1 ////////////////////////////////////////////////////////////////////////////////////////

     2 // Big Integer Library v. 5.1

     3 // Created 2000, last modified 2007

     4 // Leemon Baird

     5 // www.leemon.com

     6 //

     7 // Version history:

     8 //

     9 // v 5.1  8 Oct 2007 

    10 //   - renamed inverseModInt_ to inverseModInt since it doesn't change its parameters

    11 //   - added functions GCD and randBigInt, which call GCD_ and randBigInt_

    12 //   - fixed a bug found by Rob Visser (see comment with his name below)

    13 //   - improved comments

    14 //

    15 // This file is public domain.   You can use it for any purpose without restriction.

    16 // I do not guarantee that it is correct, so use it at your own risk.  If you use 

    17 // it for something interesting, I'd appreciate hearing about it.  If you find 

    18 // any bugs or make any improvements, I'd appreciate hearing about those too.

    19 // It would also be nice if my name and address were left in the comments.

    20 // But none of that is required.

    21 //

    22 // This code defines a bigInt library for arbitrary-precision integers.

    23 // A bigInt is an array of integers storing the value in chunks of bpe bits, 

    24 // little endian (buff[0] is the least significant word).

    25 // Negative bigInts are stored two's complement.

    26 // Some functions assume their parameters have at least one leading zero element.

    27 // Functions with an underscore at the end of the name have unpredictable behavior in case of overflow, 

    28 // so the caller must make sure the arrays must be big enough to hold the answer.

    29 // For each function where a parameter is modified, that same 

    30 // variable must not be used as another argument too.

    31 // So, you cannot square x by doing multMod_(x,x,n).  

    32 // You must use squareMod_(x,n) instead, or do y=dup(x); multMod_(x,y,n).

    33 //

    34 // These functions are designed to avoid frequent dynamic memory allocation in the inner loop.

    35 // For most functions, if it needs a BigInt as a local variable it will actually use

    36 // a global, and will only allocate to it only when it's not the right size.  This ensures

    37 // that when a function is called repeatedly with same-sized parameters, it only allocates

    38 // memory on the first call.

    39 //

    40 // Note that for cryptographic purposes, the calls to Math.random() must 

    41 // be replaced with calls to a better pseudorandom number generator.

    42 //

    43 // In the following, "bigInt" means a bigInt with at least one leading zero element,

    44 // and "integer" means a nonnegative integer less than radix.  In some cases, integer 

    45 // can be negative.  Negative bigInts are 2s complement.

    46 // 

    47 // The following functions do not modify their inputs.

    48 // Those returning a bigInt, string, or Array will dynamically allocate memory for that value.

    49 // Those returning a boolean will return the integer 0 (false) or 1 (true).

    50 // Those returning boolean or int will not allocate memory except possibly on the first time they're called with a given parameter size.

    51 // 

    52 // bigInt  add(x,y)               //return (x+y) for bigInts x and y.  

    53 // bigInt  addInt(x,n)            //return (x+n) where x is a bigInt and n is an integer.

    54 // string  bigInt2str(x,base)     //return a string form of bigInt x in a given base, with 2 <= base <= 95

    55 // int     bitSize(x)             //return how many bits long the bigInt x is, not counting leading zeros

    56 // bigInt  dup(x)                 //return a copy of bigInt x

    57 // boolean equals(x,y)            //is the bigInt x equal to the bigint y?

    58 // boolean equalsInt(x,y)         //is bigint x equal to integer y?

    59 // bigInt  expand(x,n)            //return a copy of x with at least n elements, adding leading zeros if needed

    60 // Array   findPrimes(n)          //return array of all primes less than integer n

    61 // bigInt  GCD(x,y)               //return greatest common divisor of bigInts x and y (each with same number of elements).

    62 // boolean greater(x,y)           //is x>y?  (x and y are nonnegative bigInts)

    63 // boolean greaterShift(x,y,shift)//is (x <<(shift*bpe)) > y?

    64 // bigInt  int2bigInt(t,n,m)      //return a bigInt equal to integer t, with at least n bits and m array elements

    65 // bigInt  inverseMod(x,n)        //return (x**(-1) mod n) for bigInts x and n.  If no inverse exists, it returns null

    66 // int     inverseModInt(x,n)     //return x**(-1) mod n, for integers x and n.  Return 0 if there is no inverse

    67 // boolean isZero(x)              //is the bigInt x equal to zero?

    68 // 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)?

    69 // bigInt  mod(x,n)               //return a new bigInt equal to (x mod n) for bigInts x and n.

    70 // int     modInt(x,n)            //return x mod n for bigInt x and integer n.

    71 // bigInt  mult(x,y)              //return x*y for bigInts x and y. This is faster when y<x.

    72 // bigInt  multMod(x,y,n)         //return (x*y mod n) for bigInts x,y,n.  For greater speed, let y<x.

    73 // boolean negative(x)            //is bigInt x negative?

    74 // 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.

    75 // 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.

    76 // bigInt  randTruePrime(k)       //return a new, random, k-bit, true prime bigInt using Maurer's algorithm.

    77 // 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

    78 // bigInt  sub(x,y)               //return (x-y) for bigInts x and y.  Negative answers will be 2s complement

    79 // bigInt  bigint_trim(x,k)              //return a copy of x with exactly k leading zero elements

    80 //

    81 //

    82 // The following functions each have a non-underscored version, which most users should call instead.

    83 // These functions each write to a single parameter, and the caller is responsible for ensuring the array 

    84 // passed in is large enough to hold the result. 

    85 //

    86 // void    addInt_(x,n)          //do x=x+n where x is a bigInt and n is an integer

    87 // void    add_(x,y)             //do x=x+y for bigInts x and y

    88 // void    copy_(x,y)            //do x=y on bigInts x and y

    89 // void    copyInt_(x,n)         //do x=n on bigInt x and integer n

    90 // void    GCD_(x,y)             //set x to the greatest common divisor of bigInts x and y, (y is destroyed).  (This never overflows its array).

    91 // boolean inverseMod_(x,n)      //do x=x**(-1) mod n, for bigInts x and n. Returns 1 (0) if inverse does (doesn't) exist

    92 // void    mod_(x,n)             //do x=x mod n for bigInts x and n. (This never overflows its array).

