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Big Integer Library v. 5.5\r\n// Created 2000, last modified 2013\r\n// Leemon Baird\r\n// www.leemon.com\r\n//\r\n// Version history:\r\n// v 5.5 17 Mar 2013\r\n// - two lines of a form like \"if (x<0) x+=n\" had the \"if\" changed to \"while\" to\r\n// handle the case when x<-n. (Thanks to James Ansell for finding that bug)\r\n// v 5.4 3 Oct 2009\r\n// - added \"var i\" to greaterShift() so i is not global. (Thanks to PŽter Szab— for finding that bug)\r\n//\r\n// v 5.3 21 Sep 2009\r\n// - added randProbPrime(k) for probable primes\r\n// - unrolled loop in mont_ (slightly faster)\r\n// - millerRabin now takes a bigInt parameter rather than an int\r\n//\r\n// v 5.2 15 Sep 2009\r\n// - fixed capitalization in call to int2bigInt in randBigInt\r\n// (thanks to Emili Evripidou, Reinhold Behringer, and Samuel Macaleese for finding that bug)\r\n//\r\n// v 5.1 8 Oct 2007\r\n// - renamed inverseModInt_ to inverseModInt since it doesn't change its parameters\r\n// - added functions GCD and randBigInt, which call GCD_ and randBigInt_\r\n// - fixed a bug found by Rob Visser (see comment with his name below)\r\n// - improved comments\r\n//\r\n// This file is public domain. You can use it for any purpose without restriction.\r\n// I do not guarantee that it is correct, so use it at your own risk. If you use\r\n// it for something interesting, I'd appreciate hearing about it. If you find\r\n// any bugs or make any improvements, I'd appreciate hearing about those too.\r\n// It would also be nice if my name and URL were left in the comments. But none\r\n// of that is required.\r\n//\r\n// This code defines a bigInt library for arbitrary-precision integers.\r\n// A bigInt is an array of integers storing the value in chunks of bpe bits,\r\n// little endian (buff[0] is the least significant word).\r\n// Negative bigInts are stored two's complement. Almost all the functions treat\r\n// bigInts as nonnegative. The few that view them as two's complement say so\r\n// in their comments. Some functions assume their parameters have at least one\r\n// leading zero element. Functions with an underscore at the end of the name put\r\n// their answer into one of the arrays passed in, and have unpredictable behavior\r\n// in case of overflow, so the caller must make sure the arrays are big enough to\r\n// hold the answer. But the average user should never have to call any of the\r\n// underscored functions. Each important underscored function has a wrapper function\r\n// of the same name without the underscore that takes care of the details for you.\r\n// For each underscored function where a parameter is modified, that same variable\r\n// must not be used as another argument too. So, you cannot square x by doing\r\n// multMod_(x,x,n). You must use squareMod_(x,n) instead, or do y=dup(x); multMod_(x,y,n).\r\n// Or simply use the multMod(x,x,n) function without the underscore, where\r\n// such issues never arise, because non-underscored functions never change\r\n// their parameters; they always allocate new memory for the answer that is returned.\r\n//\r\n// These functions are designed to avoid frequent dynamic memory allocation in the inner loop.\r\n// For most functions, if it needs a BigInt as a local variable it will actually use\r\n// a global, and will only allocate to it only when it's not the right size. This ensures\r\n// that when a function is called repeatedly with same-sized parameters, it only allocates\r\n// memory on the first call.