diff options
Diffstat (limited to 'lib/liboqs/src/sig/falcon/pqclean_falcon-512_clean/keygen.c')
| -rw-r--r-- | lib/liboqs/src/sig/falcon/pqclean_falcon-512_clean/keygen.c | 4231 |
1 files changed, 4231 insertions, 0 deletions
diff --git a/lib/liboqs/src/sig/falcon/pqclean_falcon-512_clean/keygen.c b/lib/liboqs/src/sig/falcon/pqclean_falcon-512_clean/keygen.c new file mode 100644 index 000000000..f72ecd991 --- /dev/null +++ b/lib/liboqs/src/sig/falcon/pqclean_falcon-512_clean/keygen.c @@ -0,0 +1,4231 @@ +#include "inner.h" + +/* + * Falcon key pair generation. + * + * ==========================(LICENSE BEGIN)============================ + * + * Copyright (c) 2017-2019 Falcon Project + * + * Permission is hereby granted, free of charge, to any person obtaining + * a copy of this software and associated documentation files (the + * "Software"), to deal in the Software without restriction, including + * without limitation the rights to use, copy, modify, merge, publish, + * distribute, sublicense, and/or sell copies of the Software, and to + * permit persons to whom the Software is furnished to do so, subject to + * the following conditions: + * + * The above copyright notice and this permission notice shall be + * included in all copies or substantial portions of the Software. + * + * THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, + * EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF + * MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. + * IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY + * CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, + * TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE + * SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE. + * + * ===========================(LICENSE END)============================= + * + * @author Thomas Pornin <thomas.pornin@nccgroup.com> + */ + + +#define MKN(logn) ((size_t)1 << (logn)) + +/* ==================================================================== */ +/* + * Modular arithmetics. + * + * We implement a few functions for computing modulo a small integer p. + * + * All functions require that 2^30 < p < 2^31. Moreover, operands must + * be in the 0..p-1 range. + * + * Modular addition and subtraction work for all such p. + * + * Montgomery multiplication requires that p is odd, and must be provided + * with an additional value p0i = -1/p mod 2^31. See below for some basics + * on Montgomery multiplication. + * + * Division computes an inverse modulo p by an exponentiation (with + * exponent p-2): this works only if p is prime. Multiplication + * requirements also apply, i.e. p must be odd and p0i must be provided. + * + * The NTT and inverse NTT need all of the above, and also that + * p = 1 mod 2048. + * + * ----------------------------------------------------------------------- + * + * We use Montgomery representation with 31-bit values: + * + * Let R = 2^31 mod p. When 2^30 < p < 2^31, R = 2^31 - p. + * Montgomery representation of an integer x modulo p is x*R mod p. + * + * Montgomery multiplication computes (x*y)/R mod p for + * operands x and y. Therefore: + * + * - if operands are x*R and y*R (Montgomery representations of x and + * y), then Montgomery multiplication computes (x*R*y*R)/R = (x*y)*R + * mod p, which is the Montgomery representation of the product x*y; + * + * - if operands are x*R and y (or x and y*R), then Montgomery + * multiplication returns x*y mod p: mixed-representation + * multiplications yield results in normal representation. + * + * To convert to Montgomery representation, we multiply by R, which is done + * by Montgomery-multiplying by R^2. Stand-alone conversion back from + * Montgomery representation is Montgomery-multiplication by 1. + */ + +/* + * Precomputed small primes. Each element contains the following: + * + * p The prime itself. + * + * g A primitive root of phi = X^N+1 (in field Z_p). + * + * s The inverse of the product of all previous primes in the array, + * computed modulo p and in Montgomery representation. + * + * All primes are such that p = 1 mod 2048, and are lower than 2^31. They + * are listed in decreasing order. + */ + +typedef struct { + uint32_t p; + uint32_t g; + uint32_t s; +} small_prime; + +static const small_prime PRIMES[] = { + { 2147473409, 383167813, 10239 }, + { 2147389441, 211808905, 471403745 }, + { 2147387393, 37672282, 1329335065 }, + { 2147377153, 1977035326, 968223422 }, + { 2147358721, 1067163706, 132460015 }, + { 2147352577, 1606082042, 598693809 }, + { 2147346433, 2033915641, 1056257184 }, + { 2147338241, 1653770625, 421286710 }, + { 2147309569, 631200819, 1111201074 }, + { 2147297281, 2038364663, 1042003613 }, + { 2147295233, 1962540515, 19440033 }, + { 2147239937, 2100082663, 353296760 }, + { 2147235841, 1991153006, 1703918027 }, + { 2147217409, 516405114, 1258919613 }, + { 2147205121, 409347988, 1089726929 }, + { 2147196929, 927788991, 1946238668 }, + { 2147178497, 1136922411, 1347028164 }, + { 2147100673, 868626236, 701164723 }, + { 2147082241, 1897279176, 617820870 }, + { 2147074049, 1888819123, 158382189 }, + { 2147051521, 25006327, 522758543 }, + { 2147043329, 327546255, 37227845 }, + { 2147039233, 766324424, 1133356428 }, + { 2146988033, 1862817362, 73861329 }, + { 2146963457, 404622040, 653019435 }, + { 2146959361, 1936581214, 995143093 }, + { 2146938881, 1559770096, 634921513 }, + { 2146908161, 422623708, 1985060172 }, + { 2146885633, 1751189170, 298238186 }, + { 2146871297, 578919515, 291810829 }, + { 2146846721, 1114060353, 915902322 }, + { 2146834433, 2069565474, 47859524 }, + { 2146818049, 1552824584, 646281055 }, + { 2146775041, 1906267847, 1597832891 }, + { 2146756609, 1847414714, 1228090888 }, + { 2146744321, 1818792070, 1176377637 }, + { 2146738177, 1118066398, 1054971214 }, + { 2146736129, 52057278, 933422153 }, + { 2146713601, 592259376, 1406621510 }, + { 2146695169, 263161877, 1514178701 }, + { 2146656257, 685363115, 384505091 }, + { 2146650113, 927727032, 537575289 }, + { 2146646017, 52575506, 1799464037 }, + { 2146643969, 1276803876, 1348954416 }, + { 2146603009, 814028633, 1521547704 }, + { 2146572289, 1846678872, 1310832121 }, + { 2146547713, 919368090, 1019041349 }, + { 2146508801, 671847612, 38582496 }, + { 2146492417, 283911680, 532424562 }, + { 2146490369, 1780044827, 896447978 }, + { 2146459649, 327980850, 1327906900 }, + { 2146447361, 1310561493, 958645253 }, + { 2146441217, 412148926, 287271128 }, + { 2146437121, 293186449, 2009822534 }, + { 2146430977, 179034356, 1359155584 }, + { 2146418689, 1517345488, 1790248672 }, + { 2146406401, 1615820390, 1584833571 }, + { 2146404353, 826651445, 607120498 }, + { 2146379777, 3816988, 1897049071 }, + { 2146363393, 1221409784, 1986921567 }, + { 2146355201, 1388081168, 849968120 }, + { 2146336769, 1803473237, 1655544036 }, + { 2146312193, 1023484977, 273671831 }, + { 2146293761, 1074591448, 467406983 }, + { 2146283521, 831604668, 1523950494 }, + { 2146203649, 712865423, 1170834574 }, + { 2146154497, 1764991362, 1064856763 }, + { 2146142209, 627386213, 1406840151 }, + { 2146127873, 1638674429, 2088393537 }, + { 2146099201, 1516001018, 690673370 }, + { 2146093057, 1294931393, 315136610 }, + { 2146091009, 1942399533, 973539425 }, + { 2146078721, 1843461814, 2132275436 }, + { 2146060289, 1098740778, 360423481 }, + { 2146048001, 1617213232, 1951981294 }, + { 2146041857, 1805783169, 2075683489 }, + { 2146019329, 272027909, 1753219918 }, + { 2145986561, 1206530344, 2034028118 }, + { 2145976321, 1243769360, 1173377644 }, + { 2145964033, 887200839, 1281344586 }, + { 2145906689, 1651026455, 906178216 }, + { 2145875969, 1673238256, 1043521212 }, + { 2145871873, 1226591210, 1399796492 }, + { 2145841153, 1465353397, 1324527802 }, + { 2145832961, 1150638905, 554084759 }, + { 2145816577, 221601706, 427340863 }, + { 2145785857, 608896761, 316590738 }, + { 2145755137, 1712054942, 1684294304 }, + { 2145742849, 1302302867, 724873116 }, + { 2145728513, 516717693, 431671476 }, + { 2145699841, 524575579, 1619722537 }, + { 2145691649, 1925625239, 982974435 }, + { 2145687553, 463795662, 1293154300 }, + { 2145673217, 771716636, 881778029 }, + { 2145630209, 1509556977, 837364988 }, + { 2145595393, 229091856, 851648427 }, + { 2145587201, 1796903241, 635342424 }, + { 2145525761, 715310882, 1677228081 }, + { 2145495041, 1040930522, 200685896 }, + { 2145466369, 949804237, 1809146322 }, + { 2145445889, 1673903706, 95316881 }, + { 2145390593, 806941852, 1428671135 }, + { 2145372161, 1402525292, 159350694 }, + { 2145361921, 2124760298, 1589134749 }, + { 2145359873, 1217503067, 1561543010 }, + { 2145355777, 338341402, 83865711 }, + { 2145343489, 1381532164, 641430002 }, + { 2145325057, 1883895478, 1528469895 }, + { 2145318913, 1335370424, 65809740 }, + { 2145312769, 2000008042, 1919775760 }, + { 2145300481, 961450962, 1229540578 }, + { 2145282049, 910466767, 1964062701 }, + { 2145232897, 816527501, 450152063 }, + { 2145218561, 1435128058, 1794509700 }, + { 2145187841, 33505311, 1272467582 }, + { 2145181697, 269767433, 1380363849 }, + { 2145175553, 56386299, 1316870546 }, + { 2145079297, 2106880293, 1391797340 }, + { 2145021953, 1347906152, 720510798 }, + { 2145015809, 206769262, 1651459955 }, + { 2145003521, 1885513236, 1393381284 }, + { 2144960513, 1810381315, 31937275 }, + { 2144944129, 1306487838, 2019419520 }, + { 2144935937, 37304730, 1841489054 }, + { 2144894977, 1601434616, 157985831 }, + { 2144888833, 98749330, 2128592228 }, + { 2144880641, 1772327002, 2076128344 }, + { 2144864257, 1404514762, 2029969964 }, + { 2144827393, 801236594, 406627220 }, + { 2144806913, 349217443, 1501080290 }, + { 2144796673, 1542656776, 2084736519 }, + { 2144778241, 1210734884, 1746416203 }, + { 2144759809, 1146598851, 716464489 }, + { 2144757761, 286328400, 1823728177 }, + { 2144729089, 1347555695, 1836644881 }, + { 2144727041, 1795703790, 520296412 }, + { 2144696321, 1302475157, 852964281 }, + { 2144667649, 1075877614, 504992927 }, + { 2144573441, 198765808, 1617144982 }, + { 2144555009, 321528767, 155821259 }, + { 2144550913, 814139516, 1819937644 }, + { 2144536577, 571143206, 962942255 }, + { 2144524289, 1746733766, 2471321 }, + { 2144512001, 1821415077, 124190939 }, + { 2144468993, 917871546, 1260072806 }, + { 2144458753, 378417981, 1569240563 }, + { 2144421889, 175229668, 1825620763 }, + { 2144409601, 1699216963, 351648117 }, + { 2144370689, 1071885991, 958186029 }, + { 2144348161, 1763151227, 540353574 }, + { 2144335873, 1060214804, 919598847 }, + { 2144329729, 663515846, 1448552668 }, + { 2144327681, 1057776305, 590222840 }, + { 2144309249, 1705149168, 1459294624 }, + { 2144296961, 325823721, 1649016934 }, + { 2144290817, 738775789, 447427206 }, + { 2144243713, 962347618, 893050215 }, + { 2144237569, 1655257077, 900860862 }, + { 2144161793, 242206694, 1567868672 }, + { 2144155649, 769415308, 1247993134 }, + { 2144137217, 320492023, 515841070 }, + { 2144120833, 1639388522, 770877302 }, + { 2144071681, 1761785233, 964296120 }, + { 2144065537, 419817825, 204564472 }, + { 2144028673, 666050597, 2091019760 }, + { 2144010241, 1413657615, 1518702610 }, + { 2143952897, 1238327946, 475672271 }, + { 2143940609, 307063413, 1176750846 }, + { 2143918081, 2062905559, 786785803 }, + { 2143899649, 1338112849, 1562292083 }, + { 2143891457, 68149545, 87166451 }, + { 2143885313, 921750778, 394460854 }, + { 2143854593, 719766593, 133877196 }, + { 2143836161, 1149399850, 1861591875 }, + { 2143762433, 1848739366, 1335934145 }, + { 2143756289, 1326674710, 102999236 }, + { 2143713281, 808061791, 1156900308 }, + { 2143690753, 388399459, 1926468019 }, + { 2143670273, 1427891374, 1756689401 }, + { 2143666177, 1912173949, 986629565 }, + { 2143645697, 2041160111, 371842865 }, + { 2143641601, 1279906897, 2023974350 }, + { 2143635457, 720473174, 1389027526 }, + { 2143621121, 1298309455, 1732632006 }, + { 2143598593, 1548762216, 1825417506 }, + { 2143567873, 620475784, 1073787233 }, + { 2143561729, 1932954575, 949167309 }, + { 2143553537, 354315656, 1652037534 }, + { 2143541249, 577424288, 1097027618 }, + { 2143531009, 357862822, 478640055 }, + { 2143522817, 2017706025, 1550531668 }, + { 2143506433, 2078127419, 1824320165 }, + { 2143488001, 613475285, 1604011510 }, + { 2143469569, 1466594987, 502095196 }, + { 2143426561, 1115430331, 1044637111 }, + { 2143383553, 9778045, 1902463734 }, + { 2143377409, 1557401276, 2056861771 }, + { 2143363073, 652036455, 1965915971 }, + { 2143260673, 1464581171, 1523257541 }, + { 2143246337, 1876119649, 764541916 }, + { 2143209473, 1614992673, 1920672844 }, + { 2143203329, 981052047, 2049774209 }, + { 2143160321, 1847355533, 728535665 }, + { 2143129601, 965558457, 603052992 }, + { 2143123457, 2140817191, 8348679 }, + { 2143100929, 1547263683, 694209023 }, + { 2143092737, 643459066, 1979934533 }, + { 2143082497, 188603778, 2026175670 }, + { 2143062017, 1657329695, 377451099 }, + { 2143051777, 114967950, 979255473 }, + { 2143025153, 1698431342, 1449196896 }, + { 2143006721, 1862741675, 1739650365 }, + { 2142996481, 756660457, 996160050 }, + { 2142976001, 927864010, 1166847574 }, + { 2142965761, 905070557, 661974566 }, + { 2142916609, 40932754, 1787161127 }, + { 2142892033, 1987985648, 675335382 }, + { 2142885889, 797497211, 1323096997 }, + { 2142871553, 2068025830, 1411877159 }, + { 2142861313, 1217177090, 1438410687 }, + { 2142830593, 409906375, 1767860634 }, + { 2142803969, 1197788993, 359782919 }, + { 2142785537, 643817365, 513932862 }, + { 2142779393, 1717046338, 218943121 }, + { 2142724097, 89336830, 416687049 }, + { 2142707713, 5944581, 1356813523 }, + { 2142658561, 887942135, 2074011722 }, + { 2142638081, 151851972, 1647339939 }, + { 2142564353, 1691505537, 1483107336 }, + { 2142533633, 1989920200, 1135938817 }, + { 2142529537, 959263126, 1531961857 }, + { 2142527489, 453251129, 1725566162 }, + { 2142502913, 1536028102, 182053257 }, + { 2142498817, 570138730, 701443447 }, + { 2142416897, 326965800, 411931819 }, + { 2142363649, 1675665410, 1517191733 }, + { 2142351361, 968529566, 1575712703 }, + { 2142330881, 1384953238, 1769087884 }, + { 2142314497, 1977173242, 1833745524 }, + { 2142289921, 95082313, 1714775493 }, + { 2142283777, 109377615, 1070584533 }, + { 2142277633, 16960510, 702157145 }, + { 2142263297, 553850819, 431364395 }, + { 2142208001, 241466367, 2053967982 }, + { 2142164993, 1795661326, 1031836848 }, + { 2142097409, 1212530046, 712772031 }, + { 2142087169, 1763869720, 822276067 }, + { 2142078977, 644065713, 1765268066 }, + { 2142074881, 112671944, 643204925 }, + { 2142044161, 1387785471, 1297890174 }, + { 2142025729, 783885537, 1000425730 }, + { 2142011393, 905662232, 1679401033 }, + { 2141974529, 799788433, 468119557 }, + { 2141943809, 1932544124, 449305555 }, + { 2141933569, 1527403256, 841867925 }, + { 2141931521, 1247076451, 743823916 }, + { 2141902849, 1199660531, 401687910 }, + { 2141890561, 150132350, 1720336972 }, + { 2141857793, 1287438162, 663880489 }, + { 2141833217, 618017731, 1819208266 }, + { 2141820929, 999578638, 1403090096 }, + { 2141786113, 81834325, 1523542501 }, + { 2141771777, 120001928, 463556492 }, + { 2141759489, 122455485, 2124928282 }, + { 2141749249, 141986041, 940339153 }, + { 2141685761, 889088734, 477141499 }, + { 2141673473, 324212681, 1122558298 }, + { 2141669377, 1175806187, 1373818177 }, + { 2141655041, 1113654822, 296887082 }, + { 2141587457, 991103258, 1585913875 }, + { 2141583361, 1401451409, 1802457360 }, + { 2141575169, 1571977166, 