diff -r a434d60e525d -r 242353360e37 includes/clientside/static/libbigint.js
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/includes/clientside/static/libbigint.js	Wed Feb 20 14:38:39 2008 -0500
@@ -0,0 +1,1400 @@
+////////////////////////////////////////////////////////////////////////////////////////
+// Big Integer Library v. 5.1
+// Created 2000, last modified 2007
+// Leemon Baird
+// www.leemon.com
+//
+// Version history:
+//
+// 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 address 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.
+// Some functions assume their parameters have at least one leading zero element.
+// Functions with an underscore at the end of the name have unpredictable behavior in case of overflow, 
+// so the caller must make sure the arrays must be big enough to hold the answer.
+// For each 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).
+//
+// 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 (as opposed to definitely composite)?
+// 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  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  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.
+////////////////////////////////////////////////////////////////////////////////////////
+
+//globals
+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.
+
+//the digits for converting to different bases
+digitsStr='0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz_=!@#$%^&*()[]{}|;:,.<>/?`~ \\\'\"+-';
+
+//initialize the global variables
+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
+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
+t=new Array(0);
+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_()
+
+////////////////////////////////////////////////////////////////////////////////////////
+
+//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
+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);
+  }
+
+  copyInt_(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;  
+}
+
+//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 bigint_trim(ans,1);
+}
+
+//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 bigint_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 bigint_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 bigint_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,bigint_trim(y,2),bigint_trim(n,2),0);  //this should work without the trim, but doesn't
+  return bigint_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 bigint_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 bigint_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 ? bigint_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 bigint_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) {
+        divisible=1;
+        break;
+      }      
+
+    if (!divisible)    //if it passes small primes check, then try a single Miller-Rabin base 2
+      if (!millerRabin(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]=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 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;
+  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)) {
+      if (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)) {
+      if (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 kx=x.length, ky=y.length;
+  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>>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) {   
+  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);
+  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) {   //ignore illegal characters
+      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=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>>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 bigint_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 must be large enough to hold the answer.
+//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;
+  var kn=n.length;
+  var ky=y.length;
+
+  if (sa.length!=kn)
+    sa=new Array(kn);
+
+  for (;kn>0 && n[kn-1]==0;kn--); //ignore leading zeros of n
+  //this function sometimes gives wrong answers when the next line is uncommented
+  //for (;ky>0 && y[ky-1]==0;ky--); //ignore leading zeros of y
+
+  copyInt_(sa,0);
+
+  //the following loop consumes 95% of the runtime for randTruePrime_() and powMod_() for large keys
+  for (i=0; i<kn; i++) {
+    t=sa[0]+x[i]*y[0];
+    ui=((t & mask) * np) & mask;  //the inner "& mask" is needed on Macintosh MSIE, but not windows MSIE
+    c=(t+ui*n[0]) >> bpe;
+    t=x[i];
+
+    //do sa=(sa+x[i]*y+ui*n)/b   where b=2**bpe
+    for (j=1;j<ky;j++) { 
+      c+=sa[j]+t*y[j]+ui*n[j];
+      sa[j-1]=c & mask;
+      c>>=bpe;
+    }    
+    for (;j<kn;j++) { 
+      c+=sa[j]+ui*n[j];
+      sa[j-1]=c & mask;
+      c>>=bpe;
+    }    
+    sa[j-1]=c & mask;
+  }
+
+  if (!greater(n,sa))
+    sub_(sa,n);
+  copy_(x,sa);
+}
+
+