\r\n//\r\n// Note that for cryptographic purposes, the calls to Math.random() must\r\n// be replaced with calls to a better pseudorandom number generator.\r\n//\r\n// In the following, \"bigInt\" means a bigInt with at least one leading zero element,\r\n// and \"integer\" means a nonnegative integer less than radix. In some cases, integer\r\n// can be negative. Negative bigInts are 2s complement.\r\n//\r\n// The following functions do not modify their inputs.\r\n// Those returning a bigInt, string, or Array will dynamically allocate memory for that value.\r\n// Those returning a boolean will return the integer 0 (false) or 1 (true).\r\n// Those returning boolean or int will not allocate memory except possibly on the first\r\n// time they're called with a given parameter size.\r\n//\r\n// bigInt add(x,y) //return (x+y) for bigInts x and y.\r\n// bigInt addInt(x,n) //return (x+n) where x is a bigInt and n is an integer.\r\n// string bigInt2str(x,base) //return a string form of bigInt x in a given base, with 2 <= base <= 95\r\n// int bitSize(x) //return how many bits long the bigInt x is, not counting leading zeros\r\n// bigInt dup(x) //return a copy of bigInt x\r\n// boolean equals(x,y) //is the bigInt x equal to the bigint y?\r\n// boolean equalsInt(x,y) //is bigint x equal to integer y?\r\n// bigInt expand(x,n) //return a copy of x with at least n elements, adding leading zeros if needed\r\n// Array findPrimes(n) //return array of all primes less than integer n\r\n// bigInt GCD(x,y) //return greatest common divisor of bigInts x and y (each with same number of elements).\r\n// boolean greater(x,y) //is x>y? (x and y are nonnegative bigInts)\r\n// boolean greaterShift(x,y,shift)//is (x <<(shift*bpe)) > y?\r\n// bigInt int2bigInt(t,n,m) //return a bigInt equal to integer t, with at least n bits and m array elements\r\n// bigInt inverseMod(x,n) //return (x**(-1) mod n) for bigInts x and n. If no inverse exists, it returns null\r\n// int inverseModInt(x,n) //return x**(-1) mod n, for integers x and n. Return 0 if there is no inverse\r\n// boolean isZero(x) //is the bigInt x equal to zero?\r\n// boolean millerRabin(x,b) //does one round of Miller-Rabin base integer b say that bigInt x is possibly prime? (b is bigInt, 1<b<x)\r\n// boolean millerRabinInt(x,b) //does one round of Miller-Rabin base integer b say that bigInt x is possibly prime? (b is int, 1<b<x)\r\n// bigInt mod(x,n) //return a new bigInt equal to (x mod n) for bigInts x and n.\r\n// int modInt(x,n) //return x mod n for bigInt x and integer n.\r\n// bigInt mult(x,y) //return x*y for bigInts x and y. This is faster when y<x.\r\n// bigInt multMod(x,y,n) //return (x*y mod n) for bigInts x,y,n. For greater speed, let y<x.\r\n// boolean negative(x) //is bigInt x negative?\r\n// 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.\r\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.\r\n// bigInt randTruePrime(k) //return a new, random, k-bit, true prime bigInt using Maurer's algorithm.\r\n// bigInt randProbPrime(k) //return a new, random, k-bit, probable prime bigInt (probability it's composite less than 2^-80).\r\n// 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\r\n// bigInt sub(x,y) //return (x-y) for bigInts x and y. Negative answers will be 2s complement\r\n// bigInt trim(x,k) //return a copy of x with exactly k leading zero elements\r\n//\r\n//\r\n// The following functions each have a non-underscored version, which most users should call instead.