function mailcloak_bigint() {

  // create a container to prevent variable name conflicts
  var bigintObj = new Object();


  /* ---------------------------------------------------------
     From: http://www.leemon.com/crypto/BigInt.html
  
     Note: These are only the functions used by Mailcloak.
     With the exception of our own character translation table 
     this code is the original.
     --------------------------------------------------------- */

  ////////////////////////////////////////////////////////////////////////////////////////
  // Big Integer Library v. 3.04
  // (c)2000, 2001, 2002, 2003 Leemon Baird
  // www.leemon.com
  //
  // I retain the copyright to this code, but you may redistribute 
  // or use it for any purpose.  If you use this code, please leave 
  // an acknowledgement and pointer to my home page in the comments.
  // I make no claims about whether it works correctly, so use it 
  // at your own risk.
  //
  // This code defines a bigInt library for arbitrary-precision integers.
  // A bigInt is an array of integers storing the value in bpe-bit chunks, 
  // 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.
  // The results of all these functions are undefined in case of overflow, 
  // so the caller must make sure overflow won't happen.
  // For each function where the first parameter x 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 use 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 allocation to it when it's not the right size.  This ensures
  // that when a function is called repeatedly with same-sized parameters, it only allocates
  // memory on the first call.  The only exceptions to this rule are:
  //     int2BigInt(), str2BigInt(), dup(), trim(), findPrimes()
  //
  // 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.
  //
  // function addInt(x,n)            //do x=x+n where x is a bigInt and n is an integer
  // function add(x,y)               //do x=x+y for bigInts x and y
  // function addShift(x,y,ys)       //do x=x+(y<<(ys*bpe))
  // function copy(x,y)              //do x=y on bigInts x and y
  // function copyInt(x,n)           //do x=y on bigInt x and integer n
  // function bigInt2str(x,base)     //convert a bigInt into a string in a given base, from base 2 up to base 95
  // function bitSize(x)             //returns how many bits long the bigInt x is, not counting leading zeros
  // function carry(x)               //do carries and borrows so each element of the bigInt x fits in bpe bits.
  // function divide(x,y,q,r)        //divide x by y giving quotient q and remainder r
  // function divInt(x,n)            //do x=floor(x/n) for bigInt x and integer n, and return the remainder
  // function dup(x)                 //returns a duplicate of bigInt x
  // function eGCD(x,y,d,a,b)        //sets a,b,d to positive integers such that d = GCD(x,y) = a*x-b*y
  // function equals(x,y)            //is the bigInt x equal to the bigint y?
  // function equalsInt(x,y)         //is bigint x equal to integer y?
  // function findPrimes(n)          //return array of all primes less than integer n
  // function GCD(x,y)               //set x to the greatest common divisor of bigInts x and y, (y is destroyed).
  // function greater(x,y)           //is x>y?  (x and y are nonnegative bigInts)
  // function greaterShift(x,y,shift)//is (x <<(shift*bpe)) > y?
  // function halve(x)               //do x=floor(|x|/2)*sgn(x) for bigInt x in 2's complement
  // function int2bigInt(t,n,m)      //convert integer t to a bigInt with at least n bits and m array elements
  // function inverseMod(x,n)        //do x=x**(-1) mod n, for bigInts x and n. Returns 1 (0) if inverse does (doesn't) exist
  // function inverseModInt(x,n)     //return x**(-1) mod n, for integers x and n.  Return 0 if there is no inverse
  // function isZero(x)              //is the bigInt x equal to zero?
  // function leftShift(x,n)         //left shift bigInt x by n bits.  n<bpe.
  // function linComb(x,y,a,b)       //do x=a*x+b*y for bigInts x and y and integers a and b
  // function linCombShift(x,y,b,ys) //do x=x+b*(y<<(ys*bpe)) for bigInts x and y, and integers b and ys
  // function millerRabin(x,b)       //does one round of Miller-Rabin base integer b say that bigInt x is possibly prime?
  // function mod(x,n)               //do x=x mod n for bigInts x and n.
  // function modInt(x,n)            //return x mod n for bigInt x and integer n.
  // function mont(x,y,n,np)         //Montgomery multiplication (see comments where the function is defined)
  // function mult(x,y)              //do x=x*y for bigInts x and y.
  // function multInt(x,n)           //do x=x*n where x is a bigInt and n is an integer.
  // function multMod(x,y,n)         //do x=x*y  mod n for bigInts x,y,n.
  // function negative(x)            //is bigInt x negative?
  // function 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.
  // function randBigInt(b,n,s)      //Generate an n-bit random BigInt. if s=1, then nth bit (most significant bit) is set to 1
  // function randTruePrime(ans,k)   //ans = a random k-bit true random prime (not just probable prime) with 1 in the msb.
  // function rightShift(x,n)        //right shift bigInt x by n bits.  n<bpe.
  // function squareMod(x,n)         //do x=x*x  mod n for bigInts x,n
  // function str2bigInt(s,b,n,m)    //convert string s in base b to a bigInt with at least n bits and m array elements
  // function sub(x,y)               //do x=x-y for bigInts x and y
  // function subShift(x,y,ys)       //do x=x-(y<<(ys*bpe))
  // function trim(x,k)              //return x with exactly k leading zeros
  //
  // 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_=!@#$%^&*()[]{}|;:,.<>/?`~ \\\'\"+-';

