Modulus of Product of two integers

问题

I have to find c,

c = (a*b) mod m

a,b,c,m are 32-bit integers. But (a*b) can be more than 32 bits. I am trying to figure out a way to compute c, without using long or any data type > 32 bit.

Any ideas?

What if m is a prime number, can things be simplified?

Note: based on a few comments,

c = ((a mod m) * (b mod m)) mod m, but in my case even this multiplication will overflow

回答1:

I happen to have this 256 by 128 bit division code with a random-based test and I'm sure you can reduce it trivially to 128 by 64 bit division and then to 64 by 32:

#include <limits.h>
#include <stdio.h>
#include <stdlib.h>
#include <time.h>

typedef unsigned long long uint64;

#define C_ASSERT(expr) extern char CAssertExtern[(expr)?1:-1]
C_ASSERT(sizeof(uint64) * CHAR_BIT == 64);

void mul64full(uint64 prod[2], uint64 a, uint64 b)
{
  uint64 p0, p1, p2, p3;

  p0 = (a & 0xFFFFFFFF) *
       (b & 0xFFFFFFFF);

  p1 = (a & 0xFFFFFFFF) *
       (b >> 32);

  p2 = (b & 0xFFFFFFFF) *
       (a >> 32);

  p3 = (a >> 32) *
       (b >> 32);

  prod[0] = p0;
  prod[1] = p3;

  if ((p1 << 32) > 0xFFFFFFFFFFFFFFFFULL - prod[0])
    prod[1]++;
  prod[0] += p1 << 32;
  prod[1] += p1 >> 32;

  if ((p2 << 32) > 0xFFFFFFFFFFFFFFFFULL - prod[0])
    prod[1]++;
  prod[0] += p2 << 32;
  prod[1] += p2 >> 32;
}

void mul128short(uint64 prod[2], const uint64 a[2], const uint64 b[2])
{
  uint64 p0[2], p1[2], p2[2];

  mul64full(p0, a[0], b[0]);
  mul64full(p1, a[0], b[1]);
  mul64full(p2, a[1], b[0]);

  prod[0] = p0[0];
  prod[1] = p0[1] + p1[0] + p2[0];
}

int lessThan128(const uint64 a[2], const uint64 b[2])
{
  return (a[1] < b[1]) || ((a[1] == b[1]) && (a[0] < b[0]));
}

int div256x128(uint64 qr[4], const uint64 dividend[4], const uint64 divisor[2])
{
  int i;

  if (!lessThan128(dividend + 2, divisor))
    return 0; // overflow

  qr[0] = dividend[0];
  qr[1] = dividend[1];
  qr[2] = dividend[2];
  qr[3] = dividend[3];

  for (i = 0; i < 128; i++)
  {
    if (qr[3] >= 0x8000000000000000ULL)
    {
      qr[3] = (qr[3] << 1) | (qr[2] >> 63);
      qr[2] = (qr[2] << 1) | (qr[1] >> 63);
      qr[1] = (qr[1] << 1) | (qr[0] >> 63);
      qr[0] <<= 1;

      if (qr[2] < divisor[0])
        qr[3]--;
      qr[2] -= divisor[0];
      qr[3] -= divisor[1];

      qr[0] |= 1;
    }
    else
    {
      qr[3] = (qr[3] << 1) | (qr[2] >> 63);
      qr[2] = (qr[2] << 1) | (qr[1] >> 63);
      qr[1] = (qr[1] << 1) | (qr[0] >> 63);
      qr[0] <<= 1;

      if (!lessThan128(qr + 2, divisor))
      {
        if (qr[2] < divisor[0])
          qr[3]--;
        qr[2] -= divisor[0];
        qr[3] -= divisor[1];

        qr[0] |= 1;
      }
    }
  }

  return 1;
}

void randfill(void* p, size_t s)
{
  unsigned char* pc = p;
  while (s--)
    *pc++ = rand();
}

int main(void)
{
  int i;

  srand(time(NULL));

  for (i = 0; i < 10000; i++)
  {
    uint64 qr[4], dividend[4], divisor[2];

