083), Elem(4197698), Elem(3943759), Elem(3978170), Elem(2354655), Elem(2982811), Elem(5424491), Elem(6253715), Elem(4699910), Elem(230467), Elem(3895436), Elem(5430378), Elem(7060438), Elem(360120), Elem(7429177), Elem(5504011), Elem(1769064), Elem(1435867), Elem(1114071), Elem(2948363), Elem(2785339), Elem(5247664), Elem(5278494), Elem(1054860), Elem(5079916), Elem(7287354), Elem(3987934), Elem(585406), Elem(1021204), Elem(5800142), Elem(5584557), Elem(7840371), Elem(6698525), Elem(50464), Elem(7755328), Elem(5788181), Elem(4580810), Elem(1894945), Elem(2113164), Elem(3553074), Elem(8106749), Elem(2501536), Elem(6862687), Elem(738176), Elem(750070), Elem(1965691), Elem(3481810), Elem(323784), Elem(4064308), Elem(7093761), Elem(5787503), Elem(1125235), Elem(2094920), Elem(2580492), Elem(5434953), Elem(34482), Elem(7669569), Elem(5595043), Elem(4365058), Elem(5484399), Elem(473521), Elem(3484335), Elem(2994151), Elem(3166175), Elem(3545598), Elem(3880667), Elem(6485048), Elem(6591377), Elem(2586600), Elem(7219181), Elem(8274160), Elem(2520168), Elem(3070870), Elem(14419), Elem(3320097), Elem(2946715), Elem(7422889), Elem(7315572), Elem(5871473), Elem(1529454), Elem(2146310), Elem(556494), Elem(2044893), Elem(6285602), Elem(6705066), Elem(518248), Elem(1593905), Elem(3303361), Elem(6081946), Elem(4584446), Elem(6627510), Elem(5799088), Elem(5396695), Elem(7449338), Elem(7976628), Elem(4419913), Elem(4193106), Elem(1632023), Elem(2928293), Elem(5211030), Elem(3002466), Elem(6002558), Elem(3543154), Elem(6262905), Elem(4084303), Elem(657724), Elem(620890), Elem(5763380), Elem(1956257), Elem(2452923), Elem(3946875), Elem(7626163), Elem(2240133), Elem(1461400), Elem(1422140), Elem(1745852), Elem(3704524), Elem(2635237), Elem(1310707), Elem(63248), Elem(7645594), Elem(6815202), Elem(2356227), Elem(3458499), Elem(4272854), Elem(8358974), Elem(2252665), Elem(4265480), Elem(8155162), Elem(409457), Elem(551899), Elem(6416799), Elem(251760), Elem(2653712), Elem(1657881), Elem(3331790), Elem(591919), Elem(3018671), Elem(8123175), Elem(2767852), Elem(5391194), Elem(6558660), Elem(1178796), Elem(5386779), Elem(2366542), Elem(6409956), Elem(7081796), Elem(2157511), Elem(7366089), Elem(7533686), Elem(197369), Elem(8129018), Elem(7551803), Elem(7133798), Elem(2886798), Elem(1274246), Elem(7411897), Elem(1949876), Elem(1466694), Elem(1860749), Elem(4072062), Elem(1394674), Elem(255285), Elem(1776636), Elem(6938899), Elem(5894598), Elem(5622640), Elem(1703122), Elem(4579330), Elem(1986384), Elem(6935808), Elem(1864956), Elem(5762531), Elem(5396051), Elem(8365977), Elem(1876112), Elem(5977448), Elem(6505457), Elem(7935440), Elem(3995755), Elem(4007258), Elem(7222551), Elem(8375513), Elem(1894949), Elem(6480400), Elem(5324070), Elem(55020), Elem(5874728), Elem(6281593), Elem(2287739), Elem(3525258), Elem(3701769), Elem(1326478), Elem(7156186), Elem(1504249), Elem(4463869), Elem(6249078), Elem(7152391), Elem(4643726), Elem(5174095), Elem(2162643), Elem(3247100), Elem(7888045), Elem(7561045)]))]))])), t1_2d_hat: NttVector(Array([NttPolynomial(Array([Elem(1314477), Elem(4507494), Elem(4982716), Elem(645712), Elem(4599215), Elem(6293752), Elem(7291952), Elem(3867876), Elem(6312458), Elem(7665664), Elem(6689962), Elem(4464660), Elem(4159913), Elem(5535340), Elem(6255995), Elem(3754243), Elem(8061675), Elem(7254527), Elem(3380734), Elem(2335717), Elem(5814394), Elem(2498801), Elem(5992460), Elem(7183190), Elem(3636733), Elem(5127471), Elem(2539099), Elem(2539632), Elem(7391128), Elem(2525777), Elem(1492819), Elem(1264460), Elem(979262), Elem(7444215), Elem(8352750), Elem(2083383), Elem(7492462), Elem(7679116), Elem(4953163), Elem(7893303), Elem(4823844), Elem(2507470), Elem(5886079), Elem(1436706), Elem(4895264), Elem(3229468), Elem(1696821), Elem(1810126), Elem(6143656), Elem(4273849), Elem(7635885), Elem(2216180), Elem(7997439), Elem(1115344), Elem(4871074), Elem(3663699), 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Elem(3020315), Elem(1169297), Elem(6266198), Elem(6606041), Elem(7762317), Elem(7249192), Elem(6607406), Elem(5491061), Elem(1683685), Elem(7916793), Elem(7350094), Elem(3904239), Elem(94745), Elem(1068850), Elem(1980589), Elem(2018582), Elem(7564692), Elem(6238828), Elem(3622217), Elem(1825158), Elem(81434), Elem(1011250), Elem(6765951), Elem(7259784), Elem(7468618), Elem(7505180), Elem(2177306), Elem(668653), Elem(1459382), Elem(5892969), Elem(394145), Elem(3382965), Elem(5392598), Elem(6060722), Elem(1255311), Elem(8220700), Elem(14606), Elem(7832893), Elem(3724937), Elem(4466715), Elem(6028697), Elem(5115945), Elem(8132757), Elem(1547685), Elem(980154), Elem(7238534), Elem(5808005), Elem(8189595), Elem(2281915), Elem(8117708), Elem(6109732), Elem(3742322), Elem(3837872), Elem(8020959), Elem(7601642), Elem(116379), Elem(7381729), Elem(3730921), Elem(6493141), Elem(1903295), Elem(6128548), Elem(3542889), Elem(4451645), Elem(1605999), Elem(7811177), Elem(8368064), Elem(2617577), 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NttPolynomial(Array([Elem(6084411), Elem(1277528), Elem(5333289), Elem(3375877), Elem(8304688), Elem(6606686), Elem(4863466), Elem(2450312), Elem(6744195), Elem(4131008), Elem(6356722), Elem(1773923), Elem(5997144), Elem(3050542), Elem(5013613), Elem(8241726), Elem(6882231), Elem(2068236), Elem(5161518), Elem(5671858), Elem(109489), Elem(5552758), Elem(7052856), Elem(5664991), Elem(6829711), Elem(1889880), Elem(7120156), Elem(1567146), Elem(5115284), Elem(1505374), Elem(7152824), Elem(1509046), Elem(7568964), Elem(2227391), Elem(7597706), Elem(1150869), Elem(2376497), Elem(1123789), Elem(1101572), Elem(4735724), Elem(6202744), Elem(7150773), Elem(2628044), Elem(1006588), Elem(6220112), Elem(2600070), Elem(5836540), Elem(7364601), Elem(4287640), Elem(1786744), Elem(4520962), Elem(5040383), Elem(33359), Elem(2422642), Elem(30519), Elem(7434666), Elem(203438), Elem(3399003), Elem(2110636), Elem(8288063), Elem(3702028), Elem(3500239), Elem(31264), Elem(7451897), Elem(2707267), Elem(2763474), Elem(531839), Elem(1170334), Elem(5217289), Elem(2389613), Elem(5905639), Elem(635231), Elem(4735108), Elem(7610376), Elem(3115011), Elem(4641078), Elem(7077081), Elem(4690658), Elem(3758739), Elem(5091901), Elem(6942970), Elem(4259160), Elem(2203336), Elem(4457