// Fast integer multiplication using FFT in a modular ring. // Bruno Haible 5.5.1996 // FFT in the complex domain has the drawback that it needs careful round-off // error analysis. So here we choose another field of characteristic 0: Q_p. // Since Q_p contains exactly the (p-1)th roots of unity, we choose // p == 1 mod N and have the Nth roots of unity (N = 2^n) in Q_p and // even in Z_p. Actually, we compute in Z/(p^m Z). // All operations the FFT algorithm needs is addition, subtraction, // multiplication, multiplication by the Nth root of unity and division // by N. Hence we can use the domain Z/(p^m Z) even if p is not a prime! // We want to compute the convolution of N 32-bit words. The resulting // words are < (2^32)^2 * N. If is safe to compute in Z/pZ with p = 2^94 + 1 // or p = 7*2^92 + 1. We choose p < 2^95 so that we can easily represent every // element of Z/pZ as three 32-bit words. #if !(intDsize==32) #error "fft mod p implemented only for intDsize==32" #endif #if 0 typedef union { #if CL_DS_BIG_ENDIAN_P struct { uint32 w2; uint32 w1; uint32 w0; }; #else struct { uint32 w0; uint32 w1; uint32 w2; }; #endif uintD _w[3]; } fftp_word; #else // typedef struct { uint32 w2; uint32 w1; uint32 w0; } fftp_word; // typedef struct { uint32 w0; uint32 w1; uint32 w2; } fftp_word; typedef struct { uintD _w[3]; } fftp_word; #endif #if CL_DS_BIG_ENDIAN_P #define w2 _w[0] #define w1 _w[1] #define w0 _w[2] #define W3(W2,W1,W0) { W2, W1, W0 } #else #define w0 _w[0] #define w1 _w[1] #define w2 _w[2] #define W3(W2,W1,W0) { W0, W1, W2 } #endif #if 0 // p = 19807040628566084398385987585 = 5 * 3761 * 7484047069 * 140737471578113 static const fftp_word p = W3( 1L<<30, 0, 1 ); // p = 2^94 + 1 #define FFT_P_94 static const fftp_word fftp_roots_of_1 [24+1] = // roots_of_1[n] is a (2^n)th root of unity in Z/pZ. // (Also roots_of_1[n-1] = roots_of_1[n]^2, but we don't need this.) // (To build this table, you need to compute roots of unity modulo the // factors of p and combine them using the Chinese Remainder Theorem. // Or ask me for "quadmod.lsp".) { W3( 0x00000000, 0x00000000, 0x00000001 ), // 1 W3( 0x0000003F, 0xFFFFFFFF, 0xFF800000 ), // 1180591620717402914816 W3( 0x20000040, 0x00004000, 0x00000001 ), // 9903521494874733285348474881 W3( 0x3688E9A7, 0xDD78E2A9, 0x1E75974D ), // 16877707849775746711303853901 W3( 0x286E6589, 0x5E86C1E0, 0x42710379 ), // 12512861726041464545960067961 W3( 0x00D79325, 0x1A884885, 0xEA46D6C5 ), // 260613923531515619478787781 W3( 0x1950B480, 0xC387CEE5, 0xA69C443F ), // 7834691712342412468047070271 W3( 0x19DC9D08, 0x11CADC6A, 0x5BA8B123 ), // 8003830486242687653832601891 W3( 0x21D6D905, 0xB8BAC7C3, 0xC3841613 ), // 10472740308573592285123712531 W3( 0x27D73986, 0x6AF6BD27, 0x7A6D7909 ), // 12330106088710388189231937801 W3( 0x20D4698B, 0x0039D457, 0xA092AECF ), // 10160311000635748689099534031 W3( 0x049BD1C4, 0xA94F001A, 0xFA76E358 ), // 1426314143682376031341568856 W3( 0x26DD7228, 0x09400257, 0x9BB49CB9 ), // 12028142067661291067236719801 W3( 0x12DAA9AD, 0xAF9435A9, 0xD50FF483 ), // 5835077289334326375656453251 W3( 0x0B7CDA03, 0x9418702E, 0x7CD934CA ), // 3555271451571910239441204426 W3( 0x2D272FCF, 0xB8644522, 0x68EAD40B ), // 13974199331913037576372147211 W3( 0x00EDA06E, 0x0114DA26, 0xE8D84BA9 ), // 287273027105701319912475561 W3( 0x2219C2C4, 0xFD3145C6, 0xDD019359 ), // 10553633252320053510122083161 W3( 0x1764F007, 0x4F5D5FD4, 0xDAB10AFC ), // 