712760980 }, + { 2141546497, 1107849376, 1250270109 }, + { 2141515777, 196544219, 356001130 }, + { 2141495297, 1733571506, 1060744866 }, + { 2141483009, 321552363, 1168297026 }, + { 2141458433, 505818251, 733225819 }, + { 2141360129, 1026840098, 948342276 }, + { 2141325313, 945133744, 2129965998 }, + { 2141317121, 1871100260, 1843844634 }, + { 2141286401, 1790639498, 1750465696 }, + { 2141267969, 1376858592, 186160720 }, + { 2141255681, 2129698296, 1876677959 }, + { 2141243393, 2138900688, 1340009628 }, + { 2141214721, 1933049835, 1087819477 }, + { 2141212673, 1898664939, 1786328049 }, + { 2141202433, 990234828, 940682169 }, + { 2141175809, 1406392421, 993089586 }, + { 2141165569, 1263518371, 289019479 }, + { 2141073409, 1485624211, 507864514 }, + { 2141052929, 1885134788, 311252465 }, + { 2141040641, 1285021247, 280941862 }, + { 2141028353, 1527610374, 375035110 }, + { 2141011969, 1400626168, 164696620 }, + { 2140999681, 632959608, 966175067 }, + { 2140997633, 2045628978, 1290889438 }, + { 2140993537, 1412755491, 375366253 }, + { 2140942337, 719477232, 785367828 }, + { 2140925953, 45224252, 836552317 }, + { 2140917761, 1157376588, 1001839569 }, + { 2140887041, 278480752, 2098732796 }, + { 2140837889, 1663139953, 924094810 }, + { 2140788737, 802501511, 2045368990 }, + { 2140766209, 1820083885, 1800295504 }, + { 2140764161, 1169561905, 2106792035 }, + { 2140696577, 127781498, 1885987531 }, + { 2140684289, 16014477, 1098116827 }, + { 2140653569, 665960598, 1796728247 }, + { 2140594177, 1043085491, 377310938 }, + { 2140579841, 1732838211, 1504505945 }, + { 2140569601, 302071939, 358291016 }, + { 2140567553, 192393733, 1909137143 }, + { 2140557313, 406595731, 1175330270 }, + { 2140549121, 1748850918, 525007007 }, + { 2140477441, 499436566, 1031159814 }, + { 2140469249, 1886004401, 1029951320 }, + { 2140426241, 1483168100, 1676273461 }, + { 2140420097, 1779917297, 846024476 }, + { 2140413953, 522948893, 1816354149 }, + { 2140383233, 1931364473, 1296921241 }, + { 2140366849, 1917356555, 147196204 }, + { 2140354561, 16466177, 1349052107 }, + { 2140348417, 1875366972, 1860485634 }, + { 2140323841, 456498717, 1790256483 }, + { 2140321793, 1629493973, 150031888 }, + { 2140315649, 1904063898, 395510935 }, + { 2140280833, 1784104328, 831417909 }, + { 2140250113, 256087139, 697349101 }, + { 2140229633, 388553070, 243875754 }, + { 2140223489, 747459608, 1396270850 }, + { 2140200961, 507423743, 1895572209 }, + { 2140162049, 580106016, 2045297469 }, + { 2140149761, 712426444, 785217995 }, + { 2140137473, 1441607584, 536866543 }, + { 2140119041, 346538902, 1740434653 }, + { 2140090369, 282642885, 21051094 }, + { 2140076033, 1407456228, 319910029 }, + { 2140047361, 1619330500, 1488632070 }, + { 2140041217, 2089408064, 2012026134 }, + { 2140008449, 1705524800, 1613440760 }, + { 2139924481, 1846208233, 1280649481 }, + { 2139906049, 989438755, 1185646076 }, + { 2139867137, 1522314850, 372783595 }, + { 2139842561, 1681587377, 216848235 }, + { 2139826177, 2066284988, 1784999464 }, + { 2139824129, 480888214, 1513323027 }, + { 2139789313, 847937200, 858192859 }, + { 2139783169, 1642000434, 1583261448 }, + { 2139770881, 940699589, 179702100 }, + { 2139768833, 315623242, 964612676 }, + { 2139666433, 331649203, 764666914 }, + { 2139641857, 2118730799, 1313764644 }, + { 2139635713, 519149027, 519212449 }, + { 2139598849, 1526413634, 1769667104 }, + { 2139574273, 551148610, 820739925 }, + { 2139568129, 1386800242, 472447405 }, + { 2139549697, 813760130, 1412328531 }, + { 2139537409, 1615286260, 1609362979 }, + { 2139475969, 1352559299, 1696720421 }, + { 2139455489, 1048691649, 1584935400 }, + { 2139432961, 836025845, 950121150 }, + { 2139424769, 1558281165, 1635486858 }, + { 2139406337, 1728402143, 1674423301 }, + { 2139396097, 1727715782, 1483470544 }, + { 2139383809, 1092853491, 1741699084 }, + { 2139369473, 690776899, 1242798709 }, + { 2139351041, 1768782380, 2120712049 }, + { 2139334657, 1739968247, 1427249225 }, + { 2139332609, 1547189119, 623011170 }, + { 2139310081, 1346827917, 1605466350 }, + { 2139303937, 369317948, 828392831 }, + { 2139301889, 1560417239, 1788073219 }, + { 2139283457, 1303121623, 595079358 }, + { 2139248641, 1354555286, 573424177 }, + { 2139240449, 60974056, 885781403 }, + { 2139222017, 355573421, 1221054839 }, + { 2139215873, 566477826, 1724006500 }, + { 2139150337, 871437673, 1609133294 }, + { 2139144193, 1478130914, 1137491905 }, + { 2139117569, 1854880922, 964728507 }, + { 2139076609, 202405335, 756508944 }, + { 2139062273, 1399715741, 884826059 }, + { 2139045889, 1051045798, 1202295476 }, + { 2139033601, 1707715206, 632234634 }, + { 2139006977, 2035853139, 231626690 }, + { 2138951681, 183867876, 838350879 }, + { 2138945537, 1403254661, 404460202 }, + { 2138920961, 310865011, 1282911681 }, + { 2138910721, 1328496553, 103472415 }, + { 2138904577, 78831681, 993513549 }, + { 2138902529, 1319697451, 1055904361 }, + { 2138816513, 384338872, 1706202469 }, + { 2138810369, 1084868275, 405677177 }, + { 2138787841, 401181788, 1964773901 }, + { 2138775553, 1850532988, 1247087473 }, + { 2138767361, 874261901, 1576073565 }, + { 2138757121, 1187474742, 993541415 }, + { 2138748929, 1782458888, 1043206483 }, + { 2138744833, 1221500487, 800141243 }, + { 2138738689, 413465368, 1450660558 }, + { 2138695681, 739045140, 342611472 }, + { 2138658817, 1355845756, 672674190 }, + { 2138644481, 608379162, 1538874380 }, + { 2138632193, 1444914034, 686911254 }, + { 2138607617, 484707818, 1435142134 }, + { 2138591233, 539460669, 1290458549 }, + { 2138572801, 2093538990, 2011138646 }, + { 2138552321, 1149786988, 1076414907 }, + { 2138546177, 840688206, 2108985273 }, + { 2138533889, 209669619, 198172413 }, + { 2138523649, 1975879426, 1277003968 }, + { 2138490881, 1351891144, 1976858109 }, + { 2138460161, 1817321013, 1979278293 }, + { 2138429441, 1950077177, 203441928 }, + { 2138400769, 908970113, 628395069 }, + { 2138398721, 219890864, 758486760 }, + { 2138376193, 1306654379, 977554090 }, + { 2138351617, 298822498, 2004708503 }, + { 2138337281, 441457816, 1049002108 }, + { 2138320897, 1517731724, 1442269609 }, + { 2138290177, 1355911197, 1647139103 }, + { 2138234881, 531313247, 1746591962 }, + { 2138214401, 1899410930, 781416444 }, + { 2138202113, 1813477173, 1622508515 }, + { 2138191873, 1086458299, 1025408615 }, + { 2138183681, 1998800427, 827063290 }, + { 2138173441, 1921308898, 749670117 }, + { 2138103809, 1620902804, 2126787647 }, + { 2138099713, 828647069, 1892961817 }, + { 2138085377, 179405355, 1525506535 }, + { 2138060801, 615683235, 1259580138 }, + { 2138044417, 2030277840, 1731266562 }, + { 2138042369, 2087222316, 1627902259 }, + { 2138032129, 126388712, 1108640984 }, + { 2138011649, 715026550, 1017980050 }, + { 2137993217, 1693714349, 1351778704 }, + { 2137888769, 1289762259, 1053090405 }, + { 2137853953, 199991890, 1254192789 }, + { 2137833473, 941421685, 896995556 }, + { 2137817089, 750416446, 1251031181 }, + { 2137792513, 798075119, 368077456 }, + { 2137786369, 878543495, 1035375025 }, + { 2137767937, 9351178, 1156563902 }, + { 2137755649, 1382297614, 1686559583 }, + { 2137724929, 1345472850, 1681096331 }, + { 2137704449, 834666929, 630551727 }, + { 2137673729, 1646165729, 1892091571 }, + { 2137620481, 778943821, 48456461 }, + { 2137618433, 1730837875, 1713336725 }, + { 2137581569, 805610339, 1378891359 }, + { 2137538561, 204342388, 1950165220 }, + { 2137526273, 1947629754, 1500789441 }, + { 2137516033, 719902645, 1499525372 }, + { 2137491457, 230451261, 556382829 }, + { 2137440257, 979573541, 412760291 }, + { 2137374721, 927841248, 1954137185 }, + { 2137362433, 1243778559, 861024672 }, + { 2137313281, 1341338501, 980638386 }, + { 2137311233, 937415182, 1793212117 }, + { 2137255937, 795331324, 1410253405 }, + { 2137243649, 150756339, 1966999887 }, + { 2137182209, 163346914, 1939301431 }, + { 2137171969, 1952552395, 758913141 }, + { 2137159681, 570788721, 218668666 }, + { 2137147393, 1896656810, 2045670345 }, + { 2137141249, 358493842, 518199643 }, + { 2137139201, 1505023029, 674695848 }, + { 2137133057, 27911103, 830956306 }, + { 2137122817, 439771337, 1555268614 }, + { 2137116673, 790988579, 1871449599 }, + { 2137110529, 432109234, 811805080 }, + { 2137102337, 1357900653, 1184997641 }, + { 2137098241, 515119035, 1715693095 }, + { 2137090049, 408575203, 2085660657 }, + { 2137085953, 2097793407, 1349626963 }, + { 2137055233, 1556739954, 1449960883 }, + { 2137030657, 1545758650, 1369303716 }, + { 2136987649, 332602570, 103875114 }, + { 2136969217, 1499989506, 1662964115 }, + { 2136924161, 857040753, 4738842 }, + { 2136895489, 1948872712, 570436091 }, + { 2136893441, 58969960, 1568349634 }, + { 2136887297, 2127193379, 273612548 }, + { 2136850433, 111208983, 1181257116 }, + { 2136809473, 1627275942, 1680317971 }, + { 2136764417, 1574888217, 14011331 }, + { 2136741889, 14011055, 1129154251 }, + { 2136727553, 35862563, 1838555253 }, + { 2136721409, 310235666, 1363928244 }, + { 2136698881, 1612429202, 1560383828 }, + { 2136649729, 1138540131, 800014364 }, + { 2136606721, 602323503, 1433096652 }, + { 2136563713, 182209265, 1919611038 }, + { 2136555521, 324156477, 165591039 }, + { 2136549377, 195513113, 217165345 }, + { 2136526849, 1050768046, 939647887 }, + { 2136508417, 1886286237, 1619926572 }, + { 2136477697, 609647664, 35065157 }, + { 2136471553, 679352216, 1452259468 }, + { 2136457217, 128630031, 824816521 }, + { 2136422401, 19787464, 1526049830 }, + { 2136420353, 698316836, 1530623527 }, + { 2136371201, 1651862373, 1804812805 }, + { 2136334337, 326596005, 336977082 }, + { 2136322049, 63253370, 1904972151 }, + { 2136297473, 312176076, 172182411 }, + { 2136248321, 381261841, 369032670 }, + { 2136242177, 358688773, 1640007994 }, + { 2136229889, 512677188, 75585225 }, + { 2136219649, 2095003250, 1970086149 }, + { 2136207361, 1909650722, 537760675 }, + { 2136176641, 1334616195, 1533487619 }, + { 2136158209, 2096285632, 1793285210 }, + { 2136143873, 1897347517, 293843959 }, + { 2136133633, 923586222, 1022655978 }, + { 2136096769, 1464868191, 1515074410 }, + { 2136094721, 2020679520, 2061636104 }, + { 2136076289, 290798503, 1814726809 }, + { 2136041473, 156415894, 1250757633 }, + { 2135996417, 297459940, 1132158924 }, + { 2135955457, 538755304, 1688831340 }, + { 0, 0, 0 } +}; + +/* + * Reduce a small signed integer modulo a small prime. The source + * value x MUST be such that -p < x < p. + */ +static inline uint32_t +modp_set(int32_t x, uint32_t p) { + uint32_t w; + + w = (uint32_t)x; + w += p & -(w >> 31); + return w; +} + +/* + * Normalize a modular integer around 0. + */ +static inline int32_t +modp_norm(uint32_t x, uint32_t p) { + return (int32_t)(x - (p & (((x - ((p + 1) >> 1)) >> 31) - 1))); +} + +/* + * Compute -1/p mod 2^31. This works for all odd integers p that fit + * on 31 bits. + */ +static uint32_t +modp_ninv31(uint32_t p) { + uint32_t y; + + y = 2 - p; + y *= 2 - p * y; + y *= 2 - p * y; + y *= 2 - p * y; + y *= 2 - p * y; + return (uint32_t)0x7FFFFFFF & -y; +} + +/* + * Compute R = 2^31 mod p. + */ +static inline uint32_t +modp_R(uint32_t p) { + /* + * Since 2^30 < p < 2^31, we know that 2^31 mod p is simply + * 2^31 - p. + */ + return ((uint32_t)1 << 31) - p; +} + +/* + * Addition modulo p. + */ +static inline uint32_t +modp_add(uint32_t a, uint32_t b, uint32_t p) { + uint32_t d; + + d = a + b - p; + d += p & -(d >> 31); + return d; +} + +/* + * Subtraction modulo p. + */ +static inline uint32_t +modp_sub(uint32_t a, uint32_t b, uint32_t p) { + uint32_t d; + + d = a - b; + d += p & -(d >> 31); + return d; +} + +/* + * Halving modulo p. + */ +/* unused +static inline uint32_t +modp_half(uint32_t a, uint32_t p) +{ + a += p & -(a & 1); + return a >> 1; +} +*/ + +/* + * Montgomery multiplication modulo p. The 'p0i' value is -1/p mod 2^31. + * It is required that p is an odd integer. + */ +static inline uint32_t +modp_montymul(uint32_t a, uint32_t b, uint32_t p, uint32_t p0i) { + uint64_t z, w; + uint32_t d; + + z = (uint64_t)a * (uint64_t)b; + w = ((z * p0i) & (uint64_t)0x7FFFFFFF) * p; + d = (uint32_t)((z + w) >> 31) - p; + d += p & -(d >> 31); + return d; +} + +/* + * Compute R2 = 2^62 mod p. + */ +static uint32_t +modp_R2(uint32_t p, uint32_t p0i) { + uint32_t z; + + /* + * Compute z = 2^31 mod p (this is the value 1 in Montgomery + * representation), then double it with an addition. + */ + z = modp_R(p); + z = modp_add(z, z, p); + + /* + * Square it five times to obtain 2^32 in Montgomery representation + * (i.e. 2^63 mod p). + */ + z = modp_montymul(z, z, p, p0i); + z = modp_montymul(z, z, p, p0i); + z = modp_montymul(z, z, p, p0i); + z = modp_montymul(z, z, p, p0i); + z = modp_montymul(z, z, p, p0i); + + /* + * Halve the value mod p to get 2^62. + */ + z = (z + (p & -(z & 1))) >> 1; + return z; +} + +/* + * Compute 2^(31*x) modulo p. This works for integers x up to 2^11. + * p must be prime such that 2^30 < p < 2^31; p0i must be equal to + * -1/p mod 2^31; R2 must be equal to 2^62 mod p. + */ +static inline uint32_t +modp_Rx(unsigned x, uint32_t p, uint32_t p0i, uint32_t R2) { + int i; + uint32_t r, z; + + /* + * 2^(31*x) = (2^31)*(2^(31*(x-1))); i.e. we want the Montgomery + * representation of (2^31)^e mod p, where e = x-1. + * R2 is 2^31 in Montgomery representation. + */ + x --; + r = R2; + z = modp_R(p); + for (i = 0; (1U << i) <= x; i ++) { + if ((x & (1U << i)) != 0) { + z = modp_montymul(z, r, p, p0i); + } + r = modp_montymul(r, r, p, p0i); + } + return z; +} + +/* + * Division modulo p. If the divisor (b) is 0, then 0 is returned. + * This function computes proper results only when p is prime. + * Parameters: + * a dividend + * b divisor + * p odd prime modulus + * p0i -1/p mod 2^31 + * R 2^31 mod R + */ +static uint32_t +modp_div(uint32_t a, uint32_t b, uint32_t p, uint32_t p0i, uint32_t R) { + uint32_t z, e; + int i; + + e = p - 2; + z = R; + for (i = 30; i >= 0; i --) { + uint32_t z2; + + z = modp_montymul(z, z, p, p0i); + z2 = modp_montymul(z, b, p, p0i); + z ^= (z ^ z2) & -(uint32_t)((e >> i) & 1); + } + + /* + * The loop above just assumed that b was in Montgomery + * representation, i.e. really contained b*R; under that + * assumption, it returns 1/b in Montgomery representation, + * which is R/b. But we gave it b in normal representation, + * so the loop really returned R/(b/R) = R^2/b. + * + * We want a/b, so we need one Montgomery multiplication with a, + * which also remove one of the R factors, and another such + * multiplication to remove the second R factor. + */ + z = modp_montymul(z, 1, p, p0i); + return modp_montymul(a, z, p, p0i); +} + +/* + * Bit-reversal index table. + */ +static const uint16_t REV10[] = { + 0, 512, 256, 768, 128, 640, 384, 896, 64, 576, 320, 832, + 192, 704, 448, 960, 32, 544, 288, 800, 160, 672, 416, 928, + 96, 608, 352, 864, 224, 736, 480, 992, 16, 528, 272, 784, + 144, 656, 400, 912, 80, 592, 336, 848, 208, 720, 464, 976, + 48, 560, 304, 816, 176, 688, 432, 944, 112, 624, 368, 880, + 240, 752, 496, 1008, 8, 520, 264, 776, 136, 648, 392, 904, + 72, 584, 328, 840, 200, 712, 456, 968, 40, 552, 296, 808, + 168, 680, 424, 936, 104, 616, 360, 872, 232, 744, 488, 1000, + 24, 536, 280, 792, 152, 664, 408, 920, 88, 600, 344, 856, + 216, 728, 472, 984, 56, 568, 312, 824, 184, 696, 440, 952, + 120, 632, 376, 888, 248, 760, 504, 1016, 4, 516, 260, 772, + 132, 644, 388, 900, 68, 580, 324, 836, 196, 708, 452, 964, + 36, 548, 292, 804, 164, 676, 420, 932, 