    93 // void    mult_(x,y)            //do x=x*y for bigInts x and y.

    94 // void    multMod_(x,y,n)       //do x=x*y  mod n for bigInts x,y,n.

    95 // 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.

    96 // 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.

    97 // void    randTruePrime_(ans,k) //do ans = a random k-bit true random prime (not just probable prime) with 1 in the msb.

    98 // void    sub_(x,y)             //do x=x-y for bigInts x and y. Negative answers will be 2s complement.

    99 //

   100 // The following functions do NOT have a non-underscored version. 

   101 // They each write a bigInt result to one or more parameters.  The caller is responsible for

   102 // ensuring the arrays passed in are large enough to hold the results. 

   103 //

   104 // void addShift_(x,y,ys)       //do x=x+(y<<(ys*bpe))

   105 // void carry_(x)               //do carries and borrows so each element of the bigInt x fits in bpe bits.

   106 // void divide_(x,y,q,r)        //divide x by y giving quotient q and remainder r

   107 // int  divInt_(x,n)            //do x=floor(x/n) for bigInt x and integer n, and return the remainder. (This never overflows its array).

   108 // int  eGCD_(x,y,d,a,b)        //sets a,b,d to positive bigInts such that d = GCD_(x,y) = a*x-b*y

   109 // void halve_(x)               //do x=floor(|x|/2)*sgn(x) for bigInt x in 2's complement.  (This never overflows its array).

   110 // void leftShift_(x,n)         //left shift bigInt x by n bits.  n<bpe.

   111 // void linComb_(x,y,a,b)       //do x=a*x+b*y for bigInts x and y and integers a and b

   112 // void linCombShift_(x,y,b,ys) //do x=x+b*(y<<(ys*bpe)) for bigInts x and y, and integers b and ys

   113 // void mont_(x,y,n,np)         //Montgomery multiplication (see comments where the function is defined)

   114 // void multInt_(x,n)           //do x=x*n where x is a bigInt and n is an integer.

   115 // void rightShift_(x,n)        //right shift bigInt x by n bits.  0 <= n < bpe. (This never overflows its array).

   116 // void squareMod_(x,n)         //do x=x*x  mod n for bigInts x,n

   117 // void subShift_(x,y,ys)       //do x=x-(y<<(ys*bpe)). Negative answers will be 2s complement.

   118 //

   119 // The following functions are based on algorithms from the _Handbook of Applied Cryptography_

   120 //    powMod_()           = algorithm 14.94, Montgomery exponentiation

   121 //    eGCD_,inverseMod_() = algorithm 14.61, Binary extended GCD_

   122 //    GCD_()              = algorothm 14.57, Lehmer's algorithm

   123 //    mont_()             = algorithm 14.36, Montgomery multiplication

   124 //    divide_()           = algorithm 14.20  Multiple-precision division

   125 //    squareMod_()        = algorithm 14.16  Multiple-precision squaring

   126 //    randTruePrime_()    = algorithm  4.62, Maurer's algorithm

   127 //    millerRabin()       = algorithm  4.24, Miller-Rabin algorithm

   128 //

   129 // Profiling shows:

   130 //     randTruePrime_() spends:

   131 //         10% of its time in calls to powMod_()

   132 //         85% of its time in calls to millerRabin()

   133 //     millerRabin() spends:

   134 //         99% of its time in calls to powMod_()   (always with a base of 2)

   135 //     powMod_() spends:

   136 //         94% of its time in calls to mont_()  (almost always with x==y)

   137 //

   138 // This suggests there are several ways to speed up this library slightly:

   139 //     - convert powMod_ to use a Montgomery form of k-ary window (or maybe a Montgomery form of sliding window)

   140 //         -- this should especially focus on being fast when raising 2 to a power mod n

   141 //     - convert randTruePrime_() to use a minimum r of 1/3 instead of 1/2 with the appropriate change to the test

   142 //     - tune the parameters in randTruePrime_(), including c, m, and recLimit

   143 //     - speed up the single loop in mont_() that takes 95% of the runtime, perhaps by reducing checking

   144 //       within the loop when all the parameters are the same length.

   145 //

   146 // There are several ideas that look like they wouldn't help much at all:

   147 //     - replacing trial division in randTruePrime_() with a sieve (that speeds up something taking almost no time anyway)

   148 //     - 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)

   149 //     - speeding up mont_(x,y,n,np) when x==y by doing a non-modular, non-Montgomery square

   150 //       followed by a Montgomery reduction.  The intermediate answer will be twice as long as x, so that

   151 //       method would be slower.  This is unfortunate because the code currently spends almost all of its time

   152 //       doing mont_(x,x,...), both for randTruePrime_() and powMod_().  A faster method for Montgomery squaring

   153 //       would have a large impact on the speed of randTruePrime_() and powMod_().  HAC has a couple of poorly-worded

   154 //       sentences that seem to imply it's faster to do a non-modular square followed by a single

   155 //       Montgomery reduction, but that's obviously wrong.

   156 ////////////////////////////////////////////////////////////////////////////////////////

   157 

   158 //globals

   159 bpe=0;         //bits stored per array element

   160 mask=0;        //AND this with an array element to chop it down to bpe bits

   161 radix=mask+1;  //equals 2^bpe.  A single 1 bit to the left of the last bit of mask.