    // divisor, 64 bits only
    randfill(&divisor[0], sizeof(divisor[0]));
    divisor[1] = 0;

    if (divisor[0] != 0)
    {
      uint64 dividend2[2];

      // dividend, 128 bits only
      randfill(&dividend[0], sizeof(dividend[0]) * 2);
      dividend[3] = dividend[2] = 0;

      // divide
      div256x128(qr, dividend, divisor);

      // test via multiplication and addition
      mul128short(dividend2, qr, divisor);
      if (dividend2[0] + qr[2] < dividend2[0])
        dividend2[1]++;
      dividend2[0] += qr[2];
      dividend2[1] += qr[3];

      printf("0x%016llX%016llX / 0x%016llX = 0x%016llX%016llX:0x%016llX%016llX\n",
             dividend[1], dividend[0], divisor[0], qr[1], qr[0], qr[3], qr[2]);

      if (dividend[0] != dividend2[0] ||
          dividend[1] != dividend2[1])
      {
        printf("0x%016llX%016llX - restored dividend - ERROR\n",
               dividend2[1], dividend2[0]);
        exit(-1);
      }
    }
  }

  return 0;
}

Output reduced to 100 iterations (ideone):

0x72238CB3A1A9F5656A1E0F3567E6CE0E / 0xC7F98F34BBA0FAD9 = 0x0000000000000000921DBA985B7AB177:0x000000000000000060D9C93D34382A2F
0xA05F7D3BF45F2CBFF745F1E97EF75A4A / 0x3CBA50635A5A27A9 = 0x0000000000000002A40EC5297059E69F:0x0000000000000000290F7F44A795E253
0xD48C5015950A24AC8C2C5242F098F28C / 0x3D98659426410873 = 0x0000000000000003736258D9691352D7:0x000000000000000005E859D02FBD03F7
0x985CA6ECCF7A9A8D8A029AFDC4029830 / 0xDC572869B7C9EB7C = 0x0000000000000000B10519E30A95B186:0x0000000000000000454594E80D549948
0x13609B86D699ECD9D5EA40A50EE97DA5 / 0x32B4B9B6DF2D66CA = 0x000000000000000061D4B3346F3C56FA:0x00000000000000000C0857D82EB34061
0x91D63DB80153785C458EDFA05C2BE6A5 / 0x4BB93F8121E3DA35 = 0x0000000000000001ED086E53A490EBA9:0x00000000000000000154DC2121A232A8
0xD5E77579A2E69A028A6F1BE52462A6A3 / 0x40C3C80BAF686F29 = 0x00000000000000034D8356E4FA79EAE5:0x00000000000000002A4ACFAD31FABCF6
0xE7289EE59D2EC9B80A671267274F9F1C / 0xA037AC77A5D63AED = 0x0000000000000001715A12BD82C8D05A:0x00000000000000002762B55566F657CA
0xDB5EB523534D10EC26C14D0A21165EAA / 0x70877083B2890E38 = 0x0000000000000001F30F41B77230648F:0x00000000000000003B161DAC42798D62
0xF22BCF06050EB9FAEDA39C433315D3AF / 0x634AA12A63798C7E = 0x00000000000000027061B6CFBDC47CAC:0x00000000000000002CA1B7EEE6E66707
0x6A5F7E6F71DBD32EA8BE5AF85B105730 / 0xACDEA42E034679E0 = 0x00000000000000009D86B0889208F92C:0x0000000000000000A900572AAF6884B0
0x86F664584EA6FE564BEEFE1B41207713 / 0x1D319A3D3E6F6D38 = 0x00000000000000049F7C4B1AC30CB248:0x00000000000000000BDEBA5D7138CF53
0x93D55D3D896F3E6E2E1EF71A01C680C4 / 0x88115750A7F8D137 = 0x00000000000000011622DE3AEA8D6745:0x00000000000000005F3FEF0A1E3DFBF1
0xAE986E3F6A4F484F4E81CF89F9BE32A9 / 0x173AD8E0B4E17B5B = 0x00000000000000078417C63DC58B3E77:0x0000000000000000141D123D47A4D15C
