4232 lines
135 KiB
C
4232 lines
135 KiB
C
#include "inner.h"
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/*
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* Falcon key pair generation.
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*
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* ==========================(LICENSE BEGIN)============================
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*
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* Copyright (c) 2017-2019 Falcon Project
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*
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* Permission is hereby granted, free of charge, to any person obtaining
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* a copy of this software and associated documentation files (the
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* "Software"), to deal in the Software without restriction, including
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* without limitation the rights to use, copy, modify, merge, publish,
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* distribute, sublicense, and/or sell copies of the Software, and to
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* permit persons to whom the Software is furnished to do so, subject to
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* the following conditions:
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*
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* The above copyright notice and this permission notice shall be
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* included in all copies or substantial portions of the Software.
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*
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* THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
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* EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF
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* MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.
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* IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY
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* CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT,
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* TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE
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* SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
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*
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* ===========================(LICENSE END)=============================
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*
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* @author Thomas Pornin <thomas.pornin@nccgroup.com>
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*/
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#define MKN(logn) ((size_t)1 << (logn))
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/* ==================================================================== */
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/*
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* Modular arithmetics.
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*
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* We implement a few functions for computing modulo a small integer p.
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*
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* All functions require that 2^30 < p < 2^31. Moreover, operands must
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* be in the 0..p-1 range.
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*
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* Modular addition and subtraction work for all such p.
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*
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* Montgomery multiplication requires that p is odd, and must be provided
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* with an additional value p0i = -1/p mod 2^31. See below for some basics
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* on Montgomery multiplication.
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*
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* Division computes an inverse modulo p by an exponentiation (with
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* exponent p-2): this works only if p is prime. Multiplication
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* requirements also apply, i.e. p must be odd and p0i must be provided.
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*
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* The NTT and inverse NTT need all of the above, and also that
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* p = 1 mod 2048.
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*
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* -----------------------------------------------------------------------
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*
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* We use Montgomery representation with 31-bit values:
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*
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* Let R = 2^31 mod p. When 2^30 < p < 2^31, R = 2^31 - p.
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* Montgomery representation of an integer x modulo p is x*R mod p.
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*
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* Montgomery multiplication computes (x*y)/R mod p for
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* operands x and y. Therefore:
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*
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* - if operands are x*R and y*R (Montgomery representations of x and
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* y), then Montgomery multiplication computes (x*R*y*R)/R = (x*y)*R
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* mod p, which is the Montgomery representation of the product x*y;
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*
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* - if operands are x*R and y (or x and y*R), then Montgomery
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* multiplication returns x*y mod p: mixed-representation
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* multiplications yield results in normal representation.
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*
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* To convert to Montgomery representation, we multiply by R, which is done
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* by Montgomery-multiplying by R^2. Stand-alone conversion back from
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* Montgomery representation is Montgomery-multiplication by 1.
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*/
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/*
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* Precomputed small primes. Each element contains the following:
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*
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* p The prime itself.
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*
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* g A primitive root of phi = X^N+1 (in field Z_p).
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*
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* s The inverse of the product of all previous primes in the array,
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* computed modulo p and in Montgomery representation.
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*
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* All primes are such that p = 1 mod 2048, and are lower than 2^31. They
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* are listed in decreasing order.
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*/
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typedef struct {
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uint32_t p;
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uint32_t g;
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uint32_t s;
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} small_prime;
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static const small_prime PRIMES[] = {
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{ 2147473409, 383167813, 10239 },
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{ 2147389441, 211808905, 471403745 },
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{ 2147387393, 37672282, 1329335065 },
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{ 2147377153, 1977035326, 968223422 },
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{ 2147358721, 1067163706, 132460015 },
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{ 2147352577, 1606082042, 598693809 },
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{ 2147346433, 2033915641, 1056257184 },
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{ 2147338241, 1653770625, 421286710 },
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{ 2147309569, 631200819, 1111201074 },
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{ 2147297281, 2038364663, 1042003613 },
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{ 2147295233, 1962540515, 19440033 },
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{ 2147239937, 2100082663, 353296760 },
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{ 2147235841, 1991153006, 1703918027 },
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{ 2147217409, 516405114, 1258919613 },
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{ 2147205121, 409347988, 1089726929 },
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{ 2147196929, 927788991, 1946238668 },
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{ 2147178497, 1136922411, 1347028164 },
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{ 2147100673, 868626236, 701164723 },
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{ 2147082241, 1897279176, 617820870 },
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{ 2147074049, 1888819123, 158382189 },
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{ 2147051521, 25006327, 522758543 },
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{ 2147043329, 327546255, 37227845 },
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{ 2147039233, 766324424, 1133356428 },
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{ 2146988033, 1862817362, 73861329 },
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{ 2146963457, 404622040, 653019435 },
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{ 2146959361, 1936581214, 995143093 },
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{ 2146938881, 1559770096, 634921513 },
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{ 2146908161, 422623708, 1985060172 },
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{ 2146885633, 1751189170, 298238186 },
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{ 2146871297, 578919515, 291810829 },
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{ 2146846721, 1114060353, 915902322 },
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{ 2146834433, 2069565474, 47859524 },
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{ 2146818049, 1552824584, 646281055 },
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{ 2146775041, 1906267847, 1597832891 },
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{ 2146756609, 1847414714, 1228090888 },
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{ 2146744321, 1818792070, 1176377637 },
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{ 2146738177, 1118066398, 1054971214 },
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{ 2146736129, 52057278, 933422153 },
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{ 2146713601, 592259376, 1406621510 },
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{ 2146695169, 263161877, 1514178701 },
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{ 2146656257, 685363115, 384505091 },
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{ 2146650113, 927727032, 537575289 },
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{ 2146646017, 52575506, 1799464037 },
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{ 2146643969, 1276803876, 1348954416 },
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{ 2146603009, 814028633, 1521547704 },
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{ 2146572289, 1846678872, 1310832121 },
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{ 2146547713, 919368090, 1019041349 },
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{ 2146508801, 671847612, 38582496 },
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{ 2146492417, 283911680, 532424562 },
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{ 2146490369, 1780044827, 896447978 },
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{ 2146459649, 327980850, 1327906900 },
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{ 2146447361, 1310561493, 958645253 },
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{ 2146441217, 412148926, 287271128 },
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{ 2146437121, 293186449, 2009822534 },
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{ 2146430977, 179034356, 1359155584 },
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{ 2146418689, 1517345488, 1790248672 },
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{ 2146406401, 1615820390, 1584833571 },
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{ 2146404353, 826651445, 607120498 },
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{ 2146379777, 3816988, 1897049071 },
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{ 2146363393, 1221409784, 1986921567 },
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{ 2146355201, 1388081168, 849968120 },
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{ 2146336769, 1803473237, 1655544036 },
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{ 2146312193, 1023484977, 273671831 },
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{ 2146293761, 1074591448, 467406983 },
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{ 2146283521, 831604668, 1523950494 },
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{ 2146203649, 712865423, 1170834574 },
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{ 2146154497, 1764991362, 1064856763 },
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{ 2146142209, 627386213, 1406840151 },
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{ 2146127873, 1638674429, 2088393537 },
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{ 2146099201, 1516001018, 690673370 },
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{ 2146093057, 1294931393, 315136610 },
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{ 2146091009, 1942399533, 973539425 },
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{ 2146078721, 1843461814, 2132275436 },
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{ 2146060289, 1098740778, 360423481 },
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{ 2146048001, 1617213232, 1951981294 },
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{ 2146041857, 1805783169, 2075683489 },
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{ 2146019329, 272027909, 1753219918 },
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{ 2145986561, 1206530344, 2034028118 },
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{ 2145976321, 1243769360, 1173377644 },
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{ 2145964033, 887200839, 1281344586 },
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{ 2145906689, 1651026455, 906178216 },
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{ 2145875969, 1673238256, 1043521212 },
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{ 2145871873, 1226591210, 1399796492 },
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{ 2145841153, 1465353397, 1324527802 },
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{ 2145832961, 1150638905, 554084759 },
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{ 2145816577, 221601706, 427340863 },
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{ 2145785857, 608896761, 316590738 },
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{ 2145755137, 1712054942, 1684294304 },
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{ 2145742849, 1302302867, 724873116 },
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{ 2145728513, 516717693, 431671476 },
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{ 2145699841, 524575579, 1619722537 },
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{ 2145691649, 1925625239, 982974435 },
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{ 2145687553, 463795662, 1293154300 },
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{ 2145673217, 771716636, 881778029 },
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{ 2145630209, 1509556977, 837364988 },
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{ 2145595393, 229091856, 851648427 },
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{ 2145587201, 1796903241, 635342424 },
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{ 2145525761, 715310882, 1677228081 },
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{ 2145495041, 1040930522, 200685896 },
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{ 2145466369, 949804237, 1809146322 },
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{ 2145445889, 1673903706, 95316881 },
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{ 2145390593, 806941852, 1428671135 },
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{ 2145372161, 1402525292, 159350694 },
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{ 2145361921, 2124760298, 1589134749 },
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{ 2145359873, 1217503067, 1561543010 },
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{ 2145355777, 338341402, 83865711 },
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{ 2145343489, 1381532164, 641430002 },
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{ 2145325057, 1883895478, 1528469895 },
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{ 2145318913, 1335370424, 65809740 },
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{ 2145312769, 2000008042, 1919775760 },
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{ 2145300481, 961450962, 1229540578 },
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{ 2145282049, 910466767, 1964062701 },
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{ 2145232897, 816527501, 450152063 },
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{ 2145218561, 1435128058, 1794509700 },
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{ 2145187841, 33505311, 1272467582 },
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{ 2145181697, 269767433, 1380363849 },
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{ 2145175553, 56386299, 1316870546 },
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{ 2145079297, 2106880293, 1391797340 },
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{ 2145021953, 1347906152, 720510798 },
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{ 2145015809, 206769262, 1651459955 },
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{ 2145003521, 1885513236, 1393381284 },
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{ 2144960513, 1810381315, 31937275 },
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{ 2144944129, 1306487838, 2019419520 },
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{ 2144935937, 37304730, 1841489054 },
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{ 2144894977, 1601434616, 157985831 },
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{ 2144888833, 98749330, 2128592228 },
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{ 2144880641, 1772327002, 2076128344 },
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{ 2144864257, 1404514762, 2029969964 },
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{ 2144827393, 801236594, 406627220 },
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{ 2144806913, 349217443, 1501080290 },
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{ 2144796673, 1542656776, 2084736519 },
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{ 2144778241, 1210734884, 1746416203 },
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{ 2144759809, 1146598851, 716464489 },
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{ 2144757761, 286328400, 1823728177 },
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{ 2144729089, 1347555695, 1836644881 },
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{ 2144727041, 1795703790, 520296412 },
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{ 2144696321, 1302475157, 852964281 },
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{ 2144667649, 1075877614, 504992927 },
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{ 2144573441, 198765808, 1617144982 },
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{ 2144555009, 321528767, 155821259 },
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{ 2144550913, 814139516, 1819937644 },
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{ 2144536577, 571143206, 962942255 },
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{ 2144524289, 1746733766, 2471321 },
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{ 2144512001, 1821415077, 124190939 },
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{ 2144468993, 917871546, 1260072806 },
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{ 2144458753, 378417981, 1569240563 },
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{ 2144421889, 175229668, 1825620763 },
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{ 2144409601, 1699216963, 351648117 },
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{ 2144370689, 1071885991, 958186029 },
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{ 2144348161, 1763151227, 540353574 },
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{ 2144335873, 1060214804, 919598847 },
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{ 2144329729, 663515846, 1448552668 },
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{ 2144327681, 1057776305, 590222840 },
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{ 2144309249, 1705149168, 1459294624 },
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{ 2144296961, 325823721, 1649016934 },
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{ 2144290817, 738775789, 447427206 },
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{ 2144243713, 962347618, 893050215 },
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{ 2144237569, 1655257077, 900860862 },
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{ 2144161793, 242206694, 1567868672 },
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{ 2144155649, 769415308, 1247993134 },
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{ 2144137217, 320492023, 515841070 },
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{ 2144120833, 1639388522, 770877302 },
|
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{ 2144071681, 1761785233, 964296120 },
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{ 2144065537, 419817825, 204564472 },
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|
{ 2144028673, 666050597, 2091019760 },
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{ 2144010241, 1413657615, 1518702610 },
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{ 2143952897, 1238327946, 475672271 },
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{ 2143940609, 307063413, 1176750846 },
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{ 2143918081, 2062905559, 786785803 },
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|
{ 2143899649, 1338112849, 1562292083 },
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{ 2143891457, 68149545, 87166451 },
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{ 2143885313, 921750778, 394460854 },
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{ 2143854593, 719766593, 133877196 },
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{ 2143836161, 1149399850, 1861591875 },
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{ 2143762433, 1848739366, 1335934145 },
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{ 2143756289, 1326674710, 102999236 },
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{ 2143713281, 808061791, 1156900308 },
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{ 2143690753, 388399459, 1926468019 },
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{ 2143670273, 1427891374, 1756689401 },
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{ 2143666177, 1912173949, 986629565 },
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{ 2143645697, 2041160111, 371842865 },
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{ 2143641601, 1279906897, 2023974350 },
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{ 2143635457, 720473174, 1389027526 },
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{ 2143621121, 1298309455, 1732632006 },
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|
{ 2143598593, 1548762216, 1825417506 },
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{ 2143567873, 620475784, 1073787233 },
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|