   162 

   163 //the digits for converting to different bases

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

   165 

   166 //initialize the global variables

   167 for (bpe=0; (1<<(bpe+1)) > (1<<bpe); bpe++);  //bpe=number of bits in the mantissa on this platform

   168 bpe>>=1;                   //bpe=number of bits in one element of the array representing the bigInt

   169 mask=(1<<bpe)-1;           //AND the mask with an integer to get its bpe least significant bits

   170 radix=mask+1;              //2^bpe.  a single 1 bit to the left of the first bit of mask

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

   172 

   173 //the following global variables are scratchpad memory to 

   174 //reduce dynamic memory allocation in the inner loop

   175 t=new Array(0);

   176 ss=t;       //used in mult_()

   177 s0=t;       //used in multMod_(), squareMod_() 

   178 s1=t;       //used in powMod_(), multMod_(), squareMod_() 

   179 s2=t;       //used in powMod_(), multMod_()

   180 s3=t;       //used in powMod_()

   181 s4=t; s5=t; //used in mod_()

   182 s6=t;       //used in bigInt2str()

   183 s7=t;       //used in powMod_()

   184 T=t;        //used in GCD_()

   185 sa=t;       //used in mont_()

   186 mr_x1=t; mr_r=t; mr_a=t;                                      //used in millerRabin()

   187 eg_v=t; eg_u=t; eg_A=t; eg_B=t; eg_C=t; eg_D=t;               //used in eGCD_(), inverseMod_()

   188 md_q1=t; md_q2=t; md_q3=t; md_r=t; md_r1=t; md_r2=t; md_tt=t; //used in mod_()

   189 

   190 primes=t; pows=t; s_i=t; s_i2=t; s_R=t; s_rm=t; s_q=t; s_n1=t; 

   191   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_()

   192 

   193 ////////////////////////////////////////////////////////////////////////////////////////

   194 

   195 //return array of all primes less than integer n

   196 function findPrimes(n) {

   197   var i,s,p,ans;

   198   s=new Array(n);

   199   for (i=0;i<n;i++)

   200     s[i]=0;

   201   s[0]=2;

   202   p=0;    //first p elements of s are primes, the rest are a sieve

   203   for(;s[p]<n;) {                  //s[p] is the pth prime

   204     for(i=s[p]*s[p]; i<n; i+=s[p]) //mark multiples of s[p]

   205       s[i]=1;

   206     p++;

   207     s[p]=s[p-1]+1;

   208     for(; s[p]<n && s[s[p]]; s[p]++); //find next prime (where s[p]==0)

   209   }

   210   ans=new Array(p);

   211   for(i=0;i<p;i++)

   212     ans[i]=s[i];

   213   return ans;

   214 }

   215 

   216 //does a single round of Miller-Rabin base b consider x to be a possible prime?

   217 //x is a bigInt, and b is an integer

   218 function millerRabin(x,b) {

   219   var i,j,k,s;

   220 

   221   if (mr_x1.length!=x.length) {

   222     mr_x1=dup(x);

   223     mr_r=dup(x);

   224     mr_a=dup(x);

   225   }

   226 

   227   copyInt_(mr_a,b);

   228   copy_(mr_r,x);

   229   copy_(mr_x1,x);

   230 

   231   addInt_(mr_r,-1);

   232   addInt_(mr_x1,-1);

   233 

   234   //s=the highest power of two that divides mr_r

   235   k=0;

   236   for (i=0;i<mr_r.length;i++)

   237     for (j=1;j<mask;j<<=1)

   238       if (x[i] & j) {

   239         s=(k<mr_r.length+bpe ? k : 0); 

   240          i=mr_r.length;

   241          j=mask;

   242       } else

   243         k++;

   244 

   245   if (s)                

   246     rightShift_(mr_r,s);

   247 

   248   powMod_(mr_a,mr_r,x);

   249 

   250   if (!equalsInt(mr_a,1) && !equals(mr_a,mr_x1)) {

   251     j=1;

   252     while (j<=s-1 && !equals(mr_a,mr_x1)) {

   253       squareMod_(mr_a,x);

   254       if (equalsInt(mr_a,1)) {

   255         return 0;

   256       }

   257       j++;

   258     }

   259     if (!equals(mr_a,mr_x1)) {

   260       return 0;

   261     }

   262   }

   263   return 1;  

   264 }

   265 

   266 //returns how many bits long the bigInt is, not counting leading zeros.

   267 function bitSize(x) {

   268   var j,z,w;

   269   for (j=x.length-1; (x[j]==0) && (j>0); j--);

   270   for (z=0,w=x[j]; w; (w>>=1),z++);

   271   z+=bpe*j;

   272   return z;

   273 }

   274 

   275 //return a copy of x with at least n elements, adding leading zeros if needed

   276 function expand(x,n) {

   277   var ans=int2bigInt(0,(x.length>n ? x.length : n)*bpe,0);

   278   copy_(ans,x);

   279   return ans;

   280 }

   281 

   282 //return a k-bit true random prime using Maurer's algorithm.

   283 function randTruePrime(k) {

   284   var ans=int2bigInt(0,k,0);

   285   randTruePrime_(ans,k);

   286   return bigint_trim(ans,1);

   287 }

   288 

   289 //return a new bigInt equal to (x mod n) for bigInts x and n.

   290 function mod(x,n) {

   291   var ans=dup(x);

   292   mod_(ans,n);

   293   return bigint_trim(ans,1);

   294 }

   295 

   296 //return (x+n) where x is a bigInt and n is an integer.

   297 function addInt(x,n) {

   298   var ans=expand(x,x.length+1);

   299   addInt_(ans,n);

   300   return bigint_trim(ans,1);

   301 }

   302 

   303 //return x*y for bigInts x and y. This is faster when y<x.

   304 function mult(x,y) {

   305   var ans=expand(x,x.length+y.length);

   306   mult_(ans,y);

   307   return bigint_trim(ans,1);

   308 }

   309 

   310 //return (x**y mod n) where x,y,n are bigInts and ** is exponentiation.  0**0=1. Faster for odd n.

   311 function powMod(x,y,n) {

   312   var ans=expand(x,n.length);  

   313   powMod_(ans,bigint_trim(y,2),bigint_trim(n,2),0);  //this should work without the trim, but doesn't

   314   return bigint_trim(ans,1);

   315 }

   316 

   317 //return (x-y) for bigInts x and y.  Negative answers will be 2s complement

   318 function sub(x,y) {

   319   var ans=expand(x,(x.length>y.length ? x.length+1 : y.length+1)); 

   320   sub_(ans,y);

   321   return bigint_trim(ans,1);

   322 }

   323 

   324 //return (x+y) for bigInts x and y.  

   325 function add(x,y) {

   326   var ans=expand(x,(x.length>y.length ? x.length+1 : y.length+1)); 

   327   add_(ans,y);

   328   return bigint_trim(ans,1);

   329 }

   330 

   331 //return (x**(-1) mod n) for bigInts x and n.  If no inverse exists, it returns null

   332 function inverseMod(x,n) {

   333   var ans=expand(x,n.length); 

   334   var s;

   335   s=inverseMod_(ans,n);

   336   return s ? bigint_trim(ans,1) : null;

   337 }

   338 

   339 //return (x*y mod n) for bigInts x,y,n.  For greater speed, let y<x.