0x5DB95A0D6BD93EE1DF7FAF3668745E88 / 0xC07C0C65E93707AC = 0x00000000000000007CA699C204812C1C:0x0000000000000000BFFBD2C9E371F7B8
0x91FC43AF1DC400E75C8C89D4CB0CC766 / 0x23905F76E2C708A2 = 0x00000000000000041AD8BD6E95B77C6B:0x00000000000000000167C96223DFB3B0
0xD33CD0ECE044630C7857441234B4B3BD / 0xD2ECB5308FFCD581 = 0x000000000000000100613A50243CC975:0x0000000000000000BF01143AB448D6C8
0xCA7D0ABE05B379F37FC540C2F2540DC2 / 0xC511ED4380F000D0 = 0x00000000000000010709E7B22B442D2E:0x0000000000000000A454F0746FCF5862
0x75418582C2C04200CEEEF30B29E21EE6 / 0x622EE6915AAAC16E = 0x000000000000000131BACD4A53406179:0x0000000000000000527DDDC9966203E8
0x82EE158220268F14FDADF6174BC831B9 / 0x6D87F8FFBD7AF5FF = 0x00000000000000013203A1EE685F58F4:0x0000000000000000619862918C6512AD
0xBABC73A5BFC6AFA87BE777C27AF0CA7A / 0x334F8534DA44AE57 = 0x0000000000000003A3AAEC85949B00F6:0x00000000000000001720D365E24442E0
0xEC80AC4404C4CD424ADD78D0AA294A76 / 0x4F06C8F784DAE202 = 0x0000000000000002FE219872DF7C4B95:0x00000000000000001C7F230AFF95294C
0x74AC5B32627EB992D390475D4141954A / 0xBCBB67BA01AB465B = 0x00000000000000009E420C0580D7DBC7:0x000000000000000050FED88BD9810B8D
0xD0FA893E26F8F7C9B7B6346DF979053F / 0x785AF91D6D797129 = 0x0000000000000001BC813A285AA730EE:0x000000000000000029780222319B2121
0x5DED4A9336945FC99AE5C45B6D6A6250 / 0xFCE9DFD374327842 = 0x00000000000000005F12BA510F26C8DF:0x0000000000000000952062F60FB410D2
0xBD6604F3CC1E2F71B1C50F61C92682F5 / 0xB1507D6F7F83E641 = 0x0000000000000001117252CA7EC0A65F:0x0000000000000000076726C74125EAD6
0x8DEAD81C39130DD74AE755543605D7C7 / 0x8553E144F6F3FA63 = 0x0000000000000001107E30EAE9EE45D4:0x00000000000000002C281D730E73CECB
0xB62F71DEE58A89915484BA8DFF66ACBA / 0x70B8C00DFB590908 = 0x00000000000000019DC1EDE4447FF324:0x00000000000000001ED8A84F8856CF9A
0x1164808010FCC683FD5FD742EF1E82E2 / 0x2329D91B9BEF427F = 0x00000000000000007E9F79392CD015F6:0x000000000000000001894BD59B9031D8
0x1F65989C6839C52549A6A367837A8D68 / 0xDE9E265FE8F6EE0A = 0x0000000000000000241ADBD0FB260F42:0x0000000000000000C9B279DBD86298D4
0x01B09EDC67F8390074BF720CE0D4E681 / 0xEBEF93E1DE2F615E = 0x000000000000000001D569310FE86723:0x000000000000000036910723B0FDC4A7
0x43150143A042D09461FCAEE011BF4D2F / 0xE01ECE82FF69E963 = 0x00000000000000004C9FC0D605116CEE:0x000000000000000076790CFC803F8F25
0xA8BC8C145B90657A7FA62D4FC60E8144 / 0x0F175AAB16AA0D3A = 0x000000000000000B2E5C8870D0E697EA:0x0000000000000000076D41A7BEB53440
0xDBB6986137F134E82B2567E7164F8D6B / 0x39E35EFFA049FE5D = 0x0000000000000003CBA4704C8552B329:0x00000000000000000652776CE2D0C986
0xC95308E127E27A40CC2AB361F1D003F1 / 0x30BAF3D3113646FD = 0x000000000000000421A383F06B1DD66E:0x0000000000000000039BD74B637D053B
