diff options
Diffstat (limited to 'lib/liboqs/src/sig/falcon/pqclean_falcon-512_clean/keygen.c')
-rw-r--r-- | lib/liboqs/src/sig/falcon/pqclean_falcon-512_clean/keygen.c | 4231 |
1 files changed, 0 insertions, 4231 deletions
diff --git a/lib/liboqs/src/sig/falcon/pqclean_falcon-512_clean/keygen.c b/lib/liboqs/src/sig/falcon/pqclean_falcon-512_clean/keygen.c deleted file mode 100644 index f72ecd991..000000000 --- a/lib/liboqs/src/sig/falcon/pqclean_falcon-512_clean/keygen.c +++ /dev/null @@ -1,4231 +0,0 @@ -#include "inner.h" - -/* - * Falcon key pair generation. - * - * ==========================(LICENSE BEGIN)============================ - * - * Copyright (c) 2017-2019 Falcon Project - * - * Permission is hereby granted, free of charge, to any person obtaining - * a copy of this software and associated documentation files (the - * "Software"), to deal in the Software without restriction, including - * without limitation the rights to use, copy, modify, merge, publish, - * distribute, sublicense, and/or sell copies of the Software, and to - * permit persons to whom the Software is furnished to do so, subject to - * the following conditions: - * - * The above copyright notice and this permission notice shall be - * included in all copies or substantial portions of the Software. - * - * THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, - * EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF - * MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. - * IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY - * CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, - * TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE - * SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE. - * - * ===========================(LICENSE END)============================= - * - * @author Thomas Pornin <thomas.pornin@nccgroup.com> - */ - - -#define MKN(logn) ((size_t)1 << (logn)) - -/* ==================================================================== */ -/* - * Modular arithmetics. - * - * We implement a few functions for computing modulo a small integer p. - * - * All functions require that 2^30 < p < 2^31. Moreover, operands must - * be in the 0..p-1 range. - * - * Modular addition and subtraction work for all such p. - * - * Montgomery multiplication requires that p is odd, and must be provided - * with an additional value p0i = -1/p mod 2^31. See below for some basics - * on Montgomery multiplication. - * - * Division computes an inverse modulo p by an exponentiation (with - * exponent p-2): this works only if p is prime. Multiplication - * requirements also apply, i.e. p must be odd and p0i must be provided. - * - * The NTT and inverse NTT need all of the above, and also that - * p = 1 mod 2048. - * - * ----------------------------------------------------------------------- - * - * We use Montgomery representation with 31-bit values: - * - * Let R = 2^31 mod p. When 2^30 < p < 2^31, R = 2^31 - p. - * Montgomery representation of an integer x modulo p is x*R mod p. - * - * Montgomery multiplication computes (x*y)/R mod p for - * operands x and y. Therefore: - * - * - if operands are x*R and y*R (Montgomery representations of x and - * y), then Montgomery multiplication computes (x*R*y*R)/R = (x*y)*R - * mod p, which is the Montgomery representation of the product x*y; - * - * - if operands are x*R and y (or x and y*R), then Montgomery - * multiplication returns x*y mod p: mixed-representation - * multiplications yield results in normal representation. - * - * To convert to Montgomery representation, we multiply by R, which is done - * by Montgomery-multiplying by R^2. Stand-alone conversion back from - * Montgomery representation is Montgomery-multiplication by 1. - */ - -/* - * Precomputed small primes. Each element contains the following: - * - * p The prime itself. - * - * g