347), Elem(76010), Elem(301238), Elem(3313437), Elem(5368386), Elem(4805646), Elem(4336154), Elem(4389127), Elem(4119305), Elem(2835922), Elem(3419925), Elem(4796704), Elem(7332674), Elem(1676393), Elem(7197118), Elem(831225), Elem(3629339), Elem(544608), Elem(8146652), Elem(3184032), Elem(3572222), Elem(7438249), Elem(6890731), Elem(3614736), Elem(5724520), Elem(5125839), Elem(6354089), Elem(7772285), Elem(851307), Elem(4176812), Elem(1341595), Elem(3948933), Elem(2782428), Elem(7339907), Elem(5505010), Elem(2435046), Elem(8307681), Elem(921603), Elem(1118167), Elem(2835470), Elem(6329505), Elem(7812278), Elem(5012715), Elem(7758759), Elem(5231293), Elem(3948899), Elem(7665406), Elem(6151300), Elem(3704613), Elem(919132), Elem(6465730), Elem(6243747), Elem(2812564), Elem(468898), Elem(4545626), Elem(3393165), Elem(7011747), Elem(7846357), Elem(6824879), Elem(5486791), Elem(8187420), Elem(7605949), Elem(2281094), Elem(1108828), Elem(7120682), Elem(2630383), Elem(1427093), Elem(5955672), Elem(2327615), Elem(3041460), Elem(1944876), Elem(5792928), Elem(4488887), Elem(4605170), Elem(7546385), Elem(2570280), Elem(8127950), Elem(1192680), Elem(3970584), Elem(151486), Elem(5374774), Elem(656061), Elem(8298586), Elem(5846325), Elem(4503911), Elem(115712), Elem(1233462), Elem(2601892), Elem(7511557), Elem(5115983), Elem(1669337), Elem(3101708), Elem(6660715), Elem(8254585), Elem(2913752), Elem(7241640), Elem(5647906), Elem(215333), Elem(5565339), Elem(3155958), Elem(4359263), Elem(8305753), Elem(3008030), Elem(4215245), Elem(4547253), Elem(1464190), Elem(610520), Elem(4299178), Elem(117015), Elem(1095126), Elem(7928911), Elem(342004), Elem(1502825), Elem(73860), Elem(7845905), Elem(7079355), Elem(6862196), Elem(5035037), Elem(5117490), Elem(3091228), Elem(2459171), Elem(2700239), Elem(2626180), Elem(7602387), Elem(7909041), Elem(1245201), Elem(129313), Elem(7765012), Elem(294193), Elem(6199114), Elem(3228870), Elem(7498635), Elem(6657813), Elem(5812342), Elem(2056224), Elem(85482), Elem(3681410), Elem(3229368), Elem(8047675), Elem(5733442), Elem(4533448), Elem(790196), Elem(1846646), Elem(3431871), Elem(2884424), Elem(4150648), Elem(4818085), Elem(7813509), Elem(3217256), Elem(5788771), Elem(6237331), Elem(7105759), Elem(7078243), Elem(3652516), Elem(2051526), Elem(2128148), Elem(5168481), Elem(3622264), Elem(1658576), Elem(1115129), Elem(4167945), Elem(2337217), Elem(4496637), Elem(3526536), Elem(6143043), Elem(5919720), Elem(992834), Elem(5681602), Elem(4169027), Elem(716206), Elem(2349838), Elem(319339), Elem(2725875)])), NttPolynomial(Array([Elem(7954175), Elem(8059542), Elem(6219622), Elem(6032423), Elem(4802566), Elem(7680593), Elem(4503900), Elem(7486360), Elem(6641066), Elem(5228345), Elem(5362277), Elem(4795218), Elem(3089441), Elem(5439878), Elem(4671885), Elem(4191656), Elem(6715882), Elem(395137), Elem(7551305), Elem(63552), Elem(806498), Elem(4598783), Elem(2348794), Elem(1140184), Elem(4701182), Elem(5494607), Elem(3169104), Elem(5707982), Elem(7941907), Elem(1918030), Elem(7330148), Elem(3110462), Elem(5700482), Elem(1509911), Elem(4984535), Elem(5751322), Elem(3859954), Elem(4