   340 function multMod(x,y,n) {

   341   var ans=expand(x,n.length);

   342   multMod_(ans,y,n);

   343   return bigint_trim(ans,1);

   344 }

   345 

   346 //generate a k-bit true random prime using Maurer's algorithm,

   347 //and put it into ans.  The bigInt ans must be large enough to hold it.

   348 function randTruePrime_(ans,k) {

   349   var c,m,pm,dd,j,r,B,divisible,z,zz,recSize;

   350 

   351   if (primes.length==0)

   352     primes=findPrimes(30000);  //check for divisibility by primes <=30000

   353 

   354   if (pows.length==0) {

   355     pows=new Array(512);

   356     for (j=0;j<512;j++) {

   357       pows[j]=Math.pow(2,j/511.-1.);

   358     }

   359   }

   360 

   361   //c and m should be tuned for a particular machine and value of k, to maximize speed

   362   c=0.1;  //c=0.1 in HAC

   363   m=20;   //generate this k-bit number by first recursively generating a number that has between k/2 and k-m bits

   364   recLimit=20; //stop recursion when k <=recLimit.  Must have recLimit >= 2

   365 

   366   if (s_i2.length!=ans.length) {

   367     s_i2=dup(ans);

   368     s_R =dup(ans);

   369     s_n1=dup(ans);

   370     s_r2=dup(ans);

   371     s_d =dup(ans);

   372     s_x1=dup(ans);

   373     s_x2=dup(ans);

   374     s_b =dup(ans);

   375     s_n =dup(ans);

   376     s_i =dup(ans);

   377     s_rm=dup(ans);

   378     s_q =dup(ans);

   379     s_a =dup(ans);

   380     s_aa=dup(ans);

   381   }

   382 

   383   if (k <= recLimit) {  //generate small random primes by trial division up to its square root

   384     pm=(1<<((k+2)>>1))-1; //pm is binary number with all ones, just over sqrt(2^k)

   385     copyInt_(ans,0);

   386     for (dd=1;dd;) {

   387       dd=0;

   388       ans[0]= 1 | (1<<(k-1)) | Math.floor(Math.random()*(1<<k));  //random, k-bit, odd integer, with msb 1

   389       for (j=1;(j<primes.length) && ((primes[j]&pm)==primes[j]);j++) { //trial division by all primes 3...sqrt(2^k)

   390         if (0==(ans[0]%primes[j])) {

   391           dd=1;

   392           break;

   393         }

   394       }

   395     }

   396     carry_(ans);

   397     return;

   398   }

   399 

   400   B=c*k*k;    //try small primes up to B (or all the primes[] array if the largest is less than B).

   401   if (k>2*m)  //generate this k-bit number by first recursively generating a number that has between k/2 and k-m bits

   402     for (r=1; k-k*r<=m; )

   403       r=pows[Math.floor(Math.random()*512)];   //r=Math.pow(2,Math.random()-1);

   404   else

   405     r=.5;

   406 

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

   408 

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

   410 

   411   randTruePrime_(s_q,recSize);

   412   copyInt_(s_i2,0);

   413   s_i2[Math.floor((k-2)/bpe)] |= (1<<((k-2)%bpe));   //s_i2=2^(k-2)

   414   divide_(s_i2,s_q,s_i,s_rm);                        //s_i=floor((2^(k-1))/(2q))

   415 

   416   z=bitSize(s_i);

   417 

   418   for (;;) {

   419     for (;;) {  //generate z-bit numbers until one falls in the range [0,s_i-1]

   420       randBigInt_(s_R,z,0);

   421       if (greater(s_i,s_R))

   422         break;

   423     }                //now s_R is in the range [0,s_i-1]

   424     addInt_(s_R,1);  //now s_R is in the range [1,s_i]

   425     add_(s_R,s_i);   //now s_R is in the range [s_i+1,2*s_i]

   426 

   427     copy_(s_n,s_q);

   428     mult_(s_n,s_R); 

   429     multInt_(s_n,2);

   430     addInt_(s_n,1);    //s_n=2*s_R*s_q+1

   431     

   432     copy_(s_r2,s_R);

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

   434 

   435     //check s_n for divisibility by small primes up to B

   436     for (divisible=0,j=0; (j<primes.length) && (primes[j]<B); j++)

   437       if (modInt(s_n,primes[j])==0) {

   438         divisible=1;

   439         break;

   440       }      

   441 

   442     if (!divisible)    //if it passes small primes check, then try a single Miller-Rabin base 2

   443       if (!millerRabin(s_n,2)) //this line represents 75% of the total runtime for randTruePrime_ 

   444         divisible=1;

   445 

   446     if (!divisible) {  //if it passes that test, continue checking s_n

   447       addInt_(s_n,-3);

   448       for (j=s_n.length-1;(s_n[j]==0) && (j>0); j--);  //strip leading zeros

   449       for (zz=0,w=s_n[j]; w; (w>>=1),zz++);

   450       zz+=bpe*j;                             //zz=number of bits in s_n, ignoring leading zeros

   451       for (;;) {  //generate z-bit numbers until one falls in the range [0,s_n-1]

   452         randBigInt_(s_a,zz,0);

   453         if (greater(s_n,s_a))

   454           break;

   455       }                //now s_a is in the range [0,s_n-1]

   456       addInt_(s_n,3);  //now s_a is in the range [0,s_n-4]

   457       addInt_(s_a,2);  //now s_a is in the range [2,s_n-2]

   458       copy_(s_b,s_a);

   459       copy_(s_n1,s_n);

   460       addInt_(s_n1,-1);

   461       powMod_(s_b,s_n1,s_n);   //s_b=s_a^(s_n-1) modulo s_n

   462       addInt_(s_b,-1);

   463       if (isZero(s_b)) {

   464         copy_(s_b,s_a);

   465         powMod_(s_b,s_r2,s_n);

   466         addInt_(s_b,-1);

   467         copy_(s_aa,s_n);

   468         copy_(s_d,s_b);

   469         GCD_(s_d,s_n);  //if s_b and s_n are relatively prime, then s_n is a prime

   470         if (equalsInt(s_d,1)) {

   471           copy_(ans,s_aa);

   472           return;     //if we've made it this far, then s_n is absolutely guaranteed to be prime

   473         }

   474       }

   475     }

   476   }

   477 }

   478 

   479 //Return an n-bit random BigInt (n>=1).  If s=1, then the most significant of those n bits is set to 1.

