BigInt.js

// cho45: Wrap all functions to OO-style and close the scope
// karlgluck: Fixed bug with int2BigInt that causes mod(smaller, bigger) to always return 0
// karlgluck: passed JSLint
// karlgluck: fixed OO-syle functions to consistently return new objects
// karlgluck: fixed OO declaration to be 'use strict' compliant
// karlgluck: declare functions in order
// karlgluck: prevent global variable clobbering
(function (Global) {

  // http://www.leemon.com/crypto/BigInt.js

  ////////////////////////////////////////////////////////////////////////////////////////
  // Big Integer Library v. 5.5
  // Created 2000, last modified 2013
  // Leemon Baird
  // www.leemon.com
  //
  // Version history:
  // v 5.5  17 Mar 2013
  //   - two lines of a form like "if (x<0) x+=n" had the "if" changed to "while" to 
  //     handle the case when x<-n. (Thanks to James Ansell for finding that bug)
  // v 5.4  3 Oct 2009
  //   - added "var i" to greaterShift() so i is not global. (Thanks to PŽter Szab— for finding that bug)
  //
  // v 5.3  21 Sep 2009
  //   - added randProbPrime(k) for probable primes
  //   - unrolled loop in mont_ (slightly faster)
  //   - millerRabin now takes a bigInt parameter rather than an int
  //
  // v 5.2  15 Sep 2009
  //   - fixed capitalization in call to int2bigInt in randBigInt
  //     (thanks to Emili Evripidou, Reinhold Behringer, and Samuel Macaleese for finding that bug)
  //
  // v 5.1  8 Oct 2007 
  //   - renamed inverseModInt_ to inverseModInt since it doesn't change its parameters
  //   - added functions GCD and randBigInt, which call GCD_ and randBigInt_
  //   - fixed a bug found by Rob Visser (see comment with his name below)
  //   - improved comments
  //
  // This file is public domain.   You can use it for any purpose without restriction.
  // I do not guarantee that it is correct, so use it at your own risk.  If you use 
  // it for something interesting, I'd appreciate hearing about it.  If you find 
  // any bugs or make any improvements, I'd appreciate hearing about those too.
  // It would also be nice if my name and URL were left in the comments.  But none 
  // of that is required.
  //
  // This code defines a bigInt library for arbitrary-precision integers.
  // A bigInt is an array of integers storing the value in chunks of bpe bits, 
  // little endian (buff[0] is the least significant word).
  // Negative bigInts are stored two's complement.  Almost all the functions treat
  // bigInts as nonnegative.  The few that view them as two's complement say so
  // in their comments.  Some functions assume their parameters have at least one 
  // leading zero element. Functions with an underscore at the end of the name put
  // their answer into one of the arrays passed in, and have unpredictable behavior 
  // in case of overflow, so the caller must make sure the arrays are big enough to 
  // hold the answer.  But the average user should never have to call any of the 
  // underscored functions.  Each important underscored function has a wrapper function 
  // of the same name without the underscore that takes care of the details for you.  
  // For each underscored function where a parameter is modified, that same variable 
  // must not be used as another argument too.  So, you cannot square x by doing 
  // multMod_(x,x,n).  You must use squareMod_(x,n) instead, or do y=dup(x); multMod_(x,y,n).
  // Or simply use the multMod(x,x,n) function without the underscore, where
  // such issues never arise, because non-underscored functions never change
  // their parameters; they always allocate new memory for the answer that is returned.
  //
  // These functions are designed to avoid frequent dynamic memory allocation in the inner loop.
  // For most functions, if it needs a BigInt as a local variable it will actually use
  // a global, and will only allocate to it only when it's not the right size.  This ensures
  // that when a function is called repeatedly with same-sized parameters, it only allocates
  // memory on the first call.
  //
  // The calls to Math.random() have been replaced with calls to the cryptographic PRNG
  // 
  //
  // In the following, "bigInt" means a bigInt with at least one leading zero element,
  // and "integer" means a nonnegative integer less than radix.  In some cases, integer 
  // can be negative.  Negative bigInts are 2s complement.
  // 
  // The following functions do not modify their inputs.
  // Those returning a bigInt, string, or Array will dynamically allocate memory for that value.
  // Those returning a boolean will return the integer 0 (false) or 1 (true).
  // Those returning boolean or int will not allocate memory except possibly on the first 
  // time they're called with a given parameter size.
  // 
  // bigInt  add(x,y)               //return (x+y) for bigInts x and y.  
  // bigInt  addInt(x,n)            //return (x+n) where x is a bigInt and n is an integer.
  // string  bigInt2str(x,base)     //return a string form of bigInt x in a given base, with 2 <= base <= 95
  // int     bitSize(x)             //return how many bits long the bigInt x is, not counting leading zeros
  // bigInt  dup(x)                 //return a copy of bigInt x
  // boolean equals(x,y)            //is the bigInt x equal to the bigint y?
  // boolean equalsInt(x,y)         //is bigint x equal to integer y?
  // bigInt  expand(x,n)            //return a copy of x with at least n elements, adding leading zeros if needed
  // Array   findPrimes(n)          //return array of all primes less than integer n
  // bigInt  GCD(x,y)               //return greatest common divisor of bigInts x and y (each with same number of elements).
  // boolean greater(x,y)           //is x>y?  (x and y are nonnegative bigInts)
  // boolean greaterShift(x,y,shift)//is (x <<(shift*bpe)) > y?
  // bigInt  int2bigInt(t,n,m)      //return a bigInt equal to integer t, with at least n bits and m array elements
  // bigInt  inverseMod(x,n)        //return (x**(-1) mod n) for bigInts x and n.  If no inverse exists, it returns null
  // int     inverseModInt(x,n)     //return x**(-1) mod n, for integers x and n.  Return 0 if there is no inverse
  // boolean isZero(x)              //is the bigInt x equal to zero?
  // boolean millerRabin(x,b)       //does one round of Miller-Rabin base integer b say that bigInt x is possibly prime? (b is bigInt, 1<b<x)
  // boolean millerRabinInt(x,b)    //does one round of Miller-Rabin base integer b say that bigInt x is possibly prime? (b is int,    1<b<x)
  // bigInt  mod(x,n)               //return a new bigInt equal to (x mod n) for bigInts x and n.
  // int     modInt(x,n)            //return x mod n for bigInt x and integer n.
  // bigInt  mult(x,y)              //return x*y for bigInts x and y. This is faster when y<x.
  // bigInt  multMod(x,y,n)         //return (x*y mod n) for bigInts x,y,n.  For greater speed, let y<x.
  // boolean negative(x)            //is bigInt x negative?
  // bigInt  powMod(x,y,n)          //return (x**y mod n) where x,y,n are bigInts and ** is exponentiation.  0**0=1. Faster for odd n.
  // bigInt  randBigInt(n,s)        //return an n-bit random BigInt (n>=1).  If s=1, then the most significant of those n bits is set to 1.
  // bigInt  randTruePrime(k)       //return a new, random, k-bit, true prime bigInt using Maurer's algorithm.
  // bigInt  randProbPrime(k)       //return a new, random, k-bit, probable prime bigInt (probability it's composite less than 2^-80).
  // bigInt  str2bigInt(s,b,n,m)    //return a bigInt for number represented in string s in base b with at least n bits and m array elements
  // bigInt  sub(x,y)               //return (x-y) for bigInts x and y.  Negative answers will be 2s complement
  // bigInt  trim(x,k)              //return a copy of x with exactly k leading zero elements
  //
  //
  // The following functions each have a non-underscored version, which most users should call instead.
  // These functions each write to a single parameter, and the caller is responsible for ensuring the array 
  // passed in is large enough to hold the result. 
  //
  // void    addInt_(x,n)          //do x=x+n where x is a bigInt and n is an integer
  // void    add_(x,y)             //do x=x+y for bigInts x and y
  // void    copy_(x,y)            //do x=y on bigInts x and y
  // void    copyInt_(x,n)         //do x=n on bigInt x and integer n
  // void    GCD_(x,y)             //set x to the greatest common divisor of bigInts x and y, (y is destroyed).  (This never overflows its array).
  // boolean inverseMod_(x,n)      //do x=x**(-1) mod n, for bigInts x and n. Returns 1 (0) if inverse does (doesn't) exist
  // void    mod_(x,n)             //do x=x mod n for bigInts x and n. (This never overflows its array).
  // void    mult_(x,y)            //do x=x*y for bigInts x and y.
  // void    multMod_(x,y,n)       //do x=x*y  mod n for bigInts x,y,n.
  // void    powMod_(x,y,n)        //do x=x**y mod n, where x,y,n are bigInts (n is odd) and ** is exponentiation.  0**0=1.
  // void    randBigInt_(b,n,s)    //do b = an n-bit random BigInt. if s=1, then nth bit (most significant bit) is set to 1. n>=1.
  // void    randTruePrime_(ans,k) //do ans = a random k-bit true random prime (not just probable prime) with 1 in the msb.
  // void    sub_(x,y)             //do x=x-y for bigInts x and y. Negative answers will be 2s complement.
  //
  // The following functions do NOT have a non-underscored version. 
  // They each write a bigInt result to one or more parameters.  The caller is responsible for
  // ensuring the arrays passed in are large enough to hold the results. 
  //
  // void addShift_(x,y,ys)       //do x=x+(y<<(ys*bpe))
  // void carry_(x)               //do carries and borrows so each element of the bigInt x fits in bpe bits.
  // void divide_(x,y,q,r)        //divide x by y giving quotient q and remainder r
  // int  divInt_(x,n)            //do x=floor(x/n) for bigInt x and integer n, and return the remainder. (This never overflows its array).
  // int  eGCD_(x,y,d,a,b)        //sets a,b,d to positive bigInts such that d = GCD_(x,y) = a*x-b*y
  // void halve_(x)               //do x=floor(|x|/2)*sgn(x) for bigInt x in 2's complement.  (This never overflows its array).
  // void leftShift_(x,n)         //left shift bigInt x by n bits.  n<bpe.
  // void linComb_(x,y,a,b)       //do x=a*x+b*y for bigInts x and y and integers a and b
  // void linCombShift_(x,y,b,ys) //do x=x+b*(y<<(ys*bpe)) for bigInts x and y, and integers b and ys
  // void mont_(x,y,n,np)         //Montgomery multiplication (see comments where the function is defined)
  // void multInt_(x,n)           //do x=x*n where x is a bigInt and n is an integer.
  // void rightShift_(x,n)        //right shift bigInt x by n bits.  0 <= n < bpe. (This never overflows its array).
  // void squareMod_(x,n)         //do x=x*x  mod n for bigInts x,n
  // void subShift_(x,y,ys)       //do x=x-(y<<(ys*bpe)). Negative answers will be 2s complement.
  //
  // The following functions are based on algorithms from the _Handbook of Applied Cryptography_
  //    powMod_()           = algorithm 14.94, Montgomery exponentiation
  //    eGCD_,inverseMod_() = algorithm 14.61, Binary extended GCD_
  //    GCD_()              = algorothm 14.57, Lehmer's algorithm
  //    mont_()             = algorithm 14.36, Montgomery multiplication
  //    divide_()           = algorithm 14.20  Multiple-precision division
  //    squareMod_()        = algorithm 14.16  Multiple-precision squaring
  //    randTruePrime_()    = algorithm  4.62, Maurer's algorithm
  //    millerRabin()       = algorithm  4.24, Miller-Rabin algorithm
  //
  // Profiling shows:
  //     randTruePrime_() spends:
  //         10% of its time in calls to powMod_()
  //         85% of its time in calls to millerRabin()
  //     millerRabin() spends:
  //         99% of its time in calls to powMod_()   (always with a base of 2)
  //     powMod_() spends:
  //         94% of its time in calls to mont_()  (almost always with x==y)
  //
  // This suggests there are several ways to speed up this library slightly:
  //     - convert powMod_ to use a Montgomery form of k-ary window (or maybe a Montgomery form of sliding window)
  //         -- this should especially focus on being fast when raising 2 to a power mod n
  //     - convert randTruePrime_() to use a minimum r of 1/3 instead of 1/2 with the appropriate change to the test
  //     - tune the parameters in randTruePrime_(), including c, m, and recLimit
  //     - speed up the single loop in mont_() that takes 95% of the runtime, perhaps by reducing checking
  //       within the loop when all the parameters are the same length.
  //
  // There are several ideas that look like they wouldn't help much at all:
  //     - replacing trial division in randTruePrime_() with a sieve (that speeds up something taking almost no time anyway)
  //     - increase bpe from 15 to 30 (that would help if we had a 32*32->64 multiplier, but not with JavaScript's 32*32->32)
  //     - speeding up mont_(x,y,n,np) when x==y by doing a non-modular, non-Montgomery square
  //       followed by a Montgomery reduction.  The intermediate answer will be twice as long as x, so that
  //       method would be slower.  This is unfortunate because the code currently spends almost all of its time
  //       doing mont_(x,x,...), both for randTruePrime_() and powMod_().  A faster method for Montgomery squaring
  //       would have a large impact on the speed of randTruePrime_() and powMod_().  HAC has a couple of poorly-worded
  //       sentences that seem to imply it's faster to do a non-modular square followed by a single
  //       Montgomery reduction, but that's obviously wrong.
  ////////////////////////////////////////////////////////////////////////////////////////