7240181310654329198595869436 W3( 0x01AA13EE, 0x2D1CD906, 0x11D5B1EB ), // 515096517694807745704079851 W3( 0x27038944, 0x5A37BAAD, 0x5CECA64C ), // 12074190385578921562318087756 W3( 0x2459CF22, 0xF625FD38, 0xADB48511 ), // 11250032926302238120667809041 W3( 0x25B6C6A8, 0xD684063F, 0x7ABAD1EF ), // 11671908005633729316324561391 W3( 0x1C1A2BC6, 0x12B253F1, 0x0D1BBCB7 ), // 8697219061868963805380983991 W3( 0x198F3FE2, 0x5EE9919F, 0x535E80D5 } // 7910303322630257758732976341 }; // Sadly, this p doesn't work because we don't find a (2^n)th root of unity w // such that w^(2^(n-1)) = -1 mod p. However, our algorithm below assumes // that w^(2^(n-1)) = -1... #else // p = 34662321099990647697175478273, a prime static const fftp_word p = W3( 7L<<28, 0, 1 ); // p = 7 * 2^92 + 1 #define FFT_P_92 static const fftp_word fftp_roots_of_1 [92+1] = // roots_of_1[n] is a (2^n)th root of unity in Z/pZ. // (Also roots_of_1[n-1] = roots_of_1[n]^2, but we don't need this.) { W3( 0x00000000, 0x00000000, 0x00000001 ), // 1 W3( 0x70000000, 0x00000000, 0x00000000 ), // 34662321099990647697175478272 W3( 0x064AF70F, 0x997E62CE, 0x77953100 ), // 1947537281862369253065568512 W3( 0x261B8E96, 0xC3AD4296, 0xDA1BFA93 ), // 11793744727492885369350519443 W3( 0x096EA949, 0x6EDCAF05, 0x47C92A4F ), // 2919146363086089454841571919 W3( 0x366A8C3F, 0x7BF1436D, 0x2333BE9E ), // 16840998969615256469762195102 W3( 0x27569FA8, 0xAE1775F1, 0xB21956A0 ), // 12174636971387721414084220576 W3( 0x16CABB8B, 0xBAA59813, 0x62FCBCD9 ), // 7053758891710792545762852057 W3( 0x1AE130A3, 0xF909B101, 0xB6BA30CF ), // 8318848263123793919933558991 W3( 0x32AE8FEE, 0x6B1A656B, 0xED02BF24 ), // 15685283280129931240441823012 W3( 0x2D1EE047, 0x5AEDC882, 0x8E96BCCC ), // 13964152342912072497719852236 W3( 0x222A18FD, 0x3BF40635, 0xBFDEA8AD ), // 10573383226491471052459124909 W3( 0x10534EE6, 0xED5A55D4, 0x06AE2155 ), // 5052473604609413010647032149 W3( 0x02F3BFA3, 0x2D816786, 0xE6C27B3C ), // 913643975905572976593107772 W3( 0x0B0CD0A5, 0x9A1FF4F7, 0x2624A5E1 ), // 3419827524917244902802499041 W3( 0x257A492F, 0x156C141C, 0xFC5D75F4 ), // 11598779914676604137587439092 W3( 0x061FB92A, 0xB1A1F41A, 0x7006920F ), // 1895261184698485907279745551 W3( 0x2A4E1471, 0xDDB96073, 0xD8DDBB71 ), // 13092763174215078887900953457 W3( 0x213B469E, 0xD72A84CA, 0xAAA477F2 ), // 10284665443205365583657072626 W3( 0x1D7EF67C, 0x3DC2DA37, 0x4C86E9DC ), // 9128553932091860654576036316 W3( 0x0CB7AA67, 0x2E087ED8, 0x2675D6E3 ), // 3935858248479385987820410595 W3( 0x00BD7B24, 0x68388052, 0x57FFFB10 ), // 229068502577238003716979472 W3( 0x1E1724A6, 0xBA587C3D, 0x0C12825B ), // 9312528669272006966417457755 W3( 0x20595EF0, 0xC89DA33B, 0x3CB5583B ), // 10011563056352601486430394427 W3( 0x15E730B2, 0x6D34E9EB, 0x71CCE555 ), // 6778677035560020292206912853 W3( 0x015EFDBB, 0xC0A80C3B, 0xE4B1E017 ), // 424322259008787317821399063 W3( 0x1B81FC63, 0x0C694944, 0x8EB481BF ), // 8513238559382277026756198847 W3( 0x1AF53421, 0x5DCAA1A4, 0xD0C15A03 ), // 8343043259718611508685527555 W3( 0x2F2B6B58, 0xBB60E464, 0x37A7DE2E ), // 14598286201875835624993840686 W3( 0x27B4AB13, 0x54617640, 0xE86E757A ), // 12288329911800070034603013498 W3( 0x041A31D2, 0xF0AC8E3C, 0x8AA4FD27 ), // 1269607397711669380834983207 W3( 0x1A52F484, 0x39AC5917, 0x34E3F1F7 ), // 8146896869111203814625767927 W3( 0x048FC120, 0x50F6ECBF, 0x268D86A8 ), // 1411728444351387120148776616 W3( 0x27A2C427, 0x001F1239, 0x93380047 ), // 