A primitive root of phi = X^N+1 (in field Z_p). - * - * s The inverse of the product of all previous primes in the array, - * computed modulo p and in Montgomery representation. - * - * All primes are such that p = 1 mod 2048, and are lower than 2^31. They - * are listed in decreasing order. - */ - -typedef struct { - uint32_t p; - uint32_t g; - uint32_t s; -} small_prime; - -static const small_prime PRIMES[] = { - { 2147473409, 383167813, 10239 }, - { 2147389441, 211808905, 471403745 }, - { 2147387393, 37672282, 1329335065 }, - { 2147377153, 1977035326, 968223422 }, - { 2147358721, 1067163706, 132460015 }, - { 2147352577, 1606082042, 598693809 }, - { 2147346433, 2033915641, 1056257184 }, - { 2147338241, 1653770625, 421286710 }, - { 2147309569, 631200819, 1111201074 }, - { 2147297281, 2038364663, 1042003613 }, - { 2147295233, 1962540515, 19440033 }, - { 2147239937, 2100082663, 353296760 }, - { 2147235841, 1991153006, 1703918027 }, - { 2147217409, 516405114, 1258919613 }, - { 2147205121, 409347988, 1089726929 }, - { 2147196929, 927788991, 1946238668 }, - { 2147178497, 1136922411, 1347028164 }, - { 2147100673, 868626236, 701164723 }, - { 2147082241, 1897279176, 617820870 }, - { 2147074049, 1888819123, 158382189 }, - { 2147051521, 25006327, 522758543 }, - { 2147043329, 327546255, 37227845 }, - { 2147039233, 766324424, 1133356428 }, - { 2146988033, 1862817362, 73861329 }, - { 2146963457, 404622040, 653019435 }, - { 2146959361, 1936581214, 995143093 }, - { 2146938881, 1559770096, 634921513 }, - { 2146908161, 422623708, 1985060172 }, - { 2146885633, 1751189170, 298238186 }, - { 2146871297, 578919515, 291810829 }, - { 2146846721, 1114060353, 915902322 }, - { 2146834433, 2069565474, 47859524 }, - { 2146818049, 1552824584, 646281055 }, - { 2146775041, 1906267847, 1597832891 }, - { 2146756609, 1847414714, 1228090888 }, - { 2146744321, 1818792070, 1176377637 }, - { 2146738177, 1118066398, 1054971214 }, - { 2146736129, 52057278, 933422153 }, - { 2146713601, 592259376, 1406621510 }, - { 2146695169, 263161877, 1514178701 }, - { 2146656257, 685363115, 384505091 }, - { 2146650113, 927727032, 537575289 }, - { 2146646017, 52575506, 1799464037 }, - { 2146643969, 1276803876, 1348954416 }, - { 2146603009, 814028633, 1521547704 }, - { 2146572289, 1846678872, 1310832121 }, - { 2146547713, 919368090, 1019041349 }, - { 2146508801, 671847612, 38582496 }, - { 2146492417, 283911680, 532424562 }, - { 2146490369, 1780044827, 896447978 }, - { 2146459649, 327980850, 1327906900 }, - { 2146447361, 1310561493, 958645253 }, - { 2146441217, 412148926, 287271128 }, - { 2146437121, 293186449, 2009822534 }, - { 2146430977, 179034356, 1359155584 }, - { 2146418689, 1517345488, 1790248672 }, - { 2146406401, 1615820390, 1584833571 }, - { 2146404353, 826651445, 607120498 }, - { 2146379777, 3816988, 1897049071 }, - { 2146363393, 1221409784, 1986921567 }, - { 2146355201, 1388081168, 849968120 }, - { 2146336769, 1803473237, 1655544036 }, - { 2146312193, 1023484977, 273671831 }, - { 2146293761, 1074591448, 467406983 }, - { 2146283521, 831604668, 1523950494 }, - { 2146203649, 712865423, 1170834574 }, - { 2146154497, 1764991362, 1064856763 }, - { 2146142209, 627386213, 1406840151 }, - { 2146127873, 1638674429, 2088393537 }, - { 2146099201, 1516001018, 690673370 }, - { 2146093057, 1294931393, 315136610 }, - { 2146091009, 1942399533, 973539425 }, - { 2146078721, 1843461814, 2132275436 }, - { 2146060289, 1098740778, 360423481 }, - { 2146048001, 1617213232, 1951981294 }, - { 2146041857, 1805783169, 2075683489 }, - { 2146019329, 272027909, 1753219918 }, - { 2145986561, 1206530344, 2034028118 }, - { 2145976321, 1243769360, 1173377644 }, - { 2145964033, 887200839, 1281344586 }, - { 2145906689, 1651026455, 906178216 }, - { 2145875969, 1673238256, 1043521212 }, - { 2145871873, 1226591210, 1399796492 }, - { 2145841153, 1465353397, 