  'use strict';

  var bpe = 0,           // bits stored per array element
      mask = 0,          // AND this with an array element to chop it down to bpe bits
      radix = mask + 1,  // equals 2^bpe.  A single 1 bit to the left of the last bit of mask.
      one = [1, 0],      // constant used in powMod_()

  // the digits for converting to different bases
      digitsStr = '0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz_=!@#$%^&*()[]{}|;:,.<>/?`~ \\\'\"+-';

  // the following global variables are scratchpad memory to 
  // reduce dynamic memory allocation in the inner loop
  var t = new Array(0);
  var ss = t,       // used in mult_()
    s0 = t,       // used in multMod_(), squareMod_() 
    s1 = t,       // used in powMod_(), multMod_(), squareMod_() 
    s2 = t,       // used in powMod_(), multMod_()
    s3 = t,       // used in powMod_()
    s4 = t,
    s5 = t,       // used in mod_()
    s6 = t,       // used in bigInt2str()
    s7 = t,       // used in powMod_()
    T = t,        // used in GCD_()
    sa = t,       // used in mont_()
    mr_x1 = t, mr_r = t, mr_a = t,                                              // used in millerRabin()
    eg_v = t, eg_u = t, eg_A = t, eg_B = t, eg_C = t, eg_D = t,                 // used in eGCD_(), inverseMod_()
    md_q1 = t, md_q2 = t, md_q3 = t, md_r = t, md_r1 = t, md_r2 = t, md_tt = t, // used in mod_()

    primes = t, pows = t, s_i = t, s_i2 = t, s_R = t, s_rm = t, s_q = t, s_n1 = t,
    s_a = t, s_r2 = t, s_n = t, s_b = t, s_d = t, s_x1 = t, s_x2 = t, s_aa = t, // used in randTruePrime_()

    rpprb = t;    // used in randProbPrimeRounds() (which also uses "primes")

  for (bpe = 0; (1 << (bpe + 1)) > (1 << bpe); bpe++) { }  // bpe=number of bits in the mantissa on this platform
  bpe >>= 1;                     // bpe=number of bits in one element of the array representing the bigInt
  mask = (1 << bpe) - 1;         // AND the mask with an integer to get its bpe least significant bits
  radix = mask + 1;              // 2^bpe.  a single 1 bit to the left of the first bit of mask

  ////////////////////////////////////////////////////////////////////////////////////////

  //do x=y on bigInts x and y.  x must be an array at least as big as y (not counting the leading zeros in y).
  function copy_(x, y) {
    var i;
    var k = x.length < y.length ? x.length : y.length;
    for (i = 0; i < k; i++)
      x[i] = y[i];
    for (i = k; i < x.length; i++)
      x[i] = 0;
  }

  //do x=y on bigInt x and integer y.  
  function copyInt_(x, n) {
    var i, c;
    for (c = n, i = 0; i < x.length; i++) {
      x[i] = c & mask;
      c >>= bpe;
    }
  }

  //returns a duplicate of bigInt x
  function dup(x) {
    var buff = new Array(x.length);
    copy_(buff, x);
    return buff;
  }


  //do x=x+n where x is a bigInt and n is an integer.
  //x must be large enough to hold the result.
  function addInt_(x, n) {
    var i, k, c, b;
    x[0] += n;
    k = x.length;
    c = 0;
    for (i = 0; i < k; i++) {
      c += x[i];
      b = 0;
      if (c < 0) {
        b = -(c >> bpe);
        c += b * radix;
      }
      x[i] = c & mask;
      c = (c >> bpe) - b;
      if (!c) return; //stop carrying as soon as the carry is zero
    }
  }