744009), Elem(7598418), Elem(108250), Elem(1634648), Elem(2809909), Elem(6662469), Elem(1954858), Elem(2859898), Elem(2223638), Elem(12105), Elem(7419561), Elem(4860522), Elem(982815), Elem(6994972), Elem(1391469), Elem(2062175), Elem(4281221), Elem(992196), Elem(3599373), Elem(7330919), Elem(2813), Elem(1581943), Elem(5624166), Elem(7415639), Elem(3577500), Elem(6389906), Elem(3768226), Elem(5639226), Elem(1185894), Elem(4370819), Elem(5495670), Elem(1895008), Elem(7752617), Elem(2818473), Elem(4764875), Elem(6920882), Elem(4080709), Elem(956912), Elem(2096600), Elem(4371731), Elem(4910650), Elem(7689124), Elem(2523147), Elem(4073516), Elem(5985773), Elem(4929329), Elem(7943721), Elem(3407781), Elem(1112381), Elem(6313775), Elem(3875658), Elem(6319172), Elem(1483571), Elem(2678394), Elem(2726556), Elem(6126747), Elem(6047508), Elem(587962), Elem(1525069), Elem(4977249), Elem(3676334), Elem(2124094), Elem(5473636), Elem(6956970), Elem(6606652), Elem(7289285), Elem(5914059), Elem(6417541), Elem(6977855), Elem(2196349), Elem(5767322), Elem(417999), Elem(5069247), Elem(5916740), Elem(7605915), Elem(428794), Elem(7524969), Elem(3126654), Elem(7732289), Elem(1445948), Elem(5157809), Elem(53073), Elem(1995652), Elem(5220620), Elem(4775554), Elem(2037663), Elem(2437317), Elem(6236664), Elem(4201108), Elem(2134228), Elem(1794007), Elem(1036650), Elem(4937070), Elem(684119), Elem(758727), Elem(1318197), Elem(1903145), Elem(7444347), Elem(8024720), Elem(2173320), Elem(2294592), Elem(656560), Elem(7046921), Elem(2143641), Elem(6938217), Elem(3891311), Elem(5618583), Elem(8280909), Elem(4601578), Elem(7329907), Elem(696864), Elem(2537201), Elem(7250965), Elem(6372903), Elem(2274858), Elem(3115584), Elem(122845), Elem(1240452), Elem(5088606), Elem(7652416), Elem(5591378), Elem(3921627), Elem(6960662), Elem(7512748), Elem(7706266), Elem(7056222), Elem(2147183), Elem(2882586), Elem(1426873), Elem(7110161), Elem(6850720), Elem(2928475), Elem(7608233), Elem(3715743), Elem(1868400), Elem(4826322), Elem(7126077), Elem(7249208), Elem(7949328), Elem(7693947), Elem(6958170), Elem(8351591), Elem(242894), Elem(655451), Elem(1023799), Elem(1143972), Elem(4242543), Elem(4819745), Elem(4592822), Elem(5218830), Elem(3878263), Elem(2328766), Elem(3031750), Elem(1095373), Elem(2349308), Elem(1790775), Elem(6162819), Elem(2747696), Elem(3873587), Elem(4951394), Elem(6758891), Elem(7722088), Elem(3356101), Elem(6319609), Elem(4526266), Elem(2830688), Elem(8178815), Elem(1390620), Elem(1156614), Elem(5189527), Elem(6824694), Elem(7289488), Elem(5399041), Elem(5531772), Elem(7778205), Elem(4542227), Elem(2125606), Elem(4861379), Elem(1098649), Elem(1007455), Elem(742736), Elem(8220225), Elem(4973408), Elem(5274208), Elem(5483019), Elem(2761789), Elem(479998), Elem(811189), Elem(1289001), Elem(3526773), Elem(2866258), Elem(5209809), Elem(86771), Elem(6797379), Elem(6514072), Elem(4062772), Elem(7512308), Elem(3084820), Elem(3832355), Elem(1600120), Elem(5696453), Elem(5841063), Elem(7844628), Elem(5820623), Elem(2040550), Elem(5739583), Elem(2956672), Elem(2937679), Elem(4367512), Elem(6168538), Elem(821697), Elem(7537047), Elem(2824837), Elem(4576880), Elem(4614935), Elem(419518), Elem(1240807), Elem(6097408), Elem(6154791)])), NttPolynomial(Array([Elem(2278158), Elem(5354673), Elem(3741109), Elem(3785349), Elem(2003174), Elem(1589519), Elem(3056588), Elem(6160152), Elem(7780530), Elem(7909325), Elem(1180161), Elem(7884362), Elem(2935205), Elem(2012586), Elem(4334908), Elem(1768868), Elem(8317663), Elem(7968918), 