sapphire-frontend/vendor/otr/bigint.js

;(function (root, factory) {

  if (typeof define === 'function' && define.amd) {
    define(factory.bind(root, root.crypto || root.msCrypto))
  } else if (typeof module !== 'undefined' && module.exports) {
    module.exports = factory(require('crypto'))
  } else {
    root.BigInt = factory(root.crypto || root.msCrypto)
  }

}(this, function (crypto) {

  ////////////////////////////////////////////////////////////////////////////////////////
  // 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 Péter Szabó 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.
  //
  // Note that for cryptographic purposes, the calls to Math.random() must 
  // be replaced with calls to a better pseudorandom number generator.
  //
  // 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. (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.
  ////////////////////////////////////////////////////////////////////////////////////////

  //globals

  // The number of significant bits in the fraction of a JavaScript
  // floating-point number is 52, independent of platform.
  // See: https://github.com/arlolra/otr/issues/41

  var bpe = 26;          // bits stored per array element
  var radix = 1 << bpe;  // equals 2^bpe
  var mask = radix - 1;  // AND this with an array element to chop it down to bpe bits

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

  var one=int2bigInt(1,1,1);     //constant used in powMod_()

  //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_()
  var s0=t;       //used in multMod_(), squareMod_()
  var s1=t;       //used in powMod_(), multMod_(), squareMod_()
  var s2=t;       //used in powMod_(), multMod_()
  var s3=t;       //used in powMod_()
  var s4=t, s5=t; //used in mod_()
  var s6=t;       //used in bigInt2str()
  var s7=t;       //used in powMod_()
  var T=t;        //used in GCD_()
  var sa=t;       //used in mont_()
  var mr_x1=t, mr_r=t, mr_a=t;                                      //used in millerRabin()
  var eg_v=t, eg_u=t, eg_A=t, eg_B=t, eg_C=t, eg_D=t;               //used in eGCD_(), inverseMod_()
  var md_q1=t, md_q2=t, md_q3=t, md_r=t, md_r1=t, md_r2=t, md_tt=t; //used in mod_()

  var primes=t, pows=t, s_i=t, s_i2=t, s_R=t, s_rm=t, s_q=t, s_n1=t;
  var 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_()
    
  var rpprb=t; //used in randProbPrimeRounds() (which also uses "primes")

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


  //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 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);
  }

  //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++;
    */

    /* http://www.javascripter.net/math/primes/millerrabinbug-bigint54.htm */
    if (isZero(mr_r)) return 0;
    for (k=0; mr_r[k]==0; k++);
    for (i=1,j=2; mr_r[k]%j==0; j*=2,i++ );
    s = k*bpe + i - 1;
    /* end */

    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;  
  }

  //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,w,m,pm,dd,j,r,B,divisible,z,zz,recSize,recLimit;

    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/511.0-1.0);
      }
    }

    //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)) | randomBitInt(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[randomBitInt(9)];   //r=Math.pow(2,Math.random()-1);
    else
      r=0.5;

    //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
          }
        }
      }
    }
  }

  //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 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]=randomBitInt(bpe);
    }
    b[a-1] &= (2<<((n-1)%bpe))-1;
    if (s==1)
      b[a-1] |= (1<<((n-1)%bpe));
  }

  //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;
  }

  //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,qp;
    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;
    }
  }

  //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;
    }
  }

  //this deprecated function is for backward compatibility only. 
  function inverseModInt_(x,n) {
     return inverseModInt(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;
    var k=((kx+shift)<ky) ? (kx+shift) : ky;
    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;
        y2=y2 & mask;
        c = (c - y2) / radix;
        y1=c+q[i-ky]*y[ky-1];
        c=y1;
        y1=y1 & mask;
        c = (c - y1) / radix;

        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 & mask;
        b = -((c - b) / radix);
        c+=b*radix;
      }
      x[i]=c & mask;
      c = ((c - x[i]) / radix) - 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) {   
    var i,k, buff;
    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;
    }