  //right shift bigInt x by n bits.  0 <= n < bpe.
  function rightShift_(x, n) {
    var i;
    var k = Math.floor(n / bpe);
    if (k) {
      for (i = 0; i < x.length - k; i++) //right shift x by k elements
        x[i] = x[i + k];
      for (; i < x.length; i++)
        x[i] = 0;
      n %= bpe;
    }
    for (i = 0; i < x.length - 1; i++) {
      x[i] = mask & ((x[i + 1] << (bpe - n)) | (x[i] >> n));
    }
    x[i] >>= n;
  }

  //do x=floor(|x|/2)*sgn(x) for bigInt x in 2's complement
  function halve_(x) {
    var i;
    for (i = 0; i < x.length - 1; i++) {
      x[i] = mask & ((x[i + 1] << (bpe - 1)) | (x[i] >> 1));
    }
    x[i] = (x[i] >> 1) | (x[i] & (radix >> 1));  //most significant bit stays the same
  }

  //left shift bigInt x by n bits.
  function leftShift_(x, n) {
    var i;
    var k = Math.floor(n / bpe);
    if (k) {
      for (i = x.length; i >= k; i--) //left shift x by k elements
        x[i] = x[i - k];
      for (; i >= 0; i--)
        x[i] = 0;
      n %= bpe;
    }
    if (!n)
      return;
    for (i = x.length - 1; i > 0; i--) {
      x[i] = mask & ((x[i] << n) | (x[i - 1] >> (bpe - n)));
    }
    x[i] = mask & (x[i] << n);
  }

  //do x=x*n where x is a bigInt and n is an integer.
  //x must be large enough to hold the result.
  function multInt_(x, n) {
    var i, k, c, b;
    if (!n)
      return;
    k = x.length;
    c = 0;
    for (i = 0; i < k; i++) {
      c += x[i] * n;
      b = 0;
      if (c < 0) {
        b = -(c >> bpe);
        c += b * radix;
      }
      x[i] = c & mask;
      c = (c >> bpe) - b;
    }
  }

  //do x=floor(x/n) for bigInt x and integer n, and return the remainder
  function divInt_(x, n) {
    var i, r = 0, s;
    for (i = x.length - 1; i >= 0; i--) {
      s = r * radix + x[i];
      x[i] = Math.floor(s / n);
      r = s % n;
    }
    return r;
  }

  //do the linear combination x=a*x+b*y for bigInts x and y, and integers a and b.
  //x must be large enough to hold the answer.
  function linComb_(x, y, a, b) {
    var i, c, k, kk;
    k = x.length < y.length ? x.length : y.length;
    kk = x.length;
    for (c = 0, i = 0; i < k; i++) {
      c += a * x[i] + b * y[i];
      x[i] = c & mask;
      c >>= bpe;
    }
    for (i = k; i < kk; i++) {
      c += a * x[i];
      x[i] = c & mask;
      c >>= bpe;
    }
  }

  //do the linear combination x=a*x+b*(y<<(ys*bpe)) for bigInts x and y, and integers a, b and ys.
  //x must be large enough to hold the answer.
  function linCombShift_(x, y, b, ys) {
    var i, c, k, kk;
    k = x.length < ys + y.length ? x.length : ys + y.length;
    kk = x.length;
    for (c = 0, i = ys; i < k; i++) {
      c += x[i] + b * y[i - ys];
      x[i] = c & mask;
      c >>= bpe;
    }
    for (i = k; c && i < kk; i++) {
      c += x[i];
      x[i] = c & mask;
      c >>= bpe;
    }
  }

  //do x=x+(y<<(ys*bpe)) for bigInts x and y, and integers a,b and ys.
  //x must be large enough to hold the answer.
  function addShift_(x, y, ys) {
    var i, c, k, kk;
    k = x.length < ys + y.length ? x.length : ys + y.length;
    kk = x.length;
    for (c = 0, i = ys; i < k; i++) {
      c += x[i] + y[i - ys];
      x[i] = c & mask;
      c >>= bpe;
    }
    for (i = k; c && i < kk; i++) {
      c += x[i];
      x[i] = c & mask;
      c >>= bpe;
    }
  }

  //do x=x-(y<<(ys*bpe)) for bigInts x and y, and integers a,b and ys.
  //x must be large enough to hold the answer.
  function subShift_(x, y, ys) {
    var i, c, k, kk;
    k = x.length < ys + y.length ? x.length : ys + y.length;
    kk = x.length;
    for (c = 0, i = ys; i < k; i++) {
      c += x[i] - y[i - ys];
      x[i] = c & mask;
      c >>= bpe;
    }
    for (i = k; c && i < kk; i++) {
      c += x[i];
      x[i] = c & mask;
      c >>= bpe;
    }
  }

  //do x=x-y for bigInts x and y.
  //x must be large enough to hold the answer.
  //negative answers will be 2s complement
  function sub_(x, y) {
    var i, c, k, kk;
    k = x.length < y.length ? x.length : y.length;
    for (c = 0, i = 0; i < k; i++) {
      c += x[i] - y[i];
      x[i] = c & mask;
      c >>= bpe;
    }
    for (i = k; c && i < x.length; i++) {
      c += x[i];
      x[i] = c & mask;
      c >>= bpe;
    }
  }

  //do x=x+y for bigInts x and y.
  //x must be large enough to hold the answer.
  function add_(x, y) {
    var i, c, k, kk;
    k = x.length < y.length ? x.length : y.length;
    for (c = 0, i = 0; i < k; i++) {
      c += x[i] + y[i];
      x[i] = c & mask;
      c >>= bpe;
    }
    for (i = k; c && i < x.length; i++) {
      c += x[i];
      x[i] = c & mask;
      c >>= bpe;
    }
  }

  //do x=x*y for bigInts x and y.  This is faster when y<x.
  function mult_(x, y) {
    var i;
    if (ss.length != 2 * x.length)
      ss = new Array(2 * x.length);
    copyInt_(ss, 0);
    for (i = 0; i < y.length; i++)
      if (y[i])
        linCombShift_(ss, x, y[i], i);   //ss=1*ss+y[i]*(x<<(i*bpe))
    copy_(x, ss);
  }

  //do x=x mod n for bigInts x and n.
  function mod_(x, n) {
    if (s4.length != x.length)
      s4 = dup(x);
    else
      copy_(s4, x);
    if (s5.length != x.length)
      s5 = dup(x);
    divide_(s4, n, s5, x);  //x = remainder of s4 / n
  }

  //do x=x*y mod n for bigInts x,y,n.
  //for greater speed, let y<x.
  function multMod_(x, y, n) {
    var i;
    if (s0.length != 2 * x.length)
      s0 = new Array(2 * x.length);
    copyInt_(s0, 0);
    for (i = 0; i < y.length; i++)
      if (y[i])
        linCombShift_(s0, x, y[i], i);   //s0=1*s0+y[i]*(x<<(i*bpe))
    mod_(s0, n);
    copy_(x, s0);
  }

  //do x=x*x mod n for bigInts x,n.
  function squareMod_(x, n) {
    var i, j, d, c, kx, kn, k;
    for (kx = x.length; kx > 0 && !x[kx - 1]; kx--);  //ignore leading zeros in x
    k = kx > n.length ? 2 * kx : 2 * n.length; //k=# elements in the product, which is twice the elements in the larger of x and n
    if (s0.length != k)
      s0 = new Array(k);
    copyInt_(s0, 0);
    for (i = 0; i < kx; i++) {
      c = s0[2 * i] + x[i] * x[i];
      s0[2 * i] = c & mask;
      c >>= bpe;
      for (j = i + 1; j < kx; j++) {
        c = s0[i + j] + 2 * x[i] * x[j] + c;
        s0[i + j] = (c & mask);
        c >>= bpe;
      }
      s0[i + kx] = c;
    }
    mod_(s0, n);
    copy_(x, s0);
  }

  //return x with exactly k leading zero elements
  function trim(x, k) {
    var i, y;
    for (i = x.length; i > 0 && !x[i - 1]; i--);
    y = new Array(i + k);
    copy_(y, x);
    return y;
  }