\r\n// These functions each write to a single parameter, and the caller is responsible for ensuring the array\r\n// passed in is large enough to hold the result.\r\n//\r\n// void addInt_(x,n) //do x=x+n where x is a bigInt and n is an integer\r\n// void add_(x,y) //do x=x+y for bigInts x and y\r\n// void copy_(x,y) //do x=y on bigInts x and y\r\n// void copyInt_(x,n) //do x=n on bigInt x and integer n\r\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).\r\n// boolean inverseMod_(x,n) //do x=x**(-1) mod n, for bigInts x and n. Returns 1 (0) if inverse does (doesn't) exist\r\n// void mod_(x,n) //do x=x mod n for bigInts x and n. (This never overflows its array).\r\n// void mult_(x,y) //do x=x*y for bigInts x and y.\r\n// void multMod_(x,y,n) //do x=x*y mod n for bigInts x,y,n.\r\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.\r\n// 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.\r\n// void randTruePrime_(ans,k) //do ans = a random k-bit true random prime (not just probable prime) with 1 in the msb.\r\n// void sub_(x,y) //do x=x-y for bigInts x and y. Negative answers will be 2s complement.\r\n//\r\n// The following functions do NOT have a non-underscored version.\r\n// They each write a bigInt result to one or more parameters. The caller is responsible for\r\n// ensuring the arrays passed in are large enough to hold the results.\r\n//\r\n// void addShift_(x,y,ys) //do x=x+(y<<(ys*bpe))\r\n// void carry_(x) //do carries and borrows so each element of the bigInt x fits in bpe bits.\r\n// void divide_(x,y,q,r) //divide x by y giving quotient q and remainder r\r\n// int divInt_(x,n) //do x=floor(x/n) for bigInt x and integer n, and return the remainder. (This never overflows its array).\r\n// int eGCD_(x,y,d,a,b) //sets a,b,d to positive bigInts such that d = GCD_(x,y) = a*x-b*y\r\n// void halve_(x) //do x=floor(|x|/2)*sgn(x) for bigInt x in 2's complement. (This never overflows its array).\r\n// void leftShift_(x,n) //left shift bigInt x by n bits. n<bpe.\r\n// void linComb_(x,y,a,b) //do x=a*x+b*y for bigInts x and y and integers a and b\r\n// void linCombShift_(x,y,b,ys) //do x=x+b*(y<<(ys*bpe)) for bigInts x and y, and integers b and ys\r\n// void mont_(x,y,n,np) //Montgomery multiplication (see comments where the function is defined)\r\n// void multInt_(x,n) //do x=x*n where x is a bigInt and n is an integer.\r\n// void rightShift_(x,n) //right shift bigInt x by n bits. 0 <= n < bpe. (This never overflows its array).\r\n// void squareMod_(x,n) //do x=x*x mod n for bigInts x,n\r\n// void subShift_(x,y,ys) //do x=x-(y<<(ys*bpe)). Negative answers will be 2s complement.\r\n//\r\n// The following functions are based on algorithms from the _Handbook of Applied Cryptography_\r\n// powMod_() = algorithm 14.94, Montgomery exponentiation\r\n// eGCD_,inverseMod_() = algorithm 14.61, Binary extended GCD_\r\n// GCD_() = algorothm 14.57, Lehmer's algorithm\r\n// mont_() = algorithm 14.36, Montgomery multiplication\r\n// divide_() = algorithm 14.20 Multiple-precision division\r\n// squareMod_() = algorithm 14.16 Multiple-precision squaring\r\n// randTruePrime_() = algorithm 4.62, Maurer's algorithm\r\n// millerRabin() = algorithm 4.24, Miller-Rabin algorithm\r\n//\r\n// Profiling shows:\r\n// randTruePrime_() spends:\r\n// 10% of its time in calls to powMod_()\r\n// 85% of its time in calls to millerRabin()\r\n// millerRabin() spends:\r\n// 99% of its time in calls to powMod_() (always with a base of 2)\r\n// powMod_() spends:\r\n// 94% of its time in calls to