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Elem(5308628), Elem(4539588), Elem(5822290), Elem(5485037), Elem(7693039), Elem(1712632), Elem(4179056), Elem(3319867), Elem(1732560), Elem(3984900), Elem(7387156), Elem(6728665), Elem(5772737), Elem(8131450), Elem(7210087), Elem(3466610), Elem(6796609), Elem(4073089), Elem(6567709), Elem(6243481), Elem(701680), Elem(2913360), Elem(1500236), Elem(7079368), Elem(2792730), Elem(8367320), Elem(6484785), Elem(525762), Elem(2118218), Elem(7638319), Elem(3749718), Elem(3142320), Elem(7024620), Elem(795866), Elem(2108137), Elem(1374204), Elem(3668967), Elem(503781), Elem(866144), Elem(5710719), Elem(6456383), Elem(2959478), Elem(6108711), Elem(7348796), Elem(6400939), Elem(773486), Elem(1455102), Elem(3656137), Elem(7087243), Elem(693902), Elem(4310587), Elem(8074357), Elem(6983239), Elem(8091416), Elem(7151318), Elem(6771456), Elem(2874067), Elem(4073440), Elem(3328445), Elem(7785851), Elem(5908423), Elem(3408174), Elem(2857393), Elem(7336539), Elem(6207817), Elem(6571229), Elem(8090370), Elem(6202846), Elem(3819758), Elem(5677212), Elem(7532392), Elem(371499), Elem(1922912), Elem(6255584), Elem(3286068), Elem(7051580), Elem(2032408), Elem(1403496), Elem(7447493), Elem(1783993), Elem(2236722), Elem(4940605), Elem(4403698), Elem(7263602), Elem(7349544), Elem(7502777), Elem(8183399), Elem(803118), Elem(339472), Elem(5422212), Elem(7634372), Elem(6727874), Elem(3503196), Elem(7032026), Elem(6888955), Elem(8054544), Elem(2242294), Elem(5286372), Elem(4042598), Elem(5596123), Elem(5575896), Elem(7102774), Elem(7061383), Elem(3409675), Elem(803413), Elem(1152967), Elem(2623996), Elem(4045259), Elem(1645497), Elem(176558), Elem(316543), Elem(6284398), Elem(5643341), Elem(2559040), Elem(2769116), Elem(257518), Elem(2672478), Elem(4710507), Elem(5239452), Elem(6681123), Elem(7108758), Elem(3779396), Elem(1121428), Elem(2598895), Elem(711155), Elem(7335869), Elem(4479957), Elem(564731), Elem(6201793), Elem(1868806), Elem(5934999), Elem(2258417), Elem(7375286), Elem(6814730), Elem(7989199), Elem(6878076), Elem(4697143), Elem(6115792), Elem(7873115), Elem(1924636), Elem(5301731), Elem(7377903), Elem(8012612), Elem(1497492), Elem(7328092), Elem(688217), Elem(4760423), Elem(3628035), Elem(4853018), Elem(3371893), Elem(4343813), Elem(2019187), Elem(4540094), Elem(3347420), Elem(4831057), Elem(4223171), Elem(564216), Elem(8146302), Elem(4125490), Elem(255767), Elem(2625475), Elem(4690484), Elem(4918018), Elem(470304), Elem(3013618), Elem(5423687), Elem(6165135), Elem(8064871), Elem(7703474), Elem(5561805), Elem(3889801)])), NttPolynomial(Array([Elem(1992004), Elem(6927786), Elem(5510203), Elem(7058790), Elem(1811353), Elem(7742498), Elem(7185793), Elem(4352947