{ 2143561729, 1932954575, 949167309 },
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|
{ 2143553537, 354315656, 1652037534 },
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{ 2143541249, 577424288, 1097027618 },
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{ 2143531009, 357862822, 478640055 },
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{ 2143522817, 2017706025, 1550531668 },
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|
{ 2143506433, 2078127419, 1824320165 },
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{ 2143488001, 613475285, 1604011510 },
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{ 2143469569, 1466594987, 502095196 },
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|
{ 2143426561, 1115430331, 1044637111 },
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|
{ 2143383553, 9778045, 1902463734 },
|
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{ 2143377409, 1557401276, 2056861771 },
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{ 2143363073, 652036455, 1965915971 },
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|
{ 2143260673, 1464581171, 1523257541 },
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|
{ 2143246337, 1876119649, 764541916 },
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|
{ 2143209473, 1614992673, 1920672844 },
|
|
{ 2143203329, 981052047, 2049774209 },
|
|
{ 2143160321, 1847355533, 728535665 },
|
|
{ 2143129601, 965558457, 603052992 },
|
|
{ 2143123457, 2140817191, 8348679 },
|
|
{ 2143100929, 1547263683, 694209023 },
|
|
{ 2143092737, 643459066, 1979934533 },
|
|
{ 2143082497, 188603778, 2026175670 },
|
|
{ 2143062017, 1657329695, 377451099 },
|
|
{ 2143051777, 114967950, 979255473 },
|
|
{ 2143025153, 1698431342, 1449196896 },
|
|
{ 2143006721, 1862741675, 1739650365 },
|
|
{ 2142996481, 756660457, 996160050 },
|
|
{ 2142976001, 927864010, 1166847574 },
|
|
{ 2142965761, 905070557, 661974566 },
|
|
{ 2142916609, 40932754, 1787161127 },
|
|
{ 2142892033, 1987985648, 675335382 },
|
|
{ 2142885889, 797497211, 1323096997 },
|
|
{ 2142871553, 2068025830, 1411877159 },
|
|
{ 2142861313, 1217177090, 1438410687 },
|
|
{ 2142830593, 409906375, 1767860634 },
|
|
{ 2142803969, 1197788993, 359782919 },
|
|
{ 2142785537, 643817365, 513932862 },
|
|
{ 2142779393, 1717046338, 218943121 },
|
|
{ 2142724097, 89336830, 416687049 },
|
|
{ 2142707713, 5944581, 1356813523 },
|
|
{ 2142658561, 887942135, 2074011722 },
|
|
{ 2142638081, 151851972, 1647339939 },
|
|
{ 2142564353, 1691505537, 1483107336 },
|
|
{ 2142533633, 1989920200, 1135938817 },
|
|
{ 2142529537, 959263126, 1531961857 },
|
|
{ 2142527489, 453251129, 1725566162 },
|
|
{ 2142502913, 1536028102, 182053257 },
|
|
{ 2142498817, 570138730, 701443447 },
|
|
{ 2142416897, 326965800, 411931819 },
|
|
{ 2142363649, 1675665410, 1517191733 },
|
|
{ 2142351361, 968529566, 1575712703 },
|
|
{ 2142330881, 1384953238, 1769087884 },
|
|
{ 2142314497, 1977173242, 1833745524 },
|
|
{ 2142289921, 95082313, 1714775493 },
|
|
{ 2142283777, 109377615, 1070584533 },
|
|
{ 2142277633, 16960510, 702157145 },
|
|
{ 2142263297, 553850819, 431364395 },
|
|
{ 2142208001, 241466367, 2053967982 },
|
|
{ 2142164993, 1795661326, 1031836848 },
|
|
{ 2142097409, 1212530046, 712772031 },
|
|
{ 2142087169, 1763869720, 822276067 },
|
|
{ 2142078977, 644065713, 1765268066 },
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|
{ 2142074881, 112671944, 643204925 },
|
|
{ 2142044161, 1387785471, 1297890174 },
|
|
{ 2142025729, 783885537, 1000425730 },
|
|
{ 2142011393, 905662232, 1679401033 },
|
|
{ 2141974529, 799788433, 468119557 },
|
|
{ 2141943809, 1932544124, 449305555 },
|
|
{ 2141933569, 1527403256, 841867925 },
|
|
{ 2141931521, 1247076451, 743823916 },
|
|
{ 2141902849, 1199660531, 401687910 },
|
|
{ 2141890561, 150132350, 1720336972 },
|
|
{ 2141857793, 1287438162, 663880489 },
|
|
{ 2141833217, 618017731, 1819208266 },
|
|
{ 2141820929, 999578638, 1403090096 },
|
|
{ 2141786113, 81834325, 1523542501 },
|
|
{ 2141771777, 120001928, 463556492 },
|
|
{ 2141759489, 122455485, 2124928282 },
|
|
{ 2141749249, 141986041, 940339153 },
|
|
{ 2141685761, 889088734, 477141499 },
|
|
{ 2141673473, 324212681, 1122558298 },
|
|
{ 2141669377, 1175806187, 1373818177 },
|
|
{ 2141655041, 1113654822, 296887082 },
|
|
{ 2141587457, 991103258, 1585913875 },
|
|
{ 2141583361, 1401451409, 1802457360 },
|
|
{ 2141575169, 1571977166, 712760980 },
|
|
{ 2141546497, 1107849376, 1250270109 },
|
|
{ 2141515777, 196544219, 356001130 },
|
|
{ 2141495297, 1733571506, 1060744866 },
|
|
{ 2141483009, 321552363, 1168297026 },
|
|
{ 2141458433, 505818251, 733225819 },
|
|
{ 2141360129, 1026840098, 948342276 },
|
|
{ 2141325313, 945133744, 2129965998 },
|
|
{ 2141317121, 1871100260, 1843844634 },
|
|
{ 2141286401, 1790639498, 1750465696 },
|
|
{ 2141267969, 1376858592, 186160720 },
|
|
{ 2141255681, 2129698296, 1876677959 },
|
|
{ 2141243393, 2138900688, 1340009628 },
|
|
{ 2141214721, 1933049835, 1087819477 },
|
|
{ 2141212673, 1898664939, 1786328049 },
|
|
{ 2141202433, 990234828, 940682169 },
|
|
{ 2141175809, 1406392421, 993089586 },
|
|
{ 2141165569, 1263518371, 289019479 },
|
|
{ 2141073409, 1485624211, 507864514 },
|
|
{ 2141052929, 1885134788, 311252465 },
|
|
{ 2141040641, 1285021247, 280941862 },
|
|
{ 2141028353, 1527610374, 375035110 },
|
|
{ 2141011969, 1400626168, 164696620 },
|
|
{ 2140999681, 632959608, 966175067 },
|
|
{ 2140997633, 2045628978, 1290889438 },
|
|
{ 2140993537, 1412755491, 375366253 },
|
|
{ 2140942337, 719477232, 785367828 },
|
|
{ 2140925953, 45224252, 836552317 },
|
|
{ 2140917761, 1157376588, 1001839569 },
|
|
{ 2140887041, 278480752, 2098732796 },
|
|
{ 2140837889, 1663139953, 924094810 },
|
|
{ 2140788737, 802501511, 2045368990 },
|
|
{ 2140766209, 1820083885, 1800295504 },
|
|
{ 2140764161, 1169561905, 2106792035 },
|
|
{ 2140696577, 127781498, 1885987531 },
|
|
{ 2140684289, 16014477, 1098116827 },
|
|
{ 2140653569, 665960598, 1796728247 },
|
|
{ 2140594177, 1043085491, 377310938 },
|
|
{ 2140579841, 1732838211, 1504505945 },
|
|
{ 2140569601, 302071939, 358291016 },
|
|
{ 2140567553, 192393733, 1909137143 },
|
|
{ 2140557313, 406595731, 1175330270 },
|
|
{ 2140549121, 1748850918, 525007007 },
|
|
{ 2140477441, 499436566, 1031159814 },
|
|
{ 2140469249, 1886004401, 1029951320 },
|
|
{ 2140426241, 1483168100, 1676273461 },
|
|
{ 2140420097, 1779917297, 846024476 },
|
|
{ 2140413953, 522948893, 1816354149 },
|
|
{ 2140383233, 1931364473, 1296921241 },
|
|
{ 2140366849, 1917356555, 147196204 },
|
|
{ 2140354561, 16466177, 1349052107 },
|
|
{ 2140348417, 1875366972, 1860485634 },
|
|
{ 2140323841, 456498717, 1790256483 },
|
|
{ 2140321793, 1629493973, 150031888 },
|
|
{ 2140315649, 1904063898, 395510935 },
|
|
{ 2140280833, 1784104328, 831417909 },
|
|
{ 2140250113, 256087139, 697349101 },
|
|
{ 2140229633, 388553070, 243875754 },
|
|
{ 2140223489, 747459608, 1396270850 },
|
|
{ 2140200961, 507423743, 1895572209 },
|
|
{ 2140162049, 580106016, 2045297469 },
|
|
{ 2140149761, 712426444, 785217995 },
|
|
{ 2140137473, 1441607584, 536866543 },
|
|
{ 2140119041, 346538902, 1740434653 },
|
|
{ 2140090369, 282642885, 21051094 },
|
|
{ 2140076033, 1407456228, 319910029 },
|
|
{ 2140047361, 1619330500, 1488632070 },
|
|
{ 2140041217, 2089408064, 2012026134 },
|
|
{ 2140008449, 1705524800, 1613440760 },
|
|
{ 2139924481, 1846208233, 1280649481 },
|
|
{ 2139906049, 989438755, 1185646076 },
|
|
{ 2139867137, 1522314850, 372783595 },
|
|
{ 2139842561, 1681587377, 216848235 },
|
|
{ 2139826177, 2066284988, 1784999464 },
|
|
{ 2139824129, 480888214, 1513323027 },
|
|
{ 2139789313, 847937200, 858192859 },
|
|
{ 2139783169, 1642000434, 1583261448 },
|
|
{ 2139770881, 940699589, 179702100 },
|
|
{ 2139768833, 315623242, 964612676 },
|
|
{ 2139666433, 331649203, 764666914 },
|
|
{ 2139641857, 2118730799, 1313764644 },
|
|
{ 2139635713, 519149027, 519212449 },
|
|
{ 2139598849, 1526413634, 1769667104 },
|
|
{ 2139574273, 551148610, 820739925 },
|
|
{ 2139568129, 1386800242, 472447405 },
|
|
{ 2139549697, 813760130, 1412328531 },
|
|
{ 2139537409, 1615286260, 1609362979 },
|
|
{ 2139475969, 1352559299, 1696720421 },
|
|
{ 2139455489, 1048691649, 1584935400 },
|
|
{ 2139432961, 836025845, 950121150 },
|
|
{ 2139424769, 1558281165, 1635486858 },
|
|
{ 2139406337, 1728402143, 1674423301 },
|
|
{ 2139396097, 1727715782, 1483470544 },
|
|
{ 2139383809, 1092853491, 1741699084 },
|
|
{ 2139369473, 690776899, 1242798709 },
|
|
{ 2139351041, 1768782380, 2120712049 },
|
|
{ 2139334657, 1739968247, 1427249225 },
|
|
{ 2139332609, 1547189119, 623011170 },
|
|
{ 2139310081, 1346827917, 1605466350 },
|
|
{ 2139303937, 369317948, 828392831 },
|
|
{ 2139301889, 1560417239, 1788073219 },
|
|
{ 2139283457, 1303121623, 595079358 },
|
|
{ 2139248641, 1354555286, 573424177 },
|
|
{ 2139240449, 60974056, 885781403 },
|
|
{ 2139222017, 355573421, 1221054839 },
|
|
{ 2139215873, 566477826, 1724006500 },
|
|
{ 2139150337, 871437673, 1609133294 },
|
|
{ 2139144193, 1478130914, 1137491905 },
|
|
{ 2139117569, 1854880922, 964728507 },
|
|
{ 2139076609, 202405335, 756508944 },
|
|
{ 2139062273, 1399715741, 884826059 },
|
|
{ 2139045889, 1051045798, 1202295476 },
|
|
{ 2139033601, 1707715206, 632234634 },
|
|
{ 2139006977, 2035853139, 231626690 },
|
|
{ 2138951681, 183867876, 838350879 },
|
|
{ 2138945537, 1403254661, 404460202 },
|
|
{ 2138920961, 310865011, 1282911681 },
|
|
{ 2138910721, 1328496553, 103472415 },
|
|
{ 2138904577, 78831681, 993513549 },
|
|
{ 2138902529, 1319697451, 1055904361 },
|
|
{ 2138816513, 384338872, 1706202469 },
|
|
{ 2138810369, 1084868275, 405677177 },
|
|
{ 2138787841, 401181788, 1964773901 },
|
|
{ 2138775553, 1850532988, 1247087473 },
|
|
{ 2138767361, 874261901, 1576073565 },
|
|
{ 2138757121, 1187474742, 993541415 },
|
|
{ 2138748929, 1782458888, 1043206483 },
|
|
{ 2138744833, 1221500487, 800141243 },
|
|
{ 2138738689, 413465368, 1450660558 },
|
|
{ 2138695681, 739045140, 342611472 },
|
|
{ 2138658817, 1355845756, 672674190 },
|
|
{ 2138644481, 608379162, 1538874380 },
|
|
{ 2138632193, 1444914034, 686911254 },
|
|
{ 2138607617, 484707818, 1435142134 },
|
|
{ 2138591233, 539460669, 1290458549 },
|
|
{ 2138572801, 2093538990, 2011138646 },
|
|
{ 2138552321, 1149786988, 1076414907 },
|
|
{ 2138546177, 840688206, 2108985273 },
|
|
{ 2138533889, 209669619, 198172413 },
|
|
{ 2138523649, 1975879426, 1277003968 },
|
|
{ 2138490881, 1351891144, 1976858109 },
|
|
{ 2138460161, 1817321013, 1979278293 },
|
|
{ 2138429441, 1950077177, 203441928 },
|
|
{ 2138400769, 908970113, 628395069 },
|
|
{ 2138398721, 219890864, 758486760 },
|
|
{ 2138376193, 1306654379, 977554090 },
|
|
{ 2138351617, 298822498, 2004708503 },
|
|
{ 2138337281, 441457816, 1049002108 },
|
|
{ 2138320897, 1517731724, 1442269609 },
|
|
{ 2138290177, 1355911197, 1647139103 },
|
|
{ 2138234881, 531313247, 1746591962 },
|
|
{ 2138214401, 1899410930, 781416444 },
|
|
{ 2138202113, 1813477173, 1622508515 },
|
|
{ 2138191873, 1086458299, 1025408615 },
|
|
{ 2138183681, 1998800427, 827063290 },
|
|
{ 2138173441, 1921308898, 749670117 },
|
|
{ 2138103809, 1620902804, 2126787647 },
|
|
{ 2138099713, 828647069, 1892961817 },
|
|
{ 2138085377, 179405355, 1525506535 },
|
|
{ 2138060801, 615683235, 1259580138 },
|
|
{ 2138044417, 2030277840, 1731266562 },
|
|
{ 2138042369, 2087222316, 1627902259 },
|
|
{ 2138032129, 126388712, 1108640984 },
|
|
{ 2138011649, 715026550, 1017980050 },
|
|
{ 2137993217, 1693714349, 1351778704 },
|
|
{ 2137888769, 1289762259, 1053090405 },
|
|
{ 2137853953, 199991890, 1254192789 },
|
|
{ 2137833473, 941421685, 896995556 },
|
|
{ 2137817089, 750416446, 1251031181 },
|
|
{ 2137792513, 798075119, 368077456 },
|
|
{ 2137786369, 878543495, 1035375025 },
|
|
{ 2137767937, 9351178, 1156563902 },
|
|
{ 2137755649, 1382297614, 1686559583 },
|
|
{ 2137724929, 1345472850, 1681096331 },
|
|
{ 2137704449, 834666929, 630551727 },
|
|
{ 2137673729, 1646165729, 1892091571 },
|
|
{ 2137620481, 778943821, 48456461 },
|
|
{ 2137618433, 1730837875, 1713336725 },
|
|
{ 2137581569, 805610339, 1378891359 },
|
|
{ 2137538561, 204342388, 1950165220 },
|
|
{ 2137526273, 1947629754, 1500789441 },
|
|
{ 2137516033, 719902645, 1499525372 },
|
|
{ 2137491457, 230451261, 556382829 },
|
|
{ 2137440257, 979573541, 412760291 },
|
|
{ 2137374721, 927841248, 1954137185 },
|
|
{ 2137362433, 1243778559, 861024672 },
|
|
{ 2137313281, 1341338501, 980638386 },
|
|
{ 2137311233, 937415182, 1793212117 },
|
|
{ 2137255937, 795331324, 1410253405 },
|
|
{ 2137243649, 150756339, 1966999887 },
|
|
{ 2137182209, 163346914, 1939301431 },
|
|
{ 2137171969, 1952552395, 758913141 },
|
|
{ 2137159681, 570788721, 218668666 },
|
|
{ 2137147393, 1896656810, 2045670345 },
|
|
{ 2137141249, 358493842, 518199643 },
|
|
{ 2137139201, 1505023029, 674695848 },
|
|
{ 2137133057, 27911103, 830956306 },
|
|
{ 2137122817, 439771337, 1555268614 },
|
|
{ 2137116673, 790988579, 1871449599 },
|
|
{ 2137110529, 432109234, 811805080 },
|
|
{ 2137102337, 1357900653, 1184997641 },
|
|
{ 2137098241, 515119035, 1715693095 },
|
|
{ 2137090049, 408575203, 2085660657 },
|
|
{ 2137085953, 2097793407, 1349626963 },
|
|
{ 2137055233, 1556739954, 1449960883 },
|
|
{ 2137030657, 1545758650, 1369303716 },
|
|
{ 2136987649, 332602570, 103875114 },
|
|
{ 2136969217, 1499989506, 1662964115 },
|
|
{ 2136924161, 857040753, 4738842 },
|
|
{ 2136895489, 1948872712, 570436091 },
|
|
{ 2136893441, 58969960, 1568349634 },
|
|
{ 2136887297, 2127193379, 273612548 },
|
|
{ 2136850433, 111208983, 1181257116 },
|
|
{ 2136809473, 1627275942, 1680317971 },
|
|
{ 2136764417, 1574888217, 14011331 },
|
|
{ 2136741889, 14011055, 1129154251 },
|
|
{ 2136727553, 35862563, 1838555253 },
|
|
{ 2136721409, 310235666, 1363928244 },
|
|
{ 2136698881, 1612429202, 1560383828 },
|
|
{ 2136649729, 1138540131, 800014364 },
|
|
{ 2136606721, 602323503, 1433096652 },
|
|
{ 2136563713, 182209265, 1919611038 },
|
|
{ 2136555521, 324156477, 165591039 },
|
|
{ 2136549377, 195513113, 217165345 },
|
|
{ 2136526849, 1050768046, 939647887 },
|
|
{ 2136508417, 1886286237, 1619926572 },
|
|
{ 2136477697, 609647664, 35065157 },
|
|
{ 2136471553, 679352216, 1452259468 },
|
|
{ 2136457217, 128630031, 824816521 },
|
|
{ 2136422401, 19787464, 1526049830 },
|
|
{ 2136420353, 698316836, 1530623527 },
|
|
{ 2136371201, 1651862373, 1804812805 },
|
|
{ 2136334337, 326596005, 336977082 },
|
|
{ 2136322049, 63253370, 1904972151 },
|
|
{ 2136297473, 312176076, 172182411 },
|
|
{ 2136248321, 381261841, 369032670 },
|
|
{ 2136242177, 358688773, 1640007994 },
|
|
{ 2136229889, 512677188, 75585225 },
|
|
{ 2136219649, 2095003250, 1970086149 },
|
|
{ 2136207361, 1909650722, 537760675 },
|
|
{ 2136176641, 1334616195, 1533487619 },
|
|
{ 2136158209, 2096285632, 1793285210 },
|
|
{ 2136143873, 1897347517, 293843959 },
|
|
{ 2136133633, 923586222, 1022655978 },
|
|
{ 2136096769, 1464868191, 1515074410 },
|
|
{ 2136094721, 2020679520, 2061636104 },
|
|
{ 2136076289, 290798503, 1814726809 },
|
|
{ 2136041473, 156415894, 1250757633 },
|
|
{ 2135996417, 297459940, 1132158924 },
|
|
{ 2135955457, 538755304, 1688831340 },
|
|
{ 0, 0, 0 }
|
|
};
|
|
|
|
/*
|
|
* Reduce a small signed integer modulo a small prime. The source
|
|
* value x MUST be such that -p < x < p.
|
|
*/
|
|
static inline uint32_t
|
|
modp_set(int32_t x, uint32_t p) {
|
|
uint32_t w;
|
|
|
|
w = (uint32_t)x;
|
|
w += p & -(w >> 31);
|
|
return w;
|
|
}
|
|
|
|
/*
|
|
* Normalize a modular integer around 0.
|
|
*/
|
|
static inline int32_t
|
|
modp_norm(uint32_t x, uint32_t p) {
|
|
return (int32_t)(x - (p & (((x - ((p + 1) >> 1)) >> 31) - 1)));
|
|
}
|
|
|
|
/*
|
|
* Compute -1/p mod 2^31. This works for all odd integers p that fit
|
|
* on 31 bits.
|
|
*/
|
|
static uint32_t
|
|
modp_ninv31(uint32_t p) {
|
|
uint32_t y;
|
|
|
|
y = 2 - p;
|
|
y *= 2 - p * y;
|
|
y *= 2 - p * y;
|
|
y *= 2 - p * y;
|
|
y *= 2 - p * y;
|
|
return (uint32_t)0x7FFFFFFF & -y;
|
|
}
|
|
|
|
/*
|
|
* Compute R = 2^31 mod p.
|
|
*/
|
|
static inline uint32_t
|
|
modp_R(uint32_t p) {
|
|
/*
|
|
* Since 2^30 < p < 2^31, we know that 2^31 mod p is simply
|
|
* 2^31 - p.
|
|
*/
|
|
return ((uint32_t)1 << 31) - p;
|
|
}
|
|
|
|
/*
|
|
* Addition modulo p.
|
|
*/
|
|
static inline uint32_t
|
|
modp_add(uint32_t a, uint32_t b, uint32_t p) {
|
|
uint32_t d;
|
|
|
|
d = a + b - p;
|
|
d += p & -(d >> 31);
|
|
return d;
|
|
}
|
|
|
|
/*
|
|
* Subtraction modulo p.
|
|
*/
|
|
static inline uint32_t
|
|
modp_sub(uint32_t a, uint32_t b, uint32_t p) {
|
|
uint32_t d;
|
|
|
|
d = a - b;
|
|
d += p & -(d >> 31);
|
|
return d;
|
|
}
|
|
|
|
/*
|
|
* Halving modulo p.
|
|
*/
|
|
/* unused
|
|
static inline uint32_t
|
|
modp_half(uint32_t a, uint32_t p)
|
|
{
|
|
a += p & -(a & 1);
|
|
return a >> 1;
|
|
}
|
|
*/
|
|
|
|
/*
|
|
* Montgomery multiplication modulo p. The 'p0i' value is -1/p mod 2^31.
|
|
* It is required that p is an odd integer.
|
|
*/
|
|
static inline uint32_t
|
|
modp_montymul(uint32_t a, uint32_t b, uint32_t p, uint32_t p0i) {
|
|
uint64_t z, w;
|
|
uint32_t d;
|
|
|
|
z = (uint64_t)a * (uint64_t)b;
|
|
w = ((z * p0i) & (uint64_t)0x7FFFFFFF) * p;
|
|
d = (uint32_t)((z + w) >> 31) - p;
|
|
d += p & -(d >> 31);
|
|
return d;
|
|
}
|
|
|
|
/*
|
|
* Compute R2 = 2^62 mod p.
|
|
*/
|
|
static uint32_t
|
|
modp_R2(uint32_t p, uint32_t p0i) {
|
|
uint32_t z;
|
|
|
|
/*
|
|
* Compute z = 2^31 mod p (this is the value 1 in Montgomery
|
|
* representation), then double it with an addition.
|
|
*/
|
|
z = modp_R(p);
|
|
z = modp_add(z, z, p);
|
|
|
|
/*
|
|
* Square it five times to obtain 2^32 in Montgomery representation
|
|
* (i.e. 2^63 mod p).
|
|
*/
|
|
z = modp_montymul(z, z, p, p0i);
|
|
z = modp_montymul(z, z, p, p0i);
|
|
z = modp_montymul(z, z, p, p0i);
|
|
z = modp_montymul(z, z, p, p0i);
|
|
z = modp_montymul(z, z, p, p0i);
|
|
|
|
/*
|
|
* Halve the value mod p to get 2^62.
|
|
*/
|
|
z = (z + (p & -(z & 1))) >> 1;
|
|
return z;
|
|
}
|
|
|
|
/*
|
|
* Compute 2^(31*x) modulo p. This works for integers x up to 2^11.
|
|
* p must be prime such that 2^30 < p < 2^31; p0i must be equal to
|
|
* -1/p mod 2^31; R2 must be equal to 2^62 mod p.
|
|
*/
|
|
static inline uint32_t
|
|
modp_Rx(unsigned x, uint32_t p, uint32_t p0i, uint32_t R2) {
|
|
int i;
|
|
uint32_t r, z;
|
|
|
|
/*
|
|
* 2^(31*x) = (2^31)*(2^(31*(x-1))); i.e. we want the Montgomery
|
|
* representation of (2^31)^e mod p, where e = x-1.
|
|
* R2 is 2^31 in Montgomery representation.
|
|
*/
|
|
x --;
|
|
r = R2;
|
|
z = modp_R(p);
|
|
for (i = 0; (1U << i) <= x; i ++) {
|
|
if ((x & (1U << i)) != 0) {
|
|
z = modp_montymul(z, r, p, p0i);
|
|
}
|
|
r = modp_montymul(r, r, p, p0i);
|
|
}
|
|
return z;
|
|
}
|
|
|
|
/*
|
|
* Division modulo p. If the divisor (b) is 0, then 0 is returned.
|
|
* This function computes proper results only when p is prime.
|
|
* Parameters:
|
|
* a dividend
|
|
* b divisor
|
|
* p odd prime modulus
|
|
* p0i -1/p mod 2^31
|
|
* R 2^31 mod R
|
|
*/
|
|
static uint32_t
|
|
modp_div(uint32_t a, uint32_t b, uint32_t p, uint32_t p0i, uint32_t R) {
|
|
uint32_t z, e;
|
|
int i;
|
|
|
|
e = p - 2;
|
|
z = R;
|
|
for (i = 30; i >= 0; i --) {
|
|
uint32_t z2;
|
|
|
|
z = modp_montymul(z, z, p, p0i);
|
|
z2 = modp_montymul(z, b, p, p0i);
|
|
z ^= (z ^ z2) & -(uint32_t)((e >> i) & 1);
|
|
}
|
|
|
|
/*
|
|
* The loop above just assumed that b was in Montgomery
|
|
* representation, i.e. really contained b*R; under that
|
|
* assumption, it returns 1/b in Montgomery representation,
|
|
* which is R/b. But we gave it b in normal representation,
|
|
* so the loop really returned R/(b/R) = R^2/b.
|
|
*
|
|
* We want a/b, so we need one Montgomery multiplication with a,
|
|
* which also remove one of the R factors, and another such
|
|
* multiplication to remove the second R factor.
|
|
*/
|
|
z = modp_montymul(z, 1, p, p0i);
|
|
return modp_montymul(a, z, p, p0i);
|
|
}
|
|
|
|
/*
|
|
* Bit-reversal index table.
|
|
*/
|
|
static const uint16_t REV10[] = {
|
|
0, 512, 256, 768, 128, 640, 384, 896, 64, 576, 320, 832,
|
|
192, 704, 448, 960, 32, 544, 288, 800, 160, 672, 416, 928,
|
|
96, 608, 352, 864, 224, 736, 480, 992, 16, 528, 272, 784,
|
|
144, 656, 400, 912, 80, 592, 336, 848, 208, 720, 464, 976,
|
|
48, 560, 304, 816, 176, 688, 432, 944, 112, 624, 368, 880,
|
|
240, 752, 496, 1008, 8, 520, 264, 776, 136, 648, 392, 904,
|
|
72, 584, 328, 840, 200, 712, 456, 968, 40, 552, 296, 808,
|
|
168, 680, 424, 936, 104, 616, 360, 872, 232, 744, 488, 1000,
|
|
24, 536, 280, 792, 152, 664, 408, 920, 88, 600, 344, 856,
|
|
216, 728, 472, 984, 56, 568, 312, 824, 184, 696, 440, 952,
|
|
120, 632, 376, 888, 248, 760, 504, 1016, 4, 516, 260, 772,
|
|
132, 644, 388, 900, 68, 580, 324, 836, 196, 708, 452, 964,
|
|
36, 548, 292, 804, 164, 676, 420, 932, 100, 612, 356, 868,
|
|
228, 740, 484, 996, 20, 532, 276, 788, 148, 660, 404, 916,
|
|
84, 596, 340, 852, 212, 724, 468, 980, 52, 564, 308, 820,
|
|
180, 692, 436, 948, 116, 628, 372, 884, 244, 756, 500, 1012,
|
|
12, 524, 268, 780, 140, 652, 396, 908, 76, 588, 332, 844,
|
|
204, 716, 460, 972, 44, 556, 300, 812, 172, 684, 428, 940,
|
|
108, 620, 364, 876, 236, 748, 492, 1004, 28, 540, 284, 796,
|
|
156, 668, 412, 924, 92, 604, 348, 860, 220, 732, 476, 988,
|
|
60, 572, 316, 828, 188, 700, 444, 956, 124, 636, 380, 892,
|
|
252, 764, 508, 1020, 2, 514, 258, 770, 130, 642, 386, 898,
|
|
66, 578, 322, 834, 194, 706, 450, 962, 34, 546, 290, 802,
|
|
162, 674, 418, 930, 98, 610, 354, 866, 226, 738, 482, 994,
|
|
18, 530, 274, 786, 146, 658, 402, 914, 82, 594, 338, 850,
|
|
210, 722, 466, 978, 50, 562, 306, 818, 178, 690, 434, 946,
|
|
114, 626, 370, 882, 242, 754, 498, 1010, 10, 522, 266, 778,
|
|
138, 650, 394, 906, 74, 586, 330, 842, 202, 714, 458, 970,
|
|
42, 554, 298, 810, 170, 682, 426, 938, 106, 618, 362, 874,
|
|
234, 746, 490, 1002, 26, 538, 282, 794, 154, 666, 410, 922,
|
|
90, 602, 346, 858, 218, 730, 474, 986, 58, 570, 314, 826,
|
|
186, 698, 442, 954, 122, 634, 378, 890, 250, 762, 506, 1018,
|
|
6, 518, 262, 774, 134, 646, 390, 902, 70, 582, 326, 838,
|
|
198, 710, 454, 966, 38, 550, 294, 806, 166, 678, 422, 934,
|
|
102, 614, 358, 870, 230, 742, 486, 998, 22, 534, 278, 790,
|
|
150, 662, 406, 918, 86, 598, 342, 854, 214, 726, 470, 982,
|
|
54, 566, 310, 822, 182, 694, 438, 950, 118, 630, 374, 886,
|
|
246, 758, 502, 1014, 14, 526, 270, 782, 142, 654, 398, 910,
|
|
78, 590, 334, 846, 206, 718, 462, 974, 46, 558, 302, 814,
|
|
174, 686, 430, 942, 110, 622, 366, 878, 238, 750, 494, 1006,
|
|
30, 542, 286, 798, 158, 670, 414, 926, 94, 606, 350, 862,
|
|
222, 734, 478, 990, 62, 574, 318, 830, 190, 702, 446, 958,
|
|
126, 638, 382, 894, 254, 766, 510, 1022, 1, 513, 257, 769,
|
|
129, 641, 385, 897, 65, 577, 321, 833, 193, 705, 449, 961,
|
|
33, 545, 289, 801, 161, 673, 417, 929, 97, 609, 353, 865,
|
|
225, 737, 481, 993, 17, 529, 273, 785, 145, 657, 401, 913,
|
|
81, 593, 337, 849, 209, 721, 465, 977, 49, 561, 305, 817,
|
|
177, 689, 433, 945, 113, 625, 369, 881, 241, 753, 497, 1009,
|
|
9, 521, 265, 777, 137, 649, 393, 905, 73, 585, 329, 841,
|
|
201, 713, 457, 969, 41, 553, 297, 809, 169, 681, 425, 937,
|
|
105, 617, 361, 873, 233, 745, 489, 1001, 25, 537, 281, 793,
|
|
153, 665, 409, 921, 89, 601, 345, 857, 217, 729, 473, 985,
|
|
57, 569, 313, 825, 185, 697, 441, 953, 121, 633, 377, 889,
|
|
249, 761, 505, 1017, 5, 517, 261, 773, 133, 645, 389, 901,
|
|
69, 581, 325, 837, 197, 709, 453, 965, 37, 549, 293, 805,
|
|
165, 677, 421, 933, 101, 613, 357, 869, 229, 741, 485, 997,
|
|
21, 533, 277, 789, 149, 661, 405, 917, 85, 597, 341, 853,
|
|
213, 725, 469, 981, 53, 565, 309, 821, 181, 693, 437, 949,
|
|
117, 629, 373, 885, 245, 757, 501, 1013, 13, 525, 269, 781,
|
|
141, 653, 397, 909, 77, 589, 333, 845, 205, 717, 461, 973,
|
|
45, 557, 301, 813, 173, 685, 429, 941, 109, 621, 365, 877,
|
|
237, 749, 493, 1005, 29, 541, 285, 797, 157, 669, 413, 925,
|
|
93, 605, 349, 861, 221, 733, 477, 989, 61, 573, 317, 829,
|
|
189, 701, 445, 957, 125, 637, 381, 893, 253, 765, 509, 1021,
|
|
3, 515, 259, 771, 131, 643, 387, 899, 67, 579, 323, 835,
|
|
195, 707, 451, 963, 35, 547, 291, 803, 163, 675, 419, 931,
|
|
99, 611, 355, 867, 227, 739, 483, 995, 19, 531, 275, 787,
|
|
147, 659, 403, 915, 83, 595, 339, 851, 211, 723, 467, 979,
|
|
51, 563, 307, 819, 179, 691, 435, 947, 115, 627, 371, 883,
|
|
243, 755, 499, 1011, 11, 523, 267, 779, 139, 651, 395, 907,
|
|
75, 587, 331, 843, 203, 715, 459, 971, 43, 555, 299, 811,
|
|
171, 683, 427, 939, 107, 619, 363, 875, 235, 747, 491, 1003,
|
|
27, 539, 283, 795, 155, 667, 411, 923, 91, 603, 347, 859,
|
|
219, 731, 475, 987, 59, 571, 315, 827, 187, 699, 443, 955,
|
|
123, 635, 379, 891, 251, 763, 507, 1019, 7, 519, 263, 775,
|
|
135, 647, 391, 903, 71, 583, 327, 839, 199, 711, 455, 967,
|
|
39, 551, 295, 807, 167, 679, 423, 935, 103, 615, 359, 871,
|
|
231, 743, 487, 999, 23, 535, 279, 791, 151, 663, 407, 919,
|
|
87, 599, 343, 855, 215, 727, 471, 983, 55, 567, 311, 823,
|
|
183, 695, 439, 951, 119, 631, 375, 887, 247, 759, 503, 1015,
|
|
15, 527, 271, 783, 143, 655, 399, 911, 79, 591, 335, 847,
|
|
207, 719, 463, 975, 47, 559, 303, 815, 175, 687, 431, 943,
|
|
111, 623, 367, 879, 239, 751, 495, 1007, 31, 543, 287, 799,
|
|
159, 671, 415, 927, 95, 607, 351, 863, 223, 735, 479, 991,
|
|
63, 575, 319, 831, 191, 703, 447, 959, 127, 639, 383, 895,
|
|
255, 767, 511, 1023
|
|
};
|
|
|
|
/*
|
|
* Compute the roots for NTT and inverse NTT (binary case). Input
|
|
* parameter g is a primitive 2048-th root of 1 modulo p (i.e. g^1024 =
|
|
* -1 mod p). This fills gm[] and igm[] with powers of g and 1/g:
|
|
* gm[rev(i)] = g^i mod p
|
|
* igm[rev(i)] = (1/g)^i mod p
|
|
* where rev() is the "bit reversal" function over 10 bits. It fills
|
|
* the arrays only up to N = 2^logn values.