  //do x=x**y mod n, where x,y,n are bigInts and ** is exponentiation.  0**0=1.
  //this is faster when n is odd.  x usually needs to have as many elements as n.
  function powMod_(x, y, n) {
    var k1, k2, kn, np;
    if (s7.length != n.length)
      s7 = dup(n);

    //for even modulus, use a simple square-and-multiply algorithm,
    //rather than using the more complex Montgomery algorithm.
    if ((n[0] & 1) == 0) {
      copy_(s7, x);
      copyInt_(x, 1);
      while (!equalsInt(y, 0)) {
        if (y[0] & 1)
          multMod_(x, s7, n);
        divInt_(y, 2);
        squareMod_(s7, n);
      }
      return;
    }

    //calculate np from n for the Montgomery multiplications
    copyInt_(s7, 0);
    for (kn = n.length; kn > 0 && !n[kn - 1]; kn--);
    np = radix - inverseModInt(modInt(n, radix), radix);
    s7[kn] = 1;
    multMod_(x, s7, n);   // x = x * 2**(kn*bp) mod n

    if (s3.length != x.length)
      s3 = dup(x);
    else
      copy_(s3, x);

    for (k1 = y.length - 1; k1 > 0 & !y[k1]; k1--);  //k1=first nonzero element of y
    if (y[k1] == 0) {  //anything to the 0th power is 1
      copyInt_(x, 1);
      return;
    }
    for (k2 = 1 << (bpe - 1); k2 && !(y[k1] & k2); k2 >>= 1);  //k2=position of first 1 bit in y[k1]
    for (; ; ) {
      if (!(k2 >>= 1)) {  //look at next bit of y
        k1--;
        if (k1 < 0) {
          mont_(x, one, n, np);
          return;
        }
        k2 = 1 << (bpe - 1);
      }
      mont_(x, x, n, np);

      if (k2 & y[k1]) //if next bit is a 1
        mont_(x, s3, n, np);
    }
  }


  //do x=x*y*Ri mod n for bigInts x,y,n, 
  //  where Ri = 2**(-kn*bpe) mod n, and kn is the 
  //  number of elements in the n array, not 
  //  counting leading zeros.  
  //x array must have at least as many elemnts as the n array
  //It's OK if x and y are the same variable.
  //must have:
  //  x,y < n
  //  n is odd
  //  np = -(n^(-1)) mod radix
  function mont_(x, y, n, np) {
    var i, j, c, ui, t, ks;
    var kn = n.length;
    var ky = y.length;

    if (sa.length != kn)
      sa = new Array(kn);

    copyInt_(sa, 0);

    for (; kn > 0 && n[kn - 1] == 0; kn--); //ignore leading zeros of n
    for (; ky > 0 && y[ky - 1] == 0; ky--); //ignore leading zeros of y
    ks = sa.length - 1; //sa will never have more than this many nonzero elements.  

    //the following loop consumes 95% of the runtime for randTruePrime_() and powMod_() for large numbers
    for (i = 0; i < kn; i++) {
      t = sa[0] + x[i] * y[0];
      ui = ((t & mask) * np) & mask;  //the inner "& mask" was needed on Safari (but not MSIE) at one time
      c = (t + ui * n[0]) >> bpe;
      t = x[i];

      //do sa=(sa+x[i]*y+ui*n)/b   where b=2**bpe.  Loop is unrolled 5-fold for speed
      j = 1;
      for (; j < ky - 4; ) {
        c += sa[j] + ui * n[j] + t * y[j]; sa[j - 1] = c & mask; c >>= bpe; j++;
        c += sa[j] + ui * n[j] + t * y[j]; sa[j - 1] = c & mask; c >>= bpe; j++;
        c += sa[j] + ui * n[j] + t * y[j]; sa[j - 1] = c & mask; c >>= bpe; j++;
        c += sa[j] + ui * n[j] + t * y[j]; sa[j - 1] = c & mask; c >>= bpe; j++;
        c += sa[j] + ui * n[j] + t * y[j]; sa[j - 1] = c & mask; c >>= bpe; j++;
      }
      for (; j < ky; ) { c += sa[j] + ui * n[j] + t * y[j]; sa[j - 1] = c & mask; c >>= bpe; j++; }
      for (; j < kn - 4; ) {
        c += sa[j] + ui * n[j]; sa[j - 1] = c & mask; c >>= bpe; j++;
        c += sa[j] + ui * n[j]; sa[j - 1] = c & mask; c >>= bpe; j++;
        c += sa[j] + ui * n[j]; sa[j - 1] = c & mask; c >>= bpe; j++;
        c += sa[j] + ui * n[j]; sa[j - 1] = c & mask; c >>= bpe; j++;
        c += sa[j] + ui * n[j]; sa[j - 1] = c & mask; c >>= bpe; j++;
      }
      for (; j < kn; ) { c += sa[j] + ui * n[j]; sa[j - 1] = c & mask; c >>= bpe; j++; }
      for (; j < ks; ) { c += sa[j]; sa[j - 1] = c & mask; c >>= bpe; j++; }
      sa[j - 1] = c & mask;
    }

    if (!greater(n, sa))
      sub_(sa, n);
    copy_(x, sa);
  }

  //return array of all primes less than integer n
  function findPrimes(n) {
    var i, s, p, ans;
    s = new Array(n);
    for (i = 0; i < n; i++)
      s[i] = 0;
    s[0] = 2;
    p = 0;    //first p elements of s are primes, the rest are a sieve
    for (; s[p] < n; ) {                  //s[p] is the pth prime
      for (i = s[p] * s[p]; i < n; i += s[p]) //mark multiples of s[p]
        s[i] = 1;
      p++;
      s[p] = s[p - 1] + 1;
      for (; s[p] < n && s[s[p]]; s[p]++); //find next prime (where s[p]==0)
    }
    ans = new Array(p);
    for (i = 0; i < p; i++)
      ans[i] = s[i];
    return ans;
  }


  //does a single round of Miller-Rabin base b consider x to be a possible prime?
  //x and b are bigInts with b<x
  function millerRabin(x, b) {
    var i, j, k, s;

    if (mr_x1.length != x.length) {
      mr_x1 = dup(x);
      mr_r = dup(x);
      mr_a = dup(x);
    }

    copy_(mr_a, b);
    copy_(mr_r, x);
    copy_(mr_x1, x);

    addInt_(mr_r, -1);
    addInt_(mr_x1, -1);

    //s=the highest power of two that divides mr_r
    k = 0;
    for (i = 0; i < mr_r.length; i++)
      for (j = 1; j < mask; j <<= 1)
        if (x[i] & j) {
          s = (k < mr_r.length + bpe ? k : 0);
          i = mr_r.length;
          j = mask;
        } else
          k++;

    if (s)
      rightShift_(mr_r, s);

    powMod_(mr_a, mr_r, x);

    if (!equalsInt(mr_a, 1) && !equals(mr_a, mr_x1)) {
      j = 1;
      while (j <= s - 1 && !equals(mr_a, mr_x1)) {
        squareMod_(mr_a, x);
        if (equalsInt(mr_a, 1)) {
          return 0;
        }
        j++;
      }
      if (!equals(mr_a, mr_x1)) {
        return 0;
      }
    }
    return 1;
  }

  //does a single round of Miller-Rabin base b consider x to be a possible prime?
  //x is a bigInt, and b is an integer, with b<x
  function millerRabinInt(x, b) {
    if (mr_x1.length != x.length) {
      mr_x1 = dup(x);
      mr_r = dup(x);
      mr_a = dup(x);
    }

    copyInt_(mr_a, b);
    return millerRabin(x, mr_a);
  }

  //returns how many bits long the bigInt is, not counting leading zeros.
  function bitSize(x) {
    var j, z, w;
    for (j = x.length - 1; (x[j] == 0) && (j > 0); j--);
    for (z = 0, w = x[j]; w; (w >>= 1), z++);
    z += bpe * j;
    return z;
  }

  //return a copy of x with at least n elements, adding leading zeros if needed
  function expand(x, n) {
    var ans = int2bigInt(0, (x.length > n ? x.length : n) * bpe, 0);
    copy_(ans, x);
    return ans;
  }

  //return a k-bit true random prime using Maurer's algorithm.
  function randTruePrime(k) {
    var ans = int2bigInt(0, k, 0);
    randTruePrime_(ans, k);
    return trim(ans, 1);
  }