1324527802 }, - { 2145832961, 1150638905, 554084759 }, - { 2145816577, 221601706, 427340863 }, - { 2145785857, 608896761, 316590738 }, - { 2145755137, 1712054942, 1684294304 }, - { 2145742849, 1302302867, 724873116 }, - { 2145728513, 516717693, 431671476 }, - { 2145699841, 524575579, 1619722537 }, - { 2145691649, 1925625239, 982974435 }, - { 2145687553, 463795662, 1293154300 }, - { 2145673217, 771716636, 881778029 }, - { 2145630209, 1509556977, 837364988 }, - { 2145595393, 229091856, 851648427 }, - { 2145587201, 1796903241, 635342424 }, - { 2145525761, 715310882, 1677228081 }, - { 2145495041, 1040930522, 200685896 }, - { 2145466369, 949804237, 1809146322 }, - { 2145445889, 1673903706, 95316881 }, - { 2145390593, 806941852, 1428671135 }, - { 2145372161, 1402525292, 159350694 }, - { 2145361921, 2124760298, 1589134749 }, - { 2145359873, 1217503067, 1561543010 }, - { 2145355777, 338341402, 83865711 }, - { 2145343489, 1381532164, 641430002 }, - { 2145325057, 1883895478, 1528469895 }, - { 2145318913, 1335370424, 65809740 }, - { 2145312769, 2000008042, 1919775760 }, - { 2145300481, 961450962, 1229540578 }, - { 2145282049, 910466767, 1964062701 }, - { 2145232897, 816527501, 450152063 }, - { 2145218561, 1435128058, 1794509700 }, - { 2145187841, 33505311, 1272467582 }, - { 2145181697, 269767433, 1380363849 }, - { 2145175553, 56386299, 1316870546 }, - { 2145079297, 2106880293, 1391797340 }, - { 2145021953, 1347906152, 720510798 }, - { 2145015809, 206769262, 1651459955 }, - { 2145003521, 1885513236, 1393381284 }, - { 2144960513, 1810381315, 31937275 }, - { 2144944129, 1306487838, 2019419520 }, - { 2144935937, 37304730, 1841489054 }, - { 2144894977, 1601434616, 157985831 }, - { 2144888833, 98749330, 2128592228 }, - { 2144880641, 1772327002, 2076128344 }, - { 2144864257, 1404514762, 2029969964 }, - { 2144827393, 801236594, 406627220 }, - { 2144806913, 349217443, 1501080290 }, - { 2144796673, 1542656776, 2084736519 }, - { 2144778241, 1210734884, 1746416203 }, - { 2144759809, 1146598851, 716464489 }, - { 2144757761, 286328400, 1823728177 }, - { 2144729089, 1347555695, 1836644881 }, - { 2144727041, 1795703790, 520296412 }, - { 2144696321, 1302475157, 852964281 }, - { 2144667649, 1075877614, 504992927 }, - { 2144573441, 198765808, 1617144982 }, - { 2144555009, 321528767, 155821259 }, - { 2144550913, 814139516, 1819937644 }, - { 2144536577, 571143206, 962942255 }, - { 2144524289, 1746733766, 2471321 }, - { 2144512001, 1821415077, 124190939 }, - { 2144468993, 917871546, 1260072806 }, - { 2144458753, 378417981, 1569240563 }, - { 2144421889, 175229668, 1825620763 }, - { 2144409601, 1699216963, 351648117 }, - { 2144370689, 1071885991, 958186029 }, - { 2144348161, 1763151227, 540353574 }, - { 2144335873, 1060214804, 919598847 }, - { 2144329729, 663515846, 1448552668 }, - { 2144327681, 1057776305, 590222840 }, - { 2144309249, 1705149168, 1459294624 }, - { 2144296961, 325823721, 1649016934 }, - { 2144290817, 738775789, 447427206 }, - { 2144243713, 962347618, 893050215 }, - { 2144237569, 1655257077, 900860862 }, - { 2144161793, 242206694, 1567868672 }, - { 2144155649, 769415308, 1247993134 }, - { 2144137217, 320492023, 515841070 }, - { 2144120833, 1639388522, 770877302 }, - { 2144071681, 1761785233, 964296120 }, - { 2144065537, 419817825, 204564472 }, - { 2144028673, 666050597, 2091019760 }, - { 2144010241, 1413657615, 1518702610 }, - { 2143952897, 1238327946, 475672271 }, - { 2143940609, 307063413, 1176750846 }, - { 2143918081, 2062905559, 786785803 }, - { 2143899649, 1338112849, 1562292083 }, - { 2143891457, 68149545, 87166451 }, - { 2143885313, 921750778, 394460854 }, - { 2143854593, 719766593, 133877196 }, - { 2143836161, 1149399850, 1861591875 }, - { 2143762433, 1848739366, 1335934145 }, - { 2143756289, 1326674710, 102999236 }, - { 2143713281, 808061791, 1156900308 }, - { 2143690753, 388399459, 1926468019 }, - { 2143670273, 1427891374, 1756689401 }, - { 2143666177, 1912173949, 986629565 }, - { 2143645697, 2041160111, 371842865 }, - { 2143641601, 1279906897, 2023974350 }, - { 2143635457, 720473174, 1389027526 }, - { 2143621121, 1298309455, 1732632006 }, - { 2143598593, 1548762216, 1825417506 }, - { 2143567873, 620475784, 1073787233 }, - { 2143561729, 1932954575, 949167309 }, - { 2143553537, 354315656, 1652037534 }, - { 2143541249, 577424288, 1097027618 }, - { 2143531009, 357862822, 478640055 }, - { 2143522817, 2017706025, 1550531668 }, - { 2143506433, 2078127419, 1824320165 }, - { 2143488001, 613475285, 1604011510 }, - { 2143469569, 1466594987, 502095196 }, - { 2143426561, 1115430331, 1044637111 }, - { 2143383553, 9778045, 1902463734 }, - { 2143377409, 1557401276, 2056861771 }, - { 2143363073, 652036455, 1965915971 }, - { 2143260673, 1464581171, 1523257541 }, - { 2143246337, 1876119649, 764541916 }, - { 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 }, - { 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_FALCON512_CLEAN_FFT(rt3, logn); - PQCLEAN_FALCON512_CLEAN_FFT(rt4, logn); - PQCLEAN_FALCON512_CLEAN_poly_invnorm2_fft(rt5, rt3, rt4, logn); - PQCLEAN_FALCON512_CLEAN_poly_adj_fft(rt3, logn); - PQCLEAN_FALCON512_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_FALCON512_CLEAN_FFT(rt1, logn); - PQCLEAN_FALCON512_CLEAN_FFT(rt2, logn); - PQCLEAN_FALCON512_CLEAN_poly_mul_fft(rt1, rt3, logn); - PQCLEAN_FALCON512_CLEAN_poly_mul_fft(rt2, rt4, logn); - PQCLEAN_FALCON512_CLEAN_poly_add(rt2, rt1, logn); - PQCLEAN_FALCON512_CLEAN_poly_mul_autoadj_fft(rt2, rt5, logn); - PQCLEAN_FALCON512_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_FALCON512_CLEAN_FFT(rt1, logn); - PQCLEAN_FALCON512_CLEAN_FFT(rt2, logn); - PQCLEAN_FALCON512_CLEAN_FFT(rt3, logn); - PQCLEAN_FALCON512_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_FALCON512_CLEAN_poly_add_muladj_fft(rt5, rt1, rt2, rt3, rt4, logn); - PQCLEAN_FALCON512_CLEAN_poly_invnorm2_fft(rt6, rt3, rt4, logn); - - /* - * Compute: - * rt5 = (F*adj(f)+G*adj(g)) / (f*adj(f)+g*adj(g)) - */ - PQCLEAN_FALCON512_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_FALCON512_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_FALCON512_CLEAN_FFT(rt5, logn); - - /* - * Subtract k*f from F, and k*g from G. - */ - PQCLEAN_FALCON512_CLEAN_poly_mul_fft(rt3, rt5, logn); - PQCLEAN_FALCON512_CLEAN_poly_mul_fft(rt4, rt5, logn); - PQCLEAN_FALCON512_CLEAN_poly_sub(rt1, rt3, logn); - PQCLEAN_FALCON512_CLEAN_poly_sub(rt2, rt4, logn); - PQCLEAN_FALCON512_CLEAN_iFFT(rt1, logn); - PQCLEAN_FALCON512_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_FALCON512_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_FALCON512_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_FALCON512_CLEAN_poly_div_autoadj_fft(rt3, rt2, logn); - PQCLEAN_FALCON512_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_FALCON512_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_FALCON512_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_FALCON512_CLEAN_FFT(rt1, logn); - PQCLEAN_FALCON512_CLEAN_FFT(rt2, logn); - PQCLEAN_FALCON512_CLEAN_poly_invnorm2_fft(rt3, rt1, rt2, logn); - PQCLEAN_FALCON512_CLEAN_poly_adj_fft(rt1, logn); - PQCLEAN_FALCON512_CLEAN_poly_adj_fft(rt2, logn); - PQCLEAN_FALCON512_CLEAN_poly_mulconst(rt1, fpr_q, logn); - PQCLEAN_FALCON512_CLEAN_poly_mulconst(rt2, fpr_q, logn); - PQCLEAN_FALCON512_CLEAN_poly_mul_autoadj_fft(rt1, rt3, logn); - PQCLEAN_FALCON512_CLEAN_poly_mul_autoadj_fft(rt2, rt3, logn); - PQCLEAN_FALCON512_CLEAN_iFFT(rt1, logn); - PQCLEAN_FALCON512_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_FALCON512_CLEAN_compute_public(h2, f, g, logn, (uint8_t *)tmp2)) { - continue; - } - - /* - * Solve the NTRU equation to get F and G. - */ - lim = (1 << (PQCLEAN_FALCON512_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; - } -} |