mont_() (almost always with x==y)\r\n//\r\n// This suggests there are several ways to speed up this library slightly:\r\n// - convert powMod_ to use a Montgomery form of k-ary window (or maybe a Montgomery form of sliding window)\r\n// -- this should especially focus on being fast when raising 2 to a power mod n\r\n// - convert randTruePrime_() to use a minimum r of 1/3 instead of 1/2 with the appropriate change to the test\r\n// - tune the parameters in randTruePrime_(), including c, m, and recLimit\r\n// - speed up the single loop in mont_() that takes 95% of the runtime, perhaps by reducing checking\r\n// within the loop when all the parameters are the same length.\r\n//\r\n// There are several ideas that look like they wouldn't help much at all:\r\n// - replacing trial division in randTruePrime_() with a sieve (that speeds up something taking almost no time anyway)\r\n// - 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)\r\n// - speeding up mont_(x,y,n,np) when x==y by doing a non-modular, non-Montgomery square\r\n// followed by a Montgomery reduction. The intermediate answer will be twice as long as x, so that\r\n// method would be slower. This is unfortunate because the code currently spends almost all of its time\r\n// doing mont_(x,x,...), both for randTruePrime_() and powMod_(). A faster method for Montgomery squaring\r\n// would have a large impact on the speed of randTruePrime_() and powMod_(). HAC has a couple of poorly-worded\r\n// sentences that seem to imply it's faster to do a non-modular square followed by a single\r\n// Montgomery reduction, but that's obviously wrong.\r\n////////////////////////////////////////////////////////////////////////////////////////\r\n\r\n//globals\r\nexport var bpe=0; //bits stored per array element\r\nvar mask=0; //AND this with an array element to chop it down to bpe bits\r\nvar radix=mask+1; //equals 2^bpe. A single 1 bit to the left of the last bit of mask.\r\n\r\n//the digits for converting to different bases\r\nvar digitsStr='0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz_=!@#$%^&*()[]{}|;:,.<>/?`~ \\\\\\'\\\"+-';\r\n\r\n//initialize the global variables\r\nfor (bpe=0; (1<<(bpe+1)) > (1<<bpe); bpe++); //bpe=number of bits in the mantissa on this platform\r\nbpe>>=1; //bpe=number of bits in one element of the array representing the bigInt\r\nmask=(1<<bpe)-1; //AND the mask with an integer to get its bpe least significant bits\r\nradix=mask+1; //2^bpe. a single 1 bit to the left of the first bit of mask\r\nexport var one=int2bigInt(1,1,1); //constant used in powMod_()\r\n\r\n//the following global variables are scratchpad memory to\r\n//reduce dynamic memory allocation in the inner loop\r\nvar t=new Array(0);\r\nvar ss=t; //used in mult_()\r\nvar s0=t; //used in multMod_(), squareMod_()\r\nvar s1=t; //used in powMod_(), multMod_(), squareMod_()\r\nvar s2=t; //used in powMod_(), multMod_()\r\nvar s3=t; //used in powMod_()\r\nvar s4=t, s5=t; //used in mod_()\r\nvar s6=t; //used in bigInt2str()\r\nvar s7=t; //used in powMod_()\r\nvar T=t; //used in GCD_()\r\nvar sa=t; //used in mont_()\r\nvar mr_x1=t, mr_r=t, mr_a=t, //used in millerRabin()\r\neg_v=t, eg_u=t, eg_A=t, eg_B=t, eg_C=t, eg_D=t, //used in eGCD_(), inverseMod_()\r\nmd_q1=t, md_q2=t, md_q3=t, md_r=t, md_r1=t, md_r2=t, md_tt=t, //used in mod_()\r\n\r\nprimes=t, pows=t, s_i=t, s_i2=t, s_R=t, s_rm=t, s_q=t, s_n1=t,\r\n 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_()\r\n\r\nrpprb=t; //used in randProbPrimeRounds() (which also