12266687669072434694473646151 W3( 0x2E7E8DFB, 0x2411A754, 0xE12A9B1D ), // 14389305591459206001391737629 W3( 0x29F14702, 0x40B3E1E2, 0xF7D71A8D ), // 12980571854778363745245010573 W3( 0x3158DCE7, 0x8B8FEB32, 0x1DE35D24 ), // 15272194145252623177165790500 W3( 0x12484C07, 0x437ED373, 0x9E45F602 ), // 5658131869639928287764805122 W3( 0x1AEAE06E, 0xB905C908, 0x4389BF5F ), // 8330558749711089231534341983 W3( 0x27BC0045, 0x43024FEB, 0xEC880258 ), // 12297194714773858676269122136 W3( 0x2EFE1CBC, 0x0D2FAA94, 0xB4EA69A6 ), // 14543513305163560781242591654 W3( 0x0B0D3D8B, 0xD779F105, 0x920367FA ), // 3420341787669373425792804858 W3( 0x2D4D7BA9, 0x0970D8CF, 0x8CE6D7EC ), // 14020496699328277892009744364 W3( 0x00DC5971, 0x0209470E, 0x713F2B27 ), // 266386055561000736260041511 W3( 0x27E54E26, 0x53BA0137, 0xDD6740B3 ), // 12347128447319282384829366451 W3( 0x2143A889, 0x8F2B57F5, 0xFB8181C1 ), // 10294799249108063706647986625 W3( 0x1125419F, 0x5C4E0608, 0xE0AC0396 ), // 5306285315793562029414679446 W3( 0x15B61D90, 0x63A27BB0, 0x26402B32 ), // 6719349317556695539371748146 W3( 0x03B582FC, 0x419EF656, 0xB06BBC35 ), // 1147889163765050226454019125 W3( 0x08FF62E1, 0xA3BB1145, 0xDA998F77 ), // 2784623116803271439773437815 W3( 0x101978AF, 0xF93CBFA1, 0xB788B5A3 ), // 4982553232749484200897852835 W3( 0x061334DE, 0x8FE5C6E9, 0x2B2309D6 ), // 1880129318103954373583505878 W3( 0x343C6E7C, 0x8019BB43, 0xD954E744 ), // 16166277816826816936484857668 W3( 0x06506A03, 0x0E6DE333, 0xF8011494 ), // 1954124751724394051182597268 W3( 0x34892A42, 0x6502DAA3, 0x8FDA6971 ), // 16259042912153157504364865905 W3( 0x0EF2C4BD, 0xF42D9711, 0xC32CEA49 ), // 4626279273705729025744104009 W3( 0x24511305, 0x4F1EAE2C, 0x62FB10F4 ), // 11239473167855288013010178292 W3( 0x14E5A052, 0xF1748A9C, 0xDD536730 ), // 6467301317787608309692589872 W3( 0x0621D0A7, 0x0A5188AF, 0x7316C352 ), // 1897789944553576071437927250 W3( 0x234498F0, 0xDF078E95, 0x6FEED50B ), // 10914904542475816633386325259 W3( 0x029E4925, 0x948D6D57, 0xD4DF93A6 ), // 810325725128913871737688998 W3( 0x11BB3805, 0x0589D746, 0x852F3E2F ), // 5487578840386649632552205871 W3( 0x1D4370CA, 0xA4441B85, 0xC9606FE0 ), // 9056595957858187419376971744 W3( 0x1C536F7D, 0x77D44926, 0x8DDB8932 ), // 8766447615182890705620797746 W3( 0x3498CE71, 0xB726A4D3, 0xF4F3C813 ), // 16277952140466335672647796755 W3( 0x1E4A297E, 0xAC13196E, 0xFACD8102 ), // 9374206759006667727054930178 W3( 0x0E7C2CCC, 0xC940C98B, 0x0BC0CA49 ), // 4482908500893894680116251209 W3( 0x124CF912, 0xD84438FD, 0x9C03585F ), // 5663784755954194195257972831 W3( 0x06180FF8, 0xD447BEBE, 0xDB8821E7 ), // 1885999704184999223512015335 W3( 0x1ED2EB11, 0x0687EC7C, 0xBE3436C8 ), // 9539534786948152514714023624 W3( 0x30EBB35C, 0x59616A3C, 0x502CBB52 ), // 15140225046175435352009653074 W3( 0x33E24883, 0xEDA36D60, 0xA25C8E5F ), // 16057295180155395438855097951 W3( 0x0D879ED9, 0x076BAB06, 0x9BE12AA2 ), // 4187260250707927256570866338 W3( 0x1A1B6C9C, 0x0966383B, 0x54123A87 ), // 8079764146434082816365050503 W3( 0x31BD863A, 0xA2A6505C, 0xD759E6CF ), // 15393886339893077529104869071 W3( 0x3209AF0A, 0x5E5055A1, 0x480AF03F ), // 15485957428841754012708171839 W3( 0x1A4CC03C, 0xC8AA650B, 0x7F4DBCE9 ), // 8139396433274519652257348841 W3( 0x3596471F, 0xB99D2EA3, 0xA3433E0C ), // 16584380266717730797139475980 W3( 0x28E87642, 0x98E21FCE, 0xDE1B53EA ), // 12660429650750886812340409322 W3( 0x20161DB9, 0xCDC199E9, 0x0A6BEDF2 ), // 9930257058416521571476434418 