   480 function randBigInt(n,s) {

   481   var a,b;

   482   a=Math.floor((n-1)/bpe)+2; //# array elements to hold the BigInt with a leading 0 element

   483   b=int2bigInt(0,0,a);

   484   randBigInt_(b,n,s);

   485   return b;

   486 }

   487 

   488 //Set b to an n-bit random BigInt.  If s=1, then the most significant of those n bits is set to 1.

   489 //Array b must be big enough to hold the result. Must have n>=1

   490 function randBigInt_(b,n,s) {

   491   var i,a;

   492   for (i=0;i<b.length;i++)

   493     b[i]=0;

   494   a=Math.floor((n-1)/bpe)+1; //# array elements to hold the BigInt

   495   for (i=0;i<a;i++) {

   496     b[i]=Math.floor(Math.random()*(1<<(bpe-1)));

   497   }

   498   b[a-1] &= (2<<((n-1)%bpe))-1;

   499   if (s==1)

   500     b[a-1] |= (1<<((n-1)%bpe));

   501 }

   502 

   503 //Return the greatest common divisor of bigInts x and y (each with same number of elements).

   504 function GCD(x,y) {

   505   var xc,yc;

   506   xc=dup(x);

   507   yc=dup(y);

   508   GCD_(xc,yc);

   509   return xc;

   510 }

   511 

   512 //set x to the greatest common divisor of bigInts x and y (each with same number of elements).

   513 //y is destroyed.

   514 function GCD_(x,y) {

   515   var i,xp,yp,A,B,C,D,q,sing;

   516   if (T.length!=x.length)

   517     T=dup(x);

   518 

   519   sing=1;

   520   while (sing) { //while y has nonzero elements other than y[0]

   521     sing=0;

   522     for (i=1;i<y.length;i++) //check if y has nonzero elements other than 0

   523       if (y[i]) {

   524         sing=1;

   525         break;

   526       }

   527     if (!sing) break; //quit when y all zero elements except possibly y[0]

   528 

   529     for (i=x.length;!x[i] && i>=0;i--);  //find most significant element of x

   530     xp=x[i];

   531     yp=y[i];

   532     A=1; B=0; C=0; D=1;

   533     while ((yp+C) && (yp+D)) {

   534       q =Math.floor((xp+A)/(yp+C));

   535       qp=Math.floor((xp+B)/(yp+D));

   536       if (q!=qp)

   537         break;

   538       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)      

   539       t= B-q*D;   B=D;   D=t;

   540       t=xp-q*yp; xp=yp; yp=t;

   541     }

   542     if (B) {

   543       copy_(T,x);

   544       linComb_(x,y,A,B); //x=A*x+B*y

   545       linComb_(y,T,D,C); //y=D*y+C*T

   546     } else {

   547       mod_(x,y);

   548       copy_(T,x);

   549       copy_(x,y);

   550       copy_(y,T);

   551     } 

   552   }

   553   if (y[0]==0)

   554     return;

   555   t=modInt(x,y[0]);

   556   copyInt_(x,y[0]);

   557   y[0]=t;

   558   while (y[0]) {

   559     x[0]%=y[0];

   560     t=x[0]; x[0]=y[0]; y[0]=t;

   561   }

   562 }

   563 

   564 //do x=x**(-1) mod n, for bigInts x and n.

   565 //If no inverse exists, it sets x to zero and returns 0, else it returns 1.

   566 //The x array must be at least as large as the n array.

   567 function inverseMod_(x,n) {

   568   var k=1+2*Math.max(x.length,n.length);

   569 

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

   571     copyInt_(x,0);

   572     return 0;

   573   }

   574 

   575   if (eg_u.length!=k) {

   576     eg_u=new Array(k);

   577     eg_v=new Array(k);

   578     eg_A=new Array(k);

   579     eg_B=new Array(k);

   580     eg_C=new Array(k);

   581     eg_D=new Array(k);

   582   }

   583 

   584   copy_(eg_u,x);

   585   copy_(eg_v,n);

   586   copyInt_(eg_A,1);

   587   copyInt_(eg_B,0);

   588   copyInt_(eg_C,0);

   589   copyInt_(eg_D,1);

   590   for (;;) {

   591     while(!(eg_u[0]&1)) {  //while eg_u is even

   592       halve_(eg_u);

   593       if (!(eg_A[0]&1) && !(eg_B[0]&1)) { //if eg_A==eg_B==0 mod 2

   594         halve_(eg_A);

   595         halve_(eg_B);      

   596       } else {

   597         add_(eg_A,n);  halve_(eg_A);

   598         sub_(eg_B,x);  halve_(eg_B);

   599       }

   600     }

   601 

   602     while (!(eg_v[0]&1)) {  //while eg_v is even

   603       halve_(eg_v);

   604       if (!(eg_C[0]&1) && !(eg_D[0]&1)) { //if eg_C==eg_D==0 mod 2

   605         halve_(eg_C);

   606         halve_(eg_D);      

   607       } else {

   608         add_(eg_C,n);  halve_(eg_C);

   609         sub_(eg_D,x);  halve_(eg_D);

   610       }

   611     }

   612 

   613     if (!greater(eg_v,eg_u)) { //eg_v <= eg_u

   614       sub_(eg_u,eg_v);

   615       sub_(eg_A,eg_C);

   616       sub_(eg_B,eg_D);

   617     } else {                   //eg_v > eg_u

   618       sub_(eg_v,eg_u);

   619       sub_(eg_C,eg_A);

   620       sub_(eg_D,eg_B);

   621     }

   622   

   623     if (equalsInt(eg_u,0)) {

   624       if (negative(eg_C)) //make sure answer is nonnegative

   625         add_(eg_C,n);

   626       copy_(x,eg_C);

   627 

   628       if (!equalsInt(eg_v,1)) { //if GCD_(x,n)!=1, then there is no inverse

   629         copyInt_(x,0);

   630         return 0;

   631       }

   632       return 1;

   633     }

   634   }

   635 }

   636 

   637 //return x**(-1) mod n, for integers x and n.  Return 0 if there is no inverse

   638 function inverseModInt(x,n) {

   639   var a=1,b=0,t;

   640   for (;;) {

   641     if (x==1) return a;

   642     if (x==0) return 0;

   643     b-=a*Math.floor(n/x);

   644     n%=x;

   645 

   646     if (n==1) return b; //to avoid negatives, change this b to n-b, and each -= to +=

   647     if (n==0) return 0;

   648     a-=b*Math.floor(x/n);

   649     x%=n;

   650   }

   651 }

   652 

   653 //this deprecated function is for backward compatibility only. 