uses \"primes\")\r\n\r\n////////////////////////////////////////////////////////////////////////////////////////\r\n\r\nvar k, buff\r\n\r\n//return array of all primes less than integer n\r\nfunction findPrimes(n) {\r\n var i,s,p,ans;\r\n s=new Array(n);\r\n for (i=0;i<n;i++)\r\n s[i]=0;\r\n s[0]=2;\r\n p=0; //first p elements of s are primes, the rest are a sieve\r\n for(;s[p]<n;) { //s[p] is the pth prime\r\n for(i=s[p]*s[p]; i<n; i+=s[p]) //mark multiples of s[p]\r\n s[i]=1;\r\n p++;\r\n s[p]=s[p-1]+1;\r\n for(; s[p]<n && s[s[p]]; s[p]++); //find next prime (where s[p]==0)\r\n }\r\n ans=new Array(p);\r\n for(i=0;i<p;i++)\r\n ans[i]=s[i];\r\n return ans;\r\n}\r\n\r\n\r\n//does a single round of Miller-Rabin base b consider x to be a possible prime?\r\n//x is a bigInt, and b is an integer, with b<x\r\nfunction millerRabinInt(x,b) {\r\n if (mr_x1.length!=x.length) {\r\n mr_x1=dup(x);\r\n mr_r=dup(x);\r\n mr_a=dup(x);\r\n }\r\n\r\n copyInt_(mr_a,b);\r\n return millerRabin(x,mr_a);\r\n}\r\n\r\n//does a single round of Miller-Rabin base b consider x to be a possible prime?\r\n//x and b are bigInts with b<x\r\nfunction millerRabin(x,b) {\r\n var i,j,k,s;\r\n\r\n if (mr_x1.length!=x.length) {\r\n mr_x1=dup(x);\r\n mr_r=dup(x);\r\n mr_a=dup(x);\r\n }\r\n\r\n copy_(mr_a,b);\r\n copy_(mr_r,x);\r\n copy_(mr_x1,x);\r\n\r\n addInt_(mr_r,-1);\r\n addInt_(mr_x1,-1);\r\n\r\n //s=the highest power of two that divides mr_r\r\n k=0;\r\n for (i=0;i<mr_r.length;i++)\r\n for (j=1;j<mask;j<<=1)\r\n if (x[i] & j) {\r\n s=(k<mr_r.length+bpe ? k : 0);\r\n i=mr_r.length;\r\n j=mask;\r\n } else\r\n k++;\r\n\r\n if (s)\r\n rightShift_(mr_r,s);\r\n\r\n powMod_(mr_a,mr_r,x);\r\n\r\n if (!equalsInt(mr_a,1) && !equals(mr_a,mr_x1)) {\r\n j=1;\r\n while (j<=s-1 && !equals(mr_a,mr_x1)) {\r\n squareMod_(mr_a,x);\r\n if (equalsInt(mr_a,1)) {\r\n return 0;\r\n }\r\n j++;\r\n }\r\n if (!equals(mr_a,mr_x1)) {\r\n return 0;\r\n }\r\n }\r\n return 1;\r\n}\r\n\r\n//returns how many bits long the bigInt is, not counting leading zeros.\r\nfunction bitSize(x) {\r\n var j,z,w;\r\n for (j=x.length-1; (x[j]==0) && (j>0); j--);\r\n for (z=0,w=x[j]; w; (w>>=1),z++);\r\n z+=bpe*j;\r\n return z;\r\n}\r\n\r\n//return a copy of x with at least n elements, adding leading zeros if needed\r\nfunction expand(x,n) {\r\n var ans=int2bigInt(0,(x.length>n ? x.length : n)*bpe,0);\r\n copy_(ans,x);\r\n return ans;\r\n}\r\n\r\n//return a k-bit true random prime using Maurer's algorithm.\r\nfunction randTruePrime(k) {\r\n var ans=int2bigInt(0,k,0);\r\n randTruePrime_(ans,k);\r\n return trim(ans,1);\r\n}\r\n\r\n//return a k-bit random probable prime with probability of error < 2^-80\r\nfunction randProbPrime(k) {\r\n if (k>=600) return randProbPrimeRounds(k,2); //numbers from HAC table 4.3\r\n if (k>=550) return randProbPrimeRounds(k,4);\r\n if (k>=500) return randProbPrimeRounds(k,5);\r\n if (k>=400) return randProbPrimeRounds(k,6);\r\n if (k>=350) return randProbPrimeRounds(k,7);\r\n if (k>=300) return randProbPrimeRounds(k,9);\r\n if (k>=250) return randProbPrimeRounds(k,12); //numbers from HAC table 4.4\r\n if (k>=200) return randProbPrimeRounds(k,15);\r\n if (k>=150) return randProbPrimeRounds(k,18);\r\n if (k>=100) return randProbPrimeRounds(k,27);\r\n return randProbPrimeRounds(k,40); //number from HAC remark 4.26 (only an estimate)\r\n}\r\n\r\n//return a k-bit probable random prime using n rounds of Miller Rabin (after trial division with small primes)\r\nfunction randProbPrimeRounds(k,n) {\r\n var ans, i, divisible, B;\r\n B=30000; //B is largest prime to use in trial division\r\n ans=int2bigInt(0,k,0);\r\n\r\n //optimization: try larger and smaller B to find the best limit.