  //return a k-bit random probable prime with probability of error < 2^-80
  function randProbPrime(k) {
    if (k >= 600) return randProbPrimeRounds(k, 2); //numbers from HAC table 4.3
    if (k >= 550) return randProbPrimeRounds(k, 4);
    if (k >= 500) return randProbPrimeRounds(k, 5);
    if (k >= 400) return randProbPrimeRounds(k, 6);
    if (k >= 350) return randProbPrimeRounds(k, 7);
    if (k >= 300) return randProbPrimeRounds(k, 9);
    if (k >= 250) return randProbPrimeRounds(k, 12); //numbers from HAC table 4.4
    if (k >= 200) return randProbPrimeRounds(k, 15);
    if (k >= 150) return randProbPrimeRounds(k, 18);
    if (k >= 100) return randProbPrimeRounds(k, 27);
    return randProbPrimeRounds(k, 40); //number from HAC remark 4.26 (only an estimate)
  }

  //return a k-bit probable random prime using n rounds of Miller Rabin (after trial division with small primes)	
  function randProbPrimeRounds(k, n) {
    var ans, i, divisible, B;
    B = 30000;  //B is largest prime to use in trial division
    ans = int2bigInt(0, k, 0);

    //optimization: try larger and smaller B to find the best limit.

    if (primes.length == 0)
      primes = findPrimes(30000);  //check for divisibility by primes <=30000

    if (rpprb.length != ans.length)
      rpprb = dup(ans);

    for (; ; ) { //keep trying random values for ans until one appears to be prime
      //optimization: pick a random number times L=2*3*5*...*p, plus a 
      //   random element of the list of all numbers in [0,L) not divisible by any prime up to p.
      //   This can reduce the amount of random number generation.

      randBigInt_(ans, k, 0); //ans = a random odd number to check
      ans[0] |= 1;
      divisible = 0;

      //check ans for divisibility by small primes up to B
      for (i = 0; (i < primes.length) && (primes[i] <= B); i++)
        if (modInt(ans, primes[i]) == 0 && !equalsInt(ans, primes[i])) {
          divisible = 1;
          break;
        }

      //optimization: change millerRabin so the base can be bigger than the number being checked, then eliminate the while here.

      //do n rounds of Miller Rabin, with random bases less than ans
      for (i = 0; i < n && !divisible; i++) {
        randBigInt_(rpprb, k, 0);
        while (!greater(ans, rpprb)) //pick a random rpprb that's < ans
          randBigInt_(rpprb, k, 0);
        if (!millerRabin(ans, rpprb))
          divisible = 1;
      }

      if (!divisible)
        return ans;
    }
  }

  //return a new bigInt equal to (x mod n) for bigInts x and n.
  function mod(x, n) {
    var ans = dup(x);
    mod_(ans, n);
    return trim(ans, 1);
  }

  //return (x+n) where x is a bigInt and n is an integer.
  function addInt(x, n) {
    var ans = expand(x, x.length + 1);
    addInt_(ans, n);
    return trim(ans, 1);
  }

  //return x*y for bigInts x and y. This is faster when y<x.
  function mult(x, y) {
    var ans = expand(x, x.length + y.length);
    mult_(ans, y);
    return trim(ans, 1);
  }

  //return (x**y mod n) where x,y,n are bigInts and ** is exponentiation.  0**0=1. Faster for odd n.
  function powMod(x, y, n) {
    var ans = expand(x, n.length);
    powMod_(ans, trim(y, 2), trim(n, 2), 0);  //this should work without the trim, but doesn't
    return trim(ans, 1);
  }

  //return (x-y) for bigInts x and y.  Negative answers will be 2s complement
  function sub(x, y) {
    var ans = expand(x, (x.length > y.length ? x.length + 1 : y.length + 1));
    sub_(ans, y);
    return trim(ans, 1);
  }

  //return (x+y) for bigInts x and y.  
  function add(x, y) {
    var ans = expand(x, (x.length > y.length ? x.length + 1 : y.length + 1));
    add_(ans, y);
    return trim(ans, 1);
  }

  //return (x**(-1) mod n) for bigInts x and n.  If no inverse exists, it returns null
  function inverseMod(x, n) {
    var ans = expand(x, n.length);
    var s;
    s = inverseMod_(ans, n);
    return s ? trim(ans, 1) : null;
  }

  //return (x*y mod n) for bigInts x,y,n.  For greater speed, let y<x.
  function multMod(x, y, n) {
    var ans = expand(x, n.length);
    multMod_(ans, y, n);
    return trim(ans, 1);
  }

  //generate a k-bit true random prime using Maurer's algorithm,
  //and put it into ans.  The bigInt ans must be large enough to hold it.
  function randTruePrime_(ans, k) {
    var c, m, pm, dd, j, r, B, divisible, z, zz, recSize;

    if (primes.length == 0)
      primes = findPrimes(30000);  //check for divisibility by primes <=30000

    if (pows.length == 0) {
      pows = new Array(512);
      for (j = 0; j < 512; j++) {
        pows[j] = Math.pow(2, j / 511. - 1.);
      }
    }

    //c and m should be tuned for a particular machine and value of k, to maximize speed
    c = 0.1;  //c=0.1 in HAC
    m = 20;   //generate this k-bit number by first recursively generating a number that has between k/2 and k-m bits
    recLimit = 20; //stop recursion when k <=recLimit.  Must have recLimit >= 2

    if (s_i2.length != ans.length) {
      s_i2 = dup(ans);
      s_R = dup(ans);
      s_n1 = dup(ans);
      s_r2 = dup(ans);
      s_d = dup(ans);
      s_x1 = dup(ans);
      s_x2 = dup(ans);
      s_b = dup(ans);
      s_n = dup(ans);
      s_i = dup(ans);
      s_rm = dup(ans);
      s_q = dup(ans);
      s_a = dup(ans);
      s_aa = dup(ans);
    }

    if (k <= recLimit) {  //generate small random primes by trial division up to its square root
      pm = (1 << ((k + 2) >> 1)) - 1; //pm is binary number with all ones, just over sqrt(2^k)
      copyInt_(ans, 0);
      for (dd = 1; dd; ) {
        dd = 0;
        ans[0] = 1 | (1 << (k - 1)) | Math.floor(Math.random() * (1 << k));  //random, k-bit, odd integer, with msb 1
        for (j = 1; (j < primes.length) && ((primes[j] & pm) == primes[j]); j++) { //trial division by all primes 3...sqrt(2^k)
          if (0 == (ans[0] % primes[j])) {
            dd = 1;
            break;
          }
        }
      }
      carry_(ans);
      return;
    }

    B = c * k * k;    //try small primes up to B (or all the primes[] array if the largest is less than B).
    if (k > 2 * m)  //generate this k-bit number by first recursively generating a number that has between k/2 and k-m bits
      for (r = 1; k - k * r <= m; )
        r = pows[Math.floor(Math.random() * 512)];   //r=Math.pow(2,Math.random()-1);
    else
      r = .5;

    //simulation suggests the more complex algorithm using r=.333 is only slightly faster.

    recSize = Math.floor(r * k) + 1;

    randTruePrime_(s_q, recSize);
    copyInt_(s_i2, 0);
    s_i2[Math.floor((k - 2) / bpe)] |= (1 << ((k - 2) % bpe));   //s_i2=2^(k-2)
    divide_(s_i2, s_q, s_i, s_rm);                        //s_i=floor((2^(k-1))/(2q))

    z = bitSize(s_i);

    for (; ; ) {
      for (; ; ) {  //generate z-bit numbers until one falls in the range [0,s_i-1]
        randBigInt_(s_R, z, 0);
        if (greater(s_i, s_R))
          break;
      }                //now s_R is in the range [0,s_i-1]
      addInt_(s_R, 1);  //now s_R is in the range [1,s_i]
      add_(s_R, s_i);   //now s_R is in the range [s_i+1,2*s_i]

      copy_(s_n, s_q);
      mult_(s_n, s_R);
      multInt_(s_n, 2);
      addInt_(s_n, 1);    //s_n=2*s_R*s_q+1

      copy_(s_r2, s_R);
      multInt_(s_r2, 2);  //s_r2=2*s_R

      //check s_n for divisibility by small primes up to B
      for (divisible = 0, j = 0; (j < primes.length) && (primes[j] < B); j++)
        if (modInt(s_n, primes[j]) == 0 && !equalsInt(s_n, primes[j])) {
          divisible = 1;
          break;
        }