   654 function inverseModInt_(x,n) {

   655    return inverseModInt(x,n);

   656 }

   657 

   658 

   659 //Given positive bigInts x and y, change the bigints v, a, and b to positive bigInts such that:

   660 //     v = GCD_(x,y) = a*x-b*y

   661 //The bigInts v, a, b, must have exactly as many elements as the larger of x and y.

   662 function eGCD_(x,y,v,a,b) {

   663   var g=0;

   664   var k=Math.max(x.length,y.length);

   665   if (eg_u.length!=k) {

   666     eg_u=new Array(k);

   667     eg_A=new Array(k);

   668     eg_B=new Array(k);

   669     eg_C=new Array(k);

   670     eg_D=new Array(k);

   671   }

   672   while(!(x[0]&1)  && !(y[0]&1)) {  //while x and y both even

   673     halve_(x);

   674     halve_(y);

   675     g++;

   676   }

   677   copy_(eg_u,x);

   678   copy_(v,y);

   679   copyInt_(eg_A,1);

   680   copyInt_(eg_B,0);

   681   copyInt_(eg_C,0);

   682   copyInt_(eg_D,1);

   683   for (;;) {

   684     while(!(eg_u[0]&1)) {  //while u is even

   685       halve_(eg_u);

   686       if (!(eg_A[0]&1) && !(eg_B[0]&1)) { //if A==B==0 mod 2

   687         halve_(eg_A);

   688         halve_(eg_B);      

   689       } else {

   690         add_(eg_A,y);  halve_(eg_A);

   691         sub_(eg_B,x);  halve_(eg_B);

   692       }

   693     }

   694 

   695     while (!(v[0]&1)) {  //while v is even

   696       halve_(v);

   697       if (!(eg_C[0]&1) && !(eg_D[0]&1)) { //if C==D==0 mod 2

   698         halve_(eg_C);

   699         halve_(eg_D);      

   700       } else {

   701         add_(eg_C,y);  halve_(eg_C);

   702         sub_(eg_D,x);  halve_(eg_D);

   703       }

   704     }

   705 

   706     if (!greater(v,eg_u)) { //v<=u

   707       sub_(eg_u,v);

   708       sub_(eg_A,eg_C);

   709       sub_(eg_B,eg_D);

   710     } else {                //v>u

   711       sub_(v,eg_u);

   712       sub_(eg_C,eg_A);

   713       sub_(eg_D,eg_B);

   714     }

   715     if (equalsInt(eg_u,0)) {

   716       if (negative(eg_C)) {   //make sure a (C)is nonnegative

   717         add_(eg_C,y);

   718         sub_(eg_D,x);

   719       }

   720       multInt_(eg_D,-1);  ///make sure b (D) is nonnegative

   721       copy_(a,eg_C);

   722       copy_(b,eg_D);

   723       leftShift_(v,g);

   724       return;

   725     }

   726   }

   727 }

   728 

   729 

   730 //is bigInt x negative?

   731 function negative(x) {

   732   return ((x[x.length-1]>>(bpe-1))&1);

   733 }

   734 

   735 

   736 //is (x << (shift*bpe)) > y?

   737 //x and y are nonnegative bigInts

   738 //shift is a nonnegative integer

   739 function greaterShift(x,y,shift) {

   740   var kx=x.length, ky=y.length;

   741   k=((kx+shift)<ky) ? (kx+shift) : ky;

   742   for (i=ky-1-shift; i<kx && i>=0; i++) 

   743     if (x[i]>0)

   744       return 1; //if there are nonzeros in x to the left of the first column of y, then x is bigger

   745   for (i=kx-1+shift; i<ky; i++)

   746     if (y[i]>0)

   747       return 0; //if there are nonzeros in y to the left of the first column of x, then x is not bigger

   748   for (i=k-1; i>=shift; i--)

   749     if      (x[i-shift]>y[i]) return 1;

   750     else if (x[i-shift]<y[i]) return 0;

   751   return 0;

   752 }

   753 

   754 //is x > y? (x and y both nonnegative)

   755 function greater(x,y) {

   756   var i;

   757   var k=(x.length<y.length) ? x.length : y.length;

   758 

   759   for (i=x.length;i<y.length;i++)

   760     if (y[i])

   761       return 0;  //y has more digits

   762 

   763   for (i=y.length;i<x.length;i++)

   764     if (x[i])

   765       return 1;  //x has more digits

   766 

   767   for (i=k-1;i>=0;i--)

   768     if (x[i]>y[i])

   769       return 1;

   770     else if (x[i]<y[i])

   771       return 0;

   772   return 0;

   773 }

   774 

   775 //divide x by y giving quotient q and remainder r.  (q=floor(x/y),  r=x mod y).  All 4 are bigints.

   776 //x must have at least one leading zero element.

   777 //y must be nonzero.

   778 //q and r must be arrays that are exactly the same length as x. (Or q can have more).

   779 //Must have x.length >= y.length >= 2.

   780 function divide_(x,y,q,r) {

   781   var kx, ky;

   782   var i,j,y1,y2,c,a,b;

   783   copy_(r,x);

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

   785 

   786   //normalize: ensure the most significant element of y has its highest bit set  

   787   b=y[ky-1];

   788   for (a=0; b; a++)

   789     b>>=1;  

   790   a=bpe-a;  //a is how many bits to shift so that the high order bit of y is leftmost in its array element

   791   leftShift_(y,a);  //multiply both by 1<<a now, then divide both by that at the end

   792   leftShift_(r,a);

   793 

   794   //Rob Visser discovered a bug: the following line was originally just before the normalization.