|
|
*
|
|
* The values stored in gm[] and igm[] are in Montgomery representation.
|
|
*
|
|
* p must be a prime such that p = 1 mod 2048.
|
|
*/
|
|
static void
|
|
modp_mkgm2(uint32_t *gm, uint32_t *igm, unsigned logn,
|
|
uint32_t g, uint32_t p, uint32_t p0i) {
|
|
size_t u, n;
|
|
unsigned k;
|
|
uint32_t ig, x1, x2, R2;
|
|
|
|
n = (size_t)1 << logn;
|
|
|
|
/*
|
|
* We want g such that g^(2N) = 1 mod p, but the provided
|
|
* generator has order 2048. We must square it a few times.
|
|
*/
|
|
R2 = modp_R2(p, p0i);
|
|
g = modp_montymul(g, R2, p, p0i);
|
|
for (k = logn; k < 10; k ++) {
|
|
g = modp_montymul(g, g, p, p0i);
|
|
}
|
|
|
|
ig = modp_div(R2, g, p, p0i, modp_R(p));
|
|
k = 10 - logn;
|
|
x1 = x2 = modp_R(p);
|
|
for (u = 0; u < n; u ++) {
|
|
size_t v;
|
|
|
|
v = REV10[u << k];
|
|
gm[v] = x1;
|
|
igm[v] = x2;
|
|
x1 = modp_montymul(x1, g, p, p0i);
|
|
x2 = modp_montymul(x2, ig, p, p0i);
|
|
}
|
|
}
|
|
|
|
/*
|
|
* Compute the NTT over a polynomial (binary case). Polynomial elements
|
|
* are a[0], a[stride], a[2 * stride]...
|
|
*/
|
|
static void
|
|
modp_NTT2_ext(uint32_t *a, size_t stride, const uint32_t *gm, unsigned logn,
|
|
uint32_t p, uint32_t p0i) {
|
|
size_t t, m, n;
|
|
|
|
if (logn == 0) {
|
|
return;
|
|
}
|
|
n = (size_t)1 << logn;
|
|
t = n;
|
|
for (m = 1; m < n; m <<= 1) {
|
|
size_t ht, u, v1;
|
|
|
|
ht = t >> 1;
|
|
for (u = 0, v1 = 0; u < m; u ++, v1 += t) {
|
|
uint32_t s;
|
|
size_t v;
|
|
uint32_t *r1, *r2;
|
|
|
|
s = gm[m + u];
|
|
r1 = a + v1 * stride;
|
|
r2 = r1 + ht * stride;
|
|
for (v = 0; v < ht; v ++, r1 += stride, r2 += stride) {
|
|
uint32_t x, y;
|
|
|
|
x = *r1;
|
|
y = modp_montymul(*r2, s, p, p0i);
|
|
*r1 = modp_add(x, y, p);
|
|
*r2 = modp_sub(x, y, p);
|
|
}
|
|
}
|
|
t = ht;
|
|
}
|
|
}
|
|
|
|
/*
|
|
* Compute the inverse NTT over a polynomial (binary case).
|
|
*/
|
|
static void
|
|
modp_iNTT2_ext(uint32_t *a, size_t stride, const uint32_t *igm, unsigned logn,
|
|
uint32_t p, uint32_t p0i) {
|
|
size_t t, m, n, k;
|
|
uint32_t ni;
|
|
uint32_t *r;
|
|
|
|
if (logn == 0) {
|
|
return;
|
|
}
|
|
n = (size_t)1 << logn;
|
|
t = 1;
|
|
for (m = n; m > 1; m >>= 1) {
|
|
size_t hm, dt, u, v1;
|
|
|
|
hm = m >> 1;
|
|
dt = t << 1;
|
|
for (u = 0, v1 = 0; u < hm; u ++, v1 += dt) {
|
|
uint32_t s;
|
|
size_t v;
|
|
uint32_t *r1, *r2;
|
|
|
|
s = igm[hm + u];
|
|
r1 = a + v1 * stride;
|
|
r2 = r1 + t * stride;
|
|
for (v = 0; v < t; v ++, r1 += stride, r2 += stride) {
|
|
uint32_t x, y;
|
|
|
|
x = *r1;
|
|
y = *r2;
|
|
*r1 = modp_add(x, y, p);
|
|
*r2 = modp_montymul(
|
|
modp_sub(x, y, p), s, p, p0i);;
|
|
}
|
|
}
|
|
t = dt;
|
|
}
|
|
|
|
/*
|
|
* We need 1/n in Montgomery representation, i.e. R/n. Since
|
|
* 1 <= logn <= 10, R/n is an integer; morever, R/n <= 2^30 < p,
|
|
* thus a simple shift will do.
|
|
*/
|
|
ni = (uint32_t)1 << (31 - logn);
|
|
for (k = 0, r = a; k < n; k ++, r += stride) {
|
|
*r = modp_montymul(*r, ni, p, p0i);
|
|
}
|
|
}
|
|
|
|
/*
|
|
* Simplified macros for NTT and iNTT (binary case) when the elements
|
|
* are consecutive in RAM.
|
|
*/
|
|
#define modp_NTT2(a, gm, logn, p, p0i) modp_NTT2_ext(a, 1, gm, logn, p, p0i)
|
|
#define modp_iNTT2(a, igm, logn, p, p0i) modp_iNTT2_ext(a, 1, igm, logn, p, p0i)
|
|
|
|
/*
|
|
* Given polynomial f in NTT representation modulo p, compute f' of degree
|
|
* less than N/2 such that f' = f0^2 - X*f1^2, where f0 and f1 are
|
|
* polynomials of degree less than N/2 such that f = f0(X^2) + X*f1(X^2).
|
|
*
|
|
* The new polynomial is written "in place" over the first N/2 elements
|
|
* of f.
|
|
*
|
|
* If applied logn times successively on a given polynomial, the resulting
|
|
* degree-0 polynomial is the resultant of f and X^N+1 modulo p.
|
|
*
|
|
* This function applies only to the binary case; it is invoked from
|
|
* solve_NTRU_binary_depth1().
|
|
*/
|
|
static void
|
|
modp_poly_rec_res(uint32_t *f, unsigned logn,
|
|
uint32_t p, uint32_t p0i, uint32_t R2) {
|
|
size_t hn, u;
|
|
|
|
hn = (size_t)1 << (logn - 1);
|
|
for (u = 0; u < hn; u ++) {
|
|
uint32_t w0, w1;
|
|
|
|
w0 = f[(u << 1) + 0];
|
|
w1 = f[(u << 1) + 1];
|
|
f[u] = modp_montymul(modp_montymul(w0, w1, p, p0i), R2, p, p0i);
|
|
}
|
|
}
|
|
|
|
/* ==================================================================== */
|
|
/*
|
|
* Custom bignum implementation.
|
|
*
|
|
* This is a very reduced set of functionalities. We need to do the
|
|
* following operations:
|
|
*
|
|
* - Rebuild the resultant and the polynomial coefficients from their
|
|
* values modulo small primes (of length 31 bits each).
|
|
*
|
|
* - Compute an extended GCD between the two computed resultants.
|
|
*
|
|
* - Extract top bits and add scaled values during the successive steps
|
|
* of Babai rounding.
|
|
*
|
|
* When rebuilding values using CRT, we must also recompute the product
|
|
* of the small prime factors. We always do it one small factor at a
|
|
* time, so the "complicated" operations can be done modulo the small
|
|
* prime with the modp_* functions. CRT coefficients (inverses) are
|
|
* precomputed.
|
|
*
|
|
* All values are positive until the last step: when the polynomial
|
|
* coefficients have been rebuilt, we normalize them around 0. But then,
|
|
* only additions and subtractions on the upper few bits are needed
|
|
* afterwards.
|
|
*
|
|
* We keep big integers as arrays of 31-bit words (in uint32_t values);
|
|
* the top bit of each uint32_t is kept equal to 0. Using 31-bit words
|
|
* makes it easier to keep track of carries. When negative values are
|
|
* used, two's complement is used.
|
|
*/
|
|
|
|
/*
|
|
* Subtract integer b from integer a. Both integers are supposed to have
|
|
* the same size. The carry (0 or 1) is returned. Source arrays a and b
|
|
* MUST be distinct.
|
|
*
|
|
* The operation is performed as described above if ctr = 1. If
|
|
* ctl = 0, the value a[] is unmodified, but all memory accesses are
|
|
* still performed, and the carry is computed and returned.
|
|
*/
|
|
static uint32_t
|
|
zint_sub(uint32_t *a, const uint32_t *b, size_t len,
|
|
uint32_t ctl) {
|
|
size_t u;
|
|
uint32_t cc, m;
|
|
|
|
cc = 0;
|
|
m = -ctl;
|
|
for (u = 0; u < len; u ++) {
|
|
uint32_t aw, w;
|
|
|
|
aw = a[u];
|
|
w = aw - b[u] - cc;
|
|
cc = w >> 31;
|
|
aw ^= ((w & 0x7FFFFFFF) ^ aw) & m;
|
|
a[u] = aw;
|
|
}
|
|
return cc;
|
|
}
|
|
|
|
/*
|
|
* Mutiply the provided big integer m with a small value x.
|
|
* This function assumes that x < 2^31. The carry word is returned.
|
|
*/
|
|
static uint32_t
|
|
zint_mul_small(uint32_t *m, size_t mlen, uint32_t x) {
|
|
size_t u;
|
|
uint32_t cc;
|
|
|
|
cc = 0;
|
|
for (u = 0; u < mlen; u ++) {
|
|
uint64_t z;
|
|
|
|
z = (uint64_t)m[u] * (uint64_t)x + cc;
|
|
m[u] = (uint32_t)z & 0x7FFFFFFF;
|
|
cc = (uint32_t)(z >> 31);
|
|
}
|
|
return cc;
|
|
}
|
|
|
|
/*
|
|
* Reduce a big integer d modulo a small integer p.
|
|
* Rules:
|
|
* d is unsigned
|
|
* p is prime
|
|
* 2^30 < p < 2^31
|
|
* p0i = -(1/p) mod 2^31
|
|
* R2 = 2^62 mod p
|
|
*/
|
|
static uint32_t
|
|
zint_mod_small_unsigned(const uint32_t *d, size_t dlen,
|
|
uint32_t p, uint32_t p0i, uint32_t R2) {
|
|
uint32_t x;
|
|
size_t u;
|
|
|
|
/*
|
|
* Algorithm: we inject words one by one, starting with the high
|
|
* word. Each step is:
|
|
* - multiply x by 2^31
|
|
* - add new word
|
|
*/
|
|
x = 0;
|
|
u = dlen;
|
|
while (u -- > 0) {
|
|
uint32_t w;
|
|
|
|
x = modp_montymul(x, R2, p, p0i);
|
|
w = d[u] - p;
|
|
w += p & -(w >> 31);
|
|
x = modp_add(x, w, p);
|
|
}
|
|
return x;
|
|
}
|
|
|
|
/*
|
|
* Similar to zint_mod_small_unsigned(), except that d may be signed.
|
|
* Extra parameter is Rx = 2^(31*dlen) mod p.
|
|
*/
|
|
static uint32_t
|
|
zint_mod_small_signed(const uint32_t *d, size_t dlen,
|
|
uint32_t p, uint32_t p0i, uint32_t R2, uint32_t Rx) {
|
|
uint32_t z;
|
|
|
|
if (dlen == 0) {
|
|
return 0;
|
|
}
|
|
z = zint_mod_small_unsigned(d, dlen, p, p0i, R2);
|
|
z = modp_sub(z, Rx & -(d[dlen - 1] >> 30), p);
|
|
return z;
|
|
}
|
|
|
|
/*
|
|
* Add y*s to x. x and y initially have length 'len' words; the new x
|
|
* has length 'len+1' words. 's' must fit on 31 bits. x[] and y[] must
|
|
* not overlap.
|
|
*/
|
|
static void
|
|
zint_add_mul_small(uint32_t *x,
|
|
const uint32_t *y, size_t len, uint32_t s) {
|
|
size_t u;
|
|
uint32_t cc;
|
|
|
|
cc = 0;
|
|
for (u = 0; u < len; u ++) {
|
|
uint32_t xw, yw;
|
|
uint64_t z;
|
|
|
|
xw = x[u];
|
|
yw = y[u];
|
|
z = (uint64_t)yw * (uint64_t)s + (uint64_t)xw + (uint64_t)cc;
|
|
x[u] = (uint32_t)z & 0x7FFFFFFF;
|
|
cc = (uint32_t)(z >> 31);
|
|
}
|
|
x[len] = cc;
|
|
}
|
|
|
|
/*
|
|
* Normalize a modular integer around 0: if x > p/2, then x is replaced
|
|
* with x - p (signed encoding with two's complement); otherwise, x is
|
|
* untouched. The two integers x and p are encoded over the same length.
|
|
*/
|
|
static void
|
|
zint_norm_zero(uint32_t *x, const uint32_t *p, size_t len) {
|
|
size_t u;
|
|
uint32_t r, bb;
|
|
|
|
/*
|
|
* Compare x with p/2. We use the shifted version of p, and p
|
|
* is odd, so we really compare with (p-1)/2; we want to perform
|
|
* the subtraction if and only if x > (p-1)/2.
|
|
*/
|
|
r = 0;
|
|
bb = 0;
|
|
u = len;
|
|
while (u -- > 0) {
|
|
uint32_t wx, wp, cc;
|
|
|
|
/*
|
|
* Get the two words to compare in wx and wp (both over
|
|
* 31 bits exactly).
|
|
*/
|
|
wx = x[u];
|
|
wp = (p[u] >> 1) | (bb << 30);
|
|
bb = p[u] & 1;
|
|
|
|
/*
|
|
* We set cc to -1, 0 or 1, depending on whether wp is
|
|
* lower than, equal to, or greater than wx.
|
|
*/
|
|
cc = wp - wx;
|
|
cc = ((-cc) >> 31) | -(cc >> 31);
|
|
|
|
/*
|
|
* If r != 0 then it is either 1 or -1, and we keep its
|
|
* value. Otherwise, if r = 0, then we replace it with cc.
|
|
*/
|
|
r |= cc & ((r & 1) - 1);
|
|
}
|
|
|
|
/*
|
|
* At this point, r = -1, 0 or 1, depending on whether (p-1)/2
|
|
* is lower than, equal to, or greater than x. We thus want to
|
|
* do the subtraction only if r = -1.
|
|
*/
|
|
zint_sub(x, p, len, r >> 31);
|
|
}
|
|
|
|
/*
|
|
* Rebuild integers from their RNS representation. There are 'num'
|
|
* integers, and each consists in 'xlen' words. 'xx' points at that
|
|
* first word of the first integer; subsequent integers are accessed
|
|
* by adding 'xstride' repeatedly.
|
|
*
|
|
* The words of an integer are the RNS representation of that integer,
|
|
* using the provided 'primes' are moduli. This function replaces
|
|
* each integer with its multi-word value (little-endian order).
|
|
*
|
|
* If "normalize_signed" is non-zero, then the returned value is
|
|
* normalized to the -m/2..m/2 interval (where m is the product of all
|
|
* small prime moduli); two's complement is used for negative values.
|
|
*/
|
|
static void
|
|
zint_rebuild_CRT(uint32_t *xx, size_t xlen, size_t xstride,
|
|
size_t num, const small_prime *primes, int normalize_signed,
|
|
uint32_t *tmp) {
|
|
size_t u;
|
|
uint32_t *x;
|
|
|
|
tmp[0] = primes[0].p;
|
|
for (u = 1; u < xlen; u ++) {
|
|
/*
|
|
* At the entry of each loop iteration:
|
|
* - the first u words of each array have been
|
|
* reassembled;
|
|
* - the first u words of tmp[] contains the
|
|
* product of the prime moduli processed so far.
|
|
*
|
|
* We call 'q' the product of all previous primes.
|
|
*/
|
|
uint32_t p, p0i, s, R2;
|
|
size_t v;
|
|
|
|
p = primes[u].p;
|
|
s = primes[u].s;
|
|
p0i = modp_ninv31(p);
|
|
R2 = modp_R2(p, p0i);
|
|
|
|
for (v = 0, x = xx; v < num; v ++, x += xstride) {
|
|
uint32_t xp, xq, xr;
|
|
/*
|
|
* xp = the integer x modulo the prime p for this
|
|
* iteration
|
|
* xq = (x mod q) mod p
|
|
*/
|
|
xp = x[u];
|
|
xq = zint_mod_small_unsigned(x, u, p, p0i, R2);
|
|
|
|
/*
|
|
* New value is (x mod q) + q * (s * (xp - xq) mod p)
|
|
*/
|
|
xr = modp_montymul(s, modp_sub(xp, xq, p), p, p0i);
|
|
zint_add_mul_small(x, tmp, u, xr);
|
|
}
|
|
|
|
/*
|
|
* Update product of primes in tmp[].
|
|
*/
|
|
tmp[u] = zint_mul_small(tmp, u, p);
|
|
}
|
|
|
|
/*
|
|
* Normalize the reconstructed values around 0.
|
|
*/
|
|
if (normalize_signed) {
|
|
for (u = 0, x = xx; u < num; u ++, x += xstride) {
|
|
zint_norm_zero(x, tmp, xlen);
|
|
}
|
|
}
|
|
}
|
|
|
|
/*
|
|
* Negate a big integer conditionally: value a is replaced with -a if
|
|
* and only if ctl = 1. Control value ctl must be 0 or 1.
|
|
*/
|
|
static void
|
|
zint_negate(uint32_t *a, size_t len, uint32_t ctl) {
|
|
size_t u;
|
|
uint32_t cc, m;
|
|
|
|
/*
|
|
* If ctl = 1 then we flip the bits of a by XORing with
|
|
* 0x7FFFFFFF, and we add 1 to the value. If ctl = 0 then we XOR
|
|
* with 0 and add 0, which leaves the value unchanged.
|
|
*/
|
|
cc = ctl;
|
|
m = -ctl >> 1;
|
|
for (u = 0; u < len; u ++) {
|
|
uint32_t aw;
|
|
|
|
aw = a[u];
|
|
aw = (aw ^ m) + cc;
|
|
a[u] = aw & 0x7FFFFFFF;
|
|
cc = aw >> 31;
|
|
}
|
|
}
|
|
|
|
/*
|
|
* Replace a with (a*xa+b*xb)/(2^31) and b with (a*ya+b*yb)/(2^31).
|
|
* The low bits are dropped (the caller should compute the coefficients
|
|
* such that these dropped bits are all zeros). If either or both
|
|
* yields a negative value, then the value is negated.
|
|
*
|
|
* Returned value is:
|
|
* 0 both values were positive
|
|
* 1 new a had to be negated
|
|
* 2 new b had to be negated
|
|
* 3 both new a and new b had to be negated
|
|
*
|
|
* Coefficients xa, xb, ya and yb may use the full signed 32-bit range.
|
|
*/
|
|
static uint32_t
|
|
zint_co_reduce(uint32_t *a, uint32_t *b, size_t len,
|
|
int64_t xa, int64_t xb, int64_t ya, int64_t yb) {
|
|
size_t u;
|
|
int64_t cca, ccb;
|
|
uint32_t nega, negb;
|
|
|
|
cca = 0;
|
|
ccb = 0;
|
|
for (u = 0; u < len; u ++) {
|
|
uint32_t wa, wb;
|
|
uint64_t za, zb;
|
|
|
|
wa = a[u];
|
|
wb = b[u];
|
|
za = wa * (uint64_t)xa + wb * (uint64_t)xb + (uint64_t)cca;
|
|
zb = wa * (uint64_t)ya + wb * (uint64_t)yb + (uint64_t)ccb;
|
|
if (u > 0) {
|
|
a[u - 1] = (uint32_t)za & 0x7FFFFFFF;
|
|
b[u - 1] = (uint32_t)zb & 0x7FFFFFFF;
|
|
}
|
|
cca = *(int64_t *)&za >> 31;
|
|
ccb = *(int64_t *)&zb >> 31;
|
|
}
|
|
a[len - 1] = (uint32_t)cca;
|
|
b[len - 1] = (uint32_t)ccb;
|
|
|
|
nega = (uint32_t)((uint64_t)cca >> 63);
|
|
negb = (uint32_t)((uint64_t)ccb >> 63);
|
|
zint_negate(a, len, nega);
|
|
zint_negate(b, len, negb);
|
|
return nega | (negb << 1);
|
|
}
|
|
|
|
/*
|
|
* Finish modular reduction. Rules on input parameters:
|
|
*
|
|
* if neg = 1, then -m <= a < 0
|
|
* if neg = 0, then 0 <= a < 2*m
|
|
*
|
|
* If neg = 0, then the top word of a[] is allowed to use 32 bits.
|
|
*
|
|
* Modulus m must be odd.
|
|
*/
|
|
static void
|
|
zint_finish_mod(uint32_t *a, size_t len, const uint32_t *m, uint32_t neg) {
|
|
size_t u;
|
|
uint32_t cc, xm, ym;
|
|
|
|
/*
|
|
* First pass: compare a (assumed nonnegative) with m. Note that
|
|
* if the top word uses 32 bits, subtracting m must yield a
|
|
* value less than 2^31 since a < 2*m.