100, 612, 356, 868, + 228, 740, 484, 996, 20, 532, 276, 788, 148, 660, 404, 916, + 84, 596, 340, 852, 212, 724, 468, 980, 52, 564, 308, 820, + 180, 692, 436, 948, 116, 628, 372, 884, 244, 756, 500, 1012, + 12, 524, 268, 780, 140, 652, 396, 908, 76, 588, 332, 844, + 204, 716, 460, 972, 44, 556, 300, 812, 172, 684, 428, 940, + 108, 620, 364, 876, 236, 748, 492, 1004, 28, 540, 284, 796, + 156, 668, 412, 924, 92, 604, 348, 860, 220, 732, 476, 988, + 60, 572, 316, 828, 188, 700, 444, 956, 124, 636, 380, 892, + 252, 764, 508, 1020, 2, 514, 258, 770, 130, 642, 386, 898, + 66, 578, 322, 834, 194, 706, 450, 962, 34, 546, 290, 802, + 162, 674, 418, 930, 98, 610, 354, 866, 226, 738, 482, 994, + 18, 530, 274, 786, 146, 658, 402, 914, 82, 594, 338, 850, + 210, 722, 466, 978, 50, 562, 306, 818, 178, 690, 434, 946, + 114, 626, 370, 882, 242, 754, 498, 1010, 10, 522, 266, 778, + 138, 650, 394, 906, 74, 586, 330, 842, 202, 714, 458, 970, + 42, 554, 298, 810, 170, 682, 426, 938, 106, 618, 362, 874, + 234, 746, 490, 1002, 26, 538, 282, 794, 154, 666, 410, 922, + 90, 602, 346, 858, 218, 730, 474, 986, 58, 570, 314, 826, + 186, 698, 442, 954, 122, 634, 378, 890, 250, 762, 506, 1018, + 6, 518, 262, 774, 134, 646, 390, 902, 70, 582, 326, 838, + 198, 710, 454, 966, 38, 550, 294, 806, 166, 678, 422, 934, + 102, 614, 358, 870, 230, 742, 486, 998, 22, 534, 278, 790, + 150, 662, 406, 918, 86, 598, 342, 854, 214, 726, 470, 982, + 54, 566, 310, 822, 182, 694, 438, 950, 118, 630, 374, 886, + 246, 758, 502, 1014, 14, 526, 270, 782, 142, 654, 398, 910, + 78, 590, 334, 846, 206, 718, 462, 974, 46, 558, 302, 814, + 174, 686, 430, 942, 110, 622, 366, 878, 238, 750, 494, 1006, + 30, 542, 286, 798, 158, 670, 414, 926, 94, 606, 350, 862, + 222, 734, 478, 990, 62, 574, 318, 830, 190, 702, 446, 958, + 126, 638, 382, 894, 254, 766, 510, 1022, 1, 513, 257, 769, + 129, 641, 385, 897, 65, 577, 321, 833, 193, 705, 449, 961, + 33, 545, 289, 801, 161, 673, 417, 929, 97, 609, 353, 865, + 225, 737, 481, 993, 17, 529, 273, 785, 145, 657, 401, 913, + 81, 593, 337, 849, 209, 721, 465, 977, 49, 561, 305, 817, + 177, 689, 433, 945, 113, 625, 369, 881, 241, 753, 497, 1009, + 9, 521, 265, 777, 137, 649, 393, 905, 73, 585, 329, 841, + 201, 713, 457, 969, 41, 553, 297, 809, 169, 681, 425, 937, + 105, 617, 361, 873, 233, 745, 489, 1001, 25, 537, 281, 793, + 153, 665, 409, 921, 89, 601, 345, 857, 217, 729, 473, 985, + 57, 569, 313, 825, 185, 697, 441, 953, 121, 633, 377, 889, + 249, 761, 505, 1017, 5, 517, 261, 773, 133, 645, 389, 901, + 69, 581, 325, 837, 197, 709, 453, 965, 37, 549, 293, 805, + 165, 677, 421, 933, 101, 613, 357, 869, 229, 741, 485, 997, + 21, 533, 277, 789, 149, 661, 405, 917, 85, 597, 341, 853, + 213, 725, 469, 981, 53, 565, 309, 821, 181, 693, 437, 949, + 117, 629, 373, 885, 245, 757, 501, 1013, 13, 525, 269, 781, + 141, 653, 397, 909, 77, 589, 333, 845, 205, 717, 461, 973, + 45, 557, 301, 813, 173, 685, 429, 941, 109, 621, 365, 877, + 237, 749, 493, 1005, 29, 541, 285, 797, 157, 669, 413, 925, + 93, 605, 349, 861, 221, 733, 477, 989, 61, 573, 317, 829, + 189, 701, 445, 957, 125, 637, 381, 893, 253, 765, 509, 1021, + 3, 515, 259, 771, 131, 643, 387, 899, 67, 579, 323, 835, + 195, 707, 451, 963, 35, 547, 291, 803, 163, 675, 419, 931, + 99, 611, 355, 867, 227, 739, 483, 995, 19, 531, 275, 787, + 147, 659, 403, 915, 83, 595, 339, 851, 211, 723, 467, 979, + 51, 563, 307, 819, 179, 691, 435, 947, 115, 627, 371, 883, + 243, 755, 499, 1011, 11, 523, 267, 779, 139, 651, 395, 907, + 75, 587, 331, 843, 203, 715, 459, 971, 43, 555, 299, 811, + 171, 683, 427, 939, 107, 619, 363, 875, 235, 747, 491, 1003, + 27, 539, 283, 795, 155, 667, 411, 923, 91, 603, 347, 859, + 219, 731, 475, 987, 59, 571, 315, 827, 187, 699, 443, 955, + 123, 635, 379, 891, 251, 763, 507, 1019, 7, 519, 263, 775, + 135, 647, 391, 903, 71, 583, 327, 839, 199, 711, 455, 967, + 39, 551, 295, 807, 167, 679, 423, 935, 103, 615, 359, 871, + 231, 743, 487, 999, 23, 535, 279, 791, 151, 663, 407, 919, + 87, 599, 343, 855, 215, 727, 471, 983, 55, 567, 311, 823, + 183, 695, 439, 951, 119, 631, 375, 887, 247, 759, 503, 1015, + 15, 527, 271, 783, 143, 655, 399, 911, 79, 591, 335, 847, + 207, 719, 463, 975, 47, 559, 303, 815, 175, 687, 431, 943, + 111, 623, 367, 879, 239, 751, 495, 1007, 31, 543, 287, 799, + 159, 671, 415, 927, 95, 607, 351, 863, 223, 735, 479, 991, + 63, 575, 319, 831, 191, 703, 447, 959, 127, 639, 383, 895, + 255, 767, 511, 1023 +}; + +/* + * Compute the roots for NTT and inverse NTT (binary case). Input + * parameter g is a primitive 2048-th root of 1 modulo p (i.e. g^1024 = + * -1 mod p). This fills gm[] and igm[] with powers of g and 1/g: + * gm[rev(i)] = g^i mod p + * igm[rev(i)] = (1/g)^i mod p + * where rev() is the "bit reversal" function over 10 bits. It fills + * the arrays only up to N = 2^logn values. + * + * The values stored in gm[] and igm[] are in Montgomery representation. + * + * p must be a prime such that p = 1 mod 2048. + */ +static void +modp_mkgm2(uint32_t *gm, uint32_t *igm, unsigned logn, + uint32_t g, uint32_t p, uint32_t p0i) { + size_t u, n; + unsigned k; + uint32_t ig, x1, x2, R2; + + n = (size_t)1 << logn; + + /* + * We want g such that g^(2N) = 1 mod p, but the provided + * generator has order 2048. We must square it a few times. + */ + R2 = modp_R2(p, p0i); + g = modp_montymul(g, R2, p, p0i); + for (k = logn; k < 10; k ++) { + g = modp_montymul(g, g, p, p0i); + } + + ig = modp_div(R2, g, p, p0i, modp_R(p)); + k = 10 - logn; + x1 = x2 = modp_R(p); + for (u = 0; u < n; u ++) { + size_t v; + + v = REV10[u << k]; + gm[v] = x1; + igm[v] = x2; + x1 = modp_montymul(x1, g, p, p0i); + x2 = modp_montymul(x2, ig, p, p0i); + } +} + +/* + * Compute the NTT over a polynomial (binary case). Polynomial elements + * are a[0], a[stride], a[2 * stride]... + */ +static void +modp_NTT2_ext(uint32_t *a, size_t stride, const uint32_t *gm, unsigned logn, + uint32_t p, uint32_t p0i) { + size_t t, m, n; + + if (logn == 0) { + return; + } + n = (size_t)1 << logn; + t = n; + for (m = 1; m < n; m <<= 1) { + size_t ht, u, v1; + + ht = t >> 1; + for (u = 0, v1 = 0; u < m; u ++, v1 += t) { + uint32_t s; + size_t v; + uint32_t *r1, *r2; + + s = gm[m + u]; + r1 = a + v1 * stride; + r2 = r1 + ht * stride; + for (v = 0; v < ht; v ++, r1 += stride, r2 += stride) { + uint32_t x, y; + + x = *r1; + y = modp_montymul(*r2, s, p, p0i); + *r1 = modp_add(x, y, p); + *r2 = modp_sub(x, y, p); + } + } + t = ht; + } +} + +/* + * Compute the inverse NTT over a polynomial (binary case). + */ +static void +modp_iNTT2_ext(uint32_t *a, size_t stride, const uint32_t *igm, unsigned logn, + uint32_t p, uint32_t p0i) { + size_t t, m, n, k; + uint32_t ni; + uint32_t *r; + + if (logn == 0) { + return; + } + n = (size_t)1 << logn; + t = 1; + for (m = n; m > 1; m >>= 1) { + size_t hm, dt, u, v1; + + hm = m >> 1; + dt = t << 1; + for (u = 0, v1 = 0; u < hm; u ++, v1 += dt) { + uint32_t s; + size_t v; + uint32_t *r1, *r2; + + s = igm[hm + u]; + r1 = a + v1 * stride; + r2 = r1 + t * stride; + for (v = 0; v < t; v ++, r1 += stride, r2 += stride) { + uint32_t x, y; + + x = *r1; + y = *r2; + *r1 = modp_add(x, y, p); + *r2 = modp_montymul( + modp_sub(x, y, p), s, p, p0i);; + } + } + t = dt; + } + + /* + * We need 1/n in Montgomery representation, i.e. R/n. Since + * 1 <= logn <= 10, R/n is an integer; morever, R/n <= 2^30 < p, + * thus a simple shift will do. + */ + ni = (uint32_t)1 << (31 - logn); + for (k = 0, r = a; k < n; k ++, r += stride) { + *r = modp_montymul(*r, ni, p, p0i); + } +} + +/* + * Simplified macros for NTT and iNTT (binary case) when the elements + * are consecutive in RAM. + */ +#define modp_NTT2(a, gm, logn, p, p0i) modp_NTT2_ext(a, 1, gm, logn, p, p0i) +#define modp_iNTT2(a, igm, logn, p, p0i) modp_iNTT2_ext(a, 1, igm, logn, p, p0i) + +/* + * Given polynomial f in NTT representation modulo p, compute f' of degree + * less than N/2 such that f' = f0^2 - X*f1^2, where f0 and f1 are + * polynomials of degree less than N/2 such that f = f0(X^2) + X*f1(X^2). + * + * The new polynomial is written "in place" over the first N/2 elements + * of f. + * + * If applied logn times successively on a given polynomial, the resulting + * degree-0 polynomial is the resultant of f and X^N+1 modulo p. + * + * This function applies only to the binary case; it is invoked from + * solve_NTRU_binary_depth1(). + */ +static void +modp_poly_rec_res(uint32_t *f, unsigned logn, + uint32_t p, uint32_t p0i, uint32_t R2) { + size_t hn, u; + + hn = (size_t)1 << (logn - 1); + for (u = 0; u < hn; u ++) { + uint32_t w0, w1; + + w0 = f[(u << 1) + 0]; + w1 = f[(u << 1) + 1]; + f[u] = modp_montymul(modp_montymul(w0, w1, p, p0i), R2, p, p0i); + } +} + +/* ==================================================================== */ +/* + * Custom bignum implementation. + * + * This is a very reduced set of functionalities. We need to do the + * following operations: + * + * - Rebuild the resultant and the polynomial coefficients from their + * values modulo small primes (of length 31 bits each). + * + * - Compute an extended GCD between the two computed resultants. + * + * - Extract top bits and add scaled values during the successive steps + * of Babai rounding. + * + * When rebuilding values using CRT, we must also recompute the product + * of the small prime factors. We always do it one small factor at a + * time, so the "complicated" operations can be done modulo the small + * prime with the modp_* functions. CRT coefficients (inverses) are + * precomputed. + * + * All values are positive until the last step: when the polynomial + * coefficients have been rebuilt, we normalize them around 0. But then, + * only additions and subtractions on the upper few bits are needed + * afterwards. + * + * We keep big integers as arrays of 31-bit words (in uint32_t values); + * the top bit of each uint32_t is kept equal to 0. Using 31-bit words + * makes it easier to keep track of carries. When negative values are + * used, two's complement is used. + */ + +/* + * Subtract integer b from integer a. Both integers are supposed to have + * the same size. The carry (0 or 1) is returned. Source arrays a and b + * MUST be distinct. + * + * The operation is performed as described above if ctr = 1. If + * ctl = 0, the value a[] is unmodified, but all memory accesses are + * still performed, and the carry is computed and returned. + */ +static uint32_t +zint_sub(uint32_t *a, const uint32_t *b, size_t len, + uint32_t ctl) { + size_t u; + uint32_t cc, m; + + cc = 0; + m = -ctl; + for (u = 0; u < len; u ++) { + uint32_t aw, w; + + aw = a[u]; + w = aw - b[u] - cc; + cc = w >> 31; + aw ^= ((w & 0x7FFFFFFF) ^ aw) & m; + a[u] = aw; + } + return cc; +} + +/* + * Mutiply the provided big integer m with a small value x. + * This function assumes that x < 2^31. The carry word is returned. + */ +static uint32_t +zint_mul_small(uint32_t *m, size_t mlen, uint32_t x) { + size_t u; + uint32_t cc; + + cc = 0; + for (u = 0; u < mlen; u ++) { + uint64_t z; + + z = (uint64_t)m[u] * (uint64_t)x + cc; + m[u] = (uint32_t)z & 0x7FFFFFFF; + cc = (uint32_t)(z >> 31); + } + return cc; +} + +/* + * Reduce a big integer d modulo a small integer p. + * Rules: + * d is unsigned + * p is prime + * 2^30 < p < 2^31 + * p0i = -(1/p) mod 2^31 + * R2 = 2^62 mod p + */ +static uint32_t +zint_mod_small_unsigned(const uint32_t *d, size_t dlen, + uint32_t p, uint32_t p0i, uint32_t R2) { + uint32_t x; + size_t u; + + /* + * Algorithm: we inject words one by one, starting with the high + * word. Each step is: + * - multiply x by 2^31 + * - add new word + */ + x = 0; + u = dlen; + while (u -- > 0) { + uint32_t w; + + x = modp_montymul(x, R2, p, p0i); + w = d[u] - p; + w += p & -(w >> 31); + x = modp_add(x, w, p); + } + return x; +} + +/* + * Similar to zint_mod_small_unsigned(), except that d may be signed. + * Extra parameter is Rx = 2^(31*dlen) mod p. + */ +static uint32_t +zint_mod_small_signed(const uint32_t *d, size_t dlen, + uint32_t p, uint32_t p0i, uint32_t R2, uint32_t Rx) { + uint32_t z; + + if (dlen == 0) { + return 0; + } + z = zint_mod_small_unsigned(d, dlen, p, p0i, R2); + z = modp_sub(z, Rx & -(d[dlen - 1] >> 30), p); + return z; +} + +/* + * Add y*s to x. x and y initially have length 'len' words; the new x + * has length 'len+1' words. 's' must fit on 31 bits. x[] and y[] must + * not overlap. + */ +static void +zint_add_mul_small(uint32_t *x, + const uint32_t *y, size_t len, uint32_t s) { + size_t u; + uint32_t cc; + + cc = 0; + for (u = 0; u < len; u ++) { + uint32_t xw, yw; + uint64_t z; + + xw = x[u]; + yw = y[u]; + z = (uint64_t)yw * (uint64_t)s + (uint64_t)xw + (uint64_t)cc; + x[u] = (uint32_t)z & 0x7FFFFFFF; + cc = (uint32_t)(z >> 31); + } + x[len] = cc; +} + +/* + * Normalize a modular integer around 0: if x > p/2, then x is replaced + * with x - p (signed encoding with two's complement); otherwise, x is + * untouched. The two integers x and p are encoded over the same length. + */ +static void +zint_norm_zero(uint32_t *x, const uint32_t *p, size_t len) { + size_t u; + uint32_t r, bb; + + /* + * Compare x with p/2. We use the shifted version of p, and p + * is odd, so we really compare with (p-1)/2; we want to perform + * the subtraction if and only if x > (p-1)/2. + */ + r = 0; + bb = 0; + u = len; + while (u -- > 0) { + uint32_t wx, wp, cc; + + /* + * Get the two words to compare in wx and wp (both over + * 31 bits exactly). + */ + wx = x[u]; + wp = (p[u] >> 1) | (bb << 30); + bb = p[u] & 1; + + /* + * We set cc to -1, 0 or 1, depending on whether wp is + * lower than, equal to, or greater than wx. + */ + cc = wp - wx; + cc = ((-cc) >> 31) | -(cc >> 31); + + /* + * If r != 0 then it is either 1 or -1, and we keep its + * value. Otherwise, if r = 0, then we replace it with cc. + */ + r |= cc & ((r & 1) - 1); + } + + /* + * At this point, r = -1, 0 or 1, depending on whether (p-1)/2 + * is lower than, equal to, or greater than x. We thus want to + * do the subtraction only if r = -1. + */ + zint_sub(x, p, len, r >> 31); +} + +/* + * Rebuild integers from their RNS representation. There are 'num' + * integers, and each consists in 'xlen' words. 