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

   796 

   797   copyInt_(q,0);                      // q=0

   798   while (!greaterShift(y,r,kx-ky)) {  // while (leftShift_(y,kx-ky) <= r) {

   799     subShift_(r,y,kx-ky);             //   r=r-leftShift_(y,kx-ky)

   800     q[kx-ky]++;                       //   q[kx-ky]++;

   801   }                                   // }

   802 

   803   for (i=kx-1; i>=ky; i--) {

   804     if (r[i]==y[ky-1])

   805       q[i-ky]=mask;

   806     else

   807       q[i-ky]=Math.floor((r[i]*radix+r[i-1])/y[ky-1]);	

   808 

   809     //The following for(;;) loop is equivalent to the commented while loop, 

   810     //except that the uncommented version avoids overflow.

   811     //The commented loop comes from HAC, which assumes r[-1]==y[-1]==0

   812     //  while (q[i-ky]*(y[ky-1]*radix+y[ky-2]) > r[i]*radix*radix+r[i-1]*radix+r[i-2])

   813     //    q[i-ky]--;    

   814     for (;;) {

   815       y2=(ky>1 ? y[ky-2] : 0)*q[i-ky];

   816       c=y2>>bpe;

   817       y2=y2 & mask;

   818       y1=c+q[i-ky]*y[ky-1];

   819       c=y1>>bpe;

   820       y1=y1 & mask;

   821 

   822       if (c==r[i] ? y1==r[i-1] ? y2>(i>1 ? r[i-2] : 0) : y1>r[i-1] : c>r[i]) 

   823         q[i-ky]--;

   824       else

   825         break;

   826     }

   827 

   828     linCombShift_(r,y,-q[i-ky],i-ky);    //r=r-q[i-ky]*leftShift_(y,i-ky)

   829     if (negative(r)) {

   830       addShift_(r,y,i-ky);         //r=r+leftShift_(y,i-ky)

   831       q[i-ky]--;

   832     }

   833   }

   834 

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

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

   837 }

   838 

   839 //do carries and borrows so each element of the bigInt x fits in bpe bits.

   840 function carry_(x) {

   841   var i,k,c,b;

   842   k=x.length;

   843   c=0;

   844   for (i=0;i<k;i++) {

   845     c+=x[i];

   846     b=0;

   847     if (c<0) {

   848       b=-(c>>bpe);

   849       c+=b*radix;

   850     }

   851     x[i]=c & mask;

   852     c=(c>>bpe)-b;

   853   }

   854 }

   855 

   856 //return x mod n for bigInt x and integer n.

   857 function modInt(x,n) {

   858   var i,c=0;

   859   for (i=x.length-1; i>=0; i--)

   860     c=(c*radix+x[i])%n;

   861   return c;

   862 }

   863 

   864 //convert the integer t into a bigInt with at least the given number of bits.

   865 //the returned array stores the bigInt in bpe-bit chunks, little endian (buff[0] is least significant word)

   866 //Pad the array with leading zeros so that it has at least minSize elements.

   867 //There will always be at least one leading 0 element.

   868 function int2bigInt(t,bits,minSize) {   

   869   var i,k;

   870   k=Math.ceil(bits/bpe)+1;

   871   k=minSize>k ? minSize : k;

   872   buff=new Array(k);

   873   copyInt_(buff,t);

   874   return buff;

   875 }

   876 

   877 //return the bigInt given a string representation in a given base.  

   878 //Pad the array with leading zeros so that it has at least minSize elements.

   879 //If base=-1, then it reads in a space-separated list of array elements in decimal.

   880 //The array will always have at least one leading zero, unless base=-1.

   881 function str2bigInt(s,base,minSize) {

   882   var d, i, j, x, y, kk;

   883   var k=s.length;

   884   if (base==-1) { //comma-separated list of array elements in decimal

   885     x=new Array(0);

   886     for (;;) {

   887       y=new Array(x.length+1);

   888       for (i=0;i<x.length;i++)

   889         y[i+1]=x[i];

   890       y[0]=parseInt(s,10);

   891       x=y;

   892       d=s.indexOf(',',0);

   893       if (d<1) 

   894         break;

   895       s=s.substring(d+1);

   896       if (s.length==0)

   897         break;

   898     }

   899     if (x.length<minSize) {

   900       y=new Array(minSize);

   901       copy_(y,x);

   902       return y;

   903     }

   904     return x;

   905   }

   906 

   907   x=int2bigInt(0,base*k,0);

   908   for (i=0;i<k;i++) {

   909     d=digitsStr.indexOf(s.substring(i,i+1),0);

   910     if (base<=36 && d>=36)  //convert lowercase to uppercase if base<=36

   911       d-=26;

   912     if (d<base && d>=0) {   //ignore illegal characters

   913       multInt_(x,base);

   914       addInt_(x,d);

   915     }

   916   }

   917 

   918   for (k=x.length;k>0 && !x[k-1];k--); //strip off leading zeros

   919   k=minSize>k+1 ? minSize : k+1;

   920   y=new Array(k);

   921   kk=k<x.length ? k : x.length;

   922   for (i=0;i<kk;i++)

   923     y[i]=x[i];

   924   for (;i<k;i++)

   925     y[i]=0;

   926   return y;

   927 }

   928 

   929 //is bigint x equal to integer y?

   930 //y must have less than bpe bits

   931 function equalsInt(x,y) {

   932   var i;

   933   if (x[0]!=y)

   934     return 0;

   935   for (i=1;i<x.length;i++)

   936     if (x[i])

   937       return 0;

   938   return 1;

   939 }

   940 

   941 //are bigints x and y equal?

   942 //this works even if x and y are different lengths and have arbitrarily many leading zeros

   943 function equals(x,y) {

   944   var i;

   945   var k=x.length<y.length ? x.length : y.length;

   946   for (i=0;i<k;i++)

   947     if (x[i]!=y[i])

   948       return 0;

   949   if (x.length>y.length) {

   950     for (;i<x.length;i++)

   951       if (x[i])

   952         return 0;

   953   } else {

   954     for (;i<y.length;i++)

   955       if (y[i])

   956         return 0;

   957   }

   958   return 1;

   959 }

   960 

   961 //is the bigInt x equal to zero?