|
|
*/
|
|
cc = 0;
|
|
for (u = 0; u < len; u ++) {
|
|
cc = (a[u] - m[u] - cc) >> 31;
|
|
}
|
|
|
|
/*
|
|
* If neg = 1 then we must add m (regardless of cc)
|
|
* If neg = 0 and cc = 0 then we must subtract m
|
|
* If neg = 0 and cc = 1 then we must do nothing
|
|
*
|
|
* In the loop below, we conditionally subtract either m or -m
|
|
* from a. Word xm is a word of m (if neg = 0) or -m (if neg = 1);
|
|
* but if neg = 0 and cc = 1, then ym = 0 and it forces mw to 0.
|
|
*/
|
|
xm = -neg >> 1;
|
|
ym = -(neg | (1 - cc));
|
|
cc = neg;
|
|
for (u = 0; u < len; u ++) {
|
|
uint32_t aw, mw;
|
|
|
|
aw = a[u];
|
|
mw = (m[u] ^ xm) & ym;
|
|
aw = aw - mw - cc;
|
|
a[u] = aw & 0x7FFFFFFF;
|
|
cc = aw >> 31;
|
|
}
|
|
}
|
|
|
|
/*
|
|
* Replace a with (a*xa+b*xb)/(2^31) mod m, and b with
|
|
* (a*ya+b*yb)/(2^31) mod m. Modulus m must be odd; m0i = -1/m[0] mod 2^31.
|
|
*/
|
|
static void
|
|
zint_co_reduce_mod(uint32_t *a, uint32_t *b, const uint32_t *m, size_t len,
|
|
uint32_t m0i, int64_t xa, int64_t xb, int64_t ya, int64_t yb) {
|
|
size_t u;
|
|
int64_t cca, ccb;
|
|
uint32_t fa, fb;
|
|
|
|
/*
|
|
* These are actually four combined Montgomery multiplications.
|
|
*/
|
|
cca = 0;
|
|
ccb = 0;
|
|
fa = ((a[0] * (uint32_t)xa + b[0] * (uint32_t)xb) * m0i) & 0x7FFFFFFF;
|
|
fb = ((a[0] * (uint32_t)ya + b[0] * (uint32_t)yb) * m0i) & 0x7FFFFFFF;
|
|
for (u = 0; u < len; u ++) {
|
|
uint32_t wa, wb;
|
|
uint64_t za, zb;
|
|
|
|
wa = a[u];
|
|
wb = b[u];
|
|
za = wa * (uint64_t)xa + wb * (uint64_t)xb
|
|
+ m[u] * (uint64_t)fa + (uint64_t)cca;
|
|
zb = wa * (uint64_t)ya + wb * (uint64_t)yb
|
|
+ m[u] * (uint64_t)fb + (uint64_t)ccb;
|
|
if (u > 0) {
|
|
a[u - 1] = (uint32_t)za & 0x7FFFFFFF;
|
|
b[u - 1] = (uint32_t)zb & 0x7FFFFFFF;
|
|
}
|
|
cca = *(int64_t *)&za >> 31;
|
|
ccb = *(int64_t *)&zb >> 31;
|
|
}
|
|
a[len - 1] = (uint32_t)cca;
|
|
b[len - 1] = (uint32_t)ccb;
|
|
|
|
/*
|
|
* At this point:
|
|
* -m <= a < 2*m
|
|
* -m <= b < 2*m
|
|
* (this is a case of Montgomery reduction)
|
|
* The top words of 'a' and 'b' may have a 32-th bit set.
|
|
* We want to add or subtract the modulus, as required.
|
|
*/
|
|
zint_finish_mod(a, len, m, (uint32_t)((uint64_t)cca >> 63));
|
|
zint_finish_mod(b, len, m, (uint32_t)((uint64_t)ccb >> 63));
|
|
}
|
|
|
|
/*
|
|
* Compute a GCD between two positive big integers x and y. The two
|
|
* integers must be odd. Returned value is 1 if the GCD is 1, 0
|
|
* otherwise. When 1 is returned, arrays u and v are filled with values
|
|
* such that:
|
|
* 0 <= u <= y
|
|
* 0 <= v <= x
|
|
* x*u - y*v = 1
|
|
* x[] and y[] are unmodified. Both input values must have the same
|
|
* encoded length. Temporary array must be large enough to accommodate 4
|
|
* extra values of that length. Arrays u, v and tmp may not overlap with
|
|
* each other, or with either x or y.
|
|
*/
|
|
static int
|
|
zint_bezout(uint32_t *u, uint32_t *v,
|
|
const uint32_t *x, const uint32_t *y,
|
|
size_t len, uint32_t *tmp) {
|
|
/*
|
|
* Algorithm is an extended binary GCD. We maintain 6 values
|
|
* a, b, u0, u1, v0 and v1 with the following invariants:
|
|
*
|
|
* a = x*u0 - y*v0
|
|
* b = x*u1 - y*v1
|
|
* 0 <= a <= x
|
|
* 0 <= b <= y
|
|
* 0 <= u0 < y
|
|
* 0 <= v0 < x
|
|
* 0 <= u1 <= y
|
|
* 0 <= v1 < x
|
|
*
|
|
* Initial values are:
|
|
*
|
|
* a = x u0 = 1 v0 = 0
|
|
* b = y u1 = y v1 = x-1
|
|
*
|
|
* Each iteration reduces either a or b, and maintains the
|
|
* invariants. Algorithm stops when a = b, at which point their
|
|
* common value is GCD(a,b) and (u0,v0) (or (u1,v1)) contains
|
|
* the values (u,v) we want to return.
|
|
*
|
|
* The formal definition of the algorithm is a sequence of steps:
|
|
*
|
|
* - If a is even, then:
|
|
* a <- a/2
|
|
* u0 <- u0/2 mod y
|
|
* v0 <- v0/2 mod x
|
|
*
|
|
* - Otherwise, if b is even, then:
|
|
* b <- b/2
|
|
* u1 <- u1/2 mod y
|
|
* v1 <- v1/2 mod x
|
|
*
|
|
* - Otherwise, if a > b, then:
|
|
* a <- (a-b)/2
|
|
* u0 <- (u0-u1)/2 mod y
|
|
* v0 <- (v0-v1)/2 mod x
|
|
*
|
|
* - Otherwise:
|
|
* b <- (b-a)/2
|
|
* u1 <- (u1-u0)/2 mod y
|
|
* v1 <- (v1-v0)/2 mod y
|
|
*
|
|
* We can show that the operations above preserve the invariants:
|
|
*
|
|
* - If a is even, then u0 and v0 are either both even or both
|
|
* odd (since a = x*u0 - y*v0, and x and y are both odd).
|
|
* If u0 and v0 are both even, then (u0,v0) <- (u0/2,v0/2).
|
|
* Otherwise, (u0,v0) <- ((u0+y)/2,(v0+x)/2). Either way,
|
|
* the a = x*u0 - y*v0 invariant is preserved.
|
|
*
|
|
* - The same holds for the case where b is even.
|
|
*
|
|
* - If a and b are odd, and a > b, then:
|
|
*
|
|
* a-b = x*(u0-u1) - y*(v0-v1)
|
|
*
|
|
* In that situation, if u0 < u1, then x*(u0-u1) < 0, but
|
|
* a-b > 0; therefore, it must be that v0 < v1, and the
|
|
* first part of the update is: (u0,v0) <- (u0-u1+y,v0-v1+x),
|
|
* which preserves the invariants. Otherwise, if u0 > u1,
|
|
* then u0-u1 >= 1, thus x*(u0-u1) >= x. But a <= x and
|
|
* b >= 0, hence a-b <= x. It follows that, in that case,
|
|
* v0-v1 >= 0. The first part of the update is then:
|
|
* (u0,v0) <- (u0-u1,v0-v1), which again preserves the
|
|
* invariants.
|
|
*
|
|
* Either way, once the subtraction is done, the new value of
|
|
* a, which is the difference of two odd values, is even,
|
|
* and the remaining of this step is a subcase of the
|
|
* first algorithm case (i.e. when a is even).
|
|
*
|
|
* - If a and b are odd, and b > a, then the a similar
|
|
* argument holds.
|
|
*
|
|
* The values a and b start at x and y, respectively. Since x
|
|
* and y are odd, their GCD is odd, and it is easily seen that
|
|
* all steps conserve the GCD (GCD(a-b,b) = GCD(a, b);
|
|
* GCD(a/2,b) = GCD(a,b) if GCD(a,b) is odd). Moreover, either a
|
|
* or b is reduced by at least one bit at each iteration, so
|
|
* the algorithm necessarily converges on the case a = b, at
|
|
* which point the common value is the GCD.
|
|
*
|
|
* In the algorithm expressed above, when a = b, the fourth case
|
|
* applies, and sets b = 0. Since a contains the GCD of x and y,
|
|
* which are both odd, a must be odd, and subsequent iterations
|
|
* (if any) will simply divide b by 2 repeatedly, which has no
|
|
* consequence. Thus, the algorithm can run for more iterations
|
|
* than necessary; the final GCD will be in a, and the (u,v)
|
|
* coefficients will be (u0,v0).
|
|
*
|
|
*
|
|
* The presentation above is bit-by-bit. It can be sped up by
|
|
* noticing that all decisions are taken based on the low bits
|
|
* and high bits of a and b. We can extract the two top words
|
|
* and low word of each of a and b, and compute reduction
|
|
* parameters pa, pb, qa and qb such that the new values for
|
|
* a and b are:
|
|
* a' = (a*pa + b*pb) / (2^31)
|
|
* b' = (a*qa + b*qb) / (2^31)
|
|
* the two divisions being exact. The coefficients are obtained
|
|
* just from the extracted words, and may be slightly off, requiring
|
|
* an optional correction: if a' < 0, then we replace pa with -pa
|
|
* and pb with -pb. Each such step will reduce the total length
|
|
* (sum of lengths of a and b) by at least 30 bits at each
|
|
* iteration.
|
|
*/
|
|
uint32_t *u0, *u1, *v0, *v1, *a, *b;
|
|
uint32_t x0i, y0i;
|
|
uint32_t num, rc;
|
|
size_t j;
|
|
|
|
if (len == 0) {
|
|
return 0;
|
|
}
|
|
|
|
/*
|
|
* u0 and v0 are the u and v result buffers; the four other
|
|
* values (u1, v1, a and b) are taken from tmp[].
|
|
*/
|
|
u0 = u;
|
|
v0 = v;
|
|
u1 = tmp;
|
|
v1 = u1 + len;
|
|
a = v1 + len;
|
|
b = a + len;
|
|
|
|
/*
|
|
* We'll need the Montgomery reduction coefficients.
|
|
*/
|
|
x0i = modp_ninv31(x[0]);
|
|
y0i = modp_ninv31(y[0]);
|
|
|
|
/*
|
|
* Initialize a, b, u0, u1, v0 and v1.
|
|
* a = x u0 = 1 v0 = 0
|
|
* b = y u1 = y v1 = x-1
|
|
* Note that x is odd, so computing x-1 is easy.
|
|
*/
|
|
memcpy(a, x, len * sizeof * x);
|
|
memcpy(b, y, len * sizeof * y);
|
|
u0[0] = 1;
|
|
memset(u0 + 1, 0, (len - 1) * sizeof * u0);
|
|
memset(v0, 0, len * sizeof * v0);
|
|
memcpy(u1, y, len * sizeof * u1);
|
|
memcpy(v1, x, len * sizeof * v1);
|
|
v1[0] --;
|
|
|
|
/*
|
|
* Each input operand may be as large as 31*len bits, and we
|
|
* reduce the total length by at least 30 bits at each iteration.
|
|
*/
|
|
for (num = 62 * (uint32_t)len + 30; num >= 30; num -= 30) {
|
|
uint32_t c0, c1;
|
|
uint32_t a0, a1, b0, b1;
|
|
uint64_t a_hi, b_hi;
|
|
uint32_t a_lo, b_lo;
|
|
int64_t pa, pb, qa, qb;
|
|
int i;
|
|
uint32_t r;
|
|
|
|
/*
|
|
* Extract the top words of a and b. If j is the highest
|
|
* index >= 1 such that a[j] != 0 or b[j] != 0, then we
|
|
* want (a[j] << 31) + a[j-1] and (b[j] << 31) + b[j-1].
|
|
* If a and b are down to one word each, then we use
|
|
* a[0] and b[0].
|
|
*/
|
|
c0 = (uint32_t) -1;
|
|
c1 = (uint32_t) -1;
|
|
a0 = 0;
|
|
a1 = 0;
|
|
b0 = 0;
|
|
b1 = 0;
|
|
j = len;
|
|
while (j -- > 0) {
|
|
uint32_t aw, bw;
|
|
|
|
aw = a[j];
|
|
bw = b[j];
|
|
a0 ^= (a0 ^ aw) & c0;
|
|
a1 ^= (a1 ^ aw) & c1;
|
|
b0 ^= (b0 ^ bw) & c0;
|
|
b1 ^= (b1 ^ bw) & c1;
|
|
c1 = c0;
|
|
c0 &= (((aw | bw) + 0x7FFFFFFF) >> 31) - (uint32_t)1;
|
|
}
|
|
|
|
/*
|
|
* If c1 = 0, then we grabbed two words for a and b.
|
|
* If c1 != 0 but c0 = 0, then we grabbed one word. It
|
|
* is not possible that c1 != 0 and c0 != 0, because that
|
|
* would mean that both integers are zero.
|
|
*/
|
|
a1 |= a0 & c1;
|
|
a0 &= ~c1;
|
|
b1 |= b0 & c1;
|
|
b0 &= ~c1;
|
|
a_hi = ((uint64_t)a0 << 31) + a1;
|
|
b_hi = ((uint64_t)b0 << 31) + b1;
|
|
a_lo = a[0];
|
|
b_lo = b[0];
|
|
|
|
/*
|
|
* Compute reduction factors:
|
|
*
|
|
* a' = a*pa + b*pb
|
|
* b' = a*qa + b*qb
|
|
*
|
|
* such that a' and b' are both multiple of 2^31, but are
|
|
* only marginally larger than a and b.
|
|
*/
|
|
pa = 1;
|
|
pb = 0;
|
|
qa = 0;
|
|
qb = 1;
|
|
for (i = 0; i < 31; i ++) {
|
|
/*
|
|
* At each iteration:
|
|
*
|
|
* a <- (a-b)/2 if: a is odd, b is odd, a_hi > b_hi
|
|
* b <- (b-a)/2 if: a is odd, b is odd, a_hi <= b_hi
|
|
* a <- a/2 if: a is even
|
|
* b <- b/2 if: a is odd, b is even
|
|
*
|
|
* We multiply a_lo and b_lo by 2 at each
|
|
* iteration, thus a division by 2 really is a
|
|
* non-multiplication by 2.
|
|
*/
|
|
uint32_t rt, oa, ob, cAB, cBA, cA;
|
|
uint64_t rz;
|
|
|
|
/*
|
|
* rt = 1 if a_hi > b_hi, 0 otherwise.
|
|
*/
|
|
rz = b_hi - a_hi;
|
|
rt = (uint32_t)((rz ^ ((a_hi ^ b_hi)
|
|
& (a_hi ^ rz))) >> 63);
|
|
|
|
/*
|
|
* cAB = 1 if b must be subtracted from a
|
|
* cBA = 1 if a must be subtracted from b
|
|
* cA = 1 if a must be divided by 2
|
|
*
|
|
* Rules:
|
|
*
|
|
* cAB and cBA cannot both be 1.
|
|
* If a is not divided by 2, b is.
|
|
*/
|
|
oa = (a_lo >> i) & 1;
|
|
ob = (b_lo >> i) & 1;
|
|
cAB = oa & ob & rt;
|
|
cBA = oa & ob & ~rt;
|
|
cA = cAB | (oa ^ 1);
|
|
|
|
/*
|
|
* Conditional subtractions.
|
|
*/
|
|
a_lo -= b_lo & -cAB;
|
|
a_hi -= b_hi & -(uint64_t)cAB;
|
|
pa -= qa & -(int64_t)cAB;
|
|
pb -= qb & -(int64_t)cAB;
|
|
b_lo -= a_lo & -cBA;
|
|
b_hi -= a_hi & -(uint64_t)cBA;
|
|
qa -= pa & -(int64_t)cBA;
|
|
qb -= pb & -(int64_t)cBA;
|
|
|
|
/*
|
|
* Shifting.
|
|
*/
|
|
a_lo += a_lo & (cA - 1);
|
|
pa += pa & ((int64_t)cA - 1);
|
|
pb += pb & ((int64_t)cA - 1);
|
|
a_hi ^= (a_hi ^ (a_hi >> 1)) & -(uint64_t)cA;
|
|
b_lo += b_lo & -cA;
|
|
qa += qa & -(int64_t)cA;
|
|
qb += qb & -(int64_t)cA;
|
|
b_hi ^= (b_hi ^ (b_hi >> 1)) & ((uint64_t)cA - 1);
|
|
}
|
|
|
|
/*
|
|
* Apply the computed parameters to our values. We
|
|
* may have to correct pa and pb depending on the
|
|
* returned value of zint_co_reduce() (when a and/or b
|
|
* had to be negated).
|
|
*/
|
|
r = zint_co_reduce(a, b, len, pa, pb, qa, qb);
|
|
pa -= (pa + pa) & -(int64_t)(r & 1);
|
|
pb -= (pb + pb) & -(int64_t)(r & 1);
|
|
qa -= (qa + qa) & -(int64_t)(r >> 1);
|
|
qb -= (qb + qb) & -(int64_t)(r >> 1);
|
|
zint_co_reduce_mod(u0, u1, y, len, y0i, pa, pb, qa, qb);
|
|
zint_co_reduce_mod(v0, v1, x, len, x0i, pa, pb, qa, qb);
|
|
}
|
|
|
|
/*
|
|
* At that point, array a[] should contain the GCD, and the
|
|
* results (u,v) should already be set. We check that the GCD
|
|
* is indeed 1. We also check that the two operands x and y
|
|
* are odd.
|
|
*/
|
|
rc = a[0] ^ 1;
|
|
for (j = 1; j < len; j ++) {
|
|
rc |= a[j];
|
|
}
|
|
return (int)((1 - ((rc | -rc) >> 31)) & x[0] & y[0]);
|
|
}
|
|
|
|
/*
|
|
* Add k*y*2^sc to x. The result is assumed to fit in the array of
|
|
* size xlen (truncation is applied if necessary).
|
|
* Scale factor 'sc' is provided as sch and scl, such that:
|
|
* sch = sc / 31
|
|
* scl = sc % 31
|
|
* xlen MUST NOT be lower than ylen.
|
|
*
|
|
* x[] and y[] are both signed integers, using two's complement for
|
|
* negative values.
|
|
*/
|
|
static void
|
|
zint_add_scaled_mul_small(uint32_t *x, size_t xlen,
|
|
const uint32_t *y, size_t ylen, int32_t k,
|
|
uint32_t sch, uint32_t scl) {
|
|
size_t u;
|
|
uint32_t ysign, tw;
|
|
int32_t cc;
|
|
|
|
if (ylen == 0) {
|
|
return;
|
|
}
|
|
|
|
ysign = -(y[ylen - 1] >> 30) >> 1;
|
|
tw = 0;
|
|
cc = 0;
|
|
for (u = sch; u < xlen; u ++) {
|
|
size_t v;
|
|
uint32_t wy, wys, ccu;
|
|
uint64_t z;
|
|
|
|
/*
|
|
* Get the next word of y (scaled).
|
|
*/
|
|
v = u - sch;
|
|
if (v < ylen) {
|
|
wy = y[v];
|
|
} else {
|
|
wy = ysign;
|
|
}
|
|
wys = ((wy << scl) & 0x7FFFFFFF) | tw;
|
|
tw = wy >> (31 - scl);
|
|
|
|
/*
|
|
* The expression below does not overflow.
|
|
*/
|
|
z = (uint64_t)((int64_t)wys * (int64_t)k + (int64_t)x[u] + cc);
|
|
x[u] = (uint32_t)z & 0x7FFFFFFF;
|
|
|
|
/*
|
|
* Right-shifting the signed value z would yield
|
|
* implementation-defined results (arithmetic shift is
|
|
* not guaranteed). However, we can cast to unsigned,
|
|
* and get the next carry as an unsigned word. We can
|
|
* then convert it back to signed by using the guaranteed
|
|
* fact that 'int32_t' uses two's complement with no
|
|
* trap representation or padding bit, and with a layout
|
|
* compatible with that of 'uint32_t'.
|
|
*/
|
|
ccu = (uint32_t)(z >> 31);
|
|
cc = *(int32_t *)&ccu;
|
|
}
|
|
}
|
|
|
|
/*
|
|
* Subtract y*2^sc from x. The result is assumed to fit in the array of
|
|
* size xlen (truncation is applied if necessary).
|
|
* Scale factor 'sc' is provided as sch and scl, such that:
|
|
* sch = sc / 31
|
|
* scl = sc % 31
|
|
* xlen MUST NOT be lower than ylen.
|
|
*
|
|
* x[] and y[] are both signed integers, using two's complement for
|
|
* negative values.
|
|
*/
|
|
static void
|
|
zint_sub_scaled(uint32_t *x, size_t xlen,
|
|
const uint32_t *y, size_t ylen, uint32_t sch, uint32_t scl) {
|
|
size_t u;
|
|
uint32_t ysign, tw;
|
|
uint32_t cc;
|
|
|
|
if (ylen == 0) {
|
|
return;
|
|
}
|
|
|
|
ysign = -(y[ylen - 1] >> 30) >> 1;
|
|
tw = 0;
|
|
cc = 0;
|
|
for (u = sch; u < xlen; u ++) {
|
|
size_t v;
|
|
uint32_t w, wy, wys;
|
|
|
|
/*
|
|
* Get the next word of y (scaled).
|
|
*/
|
|
v = u - sch;
|
|
if (v < ylen) {
|
|
wy = y[v];
|
|
} else {
|
|
wy = ysign;
|
|
}
|
|
wys = ((wy << scl) & 0x7FFFFFFF) | tw;
|
|
tw = wy >> (31 - scl);
|
|
|
|
w = x[u] - wys - cc;
|
|
x[u] = w & 0x7FFFFFFF;
|
|
cc = w >> 31;
|
|
}
|
|
}
|
|
|
|
/*
|
|
* Convert a one-word signed big integer into a signed value.
|
|
*/
|
|
static inline int32_t
|
|
zint_one_to_plain(const uint32_t *x) {
|
|
uint32_t w;
|
|
|
|
w = x[0];
|
|
w |= (w & 0x40000000) << 1;
|
|
return *(int32_t *)&w;
|
|
}
|
|
|
|
/* ==================================================================== */
|
|
|
|
/*
|
|
* Convert a polynomial to floating-point values.
|
|
*
|
|
* Each coefficient has length flen words, and starts fstride words after
|
|
* the previous.
|
|
*
|
|
* IEEE-754 binary64 values can represent values in a finite range,
|
|
* roughly 2^(-1023) to 2^(+1023); thus, if coefficients are too large,
|
|
* they should be "trimmed" by pointing not to the lowest word of each,
|
|
* but upper.
|
|
*/
|
|
static void
|
|
poly_big_to_fp(fpr *d, const uint32_t *f, size_t flen, size_t fstride,
|
|
unsigned logn) {
|
|
size_t n, u;
|
|
|
|
n = MKN(logn);
|
|
if (flen == 0) {
|
|
for (u = 0; u < n; u ++) {
|
|
d[u] = fpr_zero;
|
|
}
|
|
return;
|
|
}
|
|
for (u = 0; u < n; u ++, f += fstride) {
|
|
size_t v;
|
|
uint32_t neg, cc, xm;
|
|
fpr x, fsc;
|
|
|
|
/*
|
|
* Get sign of the integer; if it is negative, then we
|
|
* will load its absolute value instead, and negate the
|
|
* result.