'xx' points at that + * first word of the first integer; subsequent integers are accessed + * by adding 'xstride' repeatedly. + * + * The words of an integer are the RNS representation of that integer, + * using the provided 'primes' are moduli. This function replaces + * each integer with its multi-word value (little-endian order). + * + * If "normalize_signed" is non-zero, then the returned value is + * normalized to the -m/2..m/2 interval (where m is the product of all + * small prime moduli); two's complement is used for negative values. + */ +static void +zint_rebuild_CRT(uint32_t *xx, size_t xlen, size_t xstride, + size_t num, const small_prime *primes, int normalize_signed, + uint32_t *tmp) { + size_t u; + uint32_t *x; + + tmp[0] = primes[0].p; + for (u = 1; u < xlen; u ++) { + /* + * At the entry of each loop iteration: + * - the first u words of each array have been + * reassembled; + * - the first u words of tmp[] contains the + * product of the prime moduli processed so far. + * + * We call 'q' the product of all previous primes. + */ + uint32_t p, p0i, s, R2; + size_t v; + + p = primes[u].p; + s = primes[u].s; + p0i = modp_ninv31(p); + R2 = modp_R2(p, p0i); + + for (v = 0, x = xx; v < num; v ++, x += xstride) { + uint32_t xp, xq, xr; + /* + * xp = the integer x modulo the prime p for this + * iteration + * xq = (x mod q) mod p + */ + xp = x[u]; + xq = zint_mod_small_unsigned(x, u, p, p0i, R2); + + /* + * New value is (x mod q) + q * (s * (xp - xq) mod p) + */ + xr = modp_montymul(s, modp_sub(xp, xq, p), p, p0i); + zint_add_mul_small(x, tmp, u, xr); + } + + /* + * Update product of primes in tmp[]. + */ + tmp[u] = zint_mul_small(tmp, u, p); + } + + /* + * Normalize the reconstructed values around 0. + */ + if (normalize_signed) { + for (u = 0, x = xx; u < num; u ++, x += xstride) { + zint_norm_zero(x, tmp, xlen); + } + } +} + +/* + * Negate a big integer conditionally: value a is replaced with -a if + * and only if ctl = 1. Control value ctl must be 0 or 1. + */ +static void +zint_negate(uint32_t *a, size_t len, uint32_t ctl) { + size_t u; + uint32_t cc, m; + + /* + * If ctl = 1 then we flip the bits of a by XORing with + * 0x7FFFFFFF, and we add 1 to the value. If ctl = 0 then we XOR + * with 0 and add 0, which leaves the value unchanged. + */ + cc = ctl; + m = -ctl >> 1; + for (u = 0; u < len; u ++) { + uint32_t aw; + + aw = a[u]; + aw = (aw ^ m) + cc; + a[u] = aw & 0x7FFFFFFF; + cc = aw >> 31; + } +} + +/* + * Replace a with (a*xa+b*xb)/(2^31) and b with (a*ya+b*yb)/(2^31). + * The low bits are dropped (the caller should compute the coefficients + * such that these dropped bits are all zeros). If either or both + * yields a negative value, then the value is negated. + * + * Returned value is: + * 0 both values were positive + * 1 new a had to be negated + * 2 new b had to be negated + * 3 both new a and new b had to be negated + * + * Coefficients xa, xb, ya and yb may use the full signed 32-bit range. + */ +static uint32_t +zint_co_reduce(uint32_t *a, uint32_t *b, size_t len, + int64_t xa, int64_t xb, int64_t ya, int64_t yb) { + size_t u; + int64_t cca, ccb; + uint32_t nega, negb; + + cca = 0; + ccb = 0; + for (u = 0; u < len; u ++) { + uint32_t wa, wb; + uint64_t za, zb; + + wa = a[u]; + wb = b[u]; + za = wa * (uint64_t)xa + wb * (uint64_t)xb + (uint64_t)cca; + zb = wa * (uint64_t)ya + wb * (uint64_t)yb + (uint64_t)ccb; + if (u > 0) { + a[u - 1] = (uint32_t)za & 0x7FFFFFFF; + b[u - 1] = (uint32_t)zb & 0x7FFFFFFF; + } + cca = *(int64_t *)&za >> 31; + ccb = *(int64_t *)&zb >> 31; + } + a[len - 1] = (uint32_t)cca; + b[len - 1] = (uint32_t)ccb; + + nega = (uint32_t)((uint64_t)cca >> 63); + negb = (uint32_t)((uint64_t)ccb >> 63); + zint_negate(a, len, nega); + zint_negate(b, len, negb); + return nega | (negb << 1); +} + +/* + * Finish modular reduction. Rules on input parameters: + * + * if neg = 1, then -m <= a < 0 + * if neg = 0, then 0 <= a < 2*m + * + * If neg = 0, then the top word of a[] is allowed to use 32 bits. + * + * Modulus m must be odd. + */ +static void +zint_finish_mod(uint32_t *a, size_t len, const uint32_t *m, uint32_t neg) { + size_t u; + uint32_t cc, xm, ym; + + /* + * First pass: compare a (assumed nonnegative) with m. Note that + * if the top word uses 32 bits, subtracting m must yield a + * value less than 2^31 since a < 2*m. + */ + cc = 0; + for (u = 0; u < len; u ++) { + cc = (a[u] - m[u] - cc) >> 31; + } + + /* + * If neg = 1 then we must add m (regardless of cc) + * If neg = 0 and cc = 0 then we must subtract m + * If neg = 0 and cc = 1 then we must do nothing + * + * In the loop below, we conditionally subtract either m or -m + * from a. Word xm is a word of m (if neg = 0) or -m (if neg = 1); + * but if neg = 0 and cc = 1, then ym = 0 and it forces mw to 0. + */ + xm = -neg >> 1; + ym = -(neg | (1 - cc)); + cc = neg; + for (u = 0; u < len; u ++) { + uint32_t aw, mw; + + aw = a[u]; + mw = (m[u] ^ xm) & ym; + aw = aw - mw - cc; + a[u] = aw & 0x7FFFFFFF; + cc = aw >> 31; + } +} + +/* + * Replace a with (a*xa+b*xb)/(2^31) mod m, and b with + * (a*ya+b*yb)/(2^31) mod m. Modulus m must be odd; m0i = -1/m[0] mod 2^31. + */ +static void +zint_co_reduce_mod(uint32_t *a, uint32_t *b, const uint32_t *m, size_t len, + uint32_t m0i, int64_t xa, int64_t xb, int64_t ya, int64_t yb) { + size_t u; + int64_t cca, ccb; + uint32_t fa, fb; + + /* + * These are actually four combined Montgomery multiplications. + */ + cca = 0; + ccb = 0; + fa = ((a[0] * (uint32_t)xa + b[0] * (uint32_t)xb) * m0i) & 0x7FFFFFFF; + fb = ((a[0] * (uint32_t)ya + b[0] * (uint32_t)yb) * m0i) & 0x7FFFFFFF; + for (u = 0; u < len; u ++) { + uint32_t wa, wb; + uint64_t za, zb; + + wa = a[u]; + wb = b[u]; + za = wa * (uint64_t)xa + wb * (uint64_t)xb + + m[u] * (uint64_t)fa + (uint64_t)cca; + zb = wa * (uint64_t)ya + wb * (uint64_t)yb + + m[u] * (uint64_t)fb + (uint64_t)ccb; + if (u > 0) { + a[u - 1] = (uint32_t)za & 0x7FFFFFFF; + b[u - 1] = (uint32_t)zb & 0x7FFFFFFF; + } + cca = *(int64_t *)&za >> 31; + ccb = *(int64_t *)&zb >> 31; + } + a[len - 1] = (uint32_t)cca; + b[len - 1] = (uint32_t)ccb; + + /* + * At this point: + * -m <= a < 2*m + * -m <= b < 2*m + * (this is a case of Montgomery reduction) + * The top words of 'a' and 'b' may have a 32-th bit set. + * We want to add or subtract the modulus, as required. + */ + zint_finish_mod(a, len, m, (uint32_t)((uint64_t)cca >> 63)); + zint_finish_mod(b, len, m, (uint32_t)((uint64_t)ccb >> 63)); +} + +/* + * Compute a GCD between two positive big integers x and y. The two + * integers must be odd. Returned value is 1 if the GCD is 1, 0 + * otherwise. When 1 is returned, arrays u and v are filled with values + * such that: + * 0 <= u <= y + * 0 <= v <= x + * x*u - y*v = 1 + * x[] and y[] are unmodified. Both input values must have the same + * encoded length. Temporary array must be large enough to accommodate 4 + * extra values of that length. Arrays u, v and tmp may not overlap with + * each other, or with either x or y. + */ +static int +zint_bezout(uint32_t *u, uint32_t *v, + const uint32_t *x, const uint32_t *y, + size_t len, uint32_t *tmp) { + /* + * Algorithm is an extended binary GCD. We maintain 6 values + * a, b, u0, u1, v0 and v1 with the following invariants: + * + * a = x*u0 - y*v0 + * b = x*u1 - y*v1 + * 0 <= a <= x + * 0 <= b <= y + * 0 <= u0 < y + * 0 <= v0 < x + * 0 <= u1 <= y + * 0 <= v1 < x + * + * Initial values are: + * + * a = x u0 = 1 v0 = 0 + * b = y u1 = y v1 = x-1 + * + * Each iteration reduces either a or b, and maintains the + * invariants. Algorithm stops when a = b, at which point their + * common value is GCD(a,b) and (u0,v0) (or (u1,v1)) contains + * the values (u,v) we want to return. + * + * The formal definition of the algorithm is a sequence of steps: + * + * - If a is even, then: + * a <- a/2 + * u0 <- u0/2 mod y + * v0 <- v0/2 mod x + * + * - Otherwise, if b is even, then: + * b <- b/2 + * u1 <- u1/2 mod y + * v1 <- v1/2 mod x + * + * - Otherwise, if a > b, then: + * a <- (a-b)/2 + * u0 <- (u0-u1)/2 mod y + * v0 <- (v0-v1)/2 mod x + * + * - Otherwise: + * b <- (b-a)/2 + * u1 <- (u1-u0)/2 mod y + * v1 <- (v1-v0)/2 mod y + * + * We can show that the operations above preserve the invariants: + * + * - If a is even, then u0 and v0 are either both even or both + * odd (since a = x*u0 - y*v0, and x and y are both odd). + * If u0 and v0 are both even, then (u0,v0) <- (u0/2,v0/2). + * Otherwise, (u0,v0) <- ((u0+y)/2,(v0+x)/2). Either way, + * the a = x*u0 - y*v0 invariant is preserved. + * + * - The same holds for the case where b is even. + * + * - If a and b are odd, and a > b, then: + * + * a-b = x*(u0-u1) - y*(v0-v1) + * + * In that situation, if u0 < u1, then x*(u0-u1) < 0, but + * a-b > 0; therefore, it must be that v0 < v1, and the + * first part of the update is: (u0,v0) <- (u0-u1+y,v0-v1+x), + * which preserves the invariants. Otherwise, if u0 > u1, + * then u0-u1 >= 1, thus x*(u0-u1) >= x. But a <= x and + * b >= 0, hence a-b <= x. It follows that, in that case, + * v0-v1 >= 0. The first part of the update is then: + * (u0,v0) <- (u0-u1,v0-v1), which again preserves the + * invariants. + * + * Either way, once the subtraction is done, the new value of + * a, which is the difference of two odd values, is even, + * and the remaining of this step is a subcase of the + * first algorithm case (i.e. when a is even). + * + * - If a and b are odd, and b > a, then the a similar + * argument holds. + * + * The values a and b start at x and y, respectively. Since x + * and y are odd, their GCD is odd, and it is easily seen that + * all steps conserve the GCD (GCD(a-b,b) = GCD(a, b); + * GCD(a/2,b) = GCD(a,b) if GCD(a,b) is odd). Moreover, either a + * or b is reduced by at least one bit at each iteration, so + * the algorithm necessarily converges on the case a = b, at + * which point the common value is the GCD. + * + * In the algorithm expressed above, when a = b, the fourth case + * applies, and sets b = 0. Since a contains the GCD of x and y, + * which are both odd, a must be odd, and subsequent iterations + * (if any) will simply divide b by 2 repeatedly, which has no + * consequence. Thus, the algorithm can run for more iterations + * than necessary; the final GCD will be in a, and the (u,v) + * coefficients will be (u0,v0). + * + * + * The presentation above is bit-by-bit. It can be sped up by + * noticing that all decisions are taken based on the low bits + * and high bits of a and b. We can extract the two top words + * and low word of each of a and b, and compute reduction + * parameters pa, pb, qa and qb such that the new values for + * a and b are: + * a' = (a*pa + b*pb) / (2^31) + * b' = (a*qa + b*qb) / (2^31) + * the two divisions being exact. The coefficients are obtained + * just from the extracted words, and may be slightly off, requiring + * an optional correction: if a' < 0, then we replace pa with -pa + * and pb with -pb. Each such step will reduce the total length + * (sum of lengths of a and b) by at least 30 bits at each + * iteration. + */ + uint32_t *u0, *u1, *v0, *v1, *a, *b; + uint32_t x0i, y0i; + uint32_t num, rc; + size_t j; + + if (len == 0) { + return 0; + } + + /* + * u0 and v0 are the u and v result buffers; the four other + * values (u1, v1, a and b) are taken from tmp[]. + */ + u0 = u; + v0 = v; + u1 = tmp; + v1 = u1 + len; + a = v1 + len; + b = a + len; + + /* + * We'll need the Montgomery reduction coefficients. + */ + x0i = modp_ninv31(x[0]); + y0i = modp_ninv31(y[0]); + + /* + * Initialize a, b, u0, u1, v0 and v1. + * a = x u0 = 1 v0 = 0 + * b = y u1 = y v1 = x-1 + * Note that x is odd, so computing x-1 is easy. + */ + memcpy(a, x, len * sizeof * x); + memcpy(b, y, len * sizeof * y); + u0[0] = 1; + memset(u0 + 1, 0, (len - 1) * sizeof * u0); + memset(v0, 0, len * sizeof * v0); + memcpy(u1, y, len * sizeof * u1); + memcpy(v1, x, len * sizeof * v1); + v1[0] --; + + /* + * Each input operand may be as large as 31*len bits, and we + * reduce the total length by at least 30 bits at each iteration. + */ + for (num = 62 * (uint32_t)len + 30; num >= 30; num -= 30) { + uint32_t c0, c1; + uint32_t a0, a1, b0, b1; + uint64_t a_hi, b_hi; + uint32_t a_lo, b_lo; + int64_t pa, pb, qa, qb; + int i; + uint32_t r; + + /* + * Extract the top words of a and b. If j is the highest + * index >= 1 such that a[j] != 0 or b[j] != 0, then we + * want (a[j] << 31) + a[j-1] and (b[j] << 31) + b[j-1]. + * If a and b are down to one word each, then we use + * a[0] and b[0]. + */ + c0 = (uint32_t) -1; + c1 = (uint32_t) -1; + a0 = 0; + a1 = 0; + b0 = 0; + b1 = 0; + j = len; + while (j -- > 0) { + uint32_t aw, bw; + + aw = a[j]; + bw = b[j]; + a0 ^= (a0 ^ aw) & c0; + a1 ^= (a1 ^ aw) & c1; + b0 ^= (b0 ^ bw) & c0; + b1 ^= (b1 ^ bw) & c1; + c1 = c0; + c0 &= (((aw | bw) + 0x7FFFFFFF) >> 31) - (uint32_t)1; + } + + /* + * If c1 = 0, then we grabbed two words for a and b. + * If c1 != 0 but c0 = 0, then we grabbed one word. It + * is not possible that c1 != 0 and c0 != 0, because that + * would mean that both integers are zero. + */ + a1 |= a0 & c1; + a0 &= ~c1; + b1 |= b0 & c1; + b0 &= ~c1; + a_hi = ((uint64_t)a0 << 31) + a1; + b_hi = ((uint64_t)b0 << 31) + b1; + a_lo = a[0]; + b_lo = b[0]; + + /* + * Compute reduction factors: + * + * a' = a*pa + b*pb + * b' = a*qa + b*qb + * + * such that a' and b' are both multiple of 2^31, but are + * only marginally larger than a and b. + */ + pa = 1; + pb = 0; + qa = 0; + qb = 1; + for (i = 0; i < 31; i ++) { + /* + * At each iteration: + * + * a <- (a-b)/2 if: a is odd, b is odd, a_hi > b_hi + * b <- (b-a)/2 if: a is odd, b is odd, a_hi <= b_hi + * a <- a/2 if: a is even + * b <- b/2 if: a is odd, b is even + * + * We multiply a_lo and b_lo by 2 at each + * iteration, thus a division by 2 really is a + * non-multiplication by 2. + */ + uint32_t rt, oa, ob, cAB, cBA, cA; + uint64_t rz; + + /* + * rt = 1 if a_hi > b_hi, 0 otherwise. + */ + rz = b_hi - a_hi; + rt = (uint32_t)((rz ^ ((a_hi ^ b_hi) + & (a_hi ^ rz))) >> 63); + + /* + * cAB = 1 if b must be subtracted from a + * cBA = 1 if a must be subtracted from b + * cA = 1 if a must be divided by 2 + * + * Rules: + * + * cAB and cBA cannot both be 1. + * If a is not divided by 2, b is. + */ + oa = (a_lo >> i) & 1; + ob = (b_lo >> i) & 1; + cAB = oa & ob & rt; + cBA = oa & ob & ~rt; + cA = cAB | (oa ^ 1); + + /* + * Conditional subtractions. + */ + a_lo -= b_lo & -cAB; + a_hi -= b_hi & -(uint64_t)cAB; + pa -= qa & -(int64_t)cAB; + pb -= qb & -(int64_t)cAB; + b_lo -= a_lo & -cBA; + b_hi -= a_hi & -(uint64_t)cBA; + qa -= pa & -(int64_t)cBA; + qb -= pb & -(int64_t)cBA; + + /* + * Shifting. + */ + a_lo += a_lo & (cA - 1); + pa += pa & ((int64_t)cA - 1); + pb += pb & ((int64_t)cA - 1); + a_hi ^= (a_hi ^ (a_hi >> 1)) & -(uint64_t)cA; + b_lo += b_lo & -cA; + qa += qa & -(int64_t)cA; + qb += qb & -(int64_t)cA; + b_hi ^= (b_hi ^ (b_hi >> 1)) & ((uint64_t)cA - 1); + } + + /* + * Apply the computed parameters to our values. We + * may have to correct pa and pb depending on the + * returned value of zint_co_reduce() (when a and/or b + * had to be negated). + */ + r = zint_co_reduce(a, b, len, pa, pb, qa, qb); + pa -= (pa + pa) & -(int64_t)(r & 1); + pb -= (pb + pb) & -(int64_t)(r & 1); + qa -= (qa + qa) & -(int64_t)(r >> 1); + qb -= (qb + qb) & -(int64_t)(r >> 1); + zint_co_reduce_mod(u0, u1, y, len, y0i, pa, pb, qa, qb); + zint_co_reduce_mod(v0, v1, x, len, x0i, pa, pb, qa, qb); + } + + /* + * At that point, array a[] should contain the GCD, and the + * results (u,v) should already be set. We check that the GCD + * is indeed 1. We also check that the two operands x and y + * are odd. + */ + rc = a[0] ^ 1; + for (j = 1; j < len; j ++) { + rc |= a[j]; + } + return (int)((1 - ((rc | -rc) >> 31)) & x[0] & y[0]); +} + +/* + * Add k*y*2^sc to x. The result is assumed to fit in the array of + * size xlen (truncation is applied if necessary). + * Scale factor 'sc' is provided as sch and scl, such that: + * sch = sc / 31 + * scl = sc % 31 + * xlen MUST NOT be lower than ylen. + * + * x[] and y[] are both signed integers, using two's complement for + * negative values. + */ +static void +zint_add_scaled_mul_small(uint32_t *x, size_t xlen, + const uint32_t *y, size_t ylen, int32_t k, + uint32_t sch, uint32_t scl) { + size_t u; + uint32_t ysign, tw; + int32_t cc; + + if (ylen == 0) { + return; + } + + ysign = -(y[ylen - 1] >> 30) >> 1; + tw = 0; + cc = 0; + for (u = sch; u < xlen; u ++) { + size_t v; + uint32_t wy, wys, ccu; + uint64_t z; + + /* + * Get the next word of y (scaled). + */ + v = u - sch; + if (v < ylen) { + wy = y[v]; + } else { + wy = ysign; + } + wys = ((wy << scl) & 