   962 function isZero(x) {

   963   var i;

   964   for (i=0;i<x.length;i++)

   965     if (x[i])

   966       return 0;

   967   return 1;

   968 }

   969 

   970 //convert a bigInt into a string in a given base, from base 2 up to base 95.

   971 //Base -1 prints the contents of the array representing the number.

   972 function bigInt2str(x,base) {

   973   var i,t,s="";

   974 

   975   if (s6.length!=x.length) 

   976     s6=dup(x);

   977   else

   978     copy_(s6,x);

   979 

   980   if (base==-1) { //return the list of array contents

   981     for (i=x.length-1;i>0;i--)

   982       s+=x[i]+',';

   983     s+=x[0];

   984   }

   985   else { //return it in the given base

   986     while (!isZero(s6)) {

   987       t=divInt_(s6,base);  //t=s6 % base; s6=floor(s6/base);

   988       s=digitsStr.substring(t,t+1)+s;

   989     }

   990   }

   991   if (s.length==0)

   992     s="0";

   993   return s;

   994 }

   995 

   996 //returns a duplicate of bigInt x

   997 function dup(x) {

   998   var i;

   999   buff=new Array(x.length);

  1000   copy_(buff,x);

  1001   return buff;

  1002 }

  1003 

  1004 //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).

  1005 function copy_(x,y) {

  1006   var i;

  1007   var k=x.length<y.length ? x.length : y.length;

  1008   for (i=0;i<k;i++)

  1009     x[i]=y[i];

  1010   for (i=k;i<x.length;i++)

  1011     x[i]=0;

  1012 }

  1013 

  1014 //do x=y on bigInt x and integer y.  

  1015 function copyInt_(x,n) {

  1016   var i,c;

  1017   for (c=n,i=0;i<x.length;i++) {

  1018     x[i]=c & mask;

  1019     c>>=bpe;

  1020   }

  1021 }

  1022 

  1023 //do x=x+n where x is a bigInt and n is an integer.

  1024 //x must be large enough to hold the result.

  1025 function addInt_(x,n) {

  1026   var i,k,c,b;

  1027   x[0]+=n;

  1028   k=x.length;

  1029   c=0;

  1030   for (i=0;i<k;i++) {

  1031     c+=x[i];

  1032     b=0;

  1033     if (c<0) {

  1034       b=-(c>>bpe);

  1035       c+=b*radix;

  1036     }

  1037     x[i]=c & mask;

  1038     c=(c>>bpe)-b;

  1039     if (!c) return; //stop carrying as soon as the carry_ is zero

  1040   }

  1041 }

  1042 

  1043 //right shift bigInt x by n bits.  0 <= n < bpe.

  1044 function rightShift_(x,n) {

  1045   var i;

  1046   var k=Math.floor(n/bpe);

  1047   if (k) {

  1048     for (i=0;i<x.length-k;i++) //right shift x by k elements

  1049       x[i]=x[i+k];

  1050     for (;i<x.length;i++)

  1051       x[i]=0;

  1052     n%=bpe;

  1053   }

  1054   for (i=0;i<x.length-1;i++) {

  1055     x[i]=mask & ((x[i+1]<<(bpe-n)) | (x[i]>>n));

  1056   }

  1057   x[i]>>=n;

  1058 }

  1059 

  1060 //do x=floor(|x|/2)*sgn(x) for bigInt x in 2's complement

  1061 function halve_(x) {

  1062   var i;

  1063   for (i=0;i<x.length-1;i++) {

  1064     x[i]=mask & ((x[i+1]<<(bpe-1)) | (x[i]>>1));

  1065   }

  1066   x[i]=(x[i]>>1) | (x[i] & (radix>>1));  //most significant bit stays the same

  1067 }

  1068 

  1069 //left shift bigInt x by n bits.

  1070 function leftShift_(x,n) {

  1071   var i;

  1072   var k=Math.floor(n/bpe);

  1073   if (k) {

  1074     for (i=x.length; i>=k; i--) //left shift x by k elements

  1075       x[i]=x[i-k];

  1076     for (;i>=0;i--)

  1077       x[i]=0;  

  1078     n%=bpe;

  1079   }

  1080   if (!n)

  1081     return;

  1082   for (i=x.length-1;i>0;i--) {

  1083     x[i]=mask & ((x[i]<<n) | (x[i-1]>>(bpe-n)));

  1084   }

  1085   x[i]=mask & (x[i]<<n);

  1086 }

  1087 

  1088 //do x=x*n where x is a bigInt and n is an integer.

  1089 //x must be large enough to hold the result.

  1090 function multInt_(x,n) {

  1091   var i,k,c,b;

  1092   if (!n)

  1093     return;

  1094   k=x.length;

  1095   c=0;

  1096   for (i=0;i<k;i++) {

  1097     c+=x[i]*n;

  1098     b=0;

  1099     if (c<0) {

  1100       b=-(c>>bpe);

  1101       c+=b*radix;

  1102     }

  1103     x[i]=c & mask;

  1104     c=(c>>bpe)-b;

  1105   }

  1106 }

  1107 

  1108 //do x=floor(x/n) for bigInt x and integer n, and return the remainder

  1109 function divInt_(x,n) {

  1110   var i,r=0,s;

  1111   for (i=x.length-1;i>=0;i--) {

  1112     s=r*radix+x[i];

  1113     x[i]=Math.floor(s/n);

  1114     r=s%n;

  1115   }

  1116   return r;

  1117 }

  1118 

  1119 //do the linear combination x=a*x+b*y for bigInts x and y, and integers a and b.

  1120 //x must be large enough to hold the answer.

  1121 function linComb_(x,y,a,b) {

  1122   var i,c,k,kk;

  1123   k=x.length<y.length ? x.length : y.length;

  1124   kk=x.length;

  1125   for (c=0,i=0;i<k;i++) {

  1126     c+=a*x[i]+b*y[i];

  1127     x[i]=c & mask;

  1128     c>>=bpe;

  1129   }

  1130   for (i=k;i<kk;i++) {

  1131     c+=a*x[i];

  1132     x[i]=c & mask;

  1133     c>>=bpe;

  1134   }

  1135 }