|
|
*/
|
|
neg = -(f[flen - 1] >> 30);
|
|
xm = neg >> 1;
|
|
cc = neg & 1;
|
|
x = fpr_zero;
|
|
fsc = fpr_one;
|
|
for (v = 0; v < flen; v ++, fsc = fpr_mul(fsc, fpr_ptwo31)) {
|
|
uint32_t w;
|
|
|
|
w = (f[v] ^ xm) + cc;
|
|
cc = w >> 31;
|
|
w &= 0x7FFFFFFF;
|
|
w -= (w << 1) & neg;
|
|
x = fpr_add(x, fpr_mul(fpr_of(*(int32_t *)&w), fsc));
|
|
}
|
|
d[u] = x;
|
|
}
|
|
}
|
|
|
|
/*
|
|
* Convert a polynomial to small integers. Source values are supposed
|
|
* to be one-word integers, signed over 31 bits. Returned value is 0
|
|
* if any of the coefficients exceeds the provided limit (in absolute
|
|
* value), or 1 on success.
|
|
*
|
|
* This is not constant-time; this is not a problem here, because on
|
|
* any failure, the NTRU-solving process will be deemed to have failed
|
|
* and the (f,g) polynomials will be discarded.
|
|
*/
|
|
static int
|
|
poly_big_to_small(int8_t *d, const uint32_t *s, int lim, unsigned logn) {
|
|
size_t n, u;
|
|
|
|
n = MKN(logn);
|
|
for (u = 0; u < n; u ++) {
|
|
int32_t z;
|
|
|
|
z = zint_one_to_plain(s + u);
|
|
if (z < -lim || z > lim) {
|
|
return 0;
|
|
}
|
|
d[u] = (int8_t)z;
|
|
}
|
|
return 1;
|
|
}
|
|
|
|
/*
|
|
* Subtract k*f from F, where F, f and k are polynomials modulo X^N+1.
|
|
* Coefficients of polynomial k are small integers (signed values in the
|
|
* -2^31..2^31 range) scaled by 2^sc. Value sc is provided as sch = sc / 31
|
|
* and scl = sc % 31.
|
|
*
|
|
* This function implements the basic quadratic multiplication algorithm,
|
|
* which is efficient in space (no extra buffer needed) but slow at
|
|
* high degree.
|
|
*/
|
|
static void
|
|
poly_sub_scaled(uint32_t *F, size_t Flen, size_t Fstride,
|
|
const uint32_t *f, size_t flen, size_t fstride,
|
|
const int32_t *k, uint32_t sch, uint32_t scl, unsigned logn) {
|
|
size_t n, u;
|
|
|
|
n = MKN(logn);
|
|
for (u = 0; u < n; u ++) {
|
|
int32_t kf;
|
|
size_t v;
|
|
uint32_t *x;
|
|
const uint32_t *y;
|
|
|
|
kf = -k[u];
|
|
x = F + u * Fstride;
|
|
y = f;
|
|
for (v = 0; v < n; v ++) {
|
|
zint_add_scaled_mul_small(
|
|
x, Flen, y, flen, kf, sch, scl);
|
|
if (u + v == n - 1) {
|
|
x = F;
|
|
kf = -kf;
|
|
} else {
|
|
x += Fstride;
|
|
}
|
|
y += fstride;
|
|
}
|
|
}
|
|
}
|
|
|
|
/*
|
|
* Subtract k*f from F. Coefficients of polynomial k are small integers
|
|
* (signed values in the -2^31..2^31 range) scaled by 2^sc. This function
|
|
* assumes that the degree is large, and integers relatively small.
|
|
* The value sc is provided as sch = sc / 31 and scl = sc % 31.
|
|
*/
|
|
static void
|
|
poly_sub_scaled_ntt(uint32_t *F, size_t Flen, size_t Fstride,
|
|
const uint32_t *f, size_t flen, size_t fstride,
|
|
const int32_t *k, uint32_t sch, uint32_t scl, unsigned logn,
|
|
uint32_t *tmp) {
|
|
uint32_t *gm, *igm, *fk, *t1, *x;
|
|
const uint32_t *y;
|
|
size_t n, u, tlen;
|
|
const small_prime *primes;
|
|
|
|
n = MKN(logn);
|
|
tlen = flen + 1;
|
|
gm = tmp;
|
|
igm = gm + MKN(logn);
|
|
fk = igm + MKN(logn);
|
|
t1 = fk + n * tlen;
|
|
|
|
primes = PRIMES;
|
|
|
|
/*
|
|
* Compute k*f in fk[], in RNS notation.
|
|
*/
|
|
for (u = 0; u < tlen; u ++) {
|
|
uint32_t p, p0i, R2, Rx;
|
|
size_t v;
|
|
|
|
p = primes[u].p;
|
|
p0i = modp_ninv31(p);
|
|
R2 = modp_R2(p, p0i);
|
|
Rx = modp_Rx((unsigned)flen, p, p0i, R2);
|
|
modp_mkgm2(gm, igm, logn, primes[u].g, p, p0i);
|
|
|
|
for (v = 0; v < n; v ++) {
|
|
t1[v] = modp_set(k[v], p);
|
|
}
|
|
modp_NTT2(t1, gm, logn, p, p0i);
|
|
for (v = 0, y = f, x = fk + u;
|
|
v < n; v ++, y += fstride, x += tlen) {
|
|
*x = zint_mod_small_signed(y, flen, p, p0i, R2, Rx);
|
|
}
|
|
modp_NTT2_ext(fk + u, tlen, gm, logn, p, p0i);
|
|
for (v = 0, x = fk + u; v < n; v ++, x += tlen) {
|
|
*x = modp_montymul(
|
|
modp_montymul(t1[v], *x, p, p0i), R2, p, p0i);
|
|
}
|
|
modp_iNTT2_ext(fk + u, tlen, igm, logn, p, p0i);
|
|
}
|
|
|
|
/*
|
|
* Rebuild k*f.
|
|
*/
|
|
zint_rebuild_CRT(fk, tlen, tlen, n, primes, 1, t1);
|
|
|
|
/*
|
|
* Subtract k*f, scaled, from F.
|
|
*/
|
|
for (u = 0, x = F, y = fk; u < n; u ++, x += Fstride, y += tlen) {
|
|
zint_sub_scaled(x, Flen, y, tlen, sch, scl);
|
|
}
|
|
}
|
|
|
|
/* ==================================================================== */
|
|
|
|
|
|
#define RNG_CONTEXT inner_shake256_context
|
|
|
|
/*
|
|
* Get a random 8-byte integer from a SHAKE-based RNG. This function
|
|
* ensures consistent interpretation of the SHAKE output so that
|
|
* the same values will be obtained over different platforms, in case
|
|
* a known seed is used.
|
|
*/
|
|
static inline uint64_t
|
|
get_rng_u64(inner_shake256_context *rng) {
|
|
/*
|
|
* We enforce little-endian representation.
|
|
*/
|
|
|
|
uint8_t tmp[8];
|
|
|
|
inner_shake256_extract(rng, tmp, sizeof tmp);
|
|
return (uint64_t)tmp[0]
|
|
| ((uint64_t)tmp[1] << 8)
|
|
| ((uint64_t)tmp[2] << 16)
|
|
| ((uint64_t)tmp[3] << 24)
|
|
| ((uint64_t)tmp[4] << 32)
|
|
| ((uint64_t)tmp[5] << 40)
|
|
| ((uint64_t)tmp[6] << 48)
|
|
| ((uint64_t)tmp[7] << 56);
|
|
}
|
|
|
|
/*
|
|
* Table below incarnates a discrete Gaussian distribution:
|
|
* D(x) = exp(-(x^2)/(2*sigma^2))
|
|
* where sigma = 1.17*sqrt(q/(2*N)), q = 12289, and N = 1024.
|
|
* Element 0 of the table is P(x = 0).
|
|
* For k > 0, element k is P(x >= k+1 | x > 0).
|
|
* Probabilities are scaled up by 2^63.
|
|
*/
|
|
static const uint64_t gauss_1024_12289[] = {
|
|
1283868770400643928u, 6416574995475331444u, 4078260278032692663u,
|
|
2353523259288686585u, 1227179971273316331u, 575931623374121527u,
|
|
242543240509105209u, 91437049221049666u, 30799446349977173u,
|
|
9255276791179340u, 2478152334826140u, 590642893610164u,
|
|
125206034929641u, 23590435911403u, 3948334035941u,
|
|
586753615614u, 77391054539u, 9056793210u,
|
|
940121950u, 86539696u, 7062824u,
|
|
510971u, 32764u, 1862u,
|
|
94u, 4u, 0u
|
|
};
|
|
|
|
/*
|
|
* Generate a random value with a Gaussian distribution centered on 0.
|
|
* The RNG must be ready for extraction (already flipped).
|
|
*
|
|
* Distribution has standard deviation 1.17*sqrt(q/(2*N)). The
|
|
* precomputed table is for N = 1024. Since the sum of two independent
|
|
* values of standard deviation sigma has standard deviation
|
|
* sigma*sqrt(2), then we can just generate more values and add them
|
|
* together for lower dimensions.
|
|
*/
|
|
static int
|
|
mkgauss(RNG_CONTEXT *rng, unsigned logn) {
|
|
unsigned u, g;
|
|
int val;
|
|
|
|
g = 1U << (10 - logn);
|
|
val = 0;
|
|
for (u = 0; u < g; u ++) {
|
|
/*
|
|
* Each iteration generates one value with the
|
|
* Gaussian distribution for N = 1024.
|
|
*
|
|
* We use two random 64-bit values. First value
|
|
* decides on whether the generated value is 0, and,
|
|
* if not, the sign of the value. Second random 64-bit
|
|
* word is used to generate the non-zero value.
|
|
*
|
|
* For constant-time code we have to read the complete
|
|
* table. This has negligible cost, compared with the
|
|
* remainder of the keygen process (solving the NTRU
|
|
* equation).
|
|
*/
|
|
uint64_t r;
|
|
uint32_t f, v, k, neg;
|
|
|
|
/*
|
|
* First value:
|
|
* - flag 'neg' is randomly selected to be 0 or 1.
|
|
* - flag 'f' is set to 1 if the generated value is zero,
|
|
* or set to 0 otherwise.
|
|
*/
|
|
r = get_rng_u64(rng);
|
|
neg = (uint32_t)(r >> 63);
|
|
r &= ~((uint64_t)1 << 63);
|
|
f = (uint32_t)((r - gauss_1024_12289[0]) >> 63);
|
|
|
|
/*
|
|
* We produce a new random 63-bit integer r, and go over
|
|
* the array, starting at index 1. We store in v the
|
|
* index of the first array element which is not greater
|
|
* than r, unless the flag f was already 1.
|
|
*/
|
|
v = 0;
|
|
r = get_rng_u64(rng);
|
|
r &= ~((uint64_t)1 << 63);
|
|
for (k = 1; k < (uint32_t)((sizeof gauss_1024_12289)
|
|
/ (sizeof gauss_1024_12289[0])); k ++) {
|
|
uint32_t t;
|
|
|
|
t = (uint32_t)((r - gauss_1024_12289[k]) >> 63) ^ 1;
|
|
v |= k & -(t & (f ^ 1));
|
|
f |= t;
|
|
}
|
|
|
|
/*
|
|
* We apply the sign ('neg' flag). If the value is zero,
|
|
* the sign has no effect.
|
|
*/
|
|
v = (v ^ -neg) + neg;
|
|
|
|
/*
|
|
* Generated value is added to val.
|
|
*/
|
|
val += *(int32_t *)&v;
|
|
}
|
|
return val;
|
|
}
|
|
|
|
/*
|
|
* The MAX_BL_SMALL[] and MAX_BL_LARGE[] contain the lengths, in 31-bit
|
|
* words, of intermediate values in the computation:
|
|
*
|
|
* MAX_BL_SMALL[depth]: length for the input f and g at that depth
|
|
* MAX_BL_LARGE[depth]: length for the unreduced F and G at that depth
|
|
*
|
|
* Rules:
|
|
*
|
|
* - Within an array, values grow.
|
|
*
|
|
* - The 'SMALL' array must have an entry for maximum depth, corresponding
|
|
* to the size of values used in the binary GCD. There is no such value
|
|
* for the 'LARGE' array (the binary GCD yields already reduced
|
|
* coefficients).
|
|
*
|
|
* - MAX_BL_LARGE[depth] >= MAX_BL_SMALL[depth + 1].
|
|
*
|
|
* - Values must be large enough to handle the common cases, with some
|
|
* margins.
|
|
*
|
|
* - Values must not be "too large" either because we will convert some
|
|
* integers into floating-point values by considering the top 10 words,
|
|
* i.e. 310 bits; hence, for values of length more than 10 words, we
|
|
* should take care to have the length centered on the expected size.
|
|
*
|
|
* The following average lengths, in bits, have been measured on thousands
|
|
* of random keys (fg = max length of the absolute value of coefficients
|
|
* of f and g at that depth; FG = idem for the unreduced F and G; for the
|
|
* maximum depth, F and G are the output of binary GCD, multiplied by q;
|
|
* for each value, the average and standard deviation are provided).
|
|
*
|
|
* Binary case:
|
|
* depth: 10 fg: 6307.52 (24.48) FG: 6319.66 (24.51)
|
|
* depth: 9 fg: 3138.35 (12.25) FG: 9403.29 (27.55)
|
|
* depth: 8 fg: 1576.87 ( 7.49) FG: 4703.30 (14.77)
|
|
* depth: 7 fg: 794.17 ( 4.98) FG: 2361.84 ( 9.31)
|
|
* depth: 6 fg: 400.67 ( 3.10) FG: 1188.68 ( 6.04)
|
|
* depth: 5 fg: 202.22 ( 1.87) FG: 599.81 ( 3.87)
|
|
* depth: 4 fg: 101.62 ( 1.02) FG: 303.49 ( 2.38)
|
|
* depth: 3 fg: 50.37 ( 0.53) FG: 153.65 ( 1.39)
|
|
* depth: 2 fg: 24.07 ( 0.25) FG: 78.20 ( 0.73)
|
|
* depth: 1 fg: 10.99 ( 0.08) FG: 39.82 ( 0.41)
|
|
* depth: 0 fg: 4.00 ( 0.00) FG: 19.61 ( 0.49)
|
|
*
|
|
* Integers are actually represented either in binary notation over
|
|
* 31-bit words (signed, using two's complement), or in RNS, modulo
|
|
* many small primes. These small primes are close to, but slightly
|
|
* lower than, 2^31. Use of RNS loses less than two bits, even for
|
|
* the largest values.
|
|
*
|
|
* IMPORTANT: if these values are modified, then the temporary buffer
|
|
* sizes (FALCON_KEYGEN_TEMP_*, in inner.h) must be recomputed
|
|
* accordingly.
|
|
*/
|
|
|
|
static const size_t MAX_BL_SMALL[] = {
|
|
1, 1, 2, 2, 4, 7, 14, 27, 53, 106, 209
|
|
};
|
|
|
|
static const size_t MAX_BL_LARGE[] = {
|
|
2, 2, 5, 7, 12, 21, 40, 78, 157, 308
|
|
};
|
|
|
|
/*
|
|
* Average and standard deviation for the maximum size (in bits) of
|
|
* coefficients of (f,g), depending on depth. These values are used
|
|
* to compute bounds for Babai's reduction.
|
|
*/
|
|
static const struct {
|
|
int avg;
|
|
int std;
|
|
} BITLENGTH[] = {
|
|
{ 4, 0 },
|
|
{ 11, 1 },
|
|
{ 24, 1 },
|
|
{ 50, 1 },
|
|
{ 102, 1 },
|
|
{ 202, 2 },
|
|
{ 401, 4 },
|
|
{ 794, 5 },
|
|
{ 1577, 8 },
|
|
{ 3138, 13 },
|
|
{ 6308, 25 }
|
|
};
|
|
|
|
/*
|
|
* Minimal recursion depth at which we rebuild intermediate values
|
|
* when reconstructing f and g.
|
|
*/
|
|
#define DEPTH_INT_FG 4
|
|
|
|
/*
|
|
* Compute squared norm of a short vector. Returned value is saturated to
|
|
* 2^32-1 if it is not lower than 2^31.
|
|
*/
|
|
static uint32_t
|
|
poly_small_sqnorm(const int8_t *f, unsigned logn) {
|
|
size_t n, u;
|
|
uint32_t s, ng;
|
|
|
|
n = MKN(logn);
|
|
s = 0;
|
|
ng = 0;
|
|
for (u = 0; u < n; u ++) {
|
|
int32_t z;
|
|
|
|
z = f[u];
|
|
s += (uint32_t)(z * z);
|
|
ng |= s;
|
|
}
|
|
return s | -(ng >> 31);
|
|
}
|
|
|
|
/*
|
|
* Align (upwards) the provided 'data' pointer with regards to 'base'
|
|
* so that the offset is a multiple of the size of 'fpr'.
|
|
*/
|
|
static fpr *
|
|
align_fpr(void *base, void *data) {
|
|
uint8_t *cb, *cd;
|
|
size_t k, km;
|
|
|
|
cb = base;
|
|
cd = data;
|
|
k = (size_t)(cd - cb);
|
|
km = k % sizeof(fpr);
|
|
if (km) {
|
|
k += (sizeof(fpr)) - km;
|
|
}
|
|
return (fpr *)(cb + k);
|
|
}
|
|
|
|
/*
|
|
* Align (upwards) the provided 'data' pointer with regards to 'base'
|
|
* so that the offset is a multiple of the size of 'uint32_t'.
|
|
*/
|
|
static uint32_t *
|
|
align_u32(void *base, void *data) {
|
|
uint8_t *cb, *cd;
|
|
size_t k, km;
|
|
|
|
cb = base;
|
|
cd = data;
|
|
k = (size_t)(cd - cb);
|
|
km = k % sizeof(uint32_t);
|
|
if (km) {
|
|
k += (sizeof(uint32_t)) - km;
|
|
}
|
|
return (uint32_t *)(cb + k);
|
|
}
|
|
|
|
/*
|
|
* Convert a small vector to floating point.
|
|
*/
|
|
static void
|
|
poly_small_to_fp(fpr *x, const int8_t *f, unsigned logn) {
|
|
size_t n, u;
|
|
|
|
n = MKN(logn);
|
|
for (u = 0; u < n; u ++) {
|
|
x[u] = fpr_of(f[u]);
|
|
}
|
|
}
|
|
|
|
/*
|
|
* Input: f,g of degree N = 2^logn; 'depth' is used only to get their
|
|
* individual length.
|
|
*
|
|
* Output: f',g' of degree N/2, with the length for 'depth+1'.
|
|
*
|
|
* Values are in RNS; input and/or output may also be in NTT.
|
|
*/
|
|
static void
|
|
make_fg_step(uint32_t *data, unsigned logn, unsigned depth,
|
|
int in_ntt, int out_ntt) {
|
|
size_t n, hn, u;
|
|
size_t slen, tlen;
|
|
uint32_t *fd, *gd, *fs, *gs, *gm, *igm, *t1;
|
|
const small_prime *primes;
|
|
|
|
n = (size_t)1 << logn;
|
|
hn = n >> 1;
|
|
slen = MAX_BL_SMALL[depth];
|
|
tlen = MAX_BL_SMALL[depth + 1];
|
|
primes = PRIMES;
|
|
|
|
/*
|
|
* Prepare room for the result.
|
|
*/
|
|
fd = data;
|
|
gd = fd + hn * tlen;
|
|
fs = gd + hn * tlen;
|
|
gs = fs + n * slen;
|
|
gm = gs + n * slen;
|
|
igm = gm + n;
|
|
t1 = igm + n;
|
|
memmove(fs, data, 2 * n * slen * sizeof * data);
|
|
|
|
/*
|
|
* First slen words: we use the input values directly, and apply
|
|
* inverse NTT as we go.
|
|
*/
|
|
for (u = 0; u < slen; u ++) {
|
|
uint32_t p, p0i, R2;
|
|
size_t v;
|
|
uint32_t *x;
|
|
|
|
p = primes[u].p;
|
|
p0i = modp_ninv31(p);
|
|
R2 = modp_R2(p, p0i);
|
|
modp_mkgm2(gm, igm, logn, primes[u].g, p, p0i);
|
|
|
|
for (v = 0, x = fs + u; v < n; v ++, x += slen) {
|
|
t1[v] = *x;
|
|
}
|
|
if (!in_ntt) {
|
|
modp_NTT2(t1, gm, logn, p, p0i);
|
|
}
|
|
for (v = 0, x = fd + u; v < hn; v ++, x += tlen) {
|
|
uint32_t w0, w1;
|
|
|
|
w0 = t1[(v << 1) + 0];
|
|
w1 = t1[(v << 1) + 1];
|
|
*x = modp_montymul(
|
|
modp_montymul(w0, w1, p, p0i), R2, p, p0i);
|
|
}
|
|
if (in_ntt) {
|
|
modp_iNTT2_ext(fs + u, slen, igm, logn, p, p0i);
|
|
}
|
|
|
|
for (v = 0, x = gs + u; v < n; v ++, x += slen) {
|
|
t1[v] = *x;
|
|
}
|
|
if (!in_ntt) {
|
|
modp_NTT2(t1, gm, logn, p, p0i);
|
|
}
|
|
for (v = 0, x = gd + u; v < hn; v ++, x += tlen) {
|
|
uint32_t w0, w1;
|
|
|
|
w0 = t1[(v << 1) + 0];
|
|
w1 = t1[(v << 1) + 1];
|
|
*x = modp_montymul(
|
|
modp_montymul(w0, w1, p, p0i), R2, p, p0i);
|
|
}
|
|
if (in_ntt) {
|
|
modp_iNTT2_ext(gs + u, slen, igm, logn, p, p0i);
|
|
}
|
|
|
|
if (!out_ntt) {
|
|
modp_iNTT2_ext(fd + u, tlen, igm, logn - 1, p, p0i);
|
|
modp_iNTT2_ext(gd + u, tlen, igm, logn - 1, p, p0i);
|
|
}
|
|
}
|
|
|
|
/*
|
|
* Since the fs and gs words have been de-NTTized, we can use the
|
|
* CRT to rebuild the values.
|
|
*/
|
|
zint_rebuild_CRT(fs, slen, slen, n, primes, 1, gm);
|
|
zint_rebuild_CRT(gs, slen, slen, n, primes, 1, gm);
|
|
|
|
/*
|
|
* Remaining words: use modular reductions to extract the values.
|
|
*/
|
|
for (u = slen; u < tlen; u ++) {
|
|
uint32_t p, p0i, R2, Rx;
|
|
size_t v;
|
|
uint32_t *x;
|
|
|
|
p = primes[u].p;
|
|
p0i = modp_ninv31(p);
|
|
R2 = modp_R2(p, p0i);
|
|
Rx = modp_Rx((unsigned)slen, p, p0i, R2);
|
|
modp_mkgm2(gm, igm, logn, primes[u].g, p, p0i);
|
|
for (v = 0, x = fs; v < n; v ++, x += slen) {
|
|
t1[v] = zint_mod_small_signed(x, slen, p, p0i, R2, Rx);
|
|
}
|
|
modp_NTT2(t1, gm, logn, p, p0i);
|
|
for (v = 0, x = fd + u; v < hn; v ++, x += tlen) {
|
|
uint32_t w0, w1;
|
|
|
|
w0 = t1[(v << 1) + 0];
|
|
w1 = t1[(v << 1) + 1];
|
|
*x = modp_montymul(
|
|
modp_montymul(w0, w1, p, p0i), R2, p, p0i);
|
|
}
|
|
for (v = 0, x = gs; v < n; v ++, x += slen) {
|
|
t1[v] = zint_mod_small_signed(x, slen, p, p0i, R2, Rx);
|
|
}
|
|
modp_NTT2(t1, gm, logn, p, p0i);
|
|
for (v = 0, x = gd + u; v < hn; v ++, x += tlen) {
|
|
uint32_t w0, w1;
|
|
|
|
w0 = t1[(v << 1) + 0];
|
|
w1 = t1[(v << 1) + 1];
|
|
*x = modp_montymul(
|
|
modp_montymul(w0, w1, p, p0i), R2, p, p0i);
|
|
}
|
|
|
|
if (!out_ntt) {
|
|
modp_iNTT2_ext(fd + u, tlen, igm, logn - 1, p, p0i);
|
|
modp_iNTT2_ext(gd + u, tlen, igm, logn - 1, p, p0i);
|
|
}
|
|
}
|
|
}
|
|
|
|
/*
|
|
* Compute f and g at a specific depth, in RNS notation.
