/* * 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 */ #include "inner.h" #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 *restrict gm, uint32_t *restrict 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 *restrict a, const uint32_t *restrict 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 *restrict x, const uint32_t *restrict 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 *restrict x, const uint32_t *restrict 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 *restrict xx, size_t xlen, size_t xstride, size_t num, const small_prime *primes, int normalize_signed, uint32_t *restrict 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 *restrict u, uint32_t *restrict v, const uint32_t *restrict x, const uint32_t *restrict y, size_t len, uint32_t *restrict 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 *restrict x, size_t xlen, const uint32_t *restrict 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; wy = v < ylen ? y[v] : 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 *restrict x, size_t xlen, const uint32_t *restrict 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; wy = v < ylen ? y[v] : 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 *restrict F, size_t Flen, size_t Fstride, const uint32_t *restrict f, size_t flen, size_t fstride, const int32_t *restrict 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 *restrict F, size_t Flen, size_t Fstride, const uint32_t *restrict f, size_t flen, size_t fstride, const int32_t *restrict k, uint32_t sch, uint32_t scl, unsigned logn, uint32_t *restrict 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); } } /* ==================================================================== */ #if FALCON_KG_CHACHA20 // yyyKG_CHACHA20+1 #define RNG_CONTEXT prng #define get_rng_u64 prng_get_u64 #else // yyyKG_CHACHA20+0 #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. */ #if FALCON_LE // yyyLE+1 /* * On little-endian systems we just interpret the bytes "as is" * (this is correct because the exact-width types such as * 'uint64_t' are guaranteed to have no padding and no trap * representation). */ uint64_t r; inner_shake256_extract(rng, (uint8_t *)&r, sizeof r); return r; #else // yyyLE+0 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); #endif // yyyLE- } #endif // yyyKG_CHACHA20- /* * 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 < (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; } for (d = 0; d < depth; d ++) { make_fg_step(data, logn - d, d, d != 0, (d + 1) < depth || 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 > 10) ? 10 : slen; 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. */ Zf(FFT)(rt3, logn); Zf(FFT)(rt4, logn); Zf(poly_invnorm2_fft)(rt5, rt3, rt4, logn); Zf(poly_adj_fft)(rt3, logn); Zf(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 > 10) ? 10 : FGlen; 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. */ Zf(FFT)(rt1, logn); Zf(FFT)(rt2, logn); Zf(poly_mul_fft)(rt1, rt3, logn); Zf(poly_mul_fft)(rt2, rt4, logn); Zf(poly_add)(rt2, rt1, logn); Zf(poly_mul_autoadj_fft)(rt2, rt5, logn); Zf(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. */ Zf(FFT)(rt1, logn); Zf(FFT)(rt2, logn); Zf(FFT)(rt3, logn); Zf(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; Zf(poly_add_muladj_fft)(rt5, rt1, rt2, rt3, rt4, logn); Zf(poly_invnorm2_fft)(rt6, rt3, rt4, logn); /* * Compute: * rt5 = (F*adj(f)+G*adj(g)) / (f*adj(f)+g*adj(g)) */ Zf(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. */ Zf(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)); } Zf(FFT)(rt5, logn); /* * Subtract k*f from F, and k*g from G. */ Zf(poly_mul_fft)(rt3, rt5, logn); Zf(poly_mul_fft)(rt4, rt5, logn); Zf(poly_sub)(rt1, rt3, logn); Zf(poly_sub)(rt2, rt4, logn); Zf(iFFT)(rt1, logn); Zf(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]); } Zf(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]); } Zf(FFT)(rt3, logn); /* * Compute (F*adj(f)+G*adj(g))/(f*adj(f)+g*adj(g)) and get * its rounded normal representation in t1. */ Zf(poly_div_autoadj_fft)(rt3, rt2, logn); Zf(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 Zf(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; #if FALCON_KG_CHACHA20 // yyyKG_CHACHA20+1 prng p; #endif // yyyKG_CHACHA20- n = MKN(logn); #if FALCON_KG_CHACHA20 // yyyKG_CHACHA20+1 Zf(prng_init)(&p, rng); rc = &p; #else // yyyKG_CHACHA20+0 rc = rng; #endif // yyyKG_CHACHA20- /* * 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 << (Zf(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); Zf(FFT)(rt1, logn); Zf(FFT)(rt2, logn); Zf(poly_invnorm2_fft)(rt3, rt1, rt2, logn); Zf(poly_adj_fft)(rt1, logn); Zf(poly_adj_fft)(rt2, logn); Zf(poly_mulconst)(rt1, fpr_q, logn); Zf(poly_mulconst)(rt2, fpr_q, logn); Zf(poly_mul_autoadj_fft)(rt1, rt3, logn); Zf(poly_mul_autoadj_fft)(rt2, rt3, logn); Zf(iFFT)(rt1, logn); Zf(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 (!Zf(compute_public)(h2, f, g, logn, (uint8_t *)tmp2)) { continue; } /* * Solve the NTRU equation to get F and G. */ lim = (1 << (Zf(max_FG_bits)[logn] - 1)) - 1; if (!solve_NTRU(logn, F, G, f, g, lim, (uint32_t *)tmp)) { continue; } /* * Key pair is generated. */ break; } }