0xC3D5AF830984D0A89300A5A2D0EBAFFD / 0xDA78B7DC9D00443C = 0x0000000000000000E57983384327721C:0x0000000000000000C4E3AD916D5D816D
0x72F659CA625824C08839118A5F99D382 / 0xFCD33EC00AEBB829 = 0x00000000000000007467EB48186A84B8:0x0000000000000000C6A5E47AE23E520A
0x3015B4708D7B5E031CC52F99C95C5B09 / 0x52705129AD7C7A29 = 0x00000000000000009551C5B192977B2F:0x00000000000000001D95ED27B0A13A82
0x0E1F7B0B03B5DC69EC7F0EE22D9D9103 / 0xF01589C7A2936ED8 = 0x00000000000000000F0F29462804FE9E:0x0000000000000000D04177AE1344D7B3
0xEC3D468251C6746F99D7DE96E6C90D1F / 0x267AB14E499C9AD9 = 0x000000000000000623AF464370A3B267:0x00000000000000000E79D9A7DCF0DDD0
0x324766C3AE8FE518A91C0B89F691D8A8 / 0x473C0E6E2DEFF322 = 0x0000000000000000B4B0B8770B975D14:0x0000000000000000278C077555718000
0x1EEA010641E4FBB7EE6ADF4622A338B4 / 0x7649C1545D840EDB = 0x000000000000000042E78BDEEFA2D348:0x00000000000000001A6FB41E21AA8A1C
0x26CF636EE1F98F9AD6EB622275186021 / 0xBBDC13E0AACC79E6 = 0x000000000000000034E32521D0410AB4:0x00000000000000002FA92BE28D29AE69
0x7A7BE4E0A4F87E8283652261AA454ECA / 0x9B8251B56467B45E = 0x0000000000000000C9A2458FDD775882:0x0000000000000000311B005509E9670E
0x194D9A97CA3475698A2F1BBF94996EDF / 0x35072BD7E15B7473 = 0x00000000000000007A27854605B3DE17:0x0000000000000000012503425AFD3E8A
0xD63E9B6DB46C52F4D8B986F9847ECEAD / 0x9D72392AF8BE4618 = 0x00000000000000015C59F8C79AC088BA:0x00000000000000005D586814B303213D
0xF42EA2FF56E9795D65FB8FB85E1C7F34 / 0x24C1EE0CA7A07210 = 0x0000000000000006A49EF0579168219C:0x000000000000000017B3212D2322ED74
0x924A5235E557A62DF989AEC565EDD758 / 0x4637E63561A89AF4 = 0x0000000000000002155744F31100089E:0x00000000000000002F3BD6DFA70694C0
0xF5F894C7FE0B1939A62B614DE52A67B0 / 0xC9BF16D5CBE833CC = 0x0000000000000001381E0FEF42EC23E6:0x00000000000000006DFA7B799B66FA68
0xBC54C083AE952260AFF7F9FF2E39E958 / 0x6E0125A26FDA4F3A = 0x0000000000000001B647A71AF0D7D071:0x000000000000000053A8E5E784C7D0BE
0xA5371345881C4C88EE129F95A49D7002 / 0x2E96B3F3A1BB5FD9 = 0x00000000000000038BD711431BFF2BE4:0x0000000000000000163D12E3C47B9FBE
0xED919D64A38AC50EE6289DE3FA073006 / 0x6675A78DB953CC35 = 0x000000000000000251939CAFFFB1C284:0x00000000000000001BC552CAEE6CBAB2
0x1F1FF71139CF7455D56C25CE06AF2779 / 0xF6D3ACC12E75ADE9 = 0x0000000000000000204816C1394B681E:0x00000000000000001D4C796EF1FB1E2B
0xE3FB1B8E26AE6958A8B9312FC35D8302 / 0x2F0E544621C3A9BC = 0x0000000000000004D84A89285F567516:0x00000000000000000E22D0A2A6D200DA
0x410856E95F9CB830D95BADD72B9F83E5 / 0x7EC03BC1A01CCA8F = 0x00000000000000008358CE856D657B0A:0x00000000000000005C76947347C1E54F
0xE9F5F4B91B990C446D6DC188EF014D2E / 0x603063D14D68B6BF = 0x00000000000000026EAB5A279B04DF98:0x000000000000000021D3917341A86AC6
0x3F672EFBF9C03A71D13924A3D9F33F07 / 0xA6D751E7964991AC = 0x0000000000000000614908842CC78070:0x000000000000000030AE8F43843983C7