    // log2(base)*k
    var bb = base, p = 0;
    var b = base == 1 ? k : 0;
    while (bb > 1) {
      if (bb & 1) p = 1;
      b += k;
      bb >>= 1;
    }
    b += p*k;

    x=int2bigInt(0,b,0);
    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;
  }

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

  //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;
    }
  }

  //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 & mask;
        b = -((c - b) / radix);
        c+=b*radix;
      }
      x[i]=c & mask;
      c = ((c - x[i]) / radix) - b;
      if (!c) return; //stop carrying as soon as the carry is zero
    }
  }

  //right shift bigInt x by n bits.
  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 & mask;
        b = -((c - b) / radix);
        c+=b*radix;
      }
      x[i]=c & mask;
      c = ((c - x[i]) / radix) - 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 = (c - x[i]) / radix;
    }
    for (i=k;i<kk;i++) {
      c+=a*x[i];
      x[i]=c & mask;
      c = (c - x[i]) / radix;
    }
  }

  //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 = (c - x[i]) / radix;
    }
    for (i=k;c && i<kk;i++) {
      c+=x[i];
      x[i]=c & mask;
      c = (c - x[i]) / radix;
    }
  }

  //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 = (c - x[i]) / radix;
    }
    for (i=k;c && i<kk;i++) {
      c+=x[i];
      x[i]=c & mask;
      c = (c - x[i]) / radix;
    }
  }

  //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 = (c - x[i]) / radix;
    }
    for (i=k;c && i<kk;i++) {
      c+=x[i];
      x[i]=c & mask;
      c = (c - x[i]) / radix;
    }
  }

  //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 = (c - x[i]) / radix;
    }
    for (i=k;c && i<x.length;i++) {
      c+=x[i];
      x[i]=c & mask;
      c = (c - x[i]) / radix;
    }
  }

  //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 = (c - x[i]) / radix;
    }
    for (i=k;c && i<x.length;i++) {
      c+=x[i];
      x[i]=c & mask;
      c = (c - x[i]) / radix;
    }
  }

  //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 = (c - s0[2*i]) / radix;
      for (j=i+1;j<kx;j++) {
        c=s0[i+j]+2*x[i]*x[j]+c;
        s0[i+j]=(c & mask);
        c = (c - s0[i+j]) / radix;
      }
      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,t2,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]);
      c = (c - (c & mask)) / radix;
      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]; t2=sa[j-1]=c & mask; c=(c-t2)/radix; j++;
        c+=sa[j]+ui*n[j]+t*y[j]; t2=sa[j-1]=c & mask; c=(c-t2)/radix; j++;
        c+=sa[j]+ui*n[j]+t*y[j]; t2=sa[j-1]=c & mask; c=(c-t2)/radix; j++;
        c+=sa[j]+ui*n[j]+t*y[j]; t2=sa[j-1]=c & mask; c=(c-t2)/radix; j++;
        c+=sa[j]+ui*n[j]+t*y[j]; t2=sa[j-1]=c & mask; c=(c-t2)/radix; j++;
      }
      for (;j<ky;)   {
        c+=sa[j]+ui*n[j]+t*y[j]; t2=sa[j-1]=c & mask; c=(c-t2)/radix; j++;
      }
      for (;j<kn-4;) {
        c+=sa[j]+ui*n[j];        t2=sa[j-1]=c & mask; c=(c-t2)/radix; j++;
        c+=sa[j]+ui*n[j];        t2=sa[j-1]=c & mask; c=(c-t2)/radix; j++;
        c+=sa[j]+ui*n[j];        t2=sa[j-1]=c & mask; c=(c-t2)/radix; j++;
        c+=sa[j]+ui*n[j];        t2=sa[j-1]=c & mask; c=(c-t2)/radix; j++;
        c+=sa[j]+ui*n[j];        t2=sa[j-1]=c & mask; c=(c-t2)/radix; j++;
      }
      for (;j<kn;)   {
        c+=sa[j]+ui*n[j];        t2=sa[j-1]=c & mask; c=(c-t2)/radix; j++;
      }
      for (;j<ks;)   {
        c+=sa[j];                t2=sa[j-1]=c & mask; c=(c-t2)/radix; j++;
      }
      sa[j-1]=c & mask;
    }