|
|
*
|
|
* Returned values are stored in the data[] array, at slen words per integer.
|
|
*
|
|
* Conditions:
|
|
* 0 <= depth <= logn
|
|
*
|
|
* Space use in data[]: enough room for any two successive values (f', g',
|
|
* f and g).
|
|
*/
|
|
static void
|
|
make_fg(uint32_t *data, const int8_t *f, const int8_t *g,
|
|
unsigned logn, unsigned depth, int out_ntt) {
|
|
size_t n, u;
|
|
uint32_t *ft, *gt, p0;
|
|
unsigned d;
|
|
const small_prime *primes;
|
|
|
|
n = MKN(logn);
|
|
ft = data;
|
|
gt = ft + n;
|
|
primes = PRIMES;
|
|
p0 = primes[0].p;
|
|
for (u = 0; u < n; u ++) {
|
|
ft[u] = modp_set(f[u], p0);
|
|
gt[u] = modp_set(g[u], p0);
|
|
}
|
|
|
|
if (depth == 0 && out_ntt) {
|
|
uint32_t *gm, *igm;
|
|
uint32_t p, p0i;
|
|
|
|
p = primes[0].p;
|
|
p0i = modp_ninv31(p);
|
|
gm = gt + n;
|
|
igm = gm + MKN(logn);
|
|
modp_mkgm2(gm, igm, logn, primes[0].g, p, p0i);
|
|
modp_NTT2(ft, gm, logn, p, p0i);
|
|
modp_NTT2(gt, gm, logn, p, p0i);
|
|
return;
|
|
}
|
|
|
|
if (depth == 0) {
|
|
return;
|
|
}
|
|
if (depth == 1) {
|
|
make_fg_step(data, logn, 0, 0, out_ntt);
|
|
return;
|
|
}
|
|
make_fg_step(data, logn, 0, 0, 1);
|
|
for (d = 1; d + 1 < depth; d ++) {
|
|
make_fg_step(data, logn - d, d, 1, 1);
|
|
}
|
|
make_fg_step(data, logn - depth + 1, depth - 1, 1, out_ntt);
|
|
}
|
|
|
|
/*
|
|
* Solving the NTRU equation, deepest level: compute the resultants of
|
|
* f and g with X^N+1, and use binary GCD. The F and G values are
|
|
* returned in tmp[].
|
|
*
|
|
* Returned value: 1 on success, 0 on error.
|
|
*/
|
|
static int
|
|
solve_NTRU_deepest(unsigned logn_top,
|
|
const int8_t *f, const int8_t *g, uint32_t *tmp) {
|
|
size_t len;
|
|
uint32_t *Fp, *Gp, *fp, *gp, *t1, q;
|
|
const small_prime *primes;
|
|
|
|
len = MAX_BL_SMALL[logn_top];
|
|
primes = PRIMES;
|
|
|
|
Fp = tmp;
|
|
Gp = Fp + len;
|
|
fp = Gp + len;
|
|
gp = fp + len;
|
|
t1 = gp + len;
|
|
|
|
make_fg(fp, f, g, logn_top, logn_top, 0);
|
|
|
|
/*
|
|
* We use the CRT to rebuild the resultants as big integers.
|
|
* There are two such big integers. The resultants are always
|
|
* nonnegative.
|
|
*/
|
|
zint_rebuild_CRT(fp, len, len, 2, primes, 0, t1);
|
|
|
|
/*
|
|
* Apply the binary GCD. The zint_bezout() function works only
|
|
* if both inputs are odd.
|
|
*
|
|
* We can test on the result and return 0 because that would
|
|
* imply failure of the NTRU solving equation, and the (f,g)
|
|
* values will be abandoned in that case.
|
|
*/
|
|
if (!zint_bezout(Gp, Fp, fp, gp, len, t1)) {
|
|
return 0;
|
|
}
|
|
|
|
/*
|
|
* Multiply the two values by the target value q. Values must
|
|
* fit in the destination arrays.
|
|
* We can again test on the returned words: a non-zero output
|
|
* of zint_mul_small() means that we exceeded our array
|
|
* capacity, and that implies failure and rejection of (f,g).
|
|
*/
|
|
q = 12289;
|
|
if (zint_mul_small(Fp, len, q) != 0
|
|
|| zint_mul_small(Gp, len, q) != 0) {
|
|
return 0;
|
|
}
|
|
|
|
return 1;
|
|
}
|
|
|
|
/*
|
|
* Solving the NTRU equation, intermediate level. Upon entry, the F and G
|
|
* from the previous level should be in the tmp[] array.
|
|
* This function MAY be invoked for the top-level (in which case depth = 0).
|
|
*
|
|
* Returned value: 1 on success, 0 on error.
|
|
*/
|
|
static int
|
|
solve_NTRU_intermediate(unsigned logn_top,
|
|
const int8_t *f, const int8_t *g, unsigned depth, uint32_t *tmp) {
|
|
/*
|
|
* In this function, 'logn' is the log2 of the degree for
|
|
* this step. If N = 2^logn, then:
|
|
* - the F and G values already in fk->tmp (from the deeper
|
|
* levels) have degree N/2;
|
|
* - this function should return F and G of degree N.
|
|
*/
|
|
unsigned logn;
|
|
size_t n, hn, slen, dlen, llen, rlen, FGlen, u;
|
|
uint32_t *Fd, *Gd, *Ft, *Gt, *ft, *gt, *t1;
|
|
fpr *rt1, *rt2, *rt3, *rt4, *rt5;
|
|
int scale_fg, minbl_fg, maxbl_fg, maxbl_FG, scale_k;
|
|
uint32_t *x, *y;
|
|
int32_t *k;
|
|
const small_prime *primes;
|
|
|
|
logn = logn_top - depth;
|
|
n = (size_t)1 << logn;
|
|
hn = n >> 1;
|
|
|
|
/*
|
|
* slen = size for our input f and g; also size of the reduced
|
|
* F and G we return (degree N)
|
|
*
|
|
* dlen = size of the F and G obtained from the deeper level
|
|
* (degree N/2 or N/3)
|
|
*
|
|
* llen = size for intermediary F and G before reduction (degree N)
|
|
*
|
|
* We build our non-reduced F and G as two independent halves each,
|
|
* of degree N/2 (F = F0 + X*F1, G = G0 + X*G1).
|
|
*/
|
|
slen = MAX_BL_SMALL[depth];
|
|
dlen = MAX_BL_SMALL[depth + 1];
|
|
llen = MAX_BL_LARGE[depth];
|
|
primes = PRIMES;
|
|
|
|
/*
|
|
* Fd and Gd are the F and G from the deeper level.
|
|
*/
|
|
Fd = tmp;
|
|
Gd = Fd + dlen * hn;
|
|
|
|
/*
|
|
* Compute the input f and g for this level. Note that we get f
|
|
* and g in RNS + NTT representation.
|
|
*/
|
|
ft = Gd + dlen * hn;
|
|
make_fg(ft, f, g, logn_top, depth, 1);
|
|
|
|
/*
|
|
* Move the newly computed f and g to make room for our candidate
|
|
* F and G (unreduced).
|
|
*/
|
|
Ft = tmp;
|
|
Gt = Ft + n * llen;
|
|
t1 = Gt + n * llen;
|
|
memmove(t1, ft, 2 * n * slen * sizeof * ft);
|
|
ft = t1;
|
|
gt = ft + slen * n;
|
|
t1 = gt + slen * n;
|
|
|
|
/*
|
|
* Move Fd and Gd _after_ f and g.
|
|
*/
|
|
memmove(t1, Fd, 2 * hn * dlen * sizeof * Fd);
|
|
Fd = t1;
|
|
Gd = Fd + hn * dlen;
|
|
|
|
/*
|
|
* We reduce Fd and Gd modulo all the small primes we will need,
|
|
* and store the values in Ft and Gt (only n/2 values in each).
|
|
*/
|
|
for (u = 0; u < llen; u ++) {
|
|
uint32_t p, p0i, R2, Rx;
|
|
size_t v;
|
|
uint32_t *xs, *ys, *xd, *yd;
|
|
|
|
p = primes[u].p;
|
|
p0i = modp_ninv31(p);
|
|
R2 = modp_R2(p, p0i);
|
|
Rx = modp_Rx((unsigned)dlen, p, p0i, R2);
|
|
for (v = 0, xs = Fd, ys = Gd, xd = Ft + u, yd = Gt + u;
|
|
v < hn;
|
|
v ++, xs += dlen, ys += dlen, xd += llen, yd += llen) {
|
|
*xd = zint_mod_small_signed(xs, dlen, p, p0i, R2, Rx);
|
|
*yd = zint_mod_small_signed(ys, dlen, p, p0i, R2, Rx);
|
|
}
|
|
}
|
|
|
|
/*
|
|
* We do not need Fd and Gd after that point.
|
|
*/
|
|
|
|
/*
|
|
* Compute our F and G modulo sufficiently many small primes.
|
|
*/
|
|
for (u = 0; u < llen; u ++) {
|
|
uint32_t p, p0i, R2;
|
|
uint32_t *gm, *igm, *fx, *gx, *Fp, *Gp;
|
|
size_t v;
|
|
|
|
/*
|
|
* All computations are done modulo p.
|
|
*/
|
|
p = primes[u].p;
|
|
p0i = modp_ninv31(p);
|
|
R2 = modp_R2(p, p0i);
|
|
|
|
/*
|
|
* If we processed slen words, then f and g have been
|
|
* de-NTTized, and are in RNS; we can rebuild them.
|
|
*/
|
|
if (u == slen) {
|
|
zint_rebuild_CRT(ft, slen, slen, n, primes, 1, t1);
|
|
zint_rebuild_CRT(gt, slen, slen, n, primes, 1, t1);
|
|
}
|
|
|
|
gm = t1;
|
|
igm = gm + n;
|
|
fx = igm + n;
|
|
gx = fx + n;
|
|
|
|
modp_mkgm2(gm, igm, logn, primes[u].g, p, p0i);
|
|
|
|
if (u < slen) {
|
|
for (v = 0, x = ft + u, y = gt + u;
|
|
v < n; v ++, x += slen, y += slen) {
|
|
fx[v] = *x;
|
|
gx[v] = *y;
|
|
}
|
|
modp_iNTT2_ext(ft + u, slen, igm, logn, p, p0i);
|
|
modp_iNTT2_ext(gt + u, slen, igm, logn, p, p0i);
|
|
} else {
|
|
uint32_t Rx;
|
|
|
|
Rx = modp_Rx((unsigned)slen, p, p0i, R2);
|
|
for (v = 0, x = ft, y = gt;
|
|
v < n; v ++, x += slen, y += slen) {
|
|
fx[v] = zint_mod_small_signed(x, slen,
|
|
p, p0i, R2, Rx);
|
|
gx[v] = zint_mod_small_signed(y, slen,
|
|
p, p0i, R2, Rx);
|
|
}
|
|
modp_NTT2(fx, gm, logn, p, p0i);
|
|
modp_NTT2(gx, gm, logn, p, p0i);
|
|
}
|
|
|
|
/*
|
|
* Get F' and G' modulo p and in NTT representation
|
|
* (they have degree n/2). These values were computed in
|
|
* a previous step, and stored in Ft and Gt.
|
|
*/
|
|
Fp = gx + n;
|
|
Gp = Fp + hn;
|
|
for (v = 0, x = Ft + u, y = Gt + u;
|
|
v < hn; v ++, x += llen, y += llen) {
|
|
Fp[v] = *x;
|
|
Gp[v] = *y;
|
|
}
|
|
modp_NTT2(Fp, gm, logn - 1, p, p0i);
|
|
modp_NTT2(Gp, gm, logn - 1, p, p0i);
|
|
|
|
/*
|
|
* Compute our F and G modulo p.
|
|
*
|
|
* General case:
|
|
*
|
|
* we divide degree by d = 2 or 3
|
|
* f'(x^d) = N(f)(x^d) = f * adj(f)
|
|
* g'(x^d) = N(g)(x^d) = g * adj(g)
|
|
* f'*G' - g'*F' = q
|
|
* F = F'(x^d) * adj(g)
|
|
* G = G'(x^d) * adj(f)
|
|
*
|
|
* We compute things in the NTT. We group roots of phi
|
|
* such that all roots x in a group share the same x^d.
|
|
* If the roots in a group are x_1, x_2... x_d, then:
|
|
*
|
|
* N(f)(x_1^d) = f(x_1)*f(x_2)*...*f(x_d)
|
|
*
|
|
* Thus, we have:
|
|
*
|
|
* G(x_1) = f(x_2)*f(x_3)*...*f(x_d)*G'(x_1^d)
|
|
* G(x_2) = f(x_1)*f(x_3)*...*f(x_d)*G'(x_1^d)
|
|
* ...
|
|
* G(x_d) = f(x_1)*f(x_2)*...*f(x_{d-1})*G'(x_1^d)
|
|
*
|
|
* In all cases, we can thus compute F and G in NTT
|
|
* representation by a few simple multiplications.
|
|
* Moreover, in our chosen NTT representation, roots
|
|
* from the same group are consecutive in RAM.
|
|
*/
|
|
for (v = 0, x = Ft + u, y = Gt + u; v < hn;
|
|
v ++, x += (llen << 1), y += (llen << 1)) {
|
|
uint32_t ftA, ftB, gtA, gtB;
|
|
uint32_t mFp, mGp;
|
|
|
|
ftA = fx[(v << 1) + 0];
|
|
ftB = fx[(v << 1) + 1];
|
|
gtA = gx[(v << 1) + 0];
|
|
gtB = gx[(v << 1) + 1];
|
|
mFp = modp_montymul(Fp[v], R2, p, p0i);
|
|
mGp = modp_montymul(Gp[v], R2, p, p0i);
|
|
x[0] = modp_montymul(gtB, mFp, p, p0i);
|
|
x[llen] = modp_montymul(gtA, mFp, p, p0i);
|
|
y[0] = modp_montymul(ftB, mGp, p, p0i);
|
|
y[llen] = modp_montymul(ftA, mGp, p, p0i);
|
|
}
|
|
modp_iNTT2_ext(Ft + u, llen, igm, logn, p, p0i);
|
|
modp_iNTT2_ext(Gt + u, llen, igm, logn, p, p0i);
|
|
}
|
|
|
|
/*
|
|
* Rebuild F and G with the CRT.
|
|
*/
|
|
zint_rebuild_CRT(Ft, llen, llen, n, primes, 1, t1);
|
|
zint_rebuild_CRT(Gt, llen, llen, n, primes, 1, t1);
|
|
|
|
/*
|
|
* At that point, Ft, Gt, ft and gt are consecutive in RAM (in that
|
|
* order).
|
|
*/
|
|
|
|
/*
|
|
* Apply Babai reduction to bring back F and G to size slen.
|
|
*
|
|
* We use the FFT to compute successive approximations of the
|
|
* reduction coefficient. We first isolate the top bits of
|
|
* the coefficients of f and g, and convert them to floating
|
|
* point; with the FFT, we compute adj(f), adj(g), and
|
|
* 1/(f*adj(f)+g*adj(g)).
|
|
*
|
|
* Then, we repeatedly apply the following:
|
|
*
|
|
* - Get the top bits of the coefficients of F and G into
|
|
* floating point, and use the FFT to compute:
|
|
* (F*adj(f)+G*adj(g))/(f*adj(f)+g*adj(g))
|
|
*
|
|
* - Convert back that value into normal representation, and
|
|
* round it to the nearest integers, yielding a polynomial k.
|
|
* Proper scaling is applied to f, g, F and G so that the
|
|
* coefficients fit on 32 bits (signed).
|
|
*
|
|
* - Subtract k*f from F and k*g from G.
|
|
*
|
|
* Under normal conditions, this process reduces the size of F
|
|
* and G by some bits at each iteration. For constant-time
|
|
* operation, we do not want to measure the actual length of
|
|
* F and G; instead, we do the following:
|
|
*
|
|
* - f and g are converted to floating-point, with some scaling
|
|
* if necessary to keep values in the representable range.
|
|
*
|
|
* - For each iteration, we _assume_ a maximum size for F and G,
|
|
* and use the values at that size. If we overreach, then
|
|
* we get zeros, which is harmless: the resulting coefficients
|
|
* of k will be 0 and the value won't be reduced.
|
|
*
|
|
* - We conservatively assume that F and G will be reduced by
|
|
* at least 25 bits at each iteration.
|
|
*
|
|
* Even when reaching the bottom of the reduction, reduction
|
|
* coefficient will remain low. If it goes out-of-range, then
|
|
* something wrong occurred and the whole NTRU solving fails.
|
|
*/
|
|
|
|
/*
|
|
* Memory layout:
|
|
* - We need to compute and keep adj(f), adj(g), and
|
|
* 1/(f*adj(f)+g*adj(g)) (sizes N, N and N/2 fp numbers,
|
|
* respectively).
|
|
* - At each iteration we need two extra fp buffer (N fp values),
|
|
* and produce a k (N 32-bit words). k will be shared with one
|
|
* of the fp buffers.
|
|
* - To compute k*f and k*g efficiently (with the NTT), we need
|
|
* some extra room; we reuse the space of the temporary buffers.
|
|
*
|
|
* Arrays of 'fpr' are obtained from the temporary array itself.
|
|
* We ensure that the base is at a properly aligned offset (the
|
|
* source array tmp[] is supposed to be already aligned).
|
|
*/
|
|
|
|
rt3 = align_fpr(tmp, t1);
|
|
rt4 = rt3 + n;
|
|
rt5 = rt4 + n;
|
|
rt1 = rt5 + (n >> 1);
|
|
k = (int32_t *)align_u32(tmp, rt1);
|
|
rt2 = align_fpr(tmp, k + n);
|
|
if (rt2 < (rt1 + n)) {
|
|
rt2 = rt1 + n;
|
|
}
|
|
t1 = (uint32_t *)k + n;
|
|
|
|
/*
|
|
* Get f and g into rt3 and rt4 as floating-point approximations.
|
|
*
|
|
* We need to "scale down" the floating-point representation of
|
|
* coefficients when they are too big. We want to keep the value
|
|
* below 2^310 or so. Thus, when values are larger than 10 words,
|
|
* we consider only the top 10 words. Array lengths have been
|
|
* computed so that average maximum length will fall in the
|
|
* middle or the upper half of these top 10 words.
|
|
*/
|
|
rlen = slen;
|
|
if (rlen > 10) {
|
|
rlen = 10;
|
|
}
|
|
poly_big_to_fp(rt3, ft + slen - rlen, rlen, slen, logn);
|
|
poly_big_to_fp(rt4, gt + slen - rlen, rlen, slen, logn);
|
|
|
|
/*
|
|
* Values in rt3 and rt4 are downscaled by 2^(scale_fg).
|
|
*/
|
|
scale_fg = 31 * (int)(slen - rlen);
|
|
|
|
/*
|
|
* Estimated boundaries for the maximum size (in bits) of the
|
|
* coefficients of (f,g). We use the measured average, and
|
|
* allow for a deviation of at most six times the standard
|
|
* deviation.
|
|
*/
|
|
minbl_fg = BITLENGTH[depth].avg - 6 * BITLENGTH[depth].std;
|
|
maxbl_fg = BITLENGTH[depth].avg + 6 * BITLENGTH[depth].std;
|
|
|
|
/*
|
|
* Compute 1/(f*adj(f)+g*adj(g)) in rt5. We also keep adj(f)
|
|
* and adj(g) in rt3 and rt4, respectively.
|
|
*/
|
|
PQCLEAN_FALCON1024_CLEAN_FFT(rt3, logn);
|
|
PQCLEAN_FALCON1024_CLEAN_FFT(rt4, logn);
|
|
PQCLEAN_FALCON1024_CLEAN_poly_invnorm2_fft(rt5, rt3, rt4, logn);
|
|
PQCLEAN_FALCON1024_CLEAN_poly_adj_fft(rt3, logn);
|
|
PQCLEAN_FALCON1024_CLEAN_poly_adj_fft(rt4, logn);
|
|
|
|
/*
|
|
* Reduce F and G repeatedly.
|
|
*
|
|
* The expected maximum bit length of coefficients of F and G
|
|
* is kept in maxbl_FG, with the corresponding word length in
|
|
* FGlen.
|
|
*/
|
|
FGlen = llen;
|
|
maxbl_FG = 31 * (int)llen;
|
|
|
|
/*
|
|
* Each reduction operation computes the reduction polynomial
|
|
* "k". We need that polynomial to have coefficients that fit
|
|
* on 32-bit signed integers, with some scaling; thus, we use
|
|
* a descending sequence of scaling values, down to zero.
|
|
*
|
|
* The size of the coefficients of k is (roughly) the difference
|
|
* between the size of the coefficients of (F,G) and the size
|
|
* of the coefficients of (f,g). Thus, the maximum size of the
|
|
* coefficients of k is, at the start, maxbl_FG - minbl_fg;
|
|
* this is our starting scale value for k.
|
|
*
|
|
* We need to estimate the size of (F,G) during the execution of
|
|
* the algorithm; we are allowed some overestimation but not too
|
|
* much (poly_big_to_fp() uses a 310-bit window). Generally
|
|
* speaking, after applying a reduction with k scaled to
|
|
* scale_k, the size of (F,G) will be size(f,g) + scale_k + dd,
|
|
* where 'dd' is a few bits to account for the fact that the
|
|
* reduction is never perfect (intuitively, dd is on the order
|
|
* of sqrt(N), so at most 5 bits; we here allow for 10 extra
|
|
* bits).
|
|
*
|
|
* The size of (f,g) is not known exactly, but maxbl_fg is an
|
|
* upper bound.
|
|
*/
|
|
scale_k = maxbl_FG - minbl_fg;
|
|
|
|
for (;;) {
|
|
int scale_FG, dc, new_maxbl_FG;
|
|
uint32_t scl, sch;
|
|
fpr pdc, pt;
|
|
|
|
/*
|
|
* Convert current F and G into floating-point. We apply
|
|
* scaling if the current length is more than 10 words.
|
|
*/
|
|
rlen = FGlen;
|
|
if (rlen > 10) {
|
|
rlen = 10;
|
|
}
|
|
scale_FG = 31 * (int)(FGlen - rlen);
|
|
poly_big_to_fp(rt1, Ft + FGlen - rlen, rlen, llen, logn);
|
|
poly_big_to_fp(rt2, Gt + FGlen - rlen, rlen, llen, logn);
|
|
|
|
/*
|
|
* Compute (F*adj(f)+G*adj(g))/(f*adj(f)+g*adj(g)) in rt2.