0x7FFFFFFF) | tw; + tw = wy >> (31 - scl); + + /* + * The expression below does not overflow. + */ + z = (uint64_t)((int64_t)wys * (int64_t)k + (int64_t)x[u] + cc); + x[u] = (uint32_t)z & 0x7FFFFFFF; + + /* + * Right-shifting the signed value z would yield + * implementation-defined results (arithmetic shift is + * not guaranteed). However, we can cast to unsigned, + * and get the next carry as an unsigned word. We can + * then convert it back to signed by using the guaranteed + * fact that 'int32_t' uses two's complement with no + * trap representation or padding bit, and with a layout + * compatible with that of 'uint32_t'. + */ + ccu = (uint32_t)(z >> 31); + cc = *(int32_t *)&ccu; + } +} + +/* + * Subtract y*2^sc from x. The result is assumed to fit in the array of + * size xlen (truncation is applied if necessary). + * Scale factor 'sc' is provided as sch and scl, such that: + * sch = sc / 31 + * scl = sc % 31 + * xlen MUST NOT be lower than ylen. + * + * x[] and y[] are both signed integers, using two's complement for + * negative values. + */ +static void +zint_sub_scaled(uint32_t *x, size_t xlen, + const uint32_t *y, size_t ylen, uint32_t sch, uint32_t scl) { + size_t u; + uint32_t ysign, tw; + uint32_t cc; + + if (ylen == 0) { + return; + } + + ysign = -(y[ylen - 1] >> 30) >> 1; + tw = 0; + cc = 0; + for (u = sch; u < xlen; u ++) { + size_t v; + uint32_t w, wy, wys; + + /* + * Get the next word of y (scaled). + */ + v = u - sch; + if (v < ylen) { + wy = y[v]; + } else { + wy = ysign; + } + wys = ((wy << scl) & 0x7FFFFFFF) | tw; + tw = wy >> (31 - scl); + + w = x[u] - wys - cc; + x[u] = w & 0x7FFFFFFF; + cc = w >> 31; + } +} + +/* + * Convert a one-word signed big integer into a signed value. + */ +static inline int32_t +zint_one_to_plain(const uint32_t *x) { + uint32_t w; + + w = x[0]; + w |= (w & 0x40000000) << 1; + return *(int32_t *)&w; +} + +/* ==================================================================== */ + +/* + * Convert a polynomial to floating-point values. + * + * Each coefficient has length flen words, and starts fstride words after + * the previous. + * + * IEEE-754 binary64 values can represent values in a finite range, + * roughly 2^(-1023) to 2^(+1023); thus, if coefficients are too large, + * they should be "trimmed" by pointing not to the lowest word of each, + * but upper. + */ +static void +poly_big_to_fp(fpr *d, const uint32_t *f, size_t flen, size_t fstride, + unsigned logn) { + size_t n, u; + + n = MKN(logn); + if (flen == 0) { + for (u = 0; u < n; u ++) { + d[u] = fpr_zero; + } + return; + } + for (u = 0; u < n; u ++, f += fstride) { + size_t v; + uint32_t neg, cc, xm; + fpr x, fsc; + + /* + * Get sign of the integer; if it is negative, then we + * will load its absolute value instead, and negate the + * result. + */ + neg = -(f[flen - 1] >> 30); + xm = neg >> 1; + cc = neg & 1; + x = fpr_zero; + fsc = fpr_one; + for (v = 0; v < flen; v ++, fsc = fpr_mul(fsc, fpr_ptwo31)) { + uint32_t w; + + w = (f[v] ^ xm) + cc; + cc = w >> 31; + w &= 0x7FFFFFFF; + w -= (w << 1) & neg; + x = fpr_add(x, fpr_mul(fpr_of(*(int32_t *)&w), fsc)); + } + d[u] = x; + } +} + +/* + * Convert a polynomial to small integers. Source values are supposed + * to be one-word integers, signed over 31 bits. Returned value is 0 + * if any of the coefficients exceeds the provided limit (in absolute + * value), or 1 on success. + * + * This is not constant-time; this is not a problem here, because on + * any failure, the NTRU-solving process will be deemed to have failed + * and the (f,g) polynomials will be discarded. + */ +static int +poly_big_to_small(int8_t *d, const uint32_t *s, int lim, unsigned logn) { + size_t n, u; + + n = MKN(logn); + for (u = 0; u < n; u ++) { + int32_t z; + + z = zint_one_to_plain(s + u); + if (z < -lim || z > lim) { + return 0; + } + d[u] = (int8_t)z; + } + return 1; +} + +/* + * Subtract k*f from F, where F, f and k are polynomials modulo X^N+1. + * Coefficients of polynomial k are small integers (signed values in the + * -2^31..2^31 range) scaled by 2^sc. Value sc is provided as sch = sc / 31 + * and scl = sc % 31. + * + * This function implements the basic quadratic multiplication algorithm, + * which is efficient in space (no extra buffer needed) but slow at + * high degree. + */ +static void +poly_sub_scaled(uint32_t *F, size_t Flen, size_t Fstride, + const uint32_t *f, size_t flen, size_t fstride, + const int32_t *k, uint32_t sch, uint32_t scl, unsigned logn) { + size_t n, u; + + n = MKN(logn); + for (u = 0; u < n; u ++) { + int32_t kf; + size_t v; + uint32_t *x; + const uint32_t *y; + + kf = -k[u]; + x = F + u * Fstride; + y = f; + for (v = 0; v < n; v ++) { + zint_add_scaled_mul_small( + x, Flen, y, flen, kf, sch, scl); + if (u + v == n - 1) { + x = F; + kf = -kf; + } else { + x += Fstride; + } + y += fstride; + } + } +} + +/* + * Subtract k*f from F. Coefficients of polynomial k are small integers + * (signed values in the -2^31..2^31 range) scaled by 2^sc. This function + * assumes that the degree is large, and integers relatively small. + * The value sc is provided as sch = sc / 31 and scl = sc % 31. + */ +static void +poly_sub_scaled_ntt(uint32_t *F, size_t Flen, size_t Fstride, + const uint32_t *f, size_t flen, size_t fstride, + const int32_t *k, uint32_t sch, uint32_t scl, unsigned logn, + uint32_t *tmp) { + uint32_t *gm, *igm, *fk, *t1, *x; + const uint32_t *y; + size_t n, u, tlen; + const small_prime *primes; + + n = MKN(logn); + tlen = flen + 1; + gm = tmp; + igm = gm + MKN(logn); + fk = igm + MKN(logn); + t1 = fk + n * tlen; + + primes = PRIMES; + + /* + * Compute k*f in fk[], in RNS notation. + */ + for (u = 0; u < tlen; u ++) { + uint32_t p, p0i, R2, Rx; + size_t v; + + p = primes[u].p; + p0i = modp_ninv31(p); + R2 = modp_R2(p, p0i); + Rx = modp_Rx((unsigned)flen, p, p0i, R2); + modp_mkgm2(gm, igm, logn, primes[u].g, p, p0i); + + for (v = 0; v < n; v ++) { + t1[v] = modp_set(k[v], p); + } + modp_NTT2(t1, gm, logn, p, p0i); + for (v = 0, y = f, x = fk + u; + v < n; v ++, y += fstride, x += tlen) { + *x = zint_mod_small_signed(y, flen, p, p0i, R2, Rx); + } + modp_NTT2_ext(fk + u, tlen, gm, logn, p, p0i); + for (v = 0, x = fk + u; v < n; v ++, x += tlen) { + *x = modp_montymul( + modp_montymul(t1[v], *x, p, p0i), R2, p, p0i); + } + modp_iNTT2_ext(fk + u, tlen, igm, logn, p, p0i); + } + + /* + * Rebuild k*f. + */ + zint_rebuild_CRT(fk, tlen, tlen, n, primes, 1, t1); + + /* + * Subtract k*f, scaled, from F. + */ + for (u = 0, x = F, y = fk; u < n; u ++, x += Fstride, y += tlen) { + zint_sub_scaled(x, Flen, y, tlen, sch, scl); + } +} + +/* ==================================================================== */ + + +#define RNG_CONTEXT inner_shake256_context + +/* + * Get a random 8-byte integer from a SHAKE-based RNG. This function + * ensures consistent interpretation of the SHAKE output so that + * the same values will be obtained over different platforms, in case + * a known seed is used. + */ +static inline uint64_t +get_rng_u64(inner_shake256_context *rng) { + /* + * We enforce little-endian representation. + */ + + uint8_t tmp[8]; + + inner_shake256_extract(rng, tmp, sizeof tmp); + return (uint64_t)tmp[0] + | ((uint64_t)tmp[1] << 8) + | ((uint64_t)tmp[2] << 16) + | ((uint64_t)tmp[3] << 24) + | ((uint64_t)tmp[4] << 32) + | ((uint64_t)tmp[5] << 40) + | ((uint64_t)tmp[6] << 48) + | ((uint64_t)tmp[7] << 56); +} + +/* + * Table below incarnates a discrete Gaussian distribution: + * D(x) = exp(-(x^2)/(2*sigma^2)) + * where sigma = 1.17*sqrt(q/(2*N)), q = 12289, and N = 1024. + * Element 0 of the table is P(x = 0). + * For k > 0, element k is P(x >= k+1 | x > 0). + * Probabilities are scaled up by 2^63. + */ +static const uint64_t gauss_1024_12289[] = { + 1283868770400643928u, 6416574995475331444u, 4078260278032692663u, + 2353523259288686585u, 1227179971273316331u, 575931623374121527u, + 242543240509105209u, 91437049221049666u, 30799446349977173u, + 9255276791179340u, 2478152334826140u, 590642893610164u, + 125206034929641u, 23590435911403u, 3948334035941u, + 586753615614u, 77391054539u, 9056793210u, + 940121950u, 86539696u, 7062824u, + 510971u, 32764u, 1862u, + 94u, 4u, 0u +}; + +/* + * Generate a random value with a Gaussian distribution centered on 0. + * The RNG must be ready for extraction (already flipped). + * + * Distribution has standard deviation 1.17*sqrt(q/(2*N)). The + * precomputed table is for N = 1024. Since the sum of two independent + * values of standard deviation sigma has standard deviation + * sigma*sqrt(2), then we can just generate more values and add them + * together for lower dimensions. + */ +static int +mkgauss(RNG_CONTEXT *rng, unsigned logn) { + unsigned u, g; + int val; + + g = 1U << (10 - logn); + val = 0; + for (u = 0; u < g; u ++) { + /* + * Each iteration generates one value with the + * Gaussian distribution for N = 1024. + * + * We use two random 64-bit values. First value + * decides on whether the generated value is 0, and, + * if not, the sign of the value. Second random 64-bit + * word is used to generate the non-zero value. + * + * For constant-time code we have to read the complete + * table. This has negligible cost, compared with the + * remainder of the keygen process (solving the NTRU + * equation). + */ + uint64_t r; + uint32_t f, v, k, neg; + + /* + * First value: + * - flag 'neg' is randomly selected to be 0 or 1. + * - flag 'f' is set to 1 if the generated value is zero, + * or set to 0 otherwise. + */ + r = get_rng_u64(rng); + neg = (uint32_t)(r >> 63); + r &= ~((uint64_t)1 << 63); + f = (uint32_t)((r - gauss_1024_12289[0]) >> 63); + + /* + * We produce a new random 63-bit integer r, and go over + * the array, starting at index 1. We store in v the + * index of the first array element which is not greater + * than r, unless the flag f was already 1. + */ + v = 0; + r = get_rng_u64(rng); + r &= ~((uint64_t)1 << 63); + for (k = 1; k < (uint32_t)((sizeof gauss_1024_12289) + / (sizeof gauss_1024_12289[0])); k ++) { + uint32_t t; + + t = (uint32_t)((r - gauss_1024_12289[k]) >> 63) ^ 1; + v |= k & -(t & (f ^ 1)); + f |= t; + } + + /* + * We apply the sign ('neg' flag). If the value is zero, + * the sign has no effect. + */ + v = (v ^ -neg) + neg; + + /* + * Generated value is added to val. + */ + val += *(int32_t *)&v; + } + return val; +} + +/* + * The MAX_BL_SMALL[] and MAX_BL_LARGE[] contain the lengths, in 31-bit + * words, of intermediate values in the computation: + * + * MAX_BL_SMALL[depth]: length for the input f and g at that depth + * MAX_BL_LARGE[depth]: length for the unreduced F and G at that depth + * + * Rules: + * + * - Within an array, values grow. + * + * - The 'SMALL' array must have an entry for maximum depth, corresponding + * to the size of values used in the binary GCD. There is no such value + * for the 'LARGE' array (the binary GCD yields already reduced + * coefficients). + * + * - MAX_BL_LARGE[depth] >= MAX_BL_SMALL[depth + 1]. + * + * - Values must be large enough to handle the common cases, with some + * margins. + * + * - Values must not be "too large" either because we will convert some + * integers into floating-point values by considering the top 10 words, + * i.e. 310 bits; hence, for values of length more than 10 words, we + * should take care to have the length centered on the expected size. + * + * The following average lengths, in bits, have been measured on thousands + * of random keys (fg = max length of the absolute value of coefficients + * of f and g at that depth; FG = idem for the unreduced F and G; for the + * maximum depth, F and G are the output of binary GCD, multiplied by q; + * for each value, the average and standard deviation are provided). + * + * Binary case: + * depth: 10 fg: 6307.52 (24.48) FG: 6319.66 (24.51) + * depth: 9 fg: 3138.35 (12.25) FG: 9403.29 (27.55) + * depth: 8 fg: 1576.87 ( 7.49) FG: 4703.30 (14.77) + * depth: 7 fg: 794.17 ( 4.98) FG: 2361.84 ( 9.31) + * depth: 6 fg: 400.67 ( 3.10) FG: 1188.68 ( 6.04) + * depth: 5 fg: 202.22 ( 1.87) FG: 599.81 ( 3.87) + * depth: 4 fg: 101.62 ( 1.02) FG: 303.49 ( 2.38) + * depth: 3 fg: 50.37 ( 0.53) FG: 153.65 ( 1.39) + * depth: 2 fg: 24.07 ( 0.25) FG: 78.20 ( 0.73) + * depth: 1 fg: 10.99 ( 0.08) FG: 39.82 ( 0.41) + * depth: 0 fg: 4.00 ( 0.00) FG: 19.61 ( 0.49) + * + * Integers are actually represented either in binary notation over + * 31-bit words (signed, using two's complement), or in RNS, modulo + * many small primes. These small primes are close to, but slightly + * lower than, 2^31. Use of RNS loses less than two bits, even for + * the largest values. + * + * IMPORTANT: if these values are modified, then the temporary buffer + * sizes (FALCON_KEYGEN_TEMP_*, in inner.h) must be recomputed + * accordingly. + */ + +static const size_t MAX_BL_SMALL[] = { + 1, 1, 2, 2, 4, 7, 14, 27, 53, 106, 209 +}; + +static const size_t MAX_BL_LARGE[] = { + 2, 2, 5, 7, 12, 21, 40, 78, 157, 308 +}; + +/* + * Average and standard deviation for the maximum size (in bits) of + * coefficients of (f,g), depending on depth. These values are used + * to compute bounds for Babai's reduction. + */ +static const struct { + int avg; + int std; +} BITLENGTH[] = { + { 4, 0 }, + { 11, 1 }, + { 24, 1 }, + { 50, 1 }, + { 102, 1 }, + { 202, 2 }, + { 401, 4 }, + { 794, 5 }, + { 1577, 8 }, + { 3138, 13 }, + { 6308, 25 } +}; + +/* + * Minimal recursion depth at which we rebuild intermediate values + * when reconstructing f and g. + */ +#define DEPTH_INT_FG 4 + +/* + * Compute squared norm of a short vector. Returned value is saturated to + * 2^32-1 if it is not lower than 2^31. + */ +static uint32_t +poly_small_sqnorm(const int8_t *f, unsigned logn) { + size_t n, u; + uint32_t s, ng; + + n = MKN(logn); + s = 0; + ng = 0; + for (u = 0; u < n; u ++) { + int32_t z; + + z = f[u]; + s += (uint32_t)(z * z); + ng |= s; + } + return s | -(ng >> 31); +} + +/* + * Align (upwards) the provided 'data' pointer with regards to 'base' + * so that the offset is a multiple of the size of 'fpr'. + */ +static fpr * +align_fpr(void *base, void *data) { + uint8_t *cb, *cd; + size_t k, km; + + cb = base; + cd = data; + k = (size_t)(cd - cb); + km = k % sizeof(fpr); + if (km) { + k += (sizeof(fpr)) - km; + } + return (fpr *)(cb + k); +} + +/* + * Align (upwards) the provided 'data' pointer with regards to 'base' + * so that the offset is a multiple of the size of 'uint32_t'. + */ +static uint32_t * +align_u32(void *base, void *data) { + uint8_t *cb, *cd; + size_t k, km; + + cb = base; + cd = data; + k = (size_t)(cd - cb); + km = k % sizeof(uint32_t); + if (km) { + k += (sizeof(uint32_t)) - km; + } + return (uint32_t *)(cb + k); +} + +/* + * Convert a small vector to floating point. + */ +static void +poly_small_to_fp(fpr *x, const int8_t *f, unsigned logn) { + size_t n, u; + + n = MKN(logn); + for (u = 0; u < n; u ++) { + x[u] = fpr_of(f[u]); + } +} + +/* + * Input: f,g of degree N = 2^logn; 'depth' is used only to get their + * individual length. + * + * Output: f',g' of degree N/2, with the length for 'depth+1'. + * + * Values are in RNS; input and/or output may also be in NTT. + */ +static void +make_fg_step(uint32_t *data, unsigned logn, unsigned depth, + int in_ntt, int out_ntt) { + size_t n, hn, u; + size_t slen, tlen; + uint32_t *fd, *gd, *fs, *gs, *gm, *igm, *t1; + const small_prime *primes; + + n = (size_t)1 << logn; + hn = n >> 1; + slen = MAX_BL_SMALL[depth]; + tlen = MAX_BL_SMALL[depth + 1]; + primes = PRIMES; + + /* + * Prepare room for the result. + */ + fd = data; + gd = fd + hn * tlen; + fs = gd + hn * tlen; + gs = fs + n * slen; + gm = gs + n * slen; + igm = gm + n; + t1 = igm + n; + memmove(fs, data, 2 * n * slen * sizeof * data); + + /* + * First slen words: we use the input values directly, and apply + * inverse NTT as we go. + */ + for (u = 0; u < slen; u ++) { + uint32_t p, p0i, R2; + size_t v; + uint32_t *x; + + p = primes[u].p; + p0i = modp_ninv31(p); + R2 = modp_R2(p, p0i); + modp_mkgm2(gm, igm, logn, primes[u].g, p, p0i); + + for (v = 0, x = fs + u; v < n; v ++, x += slen) { + t1[v] = *x; + } + if (!in_ntt) { + modp_NTT2(t1, gm, logn, p, p0i); + } + for (v = 0, x = fd + u; v < hn; v ++, x += tlen) { + uint32_t w0, w1; + + w0 = t1[(v << 1) + 0]; + w1 = t1[(v << 1) + 1]; + *x = modp_montymul( + modp_montymul(w0, w1, p, p0i), R2, p, p0i); + } + if (in_ntt) { + modp_iNTT2_ext(fs + u, slen, igm, logn, p, p0i); + } + + for (v = 0, x = gs + u; v < n; v ++, x += slen) { + t1[v] = *x; + } + if (!in_ntt) { + modp_NTT2(t1, gm, logn, p, p0i); + } + for (v = 0, x = gd + u; v < hn; v ++, x += tlen) { + uint32_t w0, w1; + + w0 = t1[(v << 1) + 0]; + w1 = t1[(v << 1) + 1]; + *x = modp_montymul( + modp_montymul(w0, w1, p, p0i), R2, p, p0i); + } + if (in_ntt) { + modp_iNTT2_ext(gs + u, slen, igm, logn, p, p0i); + } + + if (!out_ntt) { + modp_iNTT2_ext(fd + u, tlen, igm, logn - 1, p, p0i); + modp_iNTT2_ext(gd + u, tlen, igm, logn - 1, p, p0i); + } + } + + /* + * Since the fs and gs words have been de-NTTized, we can use the + * CRT to rebuild the values. + */ + zint_rebuild_CRT(fs, slen, slen, n, primes, 1, gm); + zint_rebuild_CRT(gs, slen, slen, n, primes, 1, gm); + + /* + * Remaining words: use modular reductions to extract the values. + */ + for (u = slen; u < tlen; u ++) { + uint32_t p, p0i, R2, Rx; + size_t v; + uint32_t *x; + + p = primes[u].p; + p0i = modp_ninv31(p); + R2 = modp_R2(p, p0i); + Rx = modp_Rx((unsigned)slen, p, p0i, R2); + modp_mkgm2(gm, igm, logn, primes[u].g, p, p0i); + for (v = 0, x = fs; v < n; v ++, x += slen) { + t1[v] = zint_mod_small_signed(x, slen, p, p0i, R2, Rx); + } + modp_NTT2(t1, gm, logn, p, p0i); + for (v = 0, x = fd + u; v < hn; v ++, x += tlen) { + uint32_t w0, w1; + + w0 = t1[(v << 1) + 0]; + w1 = t1[(v << 1) + 1]; + *x = modp_montymul( + modp_montymul(w0, w1, p, p0i), R2, p, p0i); + } + for (v = 0, x = gs; v < n; v ++, x += slen) { + t1[v] = zint_mod_small_signed(x, slen, p, p0i, R2, Rx); + } + modp_NTT2(t1, gm, logn, p, p0i); + for (v = 0, x = gd + u; v < hn; v ++, x += tlen) { + uint32_t w0, w1; + + w0 = t1[(v << 1) + 0]; + w1 = t1[(v << 1) + 1]; + *x = modp_montymul( + modp_montymul(w0, w1, p, p0i), R2, p, p0i); + } + + if (!out_ntt) { + modp_iNTT2_ext(fd + u, tlen, igm, logn - 1, p, p0i); + modp_iNTT2_ext(gd + u, tlen, igm, logn - 1, p, p0i); + } + } +} + +/* + * Compute f and g at a specific depth, in RNS notation. + * + * Returned values are stored in the data[] array, at slen words per integer. + * + * Conditions: + * 0 <= depth <= logn + * + * Space use in data[]: enough room for any two successive values (f', g', + * f and g). + */ +static void +make_fg(uint32_t *data, const int8_t *f, const int8_t *g, + unsigned logn, unsigned depth, int out_ntt) { + size_t n, u; + uint32_t *ft, *gt, p0; + unsigned d; + const small_prime *primes; + + n = MKN(logn); + ft = data; + gt = ft + n; + primes = PRIMES; + p0 = primes[0].p; + for (u = 0; u < n; u ++) { + ft[u] = modp_set(f[u], p0); + gt[u] = modp_set(g[u], p0); + } + + if (depth == 0 && out_ntt) { + uint32_t *gm, *igm; + uint32_t p, p0i; + + p = primes[0].p; + p0i = modp_ninv31(p); + gm = gt + n; + igm = gm + MKN(logn); + modp_mkgm2(gm, igm, logn, primes[0].g, p, p0i); + modp_NTT2(ft, gm, logn, p, p0i); + modp_NTT2(gt, gm, logn, p, p0i); + return; + } + + if (depth == 0) { + return; + } + if (depth == 1) { + make_fg_step(data, logn, 0, 0, out_ntt); + return; + } + make_fg_step(data, logn, 0, 0, 1); + for (d = 1; d + 1 < depth; d ++) { + make_fg_step(data, logn - d, d, 1, 1); + } + make_fg_step(data, logn - depth + 1, depth - 1, 1, out_ntt); +} + +/* + * Solving the NTRU equation, deepest level: compute the resultants of + * f and g with X^N+1, and use binary GCD. The F and G values are + * returned in tmp[]. + * + * Returned value: 1 on success, 0 on error. + */ +static int +solve_NTRU_deepest(unsigned logn_top, + const int8_t *f, const int8_t *g, uint32_t *tmp) { + size_t len; + uint32_t *Fp, *Gp, *fp, *gp, *t1, q; + const small_prime *primes; + + len = MAX_BL_SMALL[logn_top]; + primes = PRIMES; + + Fp = tmp; + Gp = Fp + len; + fp = Gp + len; + gp = fp + len; + t1 = gp + len; + + make_fg(fp, f, g, logn_top, logn_top, 0); + + /* + * We use the CRT to rebuild the resultants as big integers. + * There are two such big integers. The resultants are always + * nonnegative. + */ + zint_rebuild_CRT(fp, len, len, 2, primes, 0, t1); + + /* + * Apply the binary GCD. The zint_bezout() function works only + * if both inputs are odd. + * + * We can test on the result and return 0 because that would + * imply failure of the NTRU solving equation, and the (f,g) + * values will be abandoned in that case. + */ + if (!zint_bezout(Gp, Fp, fp, gp, len, t1)) { + return 0; + } + + /* + * Multiply the two values by the target value q. Values must + * fit in the destination arrays. + * We can again test on the returned words: a non-zero output + * of zint_mul_small() means that we exceeded our array + * capacity, and that implies failure and rejection of (f,g). + */ + q = 12289; + if (zint_mul_small(Fp, len, q) != 0 + || zint_mul_small(Gp, len, q) != 0) { + return 0; + } + + return 1; +} + +/* + * Solving the NTRU equation, intermediate level. Upon entry, the F and G + * from the previous level should be in the tmp[] array. + * This function MAY be invoked for the top-level (in which case depth = 0). + * + * Returned value: 1 on success, 0 on error. + */ +static int +solve_NTRU_intermediate(unsigned logn_top, + const int8_t *f, const int8_t *g, unsigned depth, uint32_t *tmp) { + /* + * In this function, 'logn' is the log2 of the degree for + * this step. If N = 2^logn, then: + * - the F and G values already in fk->tmp (from the deeper + * levels) have degree N/2; + * - this function should return F and G of degree N. + */ + unsigned logn; + size_t n, hn, slen, dlen, llen, rlen, FGlen, u; + uint32_t *Fd, *Gd, *Ft, *Gt, *ft, *gt, *t1; + fpr *rt1, *rt2, *rt3, *rt4, *rt5; + int scale_fg, minbl_fg, maxbl_fg, maxbl_FG, scale_k; + uint32_t *x, *y; + int32_t *k; + const small_prime *primes; + + logn = logn_top - depth; + n = (size_t)1 << logn; + hn = n >> 1; + + /* + * slen = size for our input f and g; also size of the reduced + * F and G we return (degree N) + * + * dlen = size of the F and G obtained from the deeper level + * (degree N/2 or N/3) + * + * llen = size for intermediary F and G before reduction (degree N) + * + * We build our non-reduced F and G as two independent halves each, + * of degree N/2 (F = F0 + X*F1, G = G0 + X*G1). + */ + slen = MAX_BL_SMALL[depth]; + dlen = MAX_BL_SMALL[depth + 1]; + llen = MAX_BL_LARGE[depth]; + primes = PRIMES; + + /* + * Fd and Gd are the F and G from the deeper level. + */ + Fd = tmp; + Gd = Fd + dlen * hn; + + /* + * Compute the input f and g for this level. Note that we get f + * and g in RNS + NTT representation. + */ + ft = Gd + dlen * hn; + make_fg(ft, f, g, logn_top, depth, 1); + + /* + * Move the newly computed f and g to make room for our candidate + * F and G (unreduced). + */ + Ft = tmp; + Gt = Ft + n * llen; + t1 = Gt + n * llen; + memmove(t1, ft, 2 * n * slen * sizeof * ft); + ft = t1; + gt = ft + slen * n; + t1 = gt + slen * n; + + /* + * Move Fd and Gd _after_ f and g. + */ + memmove(t1, Fd, 2 * hn * dlen * sizeof * Fd); + Fd = t1; + Gd = Fd + hn * dlen; + + /* + * We reduce Fd and Gd modulo all the small primes we will need, + * and store the values in Ft and Gt (only n/2 values in each). + */ + for (u = 0; u < llen; u ++) { + uint32_t p, p0i, R2, Rx; + size_t v; + uint32_t *xs, *ys, *xd, *yd; + + p = primes[u].p; + p0i = modp_ninv31(p); + R2 = modp_R2(p, p0i); + Rx = modp_Rx((unsigned)dlen, p, p0i, R2); + for (v = 0, xs = Fd, ys = Gd, xd = Ft + u, yd = Gt + u; + v < hn; + v ++, xs += dlen, ys += dlen, xd += llen, yd += llen) { + *xd = zint_mod_small_signed(xs, dlen, p, p0i, R2, Rx); + *yd = zint_mod_small_signed(ys, dlen, p, p0i, R2, Rx); + } + } + + /* + * We do not need Fd and Gd after that point. + */ + + /* + * Compute our F and G modulo sufficiently many small primes. + */ + for (u = 0; u < llen; u ++) { + uint32_t p, p0i, R2; + uint32_t *gm, *igm, *fx, *gx, *Fp, *Gp; + size_t v; + + /* + * All computations are done modulo p. + */ + p = primes[u].p; + p0i = modp_ninv31(p); + R2 = modp_R2(p, p0i); + + /* + * If we processed slen words, then f and g have been + * de-NTTized, and are in RNS; we can rebuild them. + */ + if (u == slen) { + zint_rebuild_CRT(ft, slen, slen, n, primes, 1, t1); + zint_rebuild_CRT(gt, slen, slen, n, primes, 1, t1); + } + + gm = t1; + igm = gm + n; + fx = igm + n; + gx = fx + n; + + modp_mkgm2(gm, igm, logn, primes[u].g, p, p0i); + + if (u < slen) { + for (v = 0, x = ft + u, y = gt + u; + v < n; v ++, x += slen, y += slen) { + fx[v] = *x; + gx[v] = *y; + } + modp_iNTT2_ext(ft + u, slen, igm, logn, p, p0i); + modp_iNTT2_ext(gt + u, slen, igm, logn, p, p0i); + } else { + uint32_t Rx; + + Rx = modp_Rx((unsigned)slen, p, p0i, R2); + for (v = 0, x = ft, y = gt; + v < n; v ++, x += slen, y += slen) { + fx[v] = zint_mod_small_signed(x, slen, + p, p0i, R2, Rx); + gx[v] = zint_mod_small_signed(y, slen, + p, p0i, R2, Rx); + } + modp_NTT2(fx, gm, logn, p, p0i); + modp_NTT2(gx, gm, logn, p, p0i); + } + + /* + * Get F' and G' modulo p and in NTT representation + * (they have degree n/2). These values were computed in + * a previous step, and stored in Ft and Gt. + */ + Fp = gx + n; + Gp = Fp + hn; + for (v = 0, x = Ft + u, y = Gt + u; + v < hn; v ++, x += llen, y += llen) { + Fp[v] = *x; + Gp[v] = *y; + } + modp_NTT2(Fp, gm, logn - 1, p, p0i); + modp_NTT2(Gp, gm, logn - 1, p, p0i); + + /* + * Compute our F and G modulo p. + * + * General case: + * + * we divide degree by d = 2 or 3 + * f'(x^d) = N(f)(x^d) = f * adj(f) + * g'(x^d) = N(g)(x^d) = g * adj(g) + * f'*G' - g'*F' = q + * F = F'(x^d) * adj(g) + * G = G'(x^d) * adj(f) + * + * We compute things in the NTT. We group roots of phi + * such that all roots x in a group share the same x^d. + * If the roots in a group are x_1, x_2... x_d, then: + * + * N(f)(x_1^d) = f(x_1)*f(x_2)*...*f(x_d) + * + * Thus, we have: + * + * G(x_1) = f(x_2)*f(x_3)*...*f(x_d)*G'(x_1^d) + * G(x_2) = f(x_1)*f(x_3)*...*f(x_d)*G'(x_1^d) + * ... + * G(x_d) = f(x_1)*f(x_2)*...*f(x_{d-1})*G'(x_1^d) + * + * In all cases, we can thus compute F and G in NTT + * representation by a few simple multiplications. + * Moreover, in our chosen NTT representation, roots + * from the same group are consecutive in RAM. + */ + for (v = 0, x = Ft + u, y = Gt + u; v < hn; + v ++, x += (llen << 1), y += (llen << 1)) { + uint32_t ftA, ftB, gtA, gtB; + uint32_t mFp, mGp; + + ftA = fx[(v << 1) + 0]; + ftB = fx[(v << 1) + 1]; + gtA = gx[(v << 1) + 0]; + gtB = gx[(v << 1) + 1]; + mFp = modp_montymul(Fp[v], R2, p, p0i); + mGp = modp_montymul(Gp[v], R2, p, p0i); + x[0] = modp_montymul(gtB, mFp, p, p0i); + x[llen] = modp_montymul(gtA, mFp, p, p0i); + y[0] = modp_montymul(ftB, mGp, p, p0i); + y[llen] = modp_montymul(ftA, mGp, p, p0i); + } + modp_iNTT2_ext(Ft + u, llen, igm, logn, p, p0i); + modp_iNTT2_ext(Gt + u, llen, igm, logn, p, p0i); + } + + /* + * Rebuild F and G with the CRT. + */ + zint_rebuild_CRT(Ft, llen, llen, n, primes, 1, t1); + zint_rebuild_CRT(Gt, llen, llen, n, primes, 1, t1); + + /* + * At that point, Ft, Gt, ft and gt are consecutive in RAM (in that + * order). + */ + + /* + * Apply Babai reduction to bring back F and G to size slen. + * + * We use the FFT to compute successive approximations of the + * reduction coefficient. We first isolate the top bits of + * the coefficients of f and g, and convert them to floating + * point; with the FFT, we compute adj(f), adj(g), and + * 1/(f*adj(f)+g*adj(g)). + * + * Then, we repeatedly apply the following: + * + * - Get the top bits of the coefficients of F and G into + * floating point, and use the FFT to compute: + * (F*adj(f)+G*adj(g))/(f*adj(f)+g*adj(g)) + * + * - Convert back that value into normal representation, and + * round it to the nearest integers, yielding a polynomial k. + * Proper scaling is applied to f, g, F and G so that the + * coefficients fit on 32 bits (signed). + * + * - Subtract k*f from F and k*g from G. + * + * Under normal conditions, this process reduces the size of F + * and G by some bits at each iteration. For constant-time + * operation, we do not want to measure the actual length of + * F and G; instead, we do the following: + * + * - f and g are converted to floating-point, with some scaling + * if necessary to keep values in the representable range. + * + * - For each iteration, we _assume_ a maximum size for F and G, + * and use the values at that size. If we overreach, then + * we get zeros, which is harmless: the resulting coefficients + * of k will be 0 and the value won't be reduced. + * + * - We conservatively assume that F and G will be reduced by + * at least 25 bits at each iteration. + * + * Even when reaching the bottom of the reduction, reduction + * coefficient will remain low. If it goes out-of-range, then + * something wrong occurred and the whole NTRU solving fails. + */ + + /* + * Memory layout: + * - We need to compute and keep adj(f), adj(g), and + * 1/(f*adj(f)+g*adj(g)) (sizes N, N and N/2 fp numbers, + * respectively). + * - At each iteration we need two extra fp buffer (N fp values), + * and produce a k (N 32-bit words). k will be shared with one + * of the fp buffers. + * - To compute k*f and k*g efficiently (with the NTT), we need + * some extra room; we reuse the space of the temporary buffers. + * + * Arrays of 'fpr' are obtained from the temporary array itself. + * We ensure that the base is at a properly aligned offset (the + * source array tmp[] is supposed to be already aligned). + */ + + rt3 = align_fpr(tmp, t1); + rt4 = rt3 + n; + rt5 = rt4 + n; + rt1 = rt5 + (n >> 1); + k = (int32_t *)align_u32(tmp, rt1); + rt2 = align_fpr(tmp, k + n); + if (rt2 < (rt1 + n)) { + rt2 = rt1 + n; + } + t1 = (uint32_t *)k + n; + + /* + * Get f and g into rt3 and rt4 as floating-point approximations. + * + * We need to "scale down" the floating-point representation of + * coefficients when they are too big. We want to keep the value + * below 2^310 or so. Thus, when values are larger than 10 words, + * we consider only the top 10 words. Array lengths have been + * computed so that average maximum length will fall in the + * middle or the upper half of these top 10 words. + */ + rlen = slen; + if (rlen > 10) { + rlen = 10; + } + poly_big_to_fp(rt3, ft + slen - rlen, rlen, slen, logn); + poly_big_to_fp(rt4, gt + slen - rlen, rlen, slen, logn); + + /* + * Values in rt3 and rt4 are downscaled by 2^(scale_fg). + */ + scale_fg = 31 * (int)(slen - rlen); + + /* + * Estimated boundaries for the maximum size (in bits) of the + * coefficients of (f,g). We use the measured average, and + * allow for a deviation