      if (!divisible)    //if it passes small primes check, then try a single Miller-Rabin base 2
        if (!millerRabinInt(s_n, 2)) //this line represents 75% of the total runtime for randTruePrime_ 
          divisible = 1;

      if (!divisible) {  //if it passes that test, continue checking s_n
        addInt_(s_n, -3);
        for (j = s_n.length - 1; (s_n[j] == 0) && (j > 0); j--);  //strip leading zeros
        for (zz = 0, w = s_n[j]; w; (w >>= 1), zz++);
        zz += bpe * j;                             //zz=number of bits in s_n, ignoring leading zeros
        for (; ; ) {  //generate z-bit numbers until one falls in the range [0,s_n-1]
          randBigInt_(s_a, zz, 0);
          if (greater(s_n, s_a))
            break;
        }                //now s_a is in the range [0,s_n-1]
        addInt_(s_n, 3);  //now s_a is in the range [0,s_n-4]
        addInt_(s_a, 2);  //now s_a is in the range [2,s_n-2]
        copy_(s_b, s_a);
        copy_(s_n1, s_n);
        addInt_(s_n1, -1);
        powMod_(s_b, s_n1, s_n);   //s_b=s_a^(s_n-1) modulo s_n
        addInt_(s_b, -1);
        if (isZero(s_b)) {
          copy_(s_b, s_a);
          powMod_(s_b, s_r2, s_n);
          addInt_(s_b, -1);
          copy_(s_aa, s_n);
          copy_(s_d, s_b);
          GCD_(s_d, s_n);  //if s_b and s_n are relatively prime, then s_n is a prime
          if (equalsInt(s_d, 1)) {
            copy_(ans, s_aa);
            return;     //if we've made it this far, then s_n is absolutely guaranteed to be prime
          }
        }
      }
    }
  }

  //Set b to an n-bit random BigInt.  If s=1, then the most significant of those n bits is set to 1.
  //Array b must be big enough to hold the result. Must have n>=1
  function randBigInt_(b, n, s) {
    var i, a;
    for (i = 0; i < b.length; i++)
      b[i] = 0;
    a = Math.floor((n - 1) / bpe) + 1; //# array elements to hold the BigInt
    for (i = 0; i < a; i++) {
      b[i] = Math.floor(Math.random() * (1 << (bpe - 1)));
    }
    b[a - 1] &= (2 << ((n - 1) % bpe)) - 1;
    if (s == 1)
      b[a - 1] |= (1 << ((n - 1) % bpe));
  }

  //Return an n-bit random BigInt (n>=1).  If s=1, then the most significant of those n bits is set to 1.
  function randBigInt(n, s) {
    var a, b;
    a = Math.floor((n - 1) / bpe) + 2; //# array elements to hold the BigInt with a leading 0 element
    b = int2bigInt(0, 0, a);
    randBigInt_(b, n, s);
    return b;
  }

  //set x to the greatest common divisor of bigInts x and y (each with same number of elements).
  //y is destroyed.
  function GCD_(x, y) {
    var i, xp, yp, A, B, C, D, q, sing;
    if (T.length != x.length)
      T = dup(x);

    sing = 1;
    while (sing) { //while y has nonzero elements other than y[0]
      sing = 0;
      for (i = 1; i < y.length; i++) //check if y has nonzero elements other than 0
        if (y[i]) {
          sing = 1;
          break;
        }
      if (!sing) break; //quit when y all zero elements except possibly y[0]

      for (i = x.length; !x[i] && i >= 0; i--);  //find most significant element of x
      xp = x[i];
      yp = y[i];
      A = 1; B = 0; C = 0; D = 1;
      while ((yp + C) && (yp + D)) {
        q = Math.floor((xp + A) / (yp + C));
        qp = Math.floor((xp + B) / (yp + D));
        if (q != qp)
          break;
        t = A - q * C; A = C; C = t;    //  do (A,B,xp, C,D,yp) = (C,D,yp, A,B,xp) - q*(0,0,0, C,D,yp)      
        t = B - q * D; B = D; D = t;
        t = xp - q * yp; xp = yp; yp = t;
      }
      if (B) {
        copy_(T, x);
        linComb_(x, y, A, B); //x=A*x+B*y
        linComb_(y, T, D, C); //y=D*y+C*T
      } else {
        mod_(x, y);
        copy_(T, x);
        copy_(x, y);
        copy_(y, T);
      }
    }
    if (y[0] == 0)
      return;
    t = modInt(x, y[0]);
    copyInt_(x, y[0]);
    y[0] = t;
    while (y[0]) {
      x[0] %= y[0];
      t = x[0]; x[0] = y[0]; y[0] = t;
    }
  }

  //Return the greatest common divisor of bigInts x and y (each with same number of elements).
  function GCD(x, y) {
    var xc, yc;
    xc = dup(x);
    yc = dup(y);
    GCD_(xc, yc);
    return xc;
  }

  //this deprecated function is for backward compatibility only. 
  function inverseModInt_(x, n) {
    return inverseModInt(x, n);
  }

  //do x=x**(-1) mod n, for bigInts x and n.
  //If no inverse exists, it sets x to zero and returns 0, else it returns 1.
  //The x array must be at least as large as the n array.
  function inverseMod_(x, n) {
    var k = 1 + 2 * Math.max(x.length, n.length);

    if (!(x[0] & 1) && !(n[0] & 1)) {  //if both inputs are even, then inverse doesn't exist
      copyInt_(x, 0);
      return 0;
    }

    if (eg_u.length != k) {
      eg_u = new Array(k);
      eg_v = new Array(k);
      eg_A = new Array(k);
      eg_B = new Array(k);
      eg_C = new Array(k);
      eg_D = new Array(k);
    }

    copy_(eg_u, x);
    copy_(eg_v, n);
    copyInt_(eg_A, 1);
    copyInt_(eg_B, 0);
    copyInt_(eg_C, 0);
    copyInt_(eg_D, 1);
    for (; ; ) {
      while (!(eg_u[0] & 1)) {  //while eg_u is even
        halve_(eg_u);
        if (!(eg_A[0] & 1) && !(eg_B[0] & 1)) { //if eg_A==eg_B==0 mod 2
          halve_(eg_A);
          halve_(eg_B);
        } else {
          add_(eg_A, n); halve_(eg_A);
          sub_(eg_B, x); halve_(eg_B);
        }
      }

      while (!(eg_v[0] & 1)) {  //while eg_v is even
        halve_(eg_v);
        if (!(eg_C[0] & 1) && !(eg_D[0] & 1)) { //if eg_C==eg_D==0 mod 2
          halve_(eg_C);
          halve_(eg_D);
        } else {
          add_(eg_C, n); halve_(eg_C);
          sub_(eg_D, x); halve_(eg_D);
        }
      }

      if (!greater(eg_v, eg_u)) { //eg_v <= eg_u
        sub_(eg_u, eg_v);
        sub_(eg_A, eg_C);
        sub_(eg_B, eg_D);
      } else {                   //eg_v > eg_u
        sub_(eg_v, eg_u);
        sub_(eg_C, eg_A);
        sub_(eg_D, eg_B);
      }

      if (equalsInt(eg_u, 0)) {
        while (negative(eg_C)) //make sure answer is nonnegative
          add_(eg_C, n);
        copy_(x, eg_C);

        if (!equalsInt(eg_v, 1)) { //if GCD_(x,n)!=1, then there is no inverse
          copyInt_(x, 0);
          return 0;
        }
        return 1;
      }
    }
  }

  //return x**(-1) mod n, for integers x and n.  Return 0 if there is no inverse
  function inverseModInt(x, n) {
    var a = 1, b = 0, t;
    for (; ; ) {
      if (x == 1) return a;
      if (x == 0) return 0;
      b -= a * Math.floor(n / x);
      n %= x;

      if (n == 1) return b; //to avoid negatives, change this b to n-b, and each -= to +=
      if (n == 0) return 0;
      a -= b * Math.floor(x / n);
      x %= n;
    }
  }