|
|
*/
|
|
PQCLEAN_FALCON1024_CLEAN_FFT(rt1, logn);
|
|
PQCLEAN_FALCON1024_CLEAN_FFT(rt2, logn);
|
|
PQCLEAN_FALCON1024_CLEAN_poly_mul_fft(rt1, rt3, logn);
|
|
PQCLEAN_FALCON1024_CLEAN_poly_mul_fft(rt2, rt4, logn);
|
|
PQCLEAN_FALCON1024_CLEAN_poly_add(rt2, rt1, logn);
|
|
PQCLEAN_FALCON1024_CLEAN_poly_mul_autoadj_fft(rt2, rt5, logn);
|
|
PQCLEAN_FALCON1024_CLEAN_iFFT(rt2, logn);
|
|
|
|
/*
|
|
* (f,g) are scaled by 'scale_fg', meaning that the
|
|
* numbers in rt3/rt4 should be multiplied by 2^(scale_fg)
|
|
* to have their true mathematical value.
|
|
*
|
|
* (F,G) are similarly scaled by 'scale_FG'. Therefore,
|
|
* the value we computed in rt2 is scaled by
|
|
* 'scale_FG-scale_fg'.
|
|
*
|
|
* We want that value to be scaled by 'scale_k', hence we
|
|
* apply a corrective scaling. After scaling, the values
|
|
* should fit in -2^31-1..+2^31-1.
|
|
*/
|
|
dc = scale_k - scale_FG + scale_fg;
|
|
|
|
/*
|
|
* We will need to multiply values by 2^(-dc). The value
|
|
* 'dc' is not secret, so we can compute 2^(-dc) with a
|
|
* non-constant-time process.
|
|
* (We could use ldexp(), but we prefer to avoid any
|
|
* dependency on libm. When using FP emulation, we could
|
|
* use our fpr_ldexp(), which is constant-time.)
|
|
*/
|
|
if (dc < 0) {
|
|
dc = -dc;
|
|
pt = fpr_two;
|
|
} else {
|
|
pt = fpr_onehalf;
|
|
}
|
|
pdc = fpr_one;
|
|
while (dc != 0) {
|
|
if ((dc & 1) != 0) {
|
|
pdc = fpr_mul(pdc, pt);
|
|
}
|
|
dc >>= 1;
|
|
pt = fpr_sqr(pt);
|
|
}
|
|
|
|
for (u = 0; u < n; u ++) {
|
|
fpr xv;
|
|
|
|
xv = fpr_mul(rt2[u], pdc);
|
|
|
|
/*
|
|
* Sometimes the values can be out-of-bounds if
|
|
* the algorithm fails; we must not call
|
|
* fpr_rint() (and cast to int32_t) if the value
|
|
* is not in-bounds. Note that the test does not
|
|
* break constant-time discipline, since any
|
|
* failure here implies that we discard the current
|
|
* secret key (f,g).
|
|
*/
|
|
if (!fpr_lt(fpr_mtwo31m1, xv)
|
|
|| !fpr_lt(xv, fpr_ptwo31m1)) {
|
|
return 0;
|
|
}
|
|
k[u] = (int32_t)fpr_rint(xv);
|
|
}
|
|
|
|
/*
|
|
* Values in k[] are integers. They really are scaled
|
|
* down by maxbl_FG - minbl_fg bits.
|
|
*
|
|
* If we are at low depth, then we use the NTT to
|
|
* compute k*f and k*g.
|
|
*/
|
|
sch = (uint32_t)(scale_k / 31);
|
|
scl = (uint32_t)(scale_k % 31);
|
|
if (depth <= DEPTH_INT_FG) {
|
|
poly_sub_scaled_ntt(Ft, FGlen, llen, ft, slen, slen,
|
|
k, sch, scl, logn, t1);
|
|
poly_sub_scaled_ntt(Gt, FGlen, llen, gt, slen, slen,
|
|
k, sch, scl, logn, t1);
|
|
} else {
|
|
poly_sub_scaled(Ft, FGlen, llen, ft, slen, slen,
|
|
k, sch, scl, logn);
|
|
poly_sub_scaled(Gt, FGlen, llen, gt, slen, slen,
|
|
k, sch, scl, logn);
|
|
}
|
|
|
|
/*
|
|
* We compute the new maximum size of (F,G), assuming that
|
|
* (f,g) has _maximal_ length (i.e. that reduction is
|
|
* "late" instead of "early". We also adjust FGlen
|
|
* accordingly.
|
|
*/
|
|
new_maxbl_FG = scale_k + maxbl_fg + 10;
|
|
if (new_maxbl_FG < maxbl_FG) {
|
|
maxbl_FG = new_maxbl_FG;
|
|
if ((int)FGlen * 31 >= maxbl_FG + 31) {
|
|
FGlen --;
|
|
}
|
|
}
|
|
|
|
/*
|
|
* We suppose that scaling down achieves a reduction by
|
|
* at least 25 bits per iteration. We stop when we have
|
|
* done the loop with an unscaled k.
|
|
*/
|
|
if (scale_k <= 0) {
|
|
break;
|
|
}
|
|
scale_k -= 25;
|
|
if (scale_k < 0) {
|
|
scale_k = 0;
|
|
}
|
|
}
|
|
|
|
/*
|
|
* If (F,G) length was lowered below 'slen', then we must take
|
|
* care to re-extend the sign.
|
|
*/
|
|
if (FGlen < slen) {
|
|
for (u = 0; u < n; u ++, Ft += llen, Gt += llen) {
|
|
size_t v;
|
|
uint32_t sw;
|
|
|
|
sw = -(Ft[FGlen - 1] >> 30) >> 1;
|
|
for (v = FGlen; v < slen; v ++) {
|
|
Ft[v] = sw;
|
|
}
|
|
sw = -(Gt[FGlen - 1] >> 30) >> 1;
|
|
for (v = FGlen; v < slen; v ++) {
|
|
Gt[v] = sw;
|
|
}
|
|
}
|
|
}
|
|
|
|
/*
|
|
* Compress encoding of all values to 'slen' words (this is the
|
|
* expected output format).
|
|
*/
|
|
for (u = 0, x = tmp, y = tmp;
|
|
u < (n << 1); u ++, x += slen, y += llen) {
|
|
memmove(x, y, slen * sizeof * y);
|
|
}
|
|
return 1;
|
|
}
|
|
|
|
/*
|
|
* Solving the NTRU equation, binary case, depth = 1. Upon entry, the
|
|
* F and G from the previous level should be in the tmp[] array.
|
|
*
|
|
* Returned value: 1 on success, 0 on error.
|
|
*/
|
|
static int
|
|
solve_NTRU_binary_depth1(unsigned logn_top,
|
|
const int8_t *f, const int8_t *g, uint32_t *tmp) {
|
|
/*
|
|
* The first half of this function is a copy of the corresponding
|
|
* part in solve_NTRU_intermediate(), for the reconstruction of
|
|
* the unreduced F and G. The second half (Babai reduction) is
|
|
* done differently, because the unreduced F and G fit in 53 bits
|
|
* of precision, allowing a much simpler process with lower RAM
|
|
* usage.
|
|
*/
|
|
unsigned depth, logn;
|
|
size_t n_top, n, hn, slen, dlen, llen, u;
|
|
uint32_t *Fd, *Gd, *Ft, *Gt, *ft, *gt, *t1;
|
|
fpr *rt1, *rt2, *rt3, *rt4, *rt5, *rt6;
|
|
uint32_t *x, *y;
|
|
|
|
depth = 1;
|
|
n_top = (size_t)1 << logn_top;
|
|
logn = logn_top - depth;
|
|
n = (size_t)1 << logn;
|
|
hn = n >> 1;
|
|
|
|
/*
|
|
* Equations are:
|
|
*
|
|
* f' = f0^2 - X^2*f1^2
|
|
* g' = g0^2 - X^2*g1^2
|
|
* F' and G' are a solution to f'G' - g'F' = q (from deeper levels)
|
|
* F = F'*(g0 - X*g1)
|
|
* G = G'*(f0 - X*f1)
|
|
*
|
|
* f0, f1, g0, g1, f', g', F' and G' are all "compressed" to
|
|
* degree N/2 (their odd-indexed coefficients are all zero).
|
|
*/
|
|
|
|
/*
|
|
* slen = size for our input f and g; also size of the reduced
|
|
* F and G we return (degree N)
|
|
*
|
|
* dlen = size of the F and G obtained from the deeper level
|
|
* (degree N/2)
|
|
*
|
|
* llen = size for intermediary F and G before reduction (degree N)
|
|
*
|
|
* We build our non-reduced F and G as two independent halves each,
|
|
* of degree N/2 (F = F0 + X*F1, G = G0 + X*G1).
|
|
*/
|
|
slen = MAX_BL_SMALL[depth];
|
|
dlen = MAX_BL_SMALL[depth + 1];
|
|
llen = MAX_BL_LARGE[depth];
|
|
|
|
/*
|
|
* Fd and Gd are the F and G from the deeper level. Ft and Gt
|
|
* are the destination arrays for the unreduced F and G.
|
|
*/
|
|
Fd = tmp;
|
|
Gd = Fd + dlen * hn;
|
|
Ft = Gd + dlen * hn;
|
|
Gt = Ft + llen * n;
|
|
|
|
/*
|
|
* We reduce Fd and Gd modulo all the small primes we will need,
|
|
* and store the values in Ft and Gt.
|
|
*/
|
|
for (u = 0; u < llen; u ++) {
|
|
uint32_t p, p0i, R2, Rx;
|
|
size_t v;
|
|
uint32_t *xs, *ys, *xd, *yd;
|
|
|
|
p = PRIMES[u].p;
|
|
p0i = modp_ninv31(p);
|
|
R2 = modp_R2(p, p0i);
|
|
Rx = modp_Rx((unsigned)dlen, p, p0i, R2);
|
|
for (v = 0, xs = Fd, ys = Gd, xd = Ft + u, yd = Gt + u;
|
|
v < hn;
|
|
v ++, xs += dlen, ys += dlen, xd += llen, yd += llen) {
|
|
*xd = zint_mod_small_signed(xs, dlen, p, p0i, R2, Rx);
|
|
*yd = zint_mod_small_signed(ys, dlen, p, p0i, R2, Rx);
|
|
}
|
|
}
|
|
|
|
/*
|
|
* Now Fd and Gd are not needed anymore; we can squeeze them out.
|
|
*/
|
|
memmove(tmp, Ft, llen * n * sizeof(uint32_t));
|
|
Ft = tmp;
|
|
memmove(Ft + llen * n, Gt, llen * n * sizeof(uint32_t));
|
|
Gt = Ft + llen * n;
|
|
ft = Gt + llen * n;
|
|
gt = ft + slen * n;
|
|
|
|
t1 = gt + slen * n;
|
|
|
|
/*
|
|
* Compute our F and G modulo sufficiently many small primes.
|
|
*/
|
|
for (u = 0; u < llen; u ++) {
|
|
uint32_t p, p0i, R2;
|
|
uint32_t *gm, *igm, *fx, *gx, *Fp, *Gp;
|
|
unsigned e;
|
|
size_t v;
|
|
|
|
/*
|
|
* All computations are done modulo p.
|
|
*/
|
|
p = PRIMES[u].p;
|
|
p0i = modp_ninv31(p);
|
|
R2 = modp_R2(p, p0i);
|
|
|
|
/*
|
|
* We recompute things from the source f and g, of full
|
|
* degree. However, we will need only the n first elements
|
|
* of the inverse NTT table (igm); the call to modp_mkgm()
|
|
* below will fill n_top elements in igm[] (thus overflowing
|
|
* into fx[]) but later code will overwrite these extra
|
|
* elements.
|
|
*/
|
|
gm = t1;
|
|
igm = gm + n_top;
|
|
fx = igm + n;
|
|
gx = fx + n_top;
|
|
modp_mkgm2(gm, igm, logn_top, PRIMES[u].g, p, p0i);
|
|
|
|
/*
|
|
* Set ft and gt to f and g modulo p, respectively.
|
|
*/
|
|
for (v = 0; v < n_top; v ++) {
|
|
fx[v] = modp_set(f[v], p);
|
|
gx[v] = modp_set(g[v], p);
|
|
}
|
|
|
|
/*
|
|
* Convert to NTT and compute our f and g.
|
|
*/
|
|
modp_NTT2(fx, gm, logn_top, p, p0i);
|
|
modp_NTT2(gx, gm, logn_top, p, p0i);
|
|
for (e = logn_top; e > logn; e --) {
|
|
modp_poly_rec_res(fx, e, p, p0i, R2);
|
|
modp_poly_rec_res(gx, e, p, p0i, R2);
|
|
}
|
|
|
|
/*
|
|
* From that point onward, we only need tables for
|
|
* degree n, so we can save some space.
|
|
*/
|
|
if (depth > 0) { /* always true */
|
|
memmove(gm + n, igm, n * sizeof * igm);
|
|
igm = gm + n;
|
|
memmove(igm + n, fx, n * sizeof * ft);
|
|
fx = igm + n;
|
|
memmove(fx + n, gx, n * sizeof * gt);
|
|
gx = fx + n;
|
|
}
|
|
|
|
/*
|
|
* Get F' and G' modulo p and in NTT representation
|
|
* (they have degree n/2). These values were computed
|
|
* in a previous step, and stored in Ft and Gt.
|
|
*/
|
|
Fp = gx + n;
|
|
Gp = Fp + hn;
|
|
for (v = 0, x = Ft + u, y = Gt + u;
|
|
v < hn; v ++, x += llen, y += llen) {
|
|
Fp[v] = *x;
|
|
Gp[v] = *y;
|
|
}
|
|
modp_NTT2(Fp, gm, logn - 1, p, p0i);
|
|
modp_NTT2(Gp, gm, logn - 1, p, p0i);
|
|
|
|
/*
|
|
* Compute our F and G modulo p.
|
|
*
|
|
* Equations are:
|
|
*
|
|
* f'(x^2) = N(f)(x^2) = f * adj(f)
|
|
* g'(x^2) = N(g)(x^2) = g * adj(g)
|
|
*
|
|
* f'*G' - g'*F' = q
|
|
*
|
|
* F = F'(x^2) * adj(g)
|
|
* G = G'(x^2) * adj(f)
|
|
*
|
|
* The NTT representation of f is f(w) for all w which
|
|
* are roots of phi. In the binary case, as well as in
|
|
* the ternary case for all depth except the deepest,
|
|
* these roots can be grouped in pairs (w,-w), and we
|
|
* then have:
|
|
*
|
|
* f(w) = adj(f)(-w)
|
|
* f(-w) = adj(f)(w)
|
|
*
|
|
* and w^2 is then a root for phi at the half-degree.
|
|
*
|
|
* At the deepest level in the ternary case, this still
|
|
* holds, in the following sense: the roots of x^2-x+1
|
|
* are (w,-w^2) (for w^3 = -1, and w != -1), and we
|
|
* have:
|
|
*
|
|
* f(w) = adj(f)(-w^2)
|
|
* f(-w^2) = adj(f)(w)
|
|
*
|
|
* In all case, we can thus compute F and G in NTT
|
|
* representation by a few simple multiplications.
|
|
* Moreover, the two roots for each pair are consecutive
|
|
* in our bit-reversal encoding.
|
|
*/
|
|
for (v = 0, x = Ft + u, y = Gt + u;
|
|
v < hn; v ++, x += (llen << 1), y += (llen << 1)) {
|
|
uint32_t ftA, ftB, gtA, gtB;
|
|
uint32_t mFp, mGp;
|
|
|
|
ftA = fx[(v << 1) + 0];
|
|
ftB = fx[(v << 1) + 1];
|
|
gtA = gx[(v << 1) + 0];
|
|
gtB = gx[(v << 1) + 1];
|
|
mFp = modp_montymul(Fp[v], R2, p, p0i);
|
|
mGp = modp_montymul(Gp[v], R2, p, p0i);
|
|
x[0] = modp_montymul(gtB, mFp, p, p0i);
|
|
x[llen] = modp_montymul(gtA, mFp, p, p0i);
|
|
y[0] = modp_montymul(ftB, mGp, p, p0i);
|
|
y[llen] = modp_montymul(ftA, mGp, p, p0i);
|
|
}
|
|
modp_iNTT2_ext(Ft + u, llen, igm, logn, p, p0i);
|
|
modp_iNTT2_ext(Gt + u, llen, igm, logn, p, p0i);
|
|
|
|
/*
|
|
* Also save ft and gt (only up to size slen).
|
|
*/
|
|
if (u < slen) {
|
|
modp_iNTT2(fx, igm, logn, p, p0i);
|
|
modp_iNTT2(gx, igm, logn, p, p0i);
|
|
for (v = 0, x = ft + u, y = gt + u;
|
|
v < n; v ++, x += slen, y += slen) {
|
|
*x = fx[v];
|
|
*y = gx[v];
|
|
}
|
|
}
|
|
}
|
|
|
|
/*
|
|
* Rebuild f, g, F and G with the CRT. Note that the elements of F
|
|
* and G are consecutive, and thus can be rebuilt in a single
|
|
* loop; similarly, the elements of f and g are consecutive.
|
|
*/
|
|
zint_rebuild_CRT(Ft, llen, llen, n << 1, PRIMES, 1, t1);
|
|
zint_rebuild_CRT(ft, slen, slen, n << 1, PRIMES, 1, t1);
|
|
|
|
/*
|
|
* Here starts the Babai reduction, specialized for depth = 1.
|
|
*
|
|
* Candidates F and G (from Ft and Gt), and base f and g (ft and gt),
|
|
* are converted to floating point. There is no scaling, and a
|
|
* single pass is sufficient.
|
|
*/
|
|
|
|
/*
|
|
* Convert F and G into floating point (rt1 and rt2).
|
|
*/
|
|
rt1 = align_fpr(tmp, gt + slen * n);
|
|
rt2 = rt1 + n;
|
|
poly_big_to_fp(rt1, Ft, llen, llen, logn);
|
|
poly_big_to_fp(rt2, Gt, llen, llen, logn);
|
|
|
|
/*
|
|
* Integer representation of F and G is no longer needed, we
|
|
* can remove it.
|
|
*/
|
|
memmove(tmp, ft, 2 * slen * n * sizeof * ft);
|
|
ft = tmp;
|
|
gt = ft + slen * n;
|
|
rt3 = align_fpr(tmp, gt + slen * n);
|
|
memmove(rt3, rt1, 2 * n * sizeof * rt1);
|
|
rt1 = rt3;
|
|
rt2 = rt1 + n;
|
|
rt3 = rt2 + n;
|
|
rt4 = rt3 + n;
|
|
|
|
/*
|
|
* Convert f and g into floating point (rt3 and rt4).
|
|
*/
|
|
poly_big_to_fp(rt3, ft, slen, slen, logn);
|
|
poly_big_to_fp(rt4, gt, slen, slen, logn);
|
|
|
|
/*
|
|
* Remove unneeded ft and gt.
|
|
*/
|
|
memmove(tmp, rt1, 4 * n * sizeof * rt1);
|
|
rt1 = (fpr *)tmp;
|
|
rt2 = rt1 + n;
|
|
rt3 = rt2 + n;
|
|
rt4 = rt3 + n;
|
|
|
|
/*
|
|
* We now have:
|
|
* rt1 = F
|
|
* rt2 = G
|
|
* rt3 = f
|
|
* rt4 = g
|
|
* in that order in RAM. We convert all of them to FFT.
|
|
*/
|
|
PQCLEAN_FALCON1024_CLEAN_FFT(rt1, logn);
|
|
PQCLEAN_FALCON1024_CLEAN_FFT(rt2, logn);
|
|
PQCLEAN_FALCON1024_CLEAN_FFT(rt3, logn);
|
|
PQCLEAN_FALCON1024_CLEAN_FFT(rt4, logn);
|
|
|
|
/*
|
|
* Compute:
|
|
* rt5 = F*adj(f) + G*adj(g)
|
|
* rt6 = 1 / (f*adj(f) + g*adj(g))
|
|
* (Note that rt6 is half-length.)
|
|
*/
|
|
rt5 = rt4 + n;
|
|
rt6 = rt5 + n;
|
|
PQCLEAN_FALCON1024_CLEAN_poly_add_muladj_fft(rt5, rt1, rt2, rt3, rt4, logn);
|
|
PQCLEAN_FALCON1024_CLEAN_poly_invnorm2_fft(rt6, rt3, rt4, logn);
|
|
|
|
/*
|
|
* Compute:
|
|
* rt5 = (F*adj(f)+G*adj(g)) / (f*adj(f)+g*adj(g))
|
|
*/
|
|
PQCLEAN_FALCON1024_CLEAN_poly_mul_autoadj_fft(rt5, rt6, logn);
|
|
|
|
/*
|
|
* Compute k as the rounded version of rt5. Check that none of
|
|
* the values is larger than 2^63-1 (in absolute value)
|
|
* because that would make the fpr_rint() do something undefined;
|
|
* note that any out-of-bounds value here implies a failure and
|
|
* (f,g) will be discarded, so we can make a simple test.
|
|
*/
|
|
PQCLEAN_FALCON1024_CLEAN_iFFT(rt5, logn);
|
|
for (u = 0; u < n; u ++) {
|
|
fpr z;
|
|
|
|
z = rt5[u];
|
|
if (!fpr_lt(z, fpr_ptwo63m1) || !fpr_lt(fpr_mtwo63m1, z)) {
|
|
return 0;
|
|
}
|
|
rt5[u] = fpr_of(fpr_rint(z));
|
|
}
|
|
PQCLEAN_FALCON1024_CLEAN_FFT(rt5, logn);
|
|
|
|
/*
|
|
* Subtract k*f from F, and k*g from G.
|
|
*/
|
|
PQCLEAN_FALCON1024_CLEAN_poly_mul_fft(rt3, rt5, logn);
|
|
PQCLEAN_FALCON1024_CLEAN_poly_mul_fft(rt4, rt5, logn);
|
|
PQCLEAN_FALCON1024_CLEAN_poly_sub(rt1, rt3, logn);
|
|
PQCLEAN_FALCON1024_CLEAN_poly_sub(rt2, rt4, logn);
|
|
PQCLEAN_FALCON1024_CLEAN_iFFT(rt1, logn);
|
|
PQCLEAN_FALCON1024_CLEAN_iFFT(rt2, logn);
|
|
|
|
/*
|
|
* Convert back F and G to integers, and return.
|
|
*/
|
|
Ft = tmp;
|
|
Gt = Ft + n;
|
|
rt3 = align_fpr(tmp, Gt + n);
|
|
memmove(rt3, rt1, 2 * n * sizeof * rt1);
|
|
rt1 = rt3;
|
|
rt2 = rt1 + n;
|
|
for (u = 0; u < n; u ++) {
|
|
Ft[u] = (uint32_t)fpr_rint(rt1[u]);
|
|
Gt[u] = (uint32_t)fpr_rint(rt2[u]);
|
|
}
|
|
|
|
return 1;
|
|
}
|
|
|
|
/*
|
|
* Solving the NTRU equation, top level. Upon entry, the F and G
|
|
* from the previous level should be in the tmp[] array.
|
|
*
|
|
* Returned value: 1 on success, 0 on error.
|
|
*/
|
|
static int
|
|
solve_NTRU_binary_depth0(unsigned logn,
|
|
const int8_t *f, const int8_t *g, uint32_t *tmp) {
|
|
size_t n, hn, u;
|
|
uint32_t p, p0i, R2;
|
|
uint32_t *Fp, *Gp, *t1, *t2, *t3, *t4, *t5;
|
|
uint32_t *gm, *igm, *ft, *gt;
|
|
fpr *rt2, *rt3;
|
|
|
|
n = (size_t)1 << logn;
|
|
hn = n >> 1;
|
|
|
|
/*
|
|
* Equations are:
|
|
*
|
|
* f' = f0^2 - X^2*f1^2
|
|
* g' = g0^2 - X^2*g1^2
|
|
* F' and G' are a solution to f'G' - g'F' = q (from deeper levels)
|
|
* F = F'*(g0 - X*g1)
|
|
* G = G'*(f0 - X*f1)
|
|
*
|
|
* f0, f1, g0, g1, f', g', F' and G' are all "compressed" to
|
|
* degree N/2 (their odd-indexed coefficients are all zero).