\r\n\r\n if (primes.length==0)\r\n primes=findPrimes(30000); //check for divisibility by primes <=30000\r\n\r\n if (rpprb.length!=ans.length)\r\n rpprb=dup(ans);\r\n\r\n for (;;) { //keep trying random values for ans until one appears to be prime\r\n //optimization: pick a random number times L=2*3*5*...*p, plus a\r\n // random element of the list of all numbers in [0,L) not divisible by any prime up to p.\r\n // This can reduce the amount of random number generation.\r\n\r\n randBigInt_(ans,k,0); //ans = a random odd number to check\r\n ans[0] |= 1;\r\n divisible=0;\r\n\r\n //check ans for divisibility by small primes up to B\r\n for (i=0; (i<primes.length) && (primes[i]<=B); i++)\r\n if (modInt(ans,primes[i])==0 && !equalsInt(ans,primes[i])) {\r\n divisible=1;\r\n break;\r\n }\r\n\r\n //optimization: change millerRabin so the base can be bigger than the number being checked, then eliminate the while here.\r\n\r\n //do n rounds of Miller Rabin, with random bases less than ans\r\n for (i=0; i<n && !divisible; i++) {\r\n randBigInt_(rpprb,k,0);\r\n while(!greater(ans,rpprb)) //pick a random rpprb that's < ans\r\n randBigInt_(rpprb,k,0);\r\n if (!millerRabin(ans,rpprb))\r\n divisible=1;\r\n }\r\n\r\n if(!divisible)\r\n return ans;\r\n }\r\n}\r\n\r\n//return a new bigInt equal to (x mod n) for bigInts x and n.\r\nfunction mod(x,n) {\r\n var ans=dup(x);\r\n mod_(ans,n);\r\n return trim(ans,1);\r\n}\r\n\r\n//return (x+n) where x is a bigInt and n is an integer.\r\nfunction addInt(x,n) {\r\n var ans=expand(x,x.length+1);\r\n addInt_(ans,n);\r\n return trim(ans,1);\r\n}\r\n\r\n//return x*y for bigInts x and y. This is faster when y<x.\r\nfunction mult(x,y) {\r\n var ans=expand(x,x.length+y.length);\r\n mult_(ans,y);\r\n return trim(ans,1);\r\n}\r\n\r\n//return (x**y mod n) where x,y,n are bigInts and ** is exponentiation. 0**0=1. Faster for odd n.\r\nexport function powMod(x,y,n) {\r\n var ans=expand(x,n.length);\r\n powMod_(ans,trim(y,2),trim(n,2),0); //this should work without the trim, but doesn't\r\n return trim(ans,1);\r\n}\r\n\r\n//return (x-y) for bigInts x and y. Negative answers will be 2s complement\r\nfunction sub(x,y) {\r\n var ans=expand(x,(x.length>y.length ? x.length+1 : y.length+1));\r\n sub_(ans,y);\r\n return trim(ans,1);\r\n}\r\n\r\n//return (x+y) for bigInts x and y.\r\nfunction add(x,y) {\r\n var ans=expand(x,(x.length>y.length ? x.length+1 : y.length+1));\r\n add_(ans,y);\r\n return trim(ans,1);\r\n}\r\n\r\n//return (x**(-1) mod n) for bigInts x and n. If no inverse exists, it returns null\r\nfunction inverseMod(x,n) {\r\n var ans=expand(x,n.length);\r\n var s;\r\n s=inverseMod_(ans,n);\r\n return s ? trim(ans,1) : null;\r\n}\r\n\r\n//return (x*y mod n) for bigInts x,y,n. For greater speed, let y<x.\r\nfunction multMod(x,y,n) {\r\n var ans=expand(x,n.length);\r\n multMod_(ans,y,n);\r\n return trim(ans,1);\r\n}\r\n\r\n//generate a k-bit true random prime using Maurer's algorithm,\r\n//and put it into ans. The bigInt ans must be large enough to hold it.\r\nfunction randTruePrime_(ans,k) {\r\n var c,m,pm,dd,j,r,B,divisible,z,zz,recSize;\r\n\r\n if (primes.length==0)\r\n primes=findPrimes(30000); //check for divisibility by primes <=30000\r\n\r\n if (pows.length==0) {\r\n pows=new Array(512);\r\n for (j=0;j<512;j++) {\r\n pows[j]=Math.pow(2,j/511.-1.);\r\n }\r\n }\r\n\r\