), Elem(5399684), Elem(4691787), Elem(1004395), Elem(2887674), Elem(2760044), Elem(5958568), Elem(8226546), Elem(1348436), Elem(2948606), Elem(6451721), Elem(307243), Elem(2148358), Elem(6341456), Elem(4360353), Elem(2575125), Elem(6266291), Elem(773019), Elem(4536449), Elem(3177362), Elem(8287822), Elem(3926358), Elem(4893652), Elem(5494380), Elem(340650), Elem(242494), Elem(3175022), Elem(4880564), Elem(3422392), Elem(5945543), Elem(1423544), Elem(6142773), Elem(1079737), Elem(2710103), Elem(2728601), Elem(3506551), Elem(7617716), Elem(5057055), Elem(808361), Elem(8239408), Elem(5388315), Elem(4339915), Elem(7418579), Elem(5949709), Elem(3817932), Elem(4813150), Elem(3950529), Elem(3105888), Elem(1745920), Elem(1987814), Elem(2322800), Elem(136402), Elem(5384669), Elem(7771992), Elem(2823567), Elem(2922696), Elem(262363), Elem(7463175), Elem(6665403), Elem(4730676), Elem(1201966), Elem(6947524), Elem(883260), Elem(7511876), Elem(1500464), Elem(2385043), Elem(1475479), Elem(3319286), Elem(7357792), Elem(5230884), Elem(2069102), Elem(270087), Elem(7051953), Elem(5665064), Elem(5226631), Elem(6046879), Elem(4479250), Elem(2438489), Elem(1366834), Elem(6063493), Elem(7395936), Elem(6569070), Elem(2451199), Elem(4596031), Elem(5030057), Elem(4522915), Elem(3017742), Elem(265322), Elem(7175605), Elem(4475605), Elem(3453267), Elem(207916), Elem(7834370), Elem(6463156), Elem(1528437), Elem(7118780), Elem(4651342), Elem(2094367), Elem(7987046), Elem(7950215), Elem(3936545), Elem(4686767), Elem(7405494), Elem(1446345), Elem(7294965), Elem(2128826), Elem(3739967), Elem(1888075), Elem(1391064), Elem(4548737), Elem(5260143), Elem(4390734), Elem(6482755), Elem(239086), Elem(7786362), Elem(1382385), Elem(8138302), Elem(7681213), Elem(2927066), Elem(6886655), Elem(3980080), Elem(4621881), Elem(3685081), Elem(6931841), Elem(6559334), Elem(6970304), Elem(7702278), Elem(4831506), Elem(5287857), Elem(3382517), Elem(1413502), Elem(952523), Elem(7643868), Elem(7204747), Elem(5202334), Elem(8221636), Elem(2662582), Elem(7523380), Elem(7510001), Elem(5625890), Elem(2414568), Elem(5125949), Elem(4180112), Elem(2122362), Elem(2919732), Elem(6038525), Elem(3619527), Elem(4260330), Elem(1736716), Elem(4766002), Elem(6135203), Elem(2386494), Elem(2255264), Elem(7996719), Elem(1423652), Elem(857149), Elem(3756938), Elem(4763512), Elem(6252671), Elem(7161945), Elem(620184), Elem(3861738), Elem(7356698), Elem(4424080), Elem(3047449), Elem(1758074), Elem(7510066), Elem(3648423), Elem(872360), Elem(5278990), Elem(3301267), Elem(7300067), Elem(6009772), Elem(4039478), Elem(7070585), Elem(7559193), Elem(4785526), Elem(7688474), Elem(2903082), Elem(5063502), Elem(5054632), Elem(7632734), Elem(4018800), Elem(7439733), Elem(2614351), Elem(1623768), Elem(4507411), Elem(584881), Elem(709473), Elem(1127167), Elem(4570678), Elem(6234087), Elem(780177), Elem(7423335), Elem(3055161), Elem(3206423), Elem(5255849), Elem(8270468), Elem(5847400), Elem(2278795), Elem(4414953), Elem(7523742), Elem(7793993), Elem(3316730), Elem(5672123), Elem(3517339), Elem(2259423), Elem(129120), Elem(3689875), Elem(6722416), Elem(3993532), Elem(2428012