  //**MODIFIED**
  //substitute the Mailcloak character translation table
  digitsStr = "0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz_=";
  digitsStr += " !\"#$%&'()*+,-./:;<=>?@[\\]^_`{|}~";

  //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;
  }


  //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
    //this was:   c=primes[primes.length-1]/k/k;  //check using all the small primes.  (c=0.1 in HAC)
    c=0.1;  
    m=20;   //generate this k-bit number by first recursively generating a number that has between k/2 and k-m bits
    recLimit=20; /*must be at least 2 (was 29)*/   //stop recursion when k <=recLimit

    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
          }
        }
      }
    }
  }


  //Generate an n-bit random BigInt.  If s=1, then nth bit (most significant bit) 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)
      b[a-1] |= (1<<((n-1)%bpe));
  }


  //set x to the greatest common divisor of x and y.
  //x,y are bigInts with the 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;
    }
  }


  //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.
  //the x array must have at least as many elements as y.

  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--); //kx,ky is number of elements in x,y, not including leading zeros
    for (kx=r.length;r[kx-1]==0 && kx>ky;kx--);

    //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);

    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.  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.
  //for greater speed, let 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 zeros

  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 (n is odd) and ** is exponentiation.  0**0=1.

  function powMod(x,y,n) {
    var k1,k2,kn,np;

    //calculate np from n for the Montgomery multiplications
    for (kn=n.length;kn>0 && !n[kn-1];kn--);
    np=radix-inverseModInt(modInt(n,radix),radix);
    if(s7.length!=n.length)
      s7=dup(n);
    copyInt(s7,0);
    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*np) & mask;
      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);
  }


  // define functions as methods of object

  bigintObj.findPrimes = findPrimes;
  bigintObj.millerRabin = millerRabin;
  bigintObj.bitSize = bitSize;
  bigintObj.randTruePrime = randTruePrime;
  bigintObj.randBigInt = randBigInt;
  bigintObj.GCD = GCD;
  bigintObj.inverseMod = inverseMod;
  bigintObj.inverseModInt = inverseModInt;
  bigintObj.eGCD = eGCD;
  bigintObj.negative = negative;
  bigintObj.greaterShift = greaterShift;
  bigintObj.greater = greater;
  bigintObj.divide = divide;
  bigintObj.carry = carry;
  bigintObj.modInt = modInt;
  bigintObj.int2bigInt = int2bigInt;
  bigintObj.str2bigInt = str2bigInt;
  bigintObj.equalsInt = equalsInt;
  bigintObj.equals = equals;
  bigintObj.isZero = isZero;
  bigintObj.bigInt2str = bigInt2str;
  bigintObj.dup = dup;
  bigintObj.copy = copy;
  bigintObj.copyInt = copyInt;
  bigintObj.addInt = addInt;
  bigintObj.rightShift = rightShift;
  bigintObj.halve = halve;
  bigintObj.leftShift = leftShift;
  bigintObj.multInt = multInt;
  bigintObj.divInt = divInt;
  bigintObj.linComb = linComb;
  bigintObj.linCombShift = linCombShift;
  bigintObj.addShift = addShift;
  bigintObj.subShift = subShift;
  bigintObj.sub = sub;
  bigintObj.add = add;
  bigintObj.mult = mult;
  bigintObj.mod = mod;
  bigintObj.multMod = multMod;
  bigintObj.squareMod = squareMod;
  bigintObj.trim = trim;
  bigintObj.powMod = powMod;
  bigintObj.mont = mont;

  return bigintObj;

  // end of outer function
}