|
|
*
|
|
* Everything should fit in 31-bit integers, hence we can just use
|
|
* the first small prime p = 2147473409.
|
|
*/
|
|
p = PRIMES[0].p;
|
|
p0i = modp_ninv31(p);
|
|
R2 = modp_R2(p, p0i);
|
|
|
|
Fp = tmp;
|
|
Gp = Fp + hn;
|
|
ft = Gp + hn;
|
|
gt = ft + n;
|
|
gm = gt + n;
|
|
igm = gm + n;
|
|
|
|
modp_mkgm2(gm, igm, logn, PRIMES[0].g, p, p0i);
|
|
|
|
/*
|
|
* Convert F' anf G' in NTT representation.
|
|
*/
|
|
for (u = 0; u < hn; u ++) {
|
|
Fp[u] = modp_set(zint_one_to_plain(Fp + u), p);
|
|
Gp[u] = modp_set(zint_one_to_plain(Gp + u), p);
|
|
}
|
|
modp_NTT2(Fp, gm, logn - 1, p, p0i);
|
|
modp_NTT2(Gp, gm, logn - 1, p, p0i);
|
|
|
|
/*
|
|
* Load f and g and convert them to NTT representation.
|
|
*/
|
|
for (u = 0; u < n; u ++) {
|
|
ft[u] = modp_set(f[u], p);
|
|
gt[u] = modp_set(g[u], p);
|
|
}
|
|
modp_NTT2(ft, gm, logn, p, p0i);
|
|
modp_NTT2(gt, gm, logn, p, p0i);
|
|
|
|
/*
|
|
* Build the unreduced F,G in ft and gt.
|
|
*/
|
|
for (u = 0; u < n; u += 2) {
|
|
uint32_t ftA, ftB, gtA, gtB;
|
|
uint32_t mFp, mGp;
|
|
|
|
ftA = ft[u + 0];
|
|
ftB = ft[u + 1];
|
|
gtA = gt[u + 0];
|
|
gtB = gt[u + 1];
|
|
mFp = modp_montymul(Fp[u >> 1], R2, p, p0i);
|
|
mGp = modp_montymul(Gp[u >> 1], R2, p, p0i);
|
|
ft[u + 0] = modp_montymul(gtB, mFp, p, p0i);
|
|
ft[u + 1] = modp_montymul(gtA, mFp, p, p0i);
|
|
gt[u + 0] = modp_montymul(ftB, mGp, p, p0i);
|
|
gt[u + 1] = modp_montymul(ftA, mGp, p, p0i);
|
|
}
|
|
modp_iNTT2(ft, igm, logn, p, p0i);
|
|
modp_iNTT2(gt, igm, logn, p, p0i);
|
|
|
|
Gp = Fp + n;
|
|
t1 = Gp + n;
|
|
memmove(Fp, ft, 2 * n * sizeof * ft);
|
|
|
|
/*
|
|
* We now need to apply the Babai reduction. At that point,
|
|
* we have F and G in two n-word arrays.
|
|
*
|
|
* We can compute F*adj(f)+G*adj(g) and f*adj(f)+g*adj(g)
|
|
* modulo p, using the NTT. We still move memory around in
|
|
* order to save RAM.
|
|
*/
|
|
t2 = t1 + n;
|
|
t3 = t2 + n;
|
|
t4 = t3 + n;
|
|
t5 = t4 + n;
|
|
|
|
/*
|
|
* Compute the NTT tables in t1 and t2. We do not keep t2
|
|
* (we'll recompute it later on).
|
|
*/
|
|
modp_mkgm2(t1, t2, logn, PRIMES[0].g, p, p0i);
|
|
|
|
/*
|
|
* Convert F and G to NTT.
|
|
*/
|
|
modp_NTT2(Fp, t1, logn, p, p0i);
|
|
modp_NTT2(Gp, t1, logn, p, p0i);
|
|
|
|
/*
|
|
* Load f and adj(f) in t4 and t5, and convert them to NTT
|
|
* representation.
|
|
*/
|
|
t4[0] = t5[0] = modp_set(f[0], p);
|
|
for (u = 1; u < n; u ++) {
|
|
t4[u] = modp_set(f[u], p);
|
|
t5[n - u] = modp_set(-f[u], p);
|
|
}
|
|
modp_NTT2(t4, t1, logn, p, p0i);
|
|
modp_NTT2(t5, t1, logn, p, p0i);
|
|
|
|
/*
|
|
* Compute F*adj(f) in t2, and f*adj(f) in t3.
|
|
*/
|
|
for (u = 0; u < n; u ++) {
|
|
uint32_t w;
|
|
|
|
w = modp_montymul(t5[u], R2, p, p0i);
|
|
t2[u] = modp_montymul(w, Fp[u], p, p0i);
|
|
t3[u] = modp_montymul(w, t4[u], p, p0i);
|
|
}
|
|
|
|
/*
|
|
* Load g and adj(g) in t4 and t5, and convert them to NTT
|
|
* representation.
|
|
*/
|
|
t4[0] = t5[0] = modp_set(g[0], p);
|
|
for (u = 1; u < n; u ++) {
|
|
t4[u] = modp_set(g[u], p);
|
|
t5[n - u] = modp_set(-g[u], p);
|
|
}
|
|
modp_NTT2(t4, t1, logn, p, p0i);
|
|
modp_NTT2(t5, t1, logn, p, p0i);
|
|
|
|
/*
|
|
* Add G*adj(g) to t2, and g*adj(g) to t3.
|
|
*/
|
|
for (u = 0; u < n; u ++) {
|
|
uint32_t w;
|
|
|
|
w = modp_montymul(t5[u], R2, p, p0i);
|
|
t2[u] = modp_add(t2[u],
|
|
modp_montymul(w, Gp[u], p, p0i), p);
|
|
t3[u] = modp_add(t3[u],
|
|
modp_montymul(w, t4[u], p, p0i), p);
|
|
}
|
|
|
|
/*
|
|
* Convert back t2 and t3 to normal representation (normalized
|
|
* around 0), and then
|
|
* move them to t1 and t2. We first need to recompute the
|
|
* inverse table for NTT.
|
|
*/
|
|
modp_mkgm2(t1, t4, logn, PRIMES[0].g, p, p0i);
|
|
modp_iNTT2(t2, t4, logn, p, p0i);
|
|
modp_iNTT2(t3, t4, logn, p, p0i);
|
|
for (u = 0; u < n; u ++) {
|
|
t1[u] = (uint32_t)modp_norm(t2[u], p);
|
|
t2[u] = (uint32_t)modp_norm(t3[u], p);
|
|
}
|
|
|
|
/*
|
|
* At that point, array contents are:
|
|
*
|
|
* F (NTT representation) (Fp)
|
|
* G (NTT representation) (Gp)
|
|
* F*adj(f)+G*adj(g) (t1)
|
|
* f*adj(f)+g*adj(g) (t2)
|
|
*
|
|
* We want to divide t1 by t2. The result is not integral; it
|
|
* must be rounded. We thus need to use the FFT.
|
|
*/
|
|
|
|
/*
|
|
* Get f*adj(f)+g*adj(g) in FFT representation. Since this
|
|
* polynomial is auto-adjoint, all its coordinates in FFT
|
|
* representation are actually real, so we can truncate off
|
|
* the imaginary parts.
|
|
*/
|
|
rt3 = align_fpr(tmp, t3);
|
|
for (u = 0; u < n; u ++) {
|
|
rt3[u] = fpr_of(((int32_t *)t2)[u]);
|
|
}
|
|
PQCLEAN_FALCON1024_CLEAN_FFT(rt3, logn);
|
|
rt2 = align_fpr(tmp, t2);
|
|
memmove(rt2, rt3, hn * sizeof * rt3);
|
|
|
|
/*
|
|
* Convert F*adj(f)+G*adj(g) in FFT representation.
|
|
*/
|
|
rt3 = rt2 + hn;
|
|
for (u = 0; u < n; u ++) {
|
|
rt3[u] = fpr_of(((int32_t *)t1)[u]);
|
|
}
|
|
PQCLEAN_FALCON1024_CLEAN_FFT(rt3, logn);
|
|
|
|
/*
|
|
* Compute (F*adj(f)+G*adj(g))/(f*adj(f)+g*adj(g)) and get
|
|
* its rounded normal representation in t1.
|
|
*/
|
|
PQCLEAN_FALCON1024_CLEAN_poly_div_autoadj_fft(rt3, rt2, logn);
|
|
PQCLEAN_FALCON1024_CLEAN_iFFT(rt3, logn);
|
|
for (u = 0; u < n; u ++) {
|
|
t1[u] = modp_set((int32_t)fpr_rint(rt3[u]), p);
|
|
}
|
|
|
|
/*
|
|
* RAM contents are now:
|
|
*
|
|
* F (NTT representation) (Fp)
|
|
* G (NTT representation) (Gp)
|
|
* k (t1)
|
|
*
|
|
* We want to compute F-k*f, and G-k*g.
|
|
*/
|
|
t2 = t1 + n;
|
|
t3 = t2 + n;
|
|
t4 = t3 + n;
|
|
t5 = t4 + n;
|
|
modp_mkgm2(t2, t3, logn, PRIMES[0].g, p, p0i);
|
|
for (u = 0; u < n; u ++) {
|
|
t4[u] = modp_set(f[u], p);
|
|
t5[u] = modp_set(g[u], p);
|
|
}
|
|
modp_NTT2(t1, t2, logn, p, p0i);
|
|
modp_NTT2(t4, t2, logn, p, p0i);
|
|
modp_NTT2(t5, t2, logn, p, p0i);
|
|
for (u = 0; u < n; u ++) {
|
|
uint32_t kw;
|
|
|
|
kw = modp_montymul(t1[u], R2, p, p0i);
|
|
Fp[u] = modp_sub(Fp[u],
|
|
modp_montymul(kw, t4[u], p, p0i), p);
|
|
Gp[u] = modp_sub(Gp[u],
|
|
modp_montymul(kw, t5[u], p, p0i), p);
|
|
}
|
|
modp_iNTT2(Fp, t3, logn, p, p0i);
|
|
modp_iNTT2(Gp, t3, logn, p, p0i);
|
|
for (u = 0; u < n; u ++) {
|
|
Fp[u] = (uint32_t)modp_norm(Fp[u], p);
|
|
Gp[u] = (uint32_t)modp_norm(Gp[u], p);
|
|
}
|
|
|
|
return 1;
|
|
}
|
|
|
|
/*
|
|
* Solve the NTRU equation. Returned value is 1 on success, 0 on error.
|
|
* G can be NULL, in which case that value is computed but not returned.
|
|
* If any of the coefficients of F and G exceeds lim (in absolute value),
|
|
* then 0 is returned.
|
|
*/
|
|
static int
|
|
solve_NTRU(unsigned logn, int8_t *F, int8_t *G,
|
|
const int8_t *f, const int8_t *g, int lim, uint32_t *tmp) {
|
|
size_t n, u;
|
|
uint32_t *ft, *gt, *Ft, *Gt, *gm;
|
|
uint32_t p, p0i, r;
|
|
const small_prime *primes;
|
|
|
|
n = MKN(logn);
|
|
|
|
if (!solve_NTRU_deepest(logn, f, g, tmp)) {
|
|
return 0;
|
|
}
|
|
|
|
/*
|
|
* For logn <= 2, we need to use solve_NTRU_intermediate()
|
|
* directly, because coefficients are a bit too large and
|
|
* do not fit the hypotheses in solve_NTRU_binary_depth0().
|
|
*/
|
|
if (logn <= 2) {
|
|
unsigned depth;
|
|
|
|
depth = logn;
|
|
while (depth -- > 0) {
|
|
if (!solve_NTRU_intermediate(logn, f, g, depth, tmp)) {
|
|
return 0;
|
|
}
|
|
}
|
|
} else {
|
|
unsigned depth;
|
|
|
|
depth = logn;
|
|
while (depth -- > 2) {
|
|
if (!solve_NTRU_intermediate(logn, f, g, depth, tmp)) {
|
|
return 0;
|
|
}
|
|
}
|
|
if (!solve_NTRU_binary_depth1(logn, f, g, tmp)) {
|
|
return 0;
|
|
}
|
|
if (!solve_NTRU_binary_depth0(logn, f, g, tmp)) {
|
|
return 0;
|
|
}
|
|
}
|
|
|
|
/*
|
|
* If no buffer has been provided for G, use a temporary one.
|
|
*/
|
|
if (G == NULL) {
|
|
G = (int8_t *)(tmp + 2 * n);
|
|
}
|
|
|
|
/*
|
|
* Final F and G are in fk->tmp, one word per coefficient
|
|
* (signed value over 31 bits).
|
|
*/
|
|
if (!poly_big_to_small(F, tmp, lim, logn)
|
|
|| !poly_big_to_small(G, tmp + n, lim, logn)) {
|
|
return 0;
|
|
}
|
|
|
|
/*
|
|
* Verify that the NTRU equation is fulfilled. Since all elements
|
|
* have short lengths, verifying modulo a small prime p works, and
|
|
* allows using the NTT.
|
|
*
|
|
* We put Gt[] first in tmp[], and process it first, so that it does
|
|
* not overlap with G[] in case we allocated it ourselves.
|
|
*/
|
|
Gt = tmp;
|
|
ft = Gt + n;
|
|
gt = ft + n;
|
|
Ft = gt + n;
|
|
gm = Ft + n;
|
|
|
|
primes = PRIMES;
|
|
p = primes[0].p;
|
|
p0i = modp_ninv31(p);
|
|
modp_mkgm2(gm, tmp, logn, primes[0].g, p, p0i);
|
|
for (u = 0; u < n; u ++) {
|
|
Gt[u] = modp_set(G[u], p);
|
|
}
|
|
for (u = 0; u < n; u ++) {
|
|
ft[u] = modp_set(f[u], p);
|
|
gt[u] = modp_set(g[u], p);
|
|
Ft[u] = modp_set(F[u], p);
|
|
}
|
|
modp_NTT2(ft, gm, logn, p, p0i);
|
|
modp_NTT2(gt, gm, logn, p, p0i);
|
|
modp_NTT2(Ft, gm, logn, p, p0i);
|
|
modp_NTT2(Gt, gm, logn, p, p0i);
|
|
r = modp_montymul(12289, 1, p, p0i);
|
|
for (u = 0; u < n; u ++) {
|
|
uint32_t z;
|
|
|
|
z = modp_sub(modp_montymul(ft[u], Gt[u], p, p0i),
|
|
modp_montymul(gt[u], Ft[u], p, p0i), p);
|
|
if (z != r) {
|
|
return 0;
|
|
}
|
|
}
|
|
|
|
return 1;
|
|
}
|
|
|
|
/*
|
|
* Generate a random polynomial with a Gaussian distribution. This function
|
|
* also makes sure that the resultant of the polynomial with phi is odd.
|
|
*/
|
|
static void
|
|
poly_small_mkgauss(RNG_CONTEXT *rng, int8_t *f, unsigned logn) {
|
|
size_t n, u;
|
|
unsigned mod2;
|
|
|
|
n = MKN(logn);
|
|
mod2 = 0;
|
|
for (u = 0; u < n; u ++) {
|
|
int s;
|
|
|
|
restart:
|
|
s = mkgauss(rng, logn);
|
|
|
|
/*
|
|
* We need the coefficient to fit within -127..+127;
|
|
* realistically, this is always the case except for
|
|
* the very low degrees (N = 2 or 4), for which there
|
|
* is no real security anyway.
|
|
*/
|
|
if (s < -127 || s > 127) {
|
|
goto restart;
|
|
}
|
|
|
|
/*
|
|
* We need the sum of all coefficients to be 1; otherwise,
|
|
* the resultant of the polynomial with X^N+1 will be even,
|
|
* and the binary GCD will fail.
|
|
*/
|
|
if (u == n - 1) {
|
|
if ((mod2 ^ (unsigned)(s & 1)) == 0) {
|
|
goto restart;
|
|
}
|
|
} else {
|
|
mod2 ^= (unsigned)(s & 1);
|
|
}
|
|
f[u] = (int8_t)s;
|
|
}
|
|
}
|
|
|
|
/* see falcon.h */
|
|
void
|
|
PQCLEAN_FALCON1024_CLEAN_keygen(inner_shake256_context *rng,
|
|
int8_t *f, int8_t *g, int8_t *F, int8_t *G, uint16_t *h,
|
|
unsigned logn, uint8_t *tmp) {
|
|
/*
|
|
* Algorithm is the following:
|
|
*
|
|
* - Generate f and g with the Gaussian distribution.
|
|
*
|
|
* - If either Res(f,phi) or Res(g,phi) is even, try again.
|
|
*
|
|
* - If ||(f,g)|| is too large, try again.
|
|
*
|
|
* - If ||B~_{f,g}|| is too large, try again.
|
|
*
|
|
* - If f is not invertible mod phi mod q, try again.
|
|
*
|
|
* - Compute h = g/f mod phi mod q.
|
|
*
|
|
* - Solve the NTRU equation fG - gF = q; if the solving fails,
|
|
* try again. Usual failure condition is when Res(f,phi)
|
|
* and Res(g,phi) are not prime to each other.
|
|
*/
|
|
size_t n, u;
|
|
uint16_t *h2, *tmp2;
|
|
RNG_CONTEXT *rc;
|
|
|
|
n = MKN(logn);
|
|
rc = rng;
|
|
|
|
/*
|
|
* We need to generate f and g randomly, until we find values
|
|
* such that the norm of (g,-f), and of the orthogonalized
|
|
* vector, are satisfying. The orthogonalized vector is:
|
|
* (q*adj(f)/(f*adj(f)+g*adj(g)), q*adj(g)/(f*adj(f)+g*adj(g)))
|
|
* (it is actually the (N+1)-th row of the Gram-Schmidt basis).
|
|
*
|
|
* In the binary case, coefficients of f and g are generated
|
|
* independently of each other, with a discrete Gaussian
|
|
* distribution of standard deviation 1.17*sqrt(q/(2*N)). Then,
|
|
* the two vectors have expected norm 1.17*sqrt(q), which is
|
|
* also our acceptance bound: we require both vectors to be no
|
|
* larger than that (this will be satisfied about 1/4th of the
|
|
* time, thus we expect sampling new (f,g) about 4 times for that
|
|
* step).
|
|
*
|
|
* We require that Res(f,phi) and Res(g,phi) are both odd (the
|
|
* NTRU equation solver requires it).
|
|
*/
|
|
for (;;) {
|
|
fpr *rt1, *rt2, *rt3;
|
|
fpr bnorm;
|
|
uint32_t normf, normg, norm;
|
|
int lim;
|
|
|
|
/*
|
|
* The poly_small_mkgauss() function makes sure
|
|
* that the sum of coefficients is 1 modulo 2
|
|
* (i.e. the resultant of the polynomial with phi
|
|
* will be odd).
|
|
*/
|
|
poly_small_mkgauss(rc, f, logn);
|
|
poly_small_mkgauss(rc, g, logn);
|
|
|
|
/*
|
|
* Verify that all coefficients are within the bounds
|
|
* defined in max_fg_bits. This is the case with
|
|
* overwhelming probability; this guarantees that the
|
|
* key will be encodable with FALCON_COMP_TRIM.
|
|
*/
|
|
lim = 1 << (PQCLEAN_FALCON1024_CLEAN_max_fg_bits[logn] - 1);
|
|
for (u = 0; u < n; u ++) {
|
|
/*
|
|
* We can use non-CT tests since on any failure
|
|
* we will discard f and g.
|
|
*/
|
|
if (f[u] >= lim || f[u] <= -lim
|
|
|| g[u] >= lim || g[u] <= -lim) {
|
|
lim = -1;
|
|
break;
|
|
}
|
|
}
|
|
if (lim < 0) {
|
|
continue;
|
|
}
|
|
|
|
/*
|
|
* Bound is 1.17*sqrt(q). We compute the squared
|
|
* norms. With q = 12289, the squared bound is:
|
|
* (1.17^2)* 12289 = 16822.4121
|
|
* Since f and g are integral, the squared norm
|
|
* of (g,-f) is an integer.
|
|
*/
|
|
normf = poly_small_sqnorm(f, logn);
|
|
normg = poly_small_sqnorm(g, logn);
|
|
norm = (normf + normg) | -((normf | normg) >> 31);
|
|
if (norm >= 16823) {
|
|
continue;
|
|
}
|
|
|
|
/*
|
|
* We compute the orthogonalized vector norm.
|
|
*/
|
|
rt1 = (fpr *)tmp;
|
|
rt2 = rt1 + n;
|
|
rt3 = rt2 + n;
|
|
poly_small_to_fp(rt1, f, logn);
|
|
poly_small_to_fp(rt2, g, logn);
|
|
PQCLEAN_FALCON1024_CLEAN_FFT(rt1, logn);
|
|
PQCLEAN_FALCON1024_CLEAN_FFT(rt2, logn);
|
|
PQCLEAN_FALCON1024_CLEAN_poly_invnorm2_fft(rt3, rt1, rt2, logn);
|
|
PQCLEAN_FALCON1024_CLEAN_poly_adj_fft(rt1, logn);
|
|
PQCLEAN_FALCON1024_CLEAN_poly_adj_fft(rt2, logn);
|
|
PQCLEAN_FALCON1024_CLEAN_poly_mulconst(rt1, fpr_q, logn);
|
|
PQCLEAN_FALCON1024_CLEAN_poly_mulconst(rt2, fpr_q, logn);
|
|
PQCLEAN_FALCON1024_CLEAN_poly_mul_autoadj_fft(rt1, rt3, logn);
|
|
PQCLEAN_FALCON1024_CLEAN_poly_mul_autoadj_fft(rt2, rt3, logn);
|
|
PQCLEAN_FALCON1024_CLEAN_iFFT(rt1, logn);
|
|
PQCLEAN_FALCON1024_CLEAN_iFFT(rt2, logn);
|
|
bnorm = fpr_zero;
|
|
for (u = 0; u < n; u ++) {
|
|
bnorm = fpr_add(bnorm, fpr_sqr(rt1[u]));
|
|
bnorm = fpr_add(bnorm, fpr_sqr(rt2[u]));
|
|
}
|
|
if (!fpr_lt(bnorm, fpr_bnorm_max)) {
|
|
continue;
|
|
}
|
|
|
|
/*
|
|
* Compute public key h = g/f mod X^N+1 mod q. If this
|
|
* fails, we must restart.
|
|
*/
|
|
if (h == NULL) {
|
|
h2 = (uint16_t *)tmp;
|
|
tmp2 = h2 + n;
|
|
} else {
|
|
h2 = h;
|
|
tmp2 = (uint16_t *)tmp;
|
|
}
|
|
if (!PQCLEAN_FALCON1024_CLEAN_compute_public(h2, f, g, logn, (uint8_t *)tmp2)) {
|
|
continue;
|
|
}
|
|
|
|
/*
|
|
* Solve the NTRU equation to get F and G.
|
|
*/
|
|
lim = (1 << (PQCLEAN_FALCON1024_CLEAN_max_FG_bits[logn] - 1)) - 1;
|
|
if (!solve_NTRU(logn, F, G, f, g, lim, (uint32_t *)tmp)) {
|
|
continue;
|
|
}
|
|
|
|
/*
|
|
* Key pair is generated.
|
|
*/
|
|
break;
|
|
}
|
|
}
|