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


  // otr.js additions


  // computes num / den mod n
  function divMod(num, den, n) {
    return multMod(num, inverseMod(den, n), n)
  }

  // computes one - two mod n
  function subMod(one, two, n) {
    one = mod(one, n)
    two = mod(two, n)
    if (greater(two, one)) one = add(one, n)
    return sub(one, two)
  }

  // computes 2^m as a bigInt
  function twoToThe(m) {
    var b = Math.floor(m / bpe) + 2
    var t = new Array(b)
    for (var i = 0; i < b; i++) t[i] = 0
    t[b - 2] = 1 << (m % bpe)
    return t
  }

  // cache these results for faster lookup
  var _num2bin = (function () {
    var i = 0, _num2bin= {}
    for (; i < 0x100; ++i) {
      _num2bin[i] = String.fromCharCode(i)  // 0 -> "\00"
    }
    return _num2bin
  }())

  // serialize a bigInt to an ascii string
  // padded up to pad length
  function bigInt2bits(bi, pad) {
    pad || (pad = 0)
    bi = dup(bi)
    var ba = ''
    while (!isZero(bi)) {
      ba = _num2bin[bi[0] & 0xff] + ba
      rightShift_(bi, 8)
    }
    while (ba.length < pad) {
      ba = '\x00' + ba
    }
    return ba
  }

  // converts a byte array to a bigInt
  function ba2bigInt(data) {
    var mpi = str2bigInt('0', 10, data.length)
    data.forEach(function (d, i) {
      if (i) leftShift_(mpi, 8)
      mpi[0] |= d
    })
    return mpi
  }

  // returns a function that returns an array of n bytes
  var randomBytes = (function () {

    // in node
    if ( typeof crypto !== 'undefined' &&
      typeof crypto.randomBytes === 'function' ) {
      return function (n) {
        try {
          var buf = crypto.randomBytes(n)
        } catch (e) { throw e }
        return Array.prototype.slice.call(buf, 0)
      }
    }

    // in browser
    else if ( typeof crypto !== 'undefined' &&
      typeof crypto.getRandomValues === 'function' ) {
      return function (n) {
        var buf = new Uint8Array(n)
        crypto.getRandomValues(buf)
        return Array.prototype.slice.call(buf, 0)
      }
    }

    // err
    else {
      throw new Error('Keys should not be generated without CSPRNG.')
    }

  }())

  // Salsa 20 in webworker needs a 40 byte seed
  function getSeed() {
    return randomBytes(40)
  }

  // returns a single random byte
  function randomByte() {
    return randomBytes(1)[0]
  }

  // returns a k-bit random integer
  function randomBitInt(k) {
    if (k > 31) throw new Error("Too many bits.")
    var i = 0, r = 0
    var b = Math.floor(k / 8)
    var mask = (1 << (k % 8)) - 1
    if (mask) r = randomByte() & mask
    for (; i < b; i++)
      r = (256 * r) + randomByte()
    return r
  }

  return {
      str2bigInt    : str2bigInt
    , bigInt2str    : bigInt2str
    , int2bigInt    : int2bigInt
    , multMod       : multMod
    , powMod        : powMod
    , inverseMod    : inverseMod
    , randBigInt    : randBigInt
    , randBigInt_   : randBigInt_
    , equals        : equals
    , equalsInt     : equalsInt
    , sub           : sub
    , mod           : mod
    , modInt        : modInt
    , mult          : mult
    , divInt_       : divInt_
    , rightShift_   : rightShift_
    , dup           : dup
    , greater       : greater
    , add           : add
    , isZero        : isZero
    , bitSize       : bitSize
    , millerRabin   : millerRabin
    , divide_       : divide_
    , trim          : trim
    , primes        : primes
    , findPrimes    : findPrimes
    , getSeed       : getSeed
    , divMod        : divMod
    , subMod        : subMod
    , twoToThe      : twoToThe
    , bigInt2bits   : bigInt2bits
    , ba2bigInt     : ba2bigInt
  }

}))