), Elem(7133647), Elem(6513330), Elem(1311384), Elem(6560861), Elem(7510477), Elem(6178580), Elem(7882545), Elem(291348), Elem(3559257), Elem(430719), Elem(1603933), Elem(6523674), Elem(3347287), Elem(5274811), Elem(2074163), Elem(6768079), Elem(1710546), Elem(3901700), Elem(6475204), Elem(1812276), Elem(5910555), Elem(6024416), Elem(3693817), Elem(2929261), Elem(5290635), Elem(6224029), Elem(4743023), Elem(3090084), Elem(4225785), Elem(2973538), Elem(6542912), Elem(8294418), Elem(7693498), Elem(1010142), Elem(8132498), Elem(856154), Elem(2709686)])), NttPolynomial(Array([Elem(4949949), Elem(7841882), Elem(7096429), Elem(3873313), Elem(2647626), Elem(6954221), Elem(1879664), Elem(2474458), Elem(5761030), Elem(8082268), Elem(3123088), Elem(2667579), Elem(3326126), Elem(6384946), Elem(7020104), Elem(8352819), Elem(3585365), Elem(766493), Elem(1593333), Elem(2076082), Elem(500415), Elem(4206077), Elem(2213650), Elem(5283536), Elem(193930), Elem(750505), Elem(2298018), Elem(560522), Elem(3081275), Elem(6097152), Elem(112590), Elem(4944485), Elem(2590884), Elem(2415052), Elem(3657045), Elem(1193172), Elem(7859898), Elem(3620807), Elem(5595327), Elem(614788), Elem(3421664), Elem(2958285), Elem(212800), Elem(1274580), Elem(4873860), Elem(3666119), Elem(2425244), Elem(626375), Elem(5867453), Elem(1631587), Elem(2086365), Elem(4280354), Elem(7139237), Elem(2104489), Elem(800312), Elem(5558642), Elem(2144719), Elem(1416097), Elem(7058282), Elem(7884104), Elem(1086660), Elem(1023056), Elem(5670056), Elem(6795253), Elem(7340959), Elem(8010867), Elem(1964771), Elem(5283810), Elem(7801291), Elem(4211775), Elem(4547672), Elem(3174722), Elem(6864109), Elem(7064559), Elem(6184012), Elem(1848875), Elem(7440000), Elem(7746770), Elem(5964127), Elem(1529750), Elem(3645666), Elem(5831825), Elem(7633301), Elem(4507724), Elem(5006470), Elem(3180788), Elem(7871657), Elem(531238), Elem(3763260), Elem(1155431), Elem(5762181), Elem(3909709), Elem(5082187), Elem(3545236), Elem(786856), Elem(1792259), Elem(1658464), Elem(3121542), Elem(7440542), Elem(6242798), Elem(7170286), Elem(7401845), Elem(4479701), Elem(1436584), Elem(5797666), Elem(6171423), Elem(1458317), Elem(5429769), Elem(7518822), Elem(3337805), Elem(855547), Elem(5767137), Elem(3411582), Elem(1706313), Elem(1098894), Elem(669733), Elem(2541948), Elem(5749388), Elem(1454064), Elem(6275184), Elem(3813450), Elem(4795235), Elem(2661281), Elem(8115578), Elem(6132146), Elem(7652328), Elem(4476517), Elem(4319775), Elem(787091), Elem(3728090), Elem(5643568), Elem(6063421), Elem(2516577), Elem(3341653), Elem(2730290), Elem(6718182), Elem(2449851), Elem(5404876), Elem(7870992), Elem(6349877), Elem(3781788), Elem(1523858), Elem(2372382), Elem(7421470), Elem(5743255), Elem(2065226), Elem(6088624), Elem(2826043), Elem(4203166), Elem(4145855), Elem(3904676), Elem(6208790), Elem(2696398), Elem(3165246), Elem(2831077), Elem(2204195), Elem(3619805), Elem(4227061), Elem(932691), Elem(2594360), Elem(4543674), Elem(4345014), Elem(8071290), Elem(7409340), Elem(4046878), Elem(1379987), Elem(2538858), Elem(5511174), Elem(2945171), Elem(5933689), Elem(2774164), Elem(1521540