W3( 0x1D0DC095, 0x2C40D22B, 0x088549BA ), // 8991690766592354745340742074 W3( 0x2FCC953C, 0xA8B62408, 0x50FC4C29 ), // 14793121080372138443684989993 W3( 0x0F854B39, 0xF659B4B5, 0xD2B0A6AC ), // 4803417528030967235217499820 W3( 0x30E087D4, 0x02F3BBAB, 0xBA503373 ), // 15126721285415891216108630899 W3( 0x0DAF660C, 0x26B99C42, 0x98B8BE05 ), // 4235349051642660841298902533 W3( 0x0ED6AE0E, 0xCD02982A, 0xD233F0D9 ), // 4592322227691334993146278105 W3( 0x3415EB9B, 0x4B61C19F, 0xB21F1255 ), // 16119720573722492095181034069 W3( 0x1015A729, 0x20A1FAA2, 0x0D094529 ), // 4977936993224010619482096937 W3( 0x1D2E3AD2, 0x7093579F, 0x1C93C97B ), // 9030953651705465548198627707 W3( 0x130EAA8F, 0x859C980F, 0xD9E7E8ED ), // 5897945597894388791627999469 W3( 0x2B7CA1C8, 0xFC34C5B5, 0x9C0B1C0C ), // 13458526232475976507763399692 W3( 0x22367055, 0xA53B526A, 0x7505EABE ), // 10588302813110450634719881918 W3( 0x344FEF55, 0x0B77067F, 0x38999E77 ) // 16189855864848287589134343799 }; #endif // Define this if you want the external loops instead of inline operations. #define FFTP_EXTERNAL_LOOPS // Define this for (cheap) consistency checks. //#define DEBUG_FFTP // Define this for extensive consistency checks. //#define DEBUG_FFTP_OPERATIONS // Define the algorithm of the backward FFT: // Either FORWARD (a normal FFT followed by a permutation) // or RECIPROOT (an FFT with reciprocal root of unity) // or CLEVER (an FFT with reciprocal root of unity but clever computation // of the reciprocals). // Drawback of FORWARD: the permutation pass. // Drawback of RECIPROOT: need all the powers of the root, not only half of them. #define FORWARD 42 #define RECIPROOT 43 #define CLEVER 44 #define FFTP_BACKWARD CLEVER // r := a + b static inline void add (const fftp_word& a, const fftp_word& b, fftp_word& r) { #ifdef FFTP_EXTERNAL_LOOPS add_loop_lsp(arrayLSDptr(a._w,3),arrayLSDptr(b._w,3),arrayLSDptr(r._w,3),3); #else var uint32 tmp; tmp = a.w0 + b.w0; if (tmp >= a.w0) { // no carry r.w0 = tmp; tmp = a.w1 + b.w1; if (tmp >= a.w1) goto no_carry_1; else goto carry_1; } else { // carry r.w0 = tmp; tmp = a.w1 + b.w1 + 1; if (tmp > a.w1) goto no_carry_1; else goto carry_1; } if (1) { no_carry_1: // no carry r.w1 = tmp; tmp = a.w2 + b.w2; } else { carry_1: // carry r.w1 = tmp; tmp = a.w2 + b.w2 + 1; } r.w2 = tmp; #endif } // r := a - b static inline void sub (const fftp_word& a, const fftp_word& b, fftp_word& r) { #ifdef FFTP_EXTERNAL_LOOPS sub_loop_lsp(arrayLSDptr(a._w,3),arrayLSDptr(b._w,3),arrayLSDptr(r._w,3),3); #else var uint32 tmp; tmp = a.w0 - b.w0; if (tmp <= a.w0) { // no carry r.w0 = tmp; tmp = a.w1 - b.w1; if (tmp <= a.w1) goto no_carry_1; else goto carry_1; } else { // carry r.w0 = tmp; tmp = a.w1 - b.w1 - 1; if (tmp < a.w1) goto no_carry_1; else goto carry_1; } if (1) { no_carry_1: // no carry r.w1 = tmp; tmp = a.w2 - b.w2; } else { carry_1: // carry r.w1 = tmp; tmp = a.w2 - b.w2 - 1; } r.w2 = tmp; #endif } // b := a >> 1 static inline void shift (const fftp_word& a, fftp_word& b) { #ifdef FFTP_EXTERNAL_LOOPS #ifdef DEBUG_FFTP if (shiftrightcopy_loop_msp(arrayMSDptr(a._w,3),arrayMSDptr(b._w,3),3,1,0)) cl_abort(); #else shiftrightcopy_loop_msp(arrayMSDptr(a._w,3),arrayMSDptr(b._w,3),3,1,0); #endif #else var uint32 tmp, carry; tmp = a.w2; b.w2 = a.w2 >> 1; carry = tmp << 31; tmp = a.w1; b.w1 = (tmp >> 1) | carry; carry = tmp << 31; tmp = a.w0; b.w0 = (tmp >> 1) | carry; #ifdef DEBUG_FFTP carry = tmp << 31; if (carry) cl_abort(); #endif #endif } #ifdef DEBUG_FFTP_OPERATIONS #define check_fftp_word(x) if (compare_loop_msp(arrayMSDptr((x)._w,3),arrayMSDptr(p._w,3),3) >= 0) cl_abort() #else #define check_fftp_word(x) #endif // r := (a + b) mod p static inline void addp (const fftp_word& a, const fftp_word& b, fftp_word& r) { check_fftp_word(a); check_fftp_word(b); #ifdef FFTP_EXTERNAL_LOOPS add(a,b, r); if (compare_loop_msp(arrayMSDptr(r._w,3),arrayMSDptr(p._w,3),3) >= 0) sub(r,p, r); #else add(a,b, r); if ((r.w2 > p.w2) || ((r.w2 == p.w2) && ((r.w1 > p.w1) || ((r.w1 == p.w1) && (r.w0 >= p.w0))))) sub(r,p, r); #endif check_fftp_word(r); } // r := (a - b) mod p static inline void subp (const fftp_word& a, const fftp_word& b, fftp_word& r) { check_fftp_word(a); check_fftp_word(b); sub(a,b, r); if ((sint32)r.w2 < 0) add(r,p, r); check_fftp_word(r); } // r := (a * b) mod p static void mulp (const fftp_word& a, const fftp_word& b, fftp_word& r) { check_fftp_word(a); check_fftp_word(b); #if defined(FFT_P_94) var uintD c[6]; var uintD* const cLSDptr = arrayLSDptr(c,6); // Multiply the two words, using the standard method. mulu_2loop(arrayLSDptr(a._w,3),3, arrayLSDptr(b._w,3),3, cLSDptr); // c[0..5] now contains the product. // Divide by p. // To divide c (0 <= c < p^2) by p = 2^n+1, // we set q := floor(c/2^n) and r := c - q*p = (c mod 2^n) - q. // If this becomes negative, set r := r + p (at most twice). // (This works because floor(c/p) <= q <= floor(c/p)+2.) // (Actually, here, 0 <= c <= (p-1)^2, hence // floor(c/p) <= q <= floor(c/p)+1, so we have // to set r := r + p at most once!) // n = 94 = 3*32-2 = 2*32+30. shiftleft_loop_lsp(cLSDptr lspop 3,3,2,lspref(cLSDptr,2)>>30); lspref(cLSDptr,2) &= bit(30)-1; // c[0..2] now contains q, c[3..5] contains (c mod 2^n). #if 0 if (compare_loop_msp(cLSDptr lspop 6,arrayMSDptr(p._w,3),3) >= 0) // q >= p ? subfrom_loop_lsp(arrayLSDptr(p._w,3),cLSDptr lspop 3,3); // q -= p; #endif if (subfrom_loop_lsp(cLSDptr lspop 3,cLSDptr,3)) // (c mod 2^n) - q addto_loop_lsp(arrayLSDptr(p._w,3),cLSDptr,3); r.w2 = lspref(cLSDptr,2); r.w1 = lspref(cLSDptr,1); r.w0 = lspref(cLSDptr,0); #elif defined(FFT_P_92) var uintD c[7]; var uintD* const cLSDptr = arrayLSDptr(c,7); // Multiply the two words, using the standard method. mulu_2loop(arrayLSDptr(a._w,3),3, arrayLSDptr(b._w,3),3, cLSDptr); // c[1..6] now contains the product. // Divide by p. // To divide c (0 <= c < p^2) by p = 7*2^n+1, // we set q := floor(floor(c/2^n)/7) and // r := c - q*p = (floor(c/2^n) mod 7)*2^n + (c mod 2^n) - q. // If this becomes negative, set r := r + p. // (As above, since 0 <= c <= (p-1)^2, we have // floor(c/p) <= q <= floor(c/p)+1, so we have // to set r := r + p at most once!) // n = 92 = 3*32-4 = 2*32+28. lspref(cLSDptr,6) = shiftleft_loop_lsp(cLSDptr lspop 3,3,4,lspref(cLSDptr,2)>>28); lspref(cLSDptr,2) &= bit(28)-1; // c[0..3] now contains floor(c/2^n), c[4..6] contains (c mod 2^n). var uintD remainder = divu_loop_msp(7,cLSDptr lspop 7,4); lspref(cLSDptr,2) |= remainder << 28; // c[0..3] now contains q, c[4..6] contains (c mod 7*2^n). #ifdef DEBUG_FFTP if (lspref(cLSDptr,6) > 0) cl_abort(); #endif #if 0 if (compare_loop_msp(cLSDptr lspop 6,arrayMSDptr(p._w,3),3) >= 0) // q >= p ? subfrom_loop_lsp(arrayLSDptr(p._w,3),cLSDptr lspop 3,3); // q -= p; #endif if (subfrom_loop_lsp(cLSDptr lspop 3,cLSDptr,3)) // (c mod 2^n) - q addto_loop_lsp(arrayLSDptr(p._w,3),cLSDptr,3); r.w2 = lspref(cLSDptr,2); r.w1 = lspref(cLSDptr,1); r.w0 = lspref(cLSDptr,0); #else #error "mulp not implemented for this prime" #endif if ((sint32)r.w2 < 0) cl_abort(); check_fftp_word(r); } #ifdef DEBUG_FFTP_OPERATIONS static void mulp_doublecheck (const fftp_word& a, const fftp_word& b, fftp_word& r) { fftp_word zero, ma, mb, or; subp(a,a, zero); subp(zero,a, ma); subp(zero,b, mb); mulp(ma,mb, or); mulp(a,b, r); if (compare_loop_msp(arrayMSDptr(r._w,3),arrayMSDptr(or._w,3),3)) cl_abort(); } #define mulp mulp_doublecheck #endif /* DEBUG_FFTP_OPERATIONS */ // b := (a / 2) mod p static inline void shiftp (const fftp_word& a, fftp_word& b) { check_fftp_word(a); if (a.w0 & 1) { var fftp_word a_even; add(a,p, a_even); shift(a_even, b); } else shift(a, b); check_fftp_word(b); } #ifndef _BIT_REVERSE #define _BIT_REVERSE // Reverse an n-bit number x. n>0. static uintL bit_reverse (uintL n, uintL x) { var uintL y = 0; do { y <<= 1; y |= (x & 1); x >>= 1; } while (!(--n == 0)); return y; } #endif // Compute an convolution mod p using FFT: z[0..N-1] := x[0..N-1] * y[0..N-1]. static void fftp_convolution (const uintL n, const uintL N, // N = 2^n fftp_word * x, // N words fftp_word * y, // N words fftp_word * z // N words result ) { CL_ALLOCA_STACK; #if (FFTP_BACKWARD == RECIPROOT) || defined(DEBUG_FFTP) var fftp_word* const w = cl_alloc_array(fftp_word,N); #else var fftp_word* const w = cl_alloc_array(fftp_word,(N>>1)+1); #endif var uintL i; // Initialize w[i] to w^i, w a primitive N-th root of unity. w[0] = fftp_roots_of_1[0]; w[1] = fftp_roots_of_1[n]; #if (FFTP_BACKWARD == RECIPROOT) || defined(DEBUG_FFTP) for (i = 2; i < N; i++) mulp(w[i-1],fftp_roots_of_1[n], w[i]); #else // need only half of the roots for (i = 2; i < N>>1; i++) mulp(w[i-1],fftp_roots_of_1[n], w[i]); #endif #ifdef DEBUG_FFTP // Check that w is really a primitive N-th root of unity. { var fftp_word w_N; mulp(w[N-1],fftp_roots_of_1[n], w_N); if (!(w_N.w2 == 0 && w_N.w1 == 0 && w_N.w0 == 1)) cl_abort(); w_N = w[N>>1]; if (!(w_N.w2 == p.w2 && w_N.w1 == p.w1 && w_N.w0 == p.w0 - 1)) cl_abort(); } #endif var bool squaring = (x == y); // Do an FFT of length N on x. { var sintL l; /* l = n-1 */ { var const uintL tmax = N>>1; // tmax = 2^(n-1) for (var uintL t = 0; t < tmax; t++) { var uintL i1 = t; var uintL i2 = i1 + tmax; // Butterfly: replace (x(i1),x(i2)) by // (x(i1) + x(i2), x(i1) - x(i2)). var fftp_word tmp; tmp = x[i2]; subp(x[i1],tmp, x[i2]); addp(x[i1],tmp, x[i1]); } } for (l = n-2; l>=0; l--) { var const uintL smax = (uintL)1 << (n-1-l); var const uintL tmax = (uintL)1 << l; for (var uintL s = 0; s < smax; s++) { var uintL exp = bit_reverse(n-1-l,s) << l; for (var uintL t = 0; t < tmax; t++) { var uintL i1 = (s << (l+1)) + t; var uintL i2 = i1 + tmax; // Butterfly: replace (x(i1),x(i2)) by // (x(i1) + w^exp*x(i2), x(i1) - w^exp*x(i2)). var fftp_word tmp; mulp(x[i2],w[exp], tmp); subp(x[i1],tmp, x[i2]); addp(x[i1],tmp, x[i1]); } } } } // Do an FFT of length N on y. if (!squaring) { var sintL l; /* l = n-1 */ { var uintL const tmax = N>>1; // tmax = 2^(n-1) for (var uintL t = 0; t < tmax; t++) { var uintL i1 = t; var uintL i2 = i1 + tmax; // Butterfly: replace (y(i1),y(i2)) by // (y(i1) + y(i2), y(i1) - y(i2)). var fftp_word tmp; tmp = y[i2]; subp(y[i1],tmp, y[i2]); addp(y[i1],tmp, y[i1]); } } for (l = n-2; l>=0; l--) { var const uintL smax = (uintL)1 << (n-1-l); var const uintL tmax = (uintL)1 << l; for (var