  //Given positive bigInts x and y, change the bigints v, a, and b to positive bigInts such that:
  //     v = GCD_(x,y) = a*x-b*y
  //The bigInts v, a, b, must have exactly as many elements as the larger of x and y.
  function eGCD_(x, y, v, a, b) {
    var g = 0;
    var k = Math.max(x.length, y.length);
    if (eg_u.length != k) {
      eg_u = new Array(k);
      eg_A = new Array(k);
      eg_B = new Array(k);
      eg_C = new Array(k);
      eg_D = new Array(k);
    }
    while (!(x[0] & 1) && !(y[0] & 1)) {  //while x and y both even
      halve_(x);
      halve_(y);
      g++;
    }
    copy_(eg_u, x);
    copy_(v, y);
    copyInt_(eg_A, 1);
    copyInt_(eg_B, 0);
    copyInt_(eg_C, 0);
    copyInt_(eg_D, 1);
    for (; ; ) {
      while (!(eg_u[0] & 1)) {  //while u is even
        halve_(eg_u);
        if (!(eg_A[0] & 1) && !(eg_B[0] & 1)) { //if A==B==0 mod 2
          halve_(eg_A);
          halve_(eg_B);
        } else {
          add_(eg_A, y); halve_(eg_A);
          sub_(eg_B, x); halve_(eg_B);
        }
      }

      while (!(v[0] & 1)) {  //while v is even
        halve_(v);
        if (!(eg_C[0] & 1) && !(eg_D[0] & 1)) { //if C==D==0 mod 2
          halve_(eg_C);
          halve_(eg_D);
        } else {
          add_(eg_C, y); halve_(eg_C);
          sub_(eg_D, x); halve_(eg_D);
        }
      }

      if (!greater(v, eg_u)) { //v<=u
        sub_(eg_u, v);
        sub_(eg_A, eg_C);
        sub_(eg_B, eg_D);
      } else {                //v>u
        sub_(v, eg_u);
        sub_(eg_C, eg_A);
        sub_(eg_D, eg_B);
      }
      if (equalsInt(eg_u, 0)) {
        while (negative(eg_C)) {   //make sure a (C) is nonnegative
          add_(eg_C, y);
          sub_(eg_D, x);
        }
        multInt_(eg_D, -1);  ///make sure b (D) is nonnegative
        copy_(a, eg_C);
        copy_(b, eg_D);
        leftShift_(v, g);
        return;
      }
    }
  }


  //is bigInt x negative?
  function negative(x) {
    return ((x[x.length - 1] >> (bpe - 1)) & 1);
  }


  //is (x << (shift*bpe)) > y?
  //x and y are nonnegative bigInts
  //shift is a nonnegative integer
  function greaterShift(x, y, shift) {
    var i, kx = x.length, ky = y.length, k;
    k = ((kx + shift) < ky) ? (kx + shift) : ky; // <ogre>: this is using a global variable in the original version
    for (i = ky - 1 - shift; i < kx && i >= 0; i++)
      if (x[i] > 0)
        return 1; //if there are nonzeros in x to the left of the first column of y, then x is bigger
    for (i = kx - 1 + shift; i < ky; i++)
      if (y[i] > 0)
        return 0; //if there are nonzeros in y to the left of the first column of x, then x is not bigger
    for (i = k - 1; i >= shift; i--)
      if (x[i - shift] > y[i]) return 1;
      else if (x[i - shift] < y[i]) return 0;
    return 0;
  }

  //is x > y? (x and y both nonnegative)
  function greater(x, y) {
    var i;
    var k = (x.length < y.length) ? x.length : y.length;

    for (i = x.length; i < y.length; i++)
      if (y[i])
        return 0;  //y has more digits

    for (i = y.length; i < x.length; i++)
      if (x[i])
        return 1;  //x has more digits

    for (i = k - 1; i >= 0; i--)
      if (x[i] > y[i])
        return 1;
      else if (x[i] < y[i])
        return 0;
    return 0;
  }

  //divide x by y giving quotient q and remainder r.  (q=floor(x/y),  r=x mod y).  All 4 are bigints.
  //x must have at least one leading zero element.
  //y must be nonzero.
  //q and r must be arrays that are exactly the same length as x. (Or q can have more).
  //Must have x.length >= y.length >= 2.
  function divide_(x, y, q, r) {
    var kx, ky;
    var i, j, y1, y2, c, a, b;
    copy_(r, x);
    for (ky = y.length; y[ky - 1] == 0; ky--); //ky is number of elements in y, not including leading zeros

    //normalize: ensure the most significant element of y has its highest bit set  
    b = y[ky - 1];
    for (a = 0; b; a++)
      b >>= 1;
    a = bpe - a;  //a is how many bits to shift so that the high order bit of y is leftmost in its array element
    leftShift_(y, a);  //multiply both by 1<<a now, then divide both by that at the end
    leftShift_(r, a);

    //Rob Visser discovered a bug: the following line was originally just before the normalization.
    for (kx = r.length; r[kx - 1] == 0 && kx > ky; kx--); //kx is number of elements in normalized x, not including leading zeros

    copyInt_(q, 0);                      // q=0
    while (!greaterShift(y, r, kx - ky)) {  // while (leftShift_(y,kx-ky) <= r) {
      subShift_(r, y, kx - ky);             //   r=r-leftShift_(y,kx-ky)
      q[kx - ky]++;                       //   q[kx-ky]++;
    }                                   // }

    for (i = kx - 1; i >= ky; i--) {
      if (r[i] == y[ky - 1])
        q[i - ky] = mask;
      else
        q[i - ky] = Math.floor((r[i] * radix + r[i - 1]) / y[ky - 1]);

      //The following for(;;) loop is equivalent to the commented while loop, 
      //except that the uncommented version avoids overflow.
      //The commented loop comes from HAC, which assumes r[-1]==y[-1]==0
      //  while (q[i-ky]*(y[ky-1]*radix+y[ky-2]) > r[i]*radix*radix+r[i-1]*radix+r[i-2])
      //    q[i-ky]--;    
      for (; ; ) {
        y2 = (ky > 1 ? y[ky - 2] : 0) * q[i - ky];
        c = y2 >> bpe;
        y2 = y2 & mask;
        y1 = c + q[i - ky] * y[ky - 1];
        c = y1 >> bpe;
        y1 = y1 & mask;

        if (c == r[i] ? y1 == r[i - 1] ? y2 > (i > 1 ? r[i - 2] : 0) : y1 > r[i - 1] : c > r[i])
          q[i - ky]--;
        else
          break;
      }

      linCombShift_(r, y, -q[i - ky], i - ky);    //r=r-q[i-ky]*leftShift_(y,i-ky)
      if (negative(r)) {
        addShift_(r, y, i - ky);         //r=r+leftShift_(y,i-ky)
        q[i - ky]--;
      }
    }

    rightShift_(y, a);  //undo the normalization step
    rightShift_(r, a);  //undo the normalization step
  }

  //do carries and borrows so each element of the bigInt x fits in bpe bits.
  function carry_(x) {
    var i, k, c, b;
    k = x.length;
    c = 0;
    for (i = 0; i < k; i++) {
      c += x[i];
      b = 0;
      if (c < 0) {
        b = -(c >> bpe);
        c += b * radix;
      }
      x[i] = c & mask;
      c = (c >> bpe) - b;
    }
  }

  //return x mod n for bigInt x and integer n.
  function modInt(x, n) {
    var i, c = 0;
    for (i = x.length - 1; i >= 0; i--)
      c = (c * radix + x[i]) % n;
    return c;
  }

  //convert the integer t into a bigInt with at least the given number of bits.
  //the returned array stores the bigInt in bpe-bit chunks, little endian (buff[0] is least significant word)
  //Pad the array with leading zeros so that it has at least minSize elements.
  //There will always be at least one leading 0 element.
  function int2bigInt(t, bits, minSize) {
    // <ogre>: there is a problem with this code. It fails to add a leading [0] in cases
    //         where the minimum size / bits is not large enough to contain the int. For
    //         example:
    //            int2bigInt(17, 0, 0) ===> [17]
    //            str2bigInt(17, 10, 0) ===> [17, 0]
    //        I've fixed this by converting to string and calling str2bigInt.
    var i, k;
    k = Math.ceil(bits / bpe) + 1;
    k = minSize > k ? minSize : k;
    return str2bigInt(t.toString(), 10, k);
    // Old code
    //var i, k;
    //k = Math.ceil(bits / bpe) + 1;
    //k = minSize > k ? minSize : k;
    //buff = new Array(k);
    //copyInt_(buff, t);
    //return buff;
  }