0xF8C19187F058634D7F6F0E77C93626E1 / 0xA0DD50F4F45B003A = 0x00000000000000018BDEEE39534EDFD1:0x00000000000000009899A7EA260C7187
0x7FD9CC315A5C23E293EDBCB24CDE6258 / 0xEA0758B0C7F182CC = 0x00000000000000008BDA913F184CADE0:0x0000000000000000CA19EEBB2F9813D8
0xCA048AE8719D2CBF504D5DF863569FB3 / 0x8EAE31C2B66F312F = 0x00000000000000016A76D09445883B90:0x00000000000000000C53FDE6587D2043
0xF1A862325714D55FB17FC0FEF112CD2E / 0x5C5EFD2DDF2461AD = 0x00000000000000029DBCDFF6149E9460:0x000000000000000059AC2B766E30284E
0x82370123858263879151B962F55B65C8 / 0x7C4067CF7662458E = 0x00000000000000010C494E7DE774E5A7:0x00000000000000006BF1E0862CB00026
0xAE4AD227B9816E578C12F2C496B15DC7 / 0x3BFB82AD09DA4BD7 = 0x0000000000000002E7DD4BD12E902D85:0x00000000000000001A49F830CE032C14
0x9A58A643CB945107FE9FA93764AEB5B6 / 0xA5DB6BCE5834CC35 = 0x0000000000000000EE3BAA824701F858:0x000000000000000093DBD3146D802B7E
0x3CFF433501A48CCA40DD1589383A1E6A / 0xE1EA9EAC3C53D914 = 0x0000000000000000451E9FB6BE32F4B4:0x00000000000000002A779739A4766C5A
0xE05CE3161C060193CDC77463E58AC539 / 0x4E716039D7089394 = 0x0000000000000002DC36829F6DEA09FE:0x00000000000000002CFDECD854902461
0x8C172FF9496884E683FA2149ACB0E973 / 0xD8E1E044A5E1006F = 0x0000000000000000A55B999B02B5B8B5:0x00000000000000001E4D953B7FD0D2F8
0xDAAC01F32907D2E05A22F6E749150705 / 0x7026050046A81D30 = 0x0000000000000001F328DBBBE708E51C:0x00000000000000001E77E203F315E5C5
0xB32B29D3D107DDE9BEC8E2B858BBB357 / 0x750B3A447F231586 = 0x000000000000000187E150D9948EB8D0:0x0000000000000000580C63326C6DE677
0xDF6E9EF5B0D613F77E584427E339EC9E / 0xBAC98934EFDD32FB = 0x000000000000000132392EB5CDD7AB7A:0x000000000000000085ABCAA502F4F800
0x70E4427966EA353F07FC52694289DF0E / 0x9A02DB9F4FB0757B = 0x0000000000000000BBA681F9A127CD04:0x00000000000000003B17F3B474F78A22
0x587602196F06C706C43D26B6A324DA04 / 0xD52A8E594B20BB56 = 0x00000000000000006A3C831F9C1DBA1A:0x00000000000000004B9CAAC798F75748
0xDDC4B4D70076B04AD38C6FCFB745415E / 0xF920A3A3B0BE6137 = 0x0000000000000000E3E2DACC13655973:0x00000000000000004D3FDAADB24076A9
0xBFB7F075E4640514ACC0D34EC7B3AB23 / 0x03EDC2CC944CCA7B = 0x0000000000000030CC7EFAC33315FE5B:0x000000000000000002DDC91DC26AA76A
0xB7850EA2D84ECF8A871CDE6319F1A14D / 0x5D5583E278BF2EA1 = 0x0000000000000001F75D58F861AAD829:0x0000000000000000104585A47C115184
0x19D30C8F4CF0B1E8D7BF478A62F20780 / 0x7A4852D9899BE914 = 0x0000000000000000361050BBA6A2E574:0x000000000000000038FF75771A258670
0xD67CC4EEA505A61BA2B4142239CCCF60 / 0x4334772F7DF221DB = 0x00000000000000033108E073FA69488D:0x00000000000000002E9320CD011791C1
0x8699D56AF72156D5E789058746D11016 / 0x531EF57805226C75 = 0x00000000000000019E8CF4E75D4C2521:0x000000000000000046269F1EEFF82C01