uintL s = 0; s < smax; s++) { var uintL exp = bit_reverse(n-1-l,s) << l; for (var uintL t = 0; t < tmax; t++) { var uintL i1 = (s << (l+1)) + t; var uintL i2 = i1 + tmax; // Butterfly: replace (y(i1),y(i2)) by // (y(i1) + w^exp*y(i2), y(i1) - w^exp*y(i2)). var fftp_word tmp; mulp(y[i2],w[exp], tmp); subp(y[i1],tmp, y[i2]); addp(y[i1],tmp, y[i1]); } } } } // Multiply the transformed vectors into z. for (i = 0; i < N; i++) mulp(x[i],y[i], z[i]); // Undo an FFT of length N on z. { var uintL l; for (l = 0; l < n-1; l++) { var const uintL smax = (uintL)1 << (n-1-l); var const uintL tmax = (uintL)1 << l; #if FFTP_BACKWARD != CLEVER for (var uintL s = 0; s < smax; s++) { var uintL exp = bit_reverse(n-1-l,s) << l; #if FFTP_BACKWARD == RECIPROOT if (exp > 0) exp = N - exp; // negate exp (use w^-1 instead of w) #endif for (var uintL t = 0; t < tmax; t++) { var uintL i1 = (s << (l+1)) + t; var uintL i2 = i1 + tmax; // Inverse Butterfly: replace (z(i1),z(i2)) by // ((z(i1)+z(i2))/2, (z(i1)-z(i2))/(2*w^exp)). var fftp_word sum; var fftp_word diff; addp(z[i1],z[i2], sum); subp(z[i1],z[i2], diff); shiftp(sum, z[i1]); mulp(diff,w[exp], diff); shiftp(diff, z[i2]); } } #else // FFTP_BACKWARD == CLEVER: clever handling of negative exponents /* s = 0, exp = 0 */ { for (var uintL t = 0; t < tmax; t++) { var uintL i1 = t; var uintL i2 = i1 + tmax; // Inverse Butterfly: replace (z(i1),z(i2)) by // ((z(i1)+z(i2))/2, (z(i1)-z(i2))/(2*w^exp)), // with exp <-- 0. var fftp_word sum; var fftp_word diff; addp(z[i1],z[i2], sum); subp(z[i1],z[i2], diff); shiftp(sum, z[i1]); shiftp(diff, z[i2]); } } for (var uintL s = 1; s < smax; s++) { var uintL exp = bit_reverse(n-1-l,s) << l; exp = (N>>1) - exp; // negate exp (use w^-1 instead of w) for (var uintL t = 0; t < tmax; t++) { var uintL i1 = (s << (l+1)) + t; var uintL i2 = i1 + tmax; // Inverse Butterfly: replace (z(i1),z(i2)) by // ((z(i1)+z(i2))/2, (z(i1)-z(i2))/(2*w^exp)), // with exp <-- (N/2 - exp). var fftp_word sum; var fftp_word diff; addp(z[i1],z[i2], sum); subp(z[i2],z[i1], diff); // note that w^(N/2) = -1 shiftp(sum, z[i1]); mulp(diff,w[exp], diff); shiftp(diff, z[i2]); } } #endif } /* l = n-1 */ { var const uintL tmax = N>>1; // tmax = 2^(n-1) for (var uintL t = 0; t < tmax; t++) { var uintL i1 = t; var uintL i2 = i1 + tmax; // Inverse Butterfly: replace (z(i1),z(i2)) by // ((z(i1)+z(i2))/2, (z(i1)-z(i2))/2). var fftp_word sum; var fftp_word diff; addp(z[i1],z[i2], sum); subp(z[i1],z[i2], diff); shiftp(sum, z[i1]); shiftp(diff, z[i2]); } } } #if FFTP_BACKWARD == FORWARD // Swap z[i] and z[N-i] for 0 < i < N/2. for (i = (N>>1)-1; i > 0; i--) { var fftp_word tmp = z[i]; z[i] = z[N-i]; z[N-i] = tmp; } #endif } static void mulu_fft_modp (const uintD* sourceptr1, uintC len1, const uintD* sourceptr2, uintC len2, uintD* destptr) // Es ist 2 <= len1 <= len2. { // Methode: // source1 ist ein Stück der Länge N1, source2 ein oder mehrere Stücke // der Länge N2, mit N1+N2 <= N, wobei N Zweierpotenz ist. // sum(i=0..N-1, x_i b^i) * sum(i=0..N-1, y_i b^i) wird errechnet, // indem man die beiden Polynome // sum(i=0..N-1, x_i T^i), sum(i=0..N-1, y_i T^i) // multipliziert, und zwar durch Fourier-Transformation (s.o.). var uint32 n; integerlength32(len1-1, n=); // 2^(n-1) < len1 <= 2^n var uintL len = (uintL)1 << n; // kleinste Zweierpotenz >= len1 // Wählt man N = len, so hat man ceiling(len2/(len-len1+1)) * FFT(len). // Wählt man N = 2*len, so hat man ceiling(len2/(2*len-len1+1)) * FFT(2*len). // Wir wählen das billigere von beiden: // Bei ceiling(len2/(len-len1+1)) <= 2 * ceiling(len2/(2*len-len1+1)) // nimmt man N = len, bei ....... > ........ dagegen N = 2*len. // (Wahl von N = 4*len oder mehr bringt nur in Extremfällen etwas.) if (len2 > 2 * (len-len1+1) * (len2 <= (2*len-len1+1) ? 1 : ceiling(len2,(2*len-len1+1)))) { n = n+1; len = len << 1; } var const uintL N = len; // N = 2^n CL_ALLOCA_STACK; var fftp_word* const x = cl_alloc_array(fftp_word,N); var fftp_word* const y = cl_alloc_array(fftp_word,N); #ifdef DEBUG_FFTP var fftp_word* const z = cl_alloc_array(fftp_word,N); #else var fftp_word* const z = x; // put z in place of x - saves memory #endif var uintD* const tmpprod = cl_alloc_array(uintD,len1+1); var uintP i; var uintL destlen = len1+len2; clear_loop_lsp(destptr,destlen); do { var uintL len2p; // length of a piece of source2 len2p = N - len1 + 1; if (len2p > len2) len2p = len2; // len2p = min(N-len1+1,len2). if (len2p == 1) { // cheap case var uintD* tmpptr = arrayLSDptr(tmpprod,len1+1); mulu_loop_lsp(lspref(sourceptr2,0),sourceptr1,tmpptr,len1); if (addto_loop_lsp(tmpptr,destptr,len1+1)) if (inc_loop_lsp(destptr lspop (len1+1),destlen-(len1+1))) cl_abort(); } else { var uintL destlenp = len1 + len2p - 1; // destlenp = min(N,destlen-1). var bool squaring = ((sourceptr1 == sourceptr2) && (len1 == len2p)); // Fill factor x. { for (i = 0; i < len1; i++) { x[i].w0 = lspref(sourceptr1,i); x[i].w1 = 0; x[i].w2 = 0; } for (i = len1; i < N; i++) { x[i].w0 = 0; x[i].w1 = 0; x[i].w2 = 0; } } // Fill factor y. if (!squaring) { for (i = 0; i < len2p; i++) { y[i].w0 = lspref(sourceptr2,i); y[i].w1 = 0; y[i].w2 = 0; } for (i = len2p; i < N; i++) { y[i].w0 = 0; y[i].w1 = 0; y[i].w2 = 0; } } // Multiply. if (!squaring) fftp_convolution(n,N, &x[0], &y[0], &z[0]); else fftp_convolution(n,N, &x[0], &x[0], &z[0]); #ifdef DEBUG_FFTP // Check result. for (i = 0; i < N; i++) if (!(z[i].w2 < N)) cl_abort(); #endif // Add result to destptr[-destlen..-1]: { var uintD* ptr = destptr; // ac2|ac1|ac0 are an accumulator. var uint32 ac0 = 0; var uint32 ac1 = 0; var uint32 ac2 = 0; var uint32 tmp; for (i = 0; i < destlenp; i++) { // Add z[i] to the accumulator. tmp = z[i].w0; if ((ac0 += tmp) < tmp) { if (++ac1 == 0) ++ac2; } tmp = z[i].w1; if ((ac1 += tmp) < tmp) ++ac2; tmp = z[i].w2; ac2 += tmp; // Add the accumulator's least significant word to destptr: tmp = lspref(ptr,0); if ((ac0 += tmp) < tmp) { if (++ac1 == 0) ++ac2; } lspref(ptr,0) = ac0; lsshrink(ptr); ac0 = ac1; ac1 = ac2; ac2 = 0; } // ac2 = 0. if (ac1 > 0) { if (!((i += 2) <= destlen)) cl_abort(); tmp = lspref(ptr,0); if ((ac0 += tmp) < tmp) ++ac1; lspref(ptr,0) = ac0; lsshrink(ptr); tmp = lspref(ptr,0); ac1 += tmp; lspref(ptr,0) = ac1; lsshrink(ptr); if (ac1 < tmp) if (inc_loop_lsp(ptr,destlen-i)) cl_abort(); } else if (ac0 > 0) { if (!((i += 1) <= destlen)) cl_abort(); tmp = lspref(ptr,0); ac0 += tmp; lspref(ptr,0) = ac0; lsshrink(ptr); if (ac0 < tmp) if (inc_loop_lsp(ptr,destlen-i)) cl_abort(); } } #ifdef DEBUG_FFTP // If destlenp < N, check that the remaining z[i] are 0. for (i = destlenp; i < N; i++) if (z[i].w2 > 0 || z[i].w1 > 0 || z[i].w0 > 0) cl_abort(); #endif } // Decrement len2. destptr = destptr lspop len2p; destlen -= len2p; sourceptr2 = sourceptr2 lspop len2p; len2 -= len2p; } while (len2 > 0); } #undef FFT_P_94 #undef FFT_P_92 #undef w0 #undef w1 #undef w2 #undef W3