  //return the bigInt given a string representation in a given base.  
  //Pad the array with leading zeros so that it has at least minSize elements.
  //If base=-1, then it reads in a space-separated list of array elements in decimal.
  //The array will always have at least one leading zero, unless base=-1.
  function str2bigInt(s, base, minSize) {
    var d, i, j, x, y, kk;
    var k = s.length;
    if (base == -1) { //comma-separated list of array elements in decimal
      x = new Array(0);
      for (; ; ) {
        y = new Array(x.length + 1);
        for (i = 0; i < x.length; i++)
          y[i + 1] = x[i];
        y[0] = parseInt(s, 10);
        x = y;
        d = s.indexOf(',', 0);
        if (d < 1)
          break;
        s = s.substring(d + 1);
        if (s.length == 0)
          break;
      }
      if (x.length < minSize) {
        y = new Array(minSize);
        copy_(y, x);
        return y;
      }
      return x;
    }

    //x = int2bigInt(0, base * k, 0); // <ogre>: had to remove this to fix recursion
    x = new Array(Math.ceil(base * k / bpe)); copyInt_(x, 0); // <ogre>: using this instead

    for (i = 0; i < k; i++) {
      d = digitsStr.indexOf(s.substring(i, i + 1), 0);
      if (base <= 36 && d >= 36)  //convert lowercase to uppercase if base<=36
        d -= 26;
      if (d >= base || d < 0) {   //stop at first illegal character
        break;
      }
      multInt_(x, base);
      addInt_(x, d);
    }

    for (k = x.length; k > 0 && !x[k - 1]; k--); //strip off leading zeros
    k = minSize > k + 1 ? minSize : k + 1;
    y = new Array(k);
    kk = k < x.length ? k : x.length;
    for (i = 0; i < kk; i++)
      y[i] = x[i];
    for (; i < k; i++)
      y[i] = 0;
    return y;
  }

  //is bigint x equal to integer y?
  //y must have less than bpe bits
  function equalsInt(x, y) {
    var i;
    if (x[0] != y)
      return 0;
    for (i = 1; i < x.length; i++)
      if (x[i])
        return 0;
    return 1;
  }

  //are bigints x and y equal?
  //this works even if x and y are different lengths and have arbitrarily many leading zeros
  function equals(x, y) {
    var i;
    var k = x.length < y.length ? x.length : y.length;
    for (i = 0; i < k; i++)
      if (x[i] != y[i])
        return 0;
    if (x.length > y.length) {
      for (; i < x.length; i++)
        if (x[i])
          return 0;
    } else {
      for (; i < y.length; i++)
        if (y[i])
          return 0;
    }
    return 1;
  }

  //is the bigInt x equal to zero?
  function isZero(x) {
    var i;
    for (i = 0; i < x.length; i++)
      if (x[i])
        return 0;
    return 1;
  }

  //convert a bigInt into a string in a given base, from base 2 up to base 95.
  //Base -1 prints the contents of the array representing the number.
  function bigInt2str(x, base) {
    var i, t, s = "";

    if (s6.length != x.length)
      s6 = dup(x);
    else
      copy_(s6, x);

    if (base == -1) { //return the list of array contents
      for (i = x.length - 1; i > 0; i--)
        s += x[i] + ',';
      s += x[0];
    }
    else { //return it in the given base
      while (!isZero(s6)) {
        t = divInt_(s6, base);  //t=s6 % base; s6=floor(s6/base);
        s = digitsStr.substring(t, t + 1) + s;
      }
    }
    if (s.length == 0)
      s = "0";
    return s;
  }

  //------------------------------------------------------------------------------------------------//
  // Global OO-style interface to the BigInt library
  //------------------------------------------------------------------------------------------------//
  function BigInt() { this.init.apply(this, arguments) };

  BigInt.prototype = {
    init: function (o, base) {
      var neg, qua;
      if (typeof (base) === 'number') {
        // Use the explicit base if it is provided. Otherwise,
        // fall back on pattern recognition.
        o = o.replace(/^(-)/, '');
        neg = RegExp.$1;
        this.bigInt = str2bigInt(o, base, 0);
        this.bigInt = str2bigInt(o, base, 0);
        if (neg) this.mul(-1);
      } else if (typeof (o) === 'string') {
        o = o.replace(/^(-)?(0x|0b|0o|0|)/, '');
        neg = RegExp.$1;
        qua = RegExp.$2;
        base = { '0x': 16, '0b': 2, '0o': 8, '0': 8}[qua] || 10;
        this.bigInt = str2bigInt(o, base, 0);
        if (neg) this.mul(-1);
      } else if (typeof (o) === 'number') {
        this.bigInt = int2bigInt(o, 0, 0);
      } else if (o instanceof BigInt) {
        this.bigInt = o.clone().bigInt;
      } else if (o instanceof Array) {
        this.bigInt = o;
      } else {
        throw "Unrecognized argument type: " + typeof (o) + " (" + o + ")";
      }
    },

    add: function (other) {
      if (other instanceof BigInt) {
        return new BigInt(add(this.bigInt, other.bigInt));
      } else {
        throw "Unsupported";
      }
    },

    sub: function (other) {
      if (other instanceof BigInt) {
        return new BigInt(sub(this.bigInt, other.bigInt));
      } else {
        throw "Unsupported";
      }
    },

    mod: function (other) {
      if (other instanceof BigInt) {
        return new BigInt(mod(this.bigInt, other.bigInt));
      } else {
        throw "Unsupported";
      }
    },

    mul: function (other) {
      if (other instanceof BigInt) {
        return new BigInt(mult(this.bigInt, other.bigInt));
      } else {
        throw "Unsupported";
      }
    },

    mulMod: function (other, n) {
      if (other instanceof BigInt && n instanceof BigInt) {
        return new BigInt(multMod(this.bigInt, other.bigInt, n.bigInt));
      } else {
        throw "Unsupported";
      }
    },

    powMod: function (other, n) {
      if (other instanceof BigInt && n instanceof BigInt) {
        return new BigInt(powMod(this.bigInt, other.bigInt, n.bigInt));
      } else {
        throw "Unsupported";
      }
    },

    div: function (other) {
      if (other instanceof BigInt) {
        var q = new Array(0);
        var r = new Array(0);
        divide_(this.bigInt, new BigInt(other).bigInt, q, r);
        return new Array(q, r);
      } else {
        throw "Unsupported";
      }
    },

    inverseMod: function (other) {
      if (other instanceof BigInt) {
        return new BigInt(inverseMod(this.bigInt, other.bigInt));
      } else {
        throw "Unsupported";
      }
    },

    equals: function (other) {
      if (other instanceof BigInt) {
        return equals(this.bigInt, other.bigInt);
      } else {
        throw "Unsupported";
      }
    },

    isZero: function () {
      return isZero(this.bigInt);
    },

    isNegative: function () {
      return !!negative(this.bigInt);
    },

    isPositive: function () {
      return !isZero(this.bigInt) && !negative(this.bigInt);
    },

    millerRabin: function (other) {
      if (other instanceof BigInt) {
        return new BigInt(millerRabin(this.bigInt, other));
      } else {
        throw "Unsupported";
      }
    },

    gcd: function (other) {
      if (other instanceof BigInt) {
        return new BigInt(GCD(this.bigInt, other.bigInt));
      } else {
        throw "Unsupported";
      }
    },

    greaterThan: function (other) {
      return greater(this.bigInt, new BigInt(other).bigInt);
    },

    lessThan: function (other) {
      return !greater(this.bigInt, new BigInt(other).bigInt);
    },

    bitSize: function () {
      return bitSize(this.bigInt);
    },

    clone: function () {
      return new BigInt(dup(this.bigInt));
    },

    dup: function () {
      return new BigInt(dup(this.bigInt));
    },

    rightShift: function (n) {
      if (typeof (n) === 'number') {
        var i = dup(this.bigInt);
        rightShift_(i, n);
        return new BigInt(i);
      } else {
        throw "Unsupported";
      }
    },

    isBit0Set: function (n) {
      return (this.bigInt[0] & 1) === 1;
    },

    toString: function (base) {
      return bigInt2str(this.bigInt, base || 10);
    },

    valueOf: function () {
      throw "You must use bigint functions instead of native operators";
    }
  };

  Global.BigInt = BigInt;
  Global.BigInt.randBigInt = function () { return new BigInt(randBigInt.apply(this, arguments)) };
  Global.BigInt.random = BigInt.randBigInt;

})(window);