0x8C7B3EB794B4B10C6EE0FC588DDD6214 / 0xD840A863692B9E7B = 0x0000000000000000A64D52F5F26EDEFF:0x000000000000000093CF8DD99921DB8F
0xDCBFDAD051F9D4F96CF696581E56B0BB / 0xED841C71FE839C94 = 0x0000000000000000EDEDB656CBF05C2E:0x00000000000000002AC3B820EFAB5E23
0xCD3D3DD4D1473D7929E7C96EFA445188 / 0x46CEECB24270748B = 0x0000000000000002E6055D0FB8CCCA0B:0x000000000000000032CCDCD39CB5A18F
0xD09684576D9D8EFFA349AD1A03DF2E51 / 0x9CBEDD11E11E3711 = 0x000000000000000154AB87F3F19CF5EA:0x00000000000000002C5BA1ECC43193C7
0x4EB7FC4F14B3A2FAAFC3CB5E260D8015 / 0x2C62DE1A95A1DDC1 = 0x0000000000000001C603BFD8AC6E84DE:0x0000000000000000074354C5F869AEB7
0x4820304E706D821246759231492F522E / 0x9AB151D8E1BB548A = 0x0000000000000000775C480C24C7798D:0x0000000000000000579F64114AC6882C
0xAFB7699C70F40A3FABDBED7D413B2B68 / 0x5A7014CF215CD3D2 = 0x0000000000000001F165630D7BC529B0:0x0000000000000000279134DF8CE2E908
0xA40B7764609C77E35B3CB7F2E247DE12 / 0x26823F543A1F24EC = 0x0000000000000004428AEC3F73053FC3:0x000000000000000022C3FA3735DCAA4E
0x4B3D5C68E220B1F03E69112C438FECEF / 0x55CDCB691E140081 = 0x0000000000000000E07B315ED33F95DF:0x000000000000000008ABEE64F9196790
0x07145917D5EF05A9AC76BDBC20F0F1B7 / 0xD2DCB75038ABD9D4 = 0x000000000000000008984E14ED4515F5:0x0000000000000000576FBDC4D17715D3
0x79D36ED027F8136BD7237A20519D6900 / 0x65BE279172E9340A = 0x00000000000000013288389B33B93E78:0x00000000000000005686822B60789850
0x10B816A5E0F32BC0A28D56A228987B97 / 0x2691631C8A4FC373 = 0x00000000000000006EF9B0E3683A6778:0x000000000000000002EA49293B8398AF
0xBF6D72FFCAE5A828BC10AD25EA1CC2CE / 0x91730B1EF937B029 = 0x000000000000000150ECA60E96E99786:0x000000000000000085FFE8CA42BD5E58
0xA9B05D80F44DD3CF63B70D95259A8A07 / 0xA0525777429F609D = 0x00000000000000010EF523D97E6167AF:0x000000000000000083B23B3A994B53B4
0xCEBC9EFF59E7F2C4C2583ABAB8E86341 / 0xA5C3E00804739505 = 0x00000000000000013F46555CC330626E:0x00000000000000007B23CE99D042711B
0x9D1340D851E89E9730B634EF10700E08 / 0x6C9B7366F2C20971 = 0x0000000000000001723EA5F3B133DFF1:0x000000000000000043DA931B7F08BBA7
0xCAC50533954FFFA63F8F983623FDC3BC / 0x0B01B39932F6FB33 = 0x00000000000000126C26F2199D49C968:0x000000000000000004C39A7E9BE1AC04
0x0A8D31644EA2CC187FCF54C2C4530F2A / 0x21192FEE7C1CADA3 = 0x0000000000000000519C6B55F4BD6EEB:0x0000000000000000054E1CB7F60CA089
0xE26C37CAED68752AA4221B7A01024A13 / 0x862E65E264A0DD30 = 0x0000000000000001AFFC08C0C0FC6210:0x00000000000000005D17FA9067081713
0x4B6BAD20C68B054C8A0302777DD3129B / 0x6F72BE3F67300E03 = 0x0000000000000000AD3E5477E997550C:0x0000000000000000237CF0BDB4266B77
0x82FCD93671D634FA5860E8BDF176FFB2 / 0x0ECFA40BED38D323 = 0x0000000000000008D80913755D8AD21C:0x00000000000000000A9E9182DA2F31DE
0xF9292AFFD1C91713D8A1142692458287 / 0x584AB435FF4987DF = 0x0000000000000002D26F9202488D82E9:0x000000000000000055D41BE953869A90