of at most six times the standard + * deviation. + */ + minbl_fg = BITLENGTH[depth].avg - 6 * BITLENGTH[depth].std; + maxbl_fg = BITLENGTH[depth].avg + 6 * BITLENGTH[depth].std; + + /* + * Compute 1/(f*adj(f)+g*adj(g)) in rt5. We also keep adj(f) + * and adj(g) in rt3 and rt4, respectively. + */ + PQCLEAN_FALCON512_CLEAN_FFT(rt3, logn); + PQCLEAN_FALCON512_CLEAN_FFT(rt4, logn); + PQCLEAN_FALCON512_CLEAN_poly_invnorm2_fft(rt5, rt3, rt4, logn); + PQCLEAN_FALCON512_CLEAN_poly_adj_fft(rt3, logn); + PQCLEAN_FALCON512_CLEAN_poly_adj_fft(rt4, logn); + + /* + * Reduce F and G repeatedly. + * + * The expected maximum bit length of coefficients of F and G + * is kept in maxbl_FG, with the corresponding word length in + * FGlen. + */ + FGlen = llen; + maxbl_FG = 31 * (int)llen; + + /* + * Each reduction operation computes the reduction polynomial + * "k". We need that polynomial to have coefficients that fit + * on 32-bit signed integers, with some scaling; thus, we use + * a descending sequence of scaling values, down to zero. + * + * The size of the coefficients of k is (roughly) the difference + * between the size of the coefficients of (F,G) and the size + * of the coefficients of (f,g). Thus, the maximum size of the + * coefficients of k is, at the start, maxbl_FG - minbl_fg; + * this is our starting scale value for k. + * + * We need to estimate the size of (F,G) during the execution of + * the algorithm; we are allowed some overestimation but not too + * much (poly_big_to_fp() uses a 310-bit window). Generally + * speaking, after applying a reduction with k scaled to + * scale_k, the size of (F,G) will be size(f,g) + scale_k + dd, + * where 'dd' is a few bits to account for the fact that the + * reduction is never perfect (intuitively, dd is on the order + * of sqrt(N), so at most 5 bits; we here allow for 10 extra + * bits). + * + * The size of (f,g) is not known exactly, but maxbl_fg is an + * upper bound. + */ + scale_k = maxbl_FG - minbl_fg; + + for (;;) { + int scale_FG, dc, new_maxbl_FG; + uint32_t scl, sch; + fpr pdc, pt; + + /* + * Convert current F and G into floating-point. We apply + * scaling if the current length is more than 10 words. + */ + rlen = FGlen; + if (rlen > 10) { + rlen = 10; + } + scale_FG = 31 * (int)(FGlen - rlen); + poly_big_to_fp(rt1, Ft + FGlen - rlen, rlen, llen, logn); + poly_big_to_fp(rt2, Gt + FGlen - rlen, rlen, llen, logn); + + /* + * Compute (F*adj(f)+G*adj(g))/(f*adj(f)+g*adj(g)) in rt2. + */ + PQCLEAN_FALCON512_CLEAN_FFT(rt1, logn); + PQCLEAN_FALCON512_CLEAN_FFT(rt2, logn); + PQCLEAN_FALCON512_CLEAN_poly_mul_fft(rt1, rt3, logn); + PQCLEAN_FALCON512_CLEAN_poly_mul_fft(rt2, rt4, logn); + PQCLEAN_FALCON512_CLEAN_poly_add(rt2, rt1, logn); + PQCLEAN_FALCON512_CLEAN_poly_mul_autoadj_fft(rt2, rt5, logn); + PQCLEAN_FALCON512_CLEAN_iFFT(rt2, logn); + + /* + * (f,g) are scaled by 'scale_fg', meaning that the + * numbers in rt3/rt4 should be multiplied by 2^(scale_fg) + * to have their true mathematical value. + * + * (F,G) are similarly scaled by 'scale_FG'. Therefore, + * the value we computed in rt2 is scaled by + * 'scale_FG-scale_fg'. + * + * We want that value to be scaled by 'scale_k', hence we + * apply a corrective scaling. After scaling, the values + * should fit in -2^31-1..+2^31-1. + */ + dc = scale_k - scale_FG + scale_fg; + + /* + * We will need to multiply values by 2^(-dc). The value + * 'dc' is not secret, so we can compute 2^(-dc) with a + * non-constant-time process. + * (We could use ldexp(), but we prefer to avoid any + * dependency on libm. When using FP emulation, we could + * use our fpr_ldexp(), which is constant-time.) + */ + if (dc < 0) { + dc = -dc; + pt = fpr_two; + } else { + pt = fpr_onehalf; + } + pdc = fpr_one; + while (dc != 0) { + if ((dc & 1) != 0) { + pdc = fpr_mul(pdc, pt); + } + dc >>= 1; + pt = fpr_sqr(pt); + } + + for (u = 0; u < n; u ++) { + fpr xv; + + xv = fpr_mul(rt2[u], pdc); + + /* + * Sometimes the values can be out-of-bounds if + * the algorithm fails; we must not call + * fpr_rint() (and cast to int32_t) if the value + * is not in-bounds. Note that the test does not + * break constant-time discipline, since any + * failure here implies that we discard the current + * secret key (f,g). + */ + if (!fpr_lt(fpr_mtwo31m1, xv) + || !fpr_lt(xv, fpr_ptwo31m1)) { + return 0; + } + k[u] = (int32_t)fpr_rint(xv); + } + + /* + * Values in k[] are integers. They really are scaled + * down by maxbl_FG - minbl_fg bits. + * + * If we are at low depth, then we use the NTT to + * compute k*f and k*g. + */ + sch = (uint32_t)(scale_k / 31); + scl = (uint32_t)(scale_k % 31); + if (depth <= DEPTH_INT_FG) { + poly_sub_scaled_ntt(Ft, FGlen, llen, ft, slen, slen, + k, sch, scl, logn, t1); + poly_sub_scaled_ntt(Gt, FGlen, llen, gt, slen, slen, + k, sch, scl, logn, t1); + } else { + poly_sub_scaled(Ft, FGlen, llen, ft, slen, slen, + k, sch, scl, logn); + poly_sub_scaled(Gt, FGlen, llen, gt, slen, slen, + k, sch, scl, logn); + } + + /* + * We compute the new maximum size of (F,G), assuming that + * (f,g) has _maximal_ length (i.e. that reduction is + * "late" instead of "early". We also adjust FGlen + * accordingly. + */ + new_maxbl_FG = scale_k + maxbl_fg + 10; + if (new_maxbl_FG < maxbl_FG) { + maxbl_FG = new_maxbl_FG; + if ((int)FGlen * 31 >= maxbl_FG + 31) { + FGlen --; + } + } + + /* + * We suppose that scaling down achieves a reduction by + * at least 25 bits per iteration. We stop when we have + * done the loop with an unscaled k. + */ + if (scale_k <= 0) { + break; + } + scale_k -= 25; + if (scale_k < 0) { + scale_k = 0; + } + } + + /* + * If (F,G) length was lowered below 'slen', then we must take + * care to re-extend the sign. + */ + if (FGlen < slen) { + for (u = 0; u < n; u ++, Ft += llen, Gt += llen) { + size_t v; + uint32_t sw; + + sw = -(Ft[FGlen - 1] >> 30) >> 1; + for (v = FGlen; v < slen; v ++) { + Ft[v] = sw; + } + sw = -(Gt[FGlen - 1] >> 30) >> 1; + for (v = FGlen; v < slen; v ++) { + Gt[v] = sw; + } + } + } + + /* + * Compress encoding of all values to 'slen' words (this is the + * expected output format). + */ + for (u = 0, x = tmp, y = tmp; + u < (n << 1); u ++, x += slen, y += llen) { + memmove(x, y, slen * sizeof * y); + } + return 1; +} + +/* + * Solving the NTRU equation, binary case, depth = 1. Upon entry, the + * F and G from the previous level should be in the tmp[] array. + * + * Returned value: 1 on success, 0 on error. + */ +static int +solve_NTRU_binary_depth1(unsigned logn_top, + const int8_t *f, const int8_t *g, uint32_t *tmp) { + /* + * The first half of this function is a copy of the corresponding + * part in solve_NTRU_intermediate(), for the reconstruction of + * the unreduced F and G. The second half (Babai reduction) is + * done differently, because the unreduced F and G fit in 53 bits + * of precision, allowing a much simpler process with lower RAM + * usage. + */ + unsigned depth, logn; + size_t n_top, n, hn, slen, dlen, llen, u; + uint32_t *Fd, *Gd, *Ft, *Gt, *ft, *gt, *t1; + fpr *rt1, *rt2, *rt3, *rt4, *rt5, *rt6; + uint32_t *x, *y; + + depth = 1; + n_top = (size_t)1 << logn_top; + logn = logn_top - depth; + n = (size_t)1 << logn; + hn = n >> 1; + + /* + * Equations are: + * + * f' = f0^2 - X^2*f1^2 + * g' = g0^2 - X^2*g1^2 + * F' and G' are a solution to f'G' - g'F' = q (from deeper levels) + * F = F'*(g0 - X*g1) + * G = G'*(f0 - X*f1) + * + * f0, f1, g0, g1, f', g', F' and G' are all "compressed" to + * degree N/2 (their odd-indexed coefficients are all zero). + */ + + /* + * slen = size for our input f and g; also size of the reduced + * F and G we return (degree N) + * + * dlen = size of the F and G obtained from the deeper level + * (degree N/2) + * + * llen = size for intermediary F and G before reduction (degree N) + * + * We build our non-reduced F and G as two independent halves each, + * of degree N/2 (F = F0 + X*F1, G = G0 + X*G1). + */ + slen = MAX_BL_SMALL[depth]; + dlen = MAX_BL_SMALL[depth + 1]; + llen = MAX_BL_LARGE[depth]; + + /* + * Fd and Gd are the F and G from the deeper level. Ft and Gt + * are the destination arrays for the unreduced F and G. + */ + Fd = tmp; + Gd = Fd + dlen * hn; + Ft = Gd + dlen * hn; + Gt = Ft + llen * n; + + /* + * We reduce Fd and Gd modulo all the small primes we will need, + * and store the values in Ft and Gt. + */ + for (u = 0; u < llen; u ++) { + uint32_t p, p0i, R2, Rx; + size_t v; + uint32_t *xs, *ys, *xd, *yd; + + p = PRIMES[u].p; + p0i = modp_ninv31(p); + R2 = modp_R2(p, p0i); + Rx = modp_Rx((unsigned)dlen, p, p0i, R2); + for (v = 0, xs = Fd, ys = Gd, xd = Ft + u, yd = Gt + u; + v < hn; + v ++, xs += dlen, ys += dlen, xd += llen, yd += llen) { + *xd = zint_mod_small_signed(xs, dlen, p, p0i, R2, Rx); + *yd = zint_mod_small_signed(ys, dlen, p, p0i, R2, Rx); + } + } + + /* + * Now Fd and Gd are not needed anymore; we can squeeze them out. + */ + memmove(tmp, Ft, llen * n * sizeof(uint32_t)); + Ft = tmp; + memmove(Ft + llen * n, Gt, llen * n * sizeof(uint32_t)); + Gt = Ft + llen * n; + ft = Gt + llen * n; + gt = ft + slen * n; + + t1 = gt + slen * n; + + /* + * Compute our F and G modulo sufficiently many small primes. + */ + for (u = 0; u < llen; u ++) { + uint32_t p, p0i, R2; + uint32_t *gm, *igm, *fx, *gx, *Fp, *Gp; + unsigned e; + size_t v; + + /* + * All computations are done modulo p. + */ + p = PRIMES[u].p; + p0i = modp_ninv31(p); + R2 = modp_R2(p, p0i); + + /* + * We recompute things from the source f and g, of full + * degree. However, we will need only the n first elements + * of the inverse NTT table (igm); the call to modp_mkgm() + * below will fill n_top elements in igm[] (thus overflowing + * into fx[]) but later code will overwrite these extra + * elements. + */ + gm = t1; + igm = gm + n_top; + fx = igm + n; + gx = fx + n_top; + modp_mkgm2(gm, igm, logn_top, PRIMES[u].g, p, p0i); + + /* + * Set ft and gt to f and g modulo p, respectively. + */ + for (v = 0; v < n_top; v ++) { + fx[v] = modp_set(f[v], p); + gx[v] = modp_set(g[v], p); + } + + /* + * Convert to NTT and compute our f and g. + */ + modp_NTT2(fx, gm, logn_top, p, p0i); + modp_NTT2(gx, gm, logn_top, p, p0i); + for (e = logn_top; e > logn; e --) { + modp_poly_rec_res(fx, e, p, p0i, R2); + modp_poly_rec_res(gx, e, p, p0i, R2); + } + + /* + * From that point onward, we only need tables for + * degree n, so we can save some space. + */ + if (depth > 0) { /* always true */ + memmove(gm + n, igm, n * sizeof * igm); + igm = gm + n; + memmove(igm + n, fx, n * sizeof * ft); + fx = igm + n; + memmove(fx + n, gx, n * sizeof * gt); + gx = fx + n; + } + + /* + * Get F' and G' modulo p and in NTT representation + * (they have degree n/2). These values were computed + * in a previous step, and stored in Ft and Gt. + */ + Fp = gx + n; + Gp = Fp + hn; + for (v = 0, x = Ft + u, y = Gt + u; + v < hn; v ++, x += llen, y += llen) { + Fp[v] = *x; + Gp[v] = *y; + } + modp_NTT2(Fp, gm, logn - 1, p, p0i); + modp_NTT2(Gp, gm, logn - 1, p, p0i); + + /* + * Compute our F and G modulo p. + * + * Equations are: + * + * f'(x^2) = N(f)(x^2) = f * adj(f) + * g'(x^2) = N(g)(x^2) = g * adj(g) + * + * f'*G' - g'*F' = q + * + * F = F'(x^2) * adj(g) + * G = G'(x^2) * adj(f) + * + * The NTT representation of f is f(w) for all w which + * are roots of phi. In the binary case, as well as in + * the ternary case for all depth except the deepest, + * these roots can be grouped in pairs (w,-w), and we + * then have: + * + * f(w) = adj(f)(-w) + * f(-w) = adj(f)(w) + * + * and w^2 is then a root for phi at the half-degree. + * + * At the deepest level in the ternary case, this still + * holds, in the following sense: the roots of x^2-x+1 + * are (w,-w^2) (for w^3 = -1, and w != -1), and we + * have: + * + * f(w) = adj(f)(-w^2) + * f(-w^2) = adj(f)(w) + * + * In all case, we can thus compute F and G in NTT + * representation by a few simple multiplications. + * Moreover, the two roots for each pair are consecutive + * in our bit-reversal encoding. + */ + for (v = 0, x = Ft + u, y = Gt + u; + v < hn; v ++, x += (llen << 1), y += (llen << 1)) { + uint32_t ftA, ftB, gtA, gtB; + uint32_t mFp, mGp; + + ftA = fx[(v << 1) + 0]; + ftB = fx[(v << 1) + 1]; + gtA = gx[(v << 1) + 0]; + gtB = gx[(v << 1) + 1]; + mFp = modp_montymul(Fp[v], R2, p, p0i); + mGp = modp_montymul(Gp[v], R2, p, p0i); + x[0] = modp_montymul(gtB, mFp, p, p0i); + x[llen] = modp_montymul(gtA, mFp, p, p0i); + y[0] = modp_montymul(ftB, mGp, p, p0i); + y[llen] = modp_montymul(ftA, mGp, p, p0i); + } + modp_iNTT2_ext(Ft + u, llen, igm, logn, p, p0i); + modp_iNTT2_ext(Gt + u, llen, igm, logn, p, p0i); + + /* + * Also save ft and gt (only up to size slen). + */ + if (u < slen) { + modp_iNTT2(fx, igm, logn, p, p0i); + modp_iNTT2(gx, igm, logn, p, p0i); + for (v = 0, x = ft + u, y = gt + u; + v < n; v ++, x += slen, y += slen) { + *x = fx[v]; + *y = gx[v]; + } + } + } + + /* + * Rebuild f, g, F and G with the CRT. Note that the elements of F + * and G are consecutive, and thus can be rebuilt in a single + * loop; similarly, the elements of f and g are consecutive. + */ + zint_rebuild_CRT(Ft, llen, llen, n << 1, PRIMES, 1, t1); + zint_rebuild_CRT(ft, slen, slen, n << 1, PRIMES, 1, t1); + + /* + * Here starts the Babai reduction, specialized for depth = 1. + * + * Candidates F and G (from Ft and Gt), and base f and g (ft and gt), + * are converted to floating point. There is no scaling, and a + * single pass is sufficient. + */ + + /* + * Convert F and G into floating point (rt1 and rt2). + */ + rt1 = align_fpr(tmp, gt + slen * n); + rt2 = rt1 + n; + poly_big_to_fp(rt1, Ft, llen, llen, logn); + poly_big_to_fp(rt2, Gt, llen, llen, logn); + + /* + * Integer representation of F and G is no longer needed, we + * can remove it. + */ + memmove(tmp, ft, 2 * slen * n * sizeof * ft); + ft = tmp; + gt = ft + slen * n; + rt3 = align_fpr(tmp, gt + slen * n); + memmove(rt3, rt1, 2 * n * sizeof * rt1); + rt1 = rt3; + rt2 = rt1 + n; + rt3 = rt2 + n; + rt4 = rt3 + n; + + /* + * Convert f and g into floating point (rt3 and rt4). + */ + poly_big_to_fp(rt3, ft, slen, slen, logn); + poly_big_to_fp(rt4, gt, slen, slen, logn); + + /* + * Remove unneeded ft and gt. + */ + memmove(tmp, rt1, 4 * n * sizeof * rt1); + rt1 = (fpr *)tmp; + rt2 = rt1 + n; + rt3 = rt2 + n; + rt4 = rt3 + n; + + /* + * We now have: + * rt1 = F + * rt2 = G + * rt3 = f + * rt4 = g + * in that order in RAM. We convert all of them to FFT. + */ + PQCLEAN_FALCON512_CLEAN_FFT(rt1, logn); + PQCLEAN_FALCON512_CLEAN_FFT(rt2, logn); + PQCLEAN_FALCON512_CLEAN_FFT(rt3, logn); + PQCLEAN_FALCON512_CLEAN_FFT(rt4, logn); + + /* + * Compute: + * rt5 = F*adj(f) + G*adj(g) + * rt6 = 1 / (f*adj(f) + g*adj(g)) + * (Note that rt6 is half-length.) + */ + rt5 = rt4 + n; + rt6 = rt5 + n; + PQCLEAN_FALCON512_CLEAN_poly_add_muladj_fft(rt5, rt1, rt2, rt3, rt4, logn); + PQCLEAN_FALCON512_CLEAN_poly_invnorm2_fft(rt6, rt3, rt4, logn); + + /* + * Compute: + * rt5 = (F*adj(f)+G*adj(g)) / (f*adj(f)+g*adj(g)) + */ + PQCLEAN_FALCON512_CLEAN_poly_mul_autoadj_fft(rt5, rt6, logn); + + /* + * Compute k as the rounded version of rt5. Check that none of + * the values is larger than 2^63-1 (in absolute value) + * because that would make the fpr_rint() do something undefined; + * note that any out-of-bounds value here implies a failure and + * (f,g) will be discarded, so we can make a simple test. + */ + PQCLEAN_FALCON512_CLEAN_iFFT(rt5, logn); + for (u = 0; u < n; u ++) { + fpr z; + + z = rt5[u]; + if (!fpr_lt(z, fpr_ptwo63m1) || !fpr_lt(fpr_mtwo63m1, z)) { + return 0; + } + rt5[u] = fpr_of(fpr_rint(z)); + } + PQCLEAN_FALCON512_CLEAN_FFT(rt5, logn); + + /* + * Subtract k*f from F, and k*g from G. + */ + PQCLEAN_FALCON512_CLEAN_poly_mul_fft(rt3, rt5, logn); + PQCLEAN_FALCON512_CLEAN_poly_mul_fft(rt4, rt5, logn); + PQCLEAN_FALCON512_CLEAN_poly_sub(rt1, rt3, logn); + PQCLEAN_FALCON512_CLEAN_poly_sub(rt2, rt4, logn); + PQCLEAN_FALCON512_CLEAN_iFFT(rt1, logn); + PQCLEAN_FALCON512_CLEAN_iFFT(rt2, logn); + + /* + * Convert back F and G to integers, and return. + */ + Ft = tmp; + Gt = Ft + n; + rt3 = align_fpr(tmp, Gt + n); + memmove(rt3, rt1, 2 * n * sizeof * rt1); + rt1 = rt3; + rt2 = rt1 + n; + for (u = 0; u < n; u ++) { + Ft[u] = (uint32_t)fpr_rint(rt1[u]); + Gt[u] = (uint32_t)fpr_rint(rt2[u]); + } + + return 1; +} + +/* + * Solving the NTRU equation, top level. Upon entry, the F and G + * from the previous level should be in the tmp[] array. + * + * Returned value: 1 on success, 0 on error. + */ +static int +solve_NTRU_binary_depth0(unsigned logn, + const int8_t *f, const int8_t *g, uint32_t *tmp) { + size_t n, hn, u; + uint32_t p, p0i, R2; + uint32_t *Fp, *Gp, *t1, *t2, *t3, *t4, *t5; + uint32_t *gm, *igm, *ft, *gt; + fpr *rt2, *rt3; + + n = (size_t)1 << logn; + hn = n >> 1; + + /* + * Equations are: + * + * f' = f0^2 - X^2*f1^2 + * g' = g0^2 - X^2*g1^2 + * F' and G' are a solution to f'G' - g'F' = q (from deeper levels) + * F = F'*(g0 - X*g1) + * G = G'*(f0 - X*f1) + * + * f0, f1, g0, g1, f', g', F' and G' are all "compressed" to + * degree N/2 (their odd-indexed coefficients are all zero). + * + * Everything should fit in 31-bit integers, hence we can just use + * the first small prime p = 2147473409. + */ + p = PRIMES[0].p; + p0i = modp_ninv31(p); + R2 = modp_R2(p, p0i); + + Fp = tmp; + Gp = Fp + hn; + ft = Gp + hn; + gt = ft + n; + gm = gt + n; + igm = gm + n; + + modp_mkgm2(gm, igm, logn, PRIMES[0].g, p, p0i); + + /* + * Convert F' anf G' in NTT representation. + */ + for (u = 0; u < hn; u ++) { + Fp[u] = modp_set(zint_one_to_plain(Fp + u), p); + Gp[u] = modp_set(zint_one_to_plain(Gp + u), p); + } + modp_NTT2(Fp, gm, logn - 1, p, p0i); + modp_NTT2(Gp, gm, logn - 1, p, p0i); + + /* + * Load f and g and convert them to NTT representation. + */ + for (u = 0; u < n; u ++) { + ft[u] = modp_set(f[u], p); + gt[u] = modp_set(g[u], p); + } + modp_NTT2(ft, gm, logn, p, p0i); + modp_NTT2(gt, gm, logn, p, p0i); + + /* + * Build the unreduced F,G in ft and gt. + */ + for (u = 0; u < n; u += 2) { + uint32_t ftA, ftB, gtA, gtB; + uint32_t mFp, mGp; + + ftA = ft[u + 0]; + ftB = ft[u + 1]; + gtA = gt[u + 0]; + gtB = gt[u + 1]; + mFp = modp_montymul(Fp[u >> 1], R2, p, p0i); + mGp = modp_montymul(Gp[u >> 1], R2, p, p0i); + ft[u + 0] = modp_montymul(gtB, mFp, p, p0i); + ft[u + 1] = modp_montymul(gtA, mFp, p, p0i); + gt[u + 0] = modp_montymul(ftB, mGp, p, p0i); + gt[u + 1] = modp_montymul(ftA, mGp, p, p0i); + } + modp_iNTT2(ft, igm, logn, p, p0i); + modp_iNTT2(gt, igm, logn, p, p0i); + + Gp = Fp + n; + t1 = Gp + n; + memmove(Fp, ft, 2 * n * sizeof * ft); + + /* + * We now need to apply the Babai reduction. At that point, + * we have F and G in two n-word arrays. + * + * We can compute F*adj(f)+G*adj(g) and f*adj(f)+g*adj(g) + * modulo p, using the NTT. We still move memory around in + * order to save RAM. + */ + t2 = t1 + n; + t3 = t2 + n; + t4 = t3 + n; + t5 = t4 + n; + + /* + * Compute the NTT tables in t1 and t2. We do not keep t2 + * (we'll recompute it later on). + */ + modp_mkgm2(t1, t2, logn, PRIMES[0].g, p, p0i); + + /* + * Convert F and G to NTT. + */ + modp_NTT2(Fp, t1, logn, p, p0i); + modp_NTT2(Gp, t1, logn, p, p0i); + + /* + * Load f and adj(f) in t4 and t5, and convert them to NTT + * representation. + */ + t4[0] = t5[0] = modp_set(f[0], p); + for (u = 1; u < n; u ++) { + t4[u] = modp_set(f[u], p); + t5[n - u] = modp_set(-f[u], p); + } + modp_NTT2(t4, t1, logn, p, p0i); + modp_NTT2(t5, t1, logn, p, p0i); + + /* + * Compute F*adj(f) in t2, and f*adj(f) in t3. + */ + for (u = 0; u < n; u ++) { + uint32_t w; + + w = modp_montymul(t5[u], R2, p, p0i); + t2[u] = modp_montymul(w, Fp[u], p, p0i); + t3[u] = modp_montymul(w, t4[u], p, p0i); + } + + /* + * Load g and adj(g) in t4 and t5, and convert them to NTT + * representation. + */ + t4[0] = t5[0] = modp_set(g[0], p); + for (u = 1; u < n; u ++) { + t4[u] = modp_set(g[u], p); + t5[n - u] = modp_set(-g[u], p); + } + modp_NTT2(t4, t1, logn, p, p0i); + modp_NTT2(t5, t1, logn, p, p0i); + + /* + * Add G*adj(g) to t2, and g*adj(g) to t3. + */ + for (u = 0; u < n; u ++) { + uint32_t w; + + w = modp_montymul(t5[u], R2, p, p0i); + t2[u] = modp_add(t2[u], + modp_montymul(w, Gp[u], p, p0i), p); + t3[u] = modp_add(t3[u], + modp_montymul(w, t4[u], p, p0i), p); + } + + /* + * Convert back t2 and t3 to normal representation (normalized + * around 0), and then + * move them to t1 and t2. We first need to recompute the + * inverse table for NTT. + */ + modp_mkgm2(t1, t4, logn, PRIMES[0].g, p, p0i); + modp_iNTT2(t2, t4, logn, p, p0i); + modp_iNTT2(t3, t4, logn, p, p0i); + for (u = 0; u < n; u ++) { + t1[u] = (uint32_t)modp_norm(t2[u], p); + t2[u] = (uint32_t)modp_norm(t3[u], p); + } + + /* + * At that point, array contents are: + * + * F (NTT representation) (Fp) + * G (NTT representation) (Gp) + * F*adj(f)+G*adj(g) (t1) + * f*adj(f)+g*adj(g) (t2) + * + * We want to divide t1 by t2. The result is not integral; it + * must be rounded. We thus need to use the FFT. + */ + + /* + * Get f*adj(f)+g*adj(g) in FFT representation. Since this + * polynomial is auto-adjoint, all its coordinates in FFT + * representation are actually real, so we can truncate off + * the imaginary parts. + */ + rt3 = align_fpr(tmp, t3); + for (u = 0; u < n; u ++) { + rt3[u] = fpr_of(((int32_t *)t2)[u]); + } + PQCLEAN_FALCON512_CLEAN_FFT(rt3, logn); + rt2 = align_fpr(tmp, t2); + memmove(rt2, rt3, hn * sizeof * rt3); + + /* + * Convert F*adj(f)+G*adj(g) in FFT representation. + */ + rt3 = rt2 + hn; + for (u = 0; u < n; u ++) { + rt3[u] = fpr_of(((int32_t *)t1)[u]); + } + PQCLEAN_FALCON512_CLEAN_FFT(rt3, logn); + + /* + * Compute (F*adj(f)+G*adj(g))/(f*adj(f)+g*adj(g)) and get + * its rounded normal representation in t1. + */ + PQCLEAN_FALCON512_CLEAN_poly_div_autoadj_fft(rt3, rt2, logn); + PQCLEAN_FALCON512_CLEAN_iFFT(rt3, logn); + for (u = 0; u < n; u ++) { + t1[u] = modp_set((int32_t)fpr_rint(rt3[u]), p); + } + + /* + * RAM contents are now: + * + * F (NTT representation) (Fp) + * G (NTT representation) (Gp) + * k (t1) + * + * We want to compute F-k*f, and G-k*g. + */ + t2 = t1 + n; + t3 = t2 + n; + t4 = t3 + n; + t5 = t4 + n; + modp_mkgm2(t2, t3, logn, PRIMES[0].g, p, p0i); + for (u = 0; u < n; u ++) { + t4[u] = modp_set(f[u], p); + t5[u] = modp_set(g[u], p); + } + modp_NTT2(t1, t2, logn, p, p0i); + modp_NTT2(t4, t2, logn, p, p0i); + modp_NTT2(t5, t2, logn, p, p0i); + for (u = 0; u < n; u ++) { + uint32_t kw; + + kw = modp_montymul(t1[u], R2, p, p0i); + Fp[u] = modp_sub(Fp[u], + modp_montymul(kw, t4[u], p, p0i), p); + Gp[u] = modp_sub(Gp[u], + modp_montymul(kw, t5[u], p, p0i), p); + } + modp_iNTT2(Fp, t3, logn, p, p0i); + modp_iNTT2(Gp, t3, logn, p, p0i); + for (u = 0; u < n; u ++) { + Fp[u] = (uint32_t)modp_norm(Fp[u], p); + Gp[u] = (uint32_t)modp_norm(Gp[u], p); + } + + return 1; +} + +/* + * Solve the NTRU equation. Returned value is 1 on success, 0 on error. + * G can be NULL, in which case that value is computed but not returned. + * If any of the coefficients of F and G exceeds lim (in absolute value), + * then 0 is returned. + */ +static int +solve_NTRU(unsigned logn, int8_t *F, int8_t *G, + const int8_t *f, const int8_t *g, int lim, uint32_t *tmp) { + size_t n, u; + uint32_t *ft, *gt, *Ft, *Gt, *gm; + uint32_t p, p0i, r; + const small_prime *primes; + + n = MKN(logn); + + if (!solve_NTRU_deepest(logn, f, g, tmp)) { + return 0; + } + + /* + * For logn <= 2, we need to use solve_NTRU_intermediate() + * directly, because coefficients are a bit too large and + * do not fit the hypotheses in solve_NTRU_binary_depth0(). + */ + if (logn <= 2) { + unsigned depth; + + depth = logn; + while (depth -- > 0) { + if (!solve_NTRU_intermediate(logn, f, g, depth, tmp)) { + return 0; + } + } + } else { + unsigned depth; + + depth = logn; + while (depth -- > 2) { + if (!solve_NTRU_intermediate(logn, f, g, depth, tmp)) { + return 0; + } + } + if (!solve_NTRU_binary_depth1(logn, f, g, tmp)) { + return 0; + } + if (!solve_NTRU_binary_depth0(logn, f, g, tmp)) { + return 0; + } + } + + /* + * If no buffer has been provided for G, use a temporary one. + */ + if (G == NULL) { + G = (int8_t *)(tmp + 2 * n); + } + + /* + * Final F and G are in fk->tmp, one word per coefficient + * (signed value over 31 bits). + */ + if (!poly_big_to_small(F, tmp, lim, logn) + || !poly_big_to_small(G, tmp + n, lim, logn)) { + return 0; + } + + /* + * Verify that the NTRU equation is fulfilled. Since all elements + * have short lengths, verifying modulo a small prime p works, and + * allows using the NTT. + * + * We put Gt[] first in tmp[], and process it first, so that it does + * not overlap with G[] in case we allocated it ourselves. + */ + Gt = tmp; + ft = Gt + n; + gt = ft + n; + Ft = gt + n; + gm = Ft + n; + + primes = PRIMES; + p = primes[0].p; + p0i = modp_ninv31(p); + modp_mkgm2(gm, tmp, logn, primes[0].g, p, p0i); + for (u = 0; u < n; u ++) { + Gt[u] = modp_set(G[u], p); + } + for (u = 0; u < n; u ++) { + ft[u] = modp_set(f[u], p); + gt[u] = modp_set(g[u], p); + Ft[u] = modp_set(F[u], p); + } + modp_NTT2(ft, gm, logn, p, p0i); + modp_NTT2(gt, gm, logn, p, p0i); + modp_NTT2(Ft, gm, logn, p, p0i); + modp_NTT2(Gt, gm, logn, p, p0i); + r = modp_montymul(12289, 1, p, p0i); + for (u = 0; u < n; u ++) { + uint32_t z; + + z = modp_sub(modp_montymul(ft[u], Gt[u], p, p0i), + modp_montymul(gt[u], Ft[u], p, p0i), p); + if (z != r) { + return 0; + } + } + + return 1; +} + +/* + * Generate a random polynomial with a Gaussian distribution. This function + * also makes sure that the resultant of the polynomial with phi is odd. + */ +static void +poly_small_mkgauss(RNG_CONTEXT *rng, int8_t *f, unsigned logn) { + size_t n, u; + unsigned mod2; + + n = MKN(logn); + mod2 = 0; + for (u = 0; u < n; u ++) { + int s; + +restart: + s = mkgauss(rng, logn); + + /* + * We need the coefficient to fit within -127..+127; + * realistically, this is always the case except for + * the very low degrees (N = 2 or 4), for which there + * is no real security anyway. + */ + if (s < -127 || s > 127) { + goto restart; + } + + /* + * We need the sum of all coefficients to be 1; otherwise, + * the resultant of the polynomial with X^N+1 will be even, + * and the binary GCD will fail. + */ + if (u == n - 1) { + if ((mod2 ^ (unsigned)(s & 1)) == 0) { + goto restart; + } + } else { + mod2 ^= (unsigned)(s & 1); + } + f[u] = (int8_t)s; + } +} + +/* see falcon.h */ +void +PQCLEAN_FALCON512_CLEAN_keygen(inner_shake256_context *rng, + int8_t *f, int8_t *g, int8_t *F, int8_t *G, uint16_t *h, + unsigned logn, uint8_t *tmp) { + /* + * Algorithm is the following: + * + * - Generate f and g with the Gaussian distribution. + * + * - If either Res(f,phi) or Res(g,phi) is even, try again. + * + * - If ||(f,g)|| is too large, try again. + * + * - If ||B~_{f,g}|| is too large, try again. + * + * - If f is not invertible mod phi mod q, try again. + * + * - Compute h = g/f mod phi mod q. + * + * - Solve the NTRU equation fG - gF = q; if the solving fails, + * try again. Usual failure condition is when Res(f,phi) + * and Res(g,phi) are not prime to each other. + */ + size_t n, u; + uint16_t *h2, *tmp2; + RNG_CONTEXT *rc; + + n = MKN(logn); + rc = rng; + + /* + * We need to generate f and g randomly, until we find values + * such that the norm of (g,-f), and of the orthogonalized + * vector, are satisfying. The orthogonalized vector is: + * (q*adj(f)/(f*adj(f)+g*adj(g)), q*adj(g)/(f*adj(f)+g*adj(g))) + * (it is actually the (N+1)-th row of the Gram-Schmidt basis). + * + * In the binary case, coefficients of f and g are generated + * independently of each other, with a discrete Gaussian + * distribution of standard deviation 1.17*sqrt(q/(2*N)). Then, + * the two vectors have expected norm 1.17*sqrt(q), which is + * also our acceptance bound: we require both vectors to be no + * larger than that (this will be satisfied about 1/4th of the + * time, thus we expect sampling new (f,g) about 4 times for that + * step). + * + * We require that Res(f,phi) and Res(g,phi) are both odd (the + * NTRU equation solver requires it). + */ + for (;;) { + fpr *rt1, *rt2, *rt3; + fpr bnorm; + uint32_t normf, normg, norm; + int lim; + + /* + * The poly_small_mkgauss() function makes sure + * that the sum of coefficients is 1 modulo 2 + * (i.e. the resultant of the polynomial with phi + * will be odd). + */ + poly_small_mkgauss(rc, f, logn); + poly_small_mkgauss(rc, g, logn); + + /* + * Verify that all coefficients are within the bounds + * defined in max_fg_bits. This is the case with + * overwhelming probability; this guarantees that the + * key will be encodable with FALCON_COMP_TRIM. + */ + lim = 1 << (PQCLEAN_FALCON512_CLEAN_max_fg_bits[logn] - 1); + for (u = 0; u < n; u ++) { + /* + * We can use non-CT tests since on any failure + * we will discard f and g. + */ + if (f[u] >= lim || f[u] <= -lim + || g[u] >= lim || g[u] <= -lim) { + lim = -1; + break; + } + } + if (lim < 0) { + continue; + } + + /* + * Bound is 1.17*sqrt(q). We compute the squared + * norms. With q = 12289, the squared bound is: + * (1.17^2)* 12289 = 16822.4121 + * Since f and g are integral, the squared norm + * of (g,-f) is an integer. + */ + normf = poly_small_sqnorm(f, logn); + normg = poly_small_sqnorm(g, logn); + norm = (normf + normg) | -((normf | normg) >> 31); + if (norm >= 16823) { + continue; + } + + /* + * We compute the orthogonalized vector norm. + */ + rt1 = (fpr *)tmp; + rt2 = rt1 + n; + rt3 = rt2 + n; + poly_small_to_fp(rt1, f, logn); + poly_small_to_fp(rt2, g, logn); + PQCLEAN_FALCON512_CLEAN_FFT(rt1, logn); + PQCLEAN_FALCON512_CLEAN_FFT(rt2, logn); + PQCLEAN_FALCON512_CLEAN_poly_invnorm2_fft(rt3, rt1, rt2, logn); + PQCLEAN_FALCON512_CLEAN_poly_adj_fft(rt1, logn); + PQCLEAN_FALCON512_CLEAN_poly_adj_fft(rt2, logn); + PQCLEAN_FALCON512_CLEAN_poly_mulconst(rt1, fpr_q, logn); + PQCLEAN_FALCON512_CLEAN_poly_mulconst(rt2, fpr_q, logn); + PQCLEAN_FALCON512_CLEAN_poly_mul_autoadj_fft(rt1, rt3, logn); + PQCLEAN_FALCON512_CLEAN_poly_mul_autoadj_fft(rt2, rt3, logn); + PQCLEAN_FALCON512_CLEAN_iFFT(rt1, logn); + PQCLEAN_FALCON512_CLEAN_iFFT(rt2, logn); + bnorm = fpr_zero; + for (u = 0; u < n; u ++) { + bnorm = fpr_add(bnorm, fpr_sqr(rt1[u])); + bnorm = fpr_add(bnorm, fpr_sqr(rt2[u])); + } + if (!fpr_lt(bnorm, fpr_bnorm_max)) { + continue; + } + + /* + * Compute public key h = g/f mod X^N+1 mod q. If this + * fails, we must restart. + */ + if (h == NULL) { + h2 = (uint16_t *)tmp; + tmp2 = h2 + n; + } else { + h2 = h; + tmp2 = (uint16_t *)tmp; + } + if (!PQCLEAN_FALCON512_CLEAN_compute_public(h2, f, g, logn, (uint8_t *)tmp2)) { + continue; + } + + /* + * Solve the NTRU equation to get F and G. + */ + lim = (1 << (PQCLEAN_FALCON512_CLEAN_max_FG_bits[logn] - 1)) - 1; + if (!solve_NTRU(logn, F, G, f, g, lim, (uint32_t *)tmp)) { + continue; + } + + /* + * Key pair is generated. + */ + break; + } +} |