), Elem(2852636), Elem(6973894), Elem(8030910), Elem(5733901), Elem(267655), Elem(1373883), Elem(7080537), Elem(784084), Elem(4152510), Elem(6212263), Elem(5316686), Elem(678994), Elem(1090565), Elem(1746315), Elem(7798580), Elem(7212837), Elem(5764182), Elem(2001774), Elem(1180324), Elem(6029355), Elem(32647), Elem(5368905), Elem(7521172), Elem(4162232), Elem(3880960), Elem(2429720), Elem(2475424), Elem(2705187), Elem(4118407), Elem(6332335), Elem(4243291), Elem(2430630), Elem(5455639), Elem(2796686), Elem(6417514), Elem(5607921), Elem(2131015), Elem(6737074), Elem(1991740), Elem(1480257), Elem(1283898), Elem(8087821), Elem(203589), Elem(6979038), Elem(2469124), Elem(2768025), Elem(829388), Elem(633268), Elem(1132614), Elem(6770655), Elem(1462119), Elem(7989647), Elem(6001182), Elem(1196162), Elem(6866591), Elem(7455721), Elem(66643), Elem(5224606), Elem(5854749), Elem(6666279), Elem(5743434), Elem(4203176), Elem(3230623), Elem(1684810), Elem(6017963), Elem(3825962), Elem(7337299), Elem(5670718), Elem(6113191), Elem(7047072), Elem(4777531), Elem(760928), Elem(5404387), Elem(6040203), Elem(3996665), Elem(6485222), Elem(7481132), Elem(2279904), Elem(2150787), Elem(3152409), Elem(5739771), Elem(1073320), Elem(5708376), Elem(3400071)]))])), tr: Array([34, 100, 230, 139, 2, 213, 62, 218, 224, 36, 135, 98, 233, 29, 38, 120, 199, 216, 252, 176, 157, 131, 178, 44, 188, 215, 157, 31, 235, 140, 153, 22, 142, 47, 189, 123, 85, 98, 250, 114, 112, 114, 119, 94, 218, 66, 9, 26, 29, 25, 21, 240, 100, 154, 16, 50, 51, 153, 42, 1, 67, 244, 100, 97]) }, .. } diff --git a/ml-dsa/tests/proptests.rs b/ml-dsa/tests/proptests.rs new file mode 100644 index 00000000..07894ce6 --- /dev/null +++ b/ml-dsa/tests/proptests.rs @@ -0,0 +1,56 @@ +use ml_dsa::{ + signature::{Signer, Verifier}, + KeyGen, KeyPair, MlDsa44, MlDsa65, MlDsa87, Signature, +}; +use proptest::prelude::*; + +/// Example message +const MSG: &[u8] = b"Hello world"; + +// Keypairs +prop_compose! { + fn mldsa44_keypair()(seed_bytes in any::<[u8; 32]>()) -> KeyPair { + MlDsa44::key_gen_internal(seed_bytes.as_ref()) + } +} +prop_compose! { + fn mldsa65_keypair()(seed_bytes in any::<[u8; 32]>()) -> KeyPair { + MlDsa65::key_gen_internal(seed_bytes.as_ref()) + } +} +prop_compose! { + fn mldsa87_keypair()(seed_bytes in any::<[u8; 32]>()) -> KeyPair { + MlDsa87::key_gen_internal(seed_bytes.as_ref()) + } +} + +macro_rules! round_trip_test { + ($params:path, $keypair:expr) => { + let sig = $keypair.signing_key().sign(MSG); + + // Check signature verification + let verify_result = $keypair.verifying_key().verify(MSG, &sig); + prop_assert!(verify_result.is_ok()); + + // Check signature encoding round trip + let sig_decoded = Signature::<$params>::decode(&sig.encode()); + prop_assert_eq!(Some(sig), sig_decoded); + }; +} + +proptest! { + #[test] + fn mldsa44_round_trip(keypair in mldsa44_keypair()) { + round_trip_test!(MlDsa44, keypair); + } + + #[test] + fn mldsa65_round_trip(keypair in mldsa65_keypair()) { + round_trip_test!(MlDsa65, keypair); + } + + #[test] + fn mldsa87_round_trip(keypair in mldsa87_keypair()) { + round_trip_test!(MlDsa87, keypair); + } +}