If you were to use x86 assembly, it would be much simpler:

#include <stdio.h>

unsigned mulmod(unsigned a, unsigned b, unsigned m)
{
  unsigned result = 0;

  __asm
  {
    mov  eax, a
    mov  ebx, b
    mul  ebx // edx:eax = 64-bit product of a*b
    push eax // store away low 32 bits of dividend

    mov  ecx, m
    mov  eax, edx
    xor  edx, edx
    div  ecx // get high 32 bits of quotient
    xchg eax, [esp] // store them on stack, get low 32 bits of dividend
    div  ecx // get low 32 bits of quotient and the remainder
    mov  result, edx
  }

  return result;
}

int main(void)
{
  unsigned a = 4294967231, b = 4294967279, m = 4294967291;
  printf("Assembly:\n(%u * %u) mod %u = %u\n", a, b, m, mulmod(a, b, m));
  printf("Validation:\n(%u * %u) mod %u = %u\n", a, b, m, (unsigned)((unsigned long long)a * b % m));
  return 0;
}

Output (compiled with Open Watcom C/C++ 1.9):

Assembly:
(4294967231 * 4294967279) mod 4294967291 = 720
Validation:
(4294967231 * 4294967279) mod 4294967291 = 720

回答2:

Look here: Modular arithmetics and NTT (finite field DFT) optimizations

its my still unanswered question
find modmul member function
all mod functions are optimized and tested ...
if you have problems with asm then convert it co C
in rems is info about what line do what so it should be easy
even for non asm user

Above modmul implementation uses double precision

but still all variables are in single precision
so if you cannot use that for some reason
then you have to stick with loop of modadd which is slow
or binary modmul ... multiplicate a by only bits of b ... much faster
this leads to loop through all bits with some modadds and bitshift and bittest functions

For single precision only I use something like this 'template' code:

_mod_type modadd(_mod_type a,_mod_type b,_mod_type c)
    {
    _mod_type d,cy;
    a=mod(a,c);
    b=mod(b,c);
    d=a+b;
    cy=(shr(a)+shr(b)+shr((a&1)+(b&1)))&_mod_mask;
    if (cy) d-=c;
    if (_mod_type(d)>=_mod_type(c)) d-=c;
    return d;
    }

_mod_type modmul(_mod_type a,_mod_type b,_mod_type c)
    {   // b is not modded !
    int i;
    _mod_type d;
    a=mod(a,c);
    for (d=0,i=0;i<_mod_bits;i++)
        {
        if (_mod_type(a&1)) d=modadd(d,b,c);
        a=shr(a);
        b=modadd(b,b,c);
        }
    return d;
    }

_mod_type is you variable type (for example 32 bit unsigned int or DWORD)
_mod_bits is number of bits constant (32)
_mod_mask is msb bit set constant (0x80000000)
mod(a,b) is modulo a%b
shr(a) is bit shift a>>1
modadd(a,b,c) is (a+b)%c
modmul(a,b,c) is (a*b)%c

Hope it helps...

来源：https://stackoverflow.com/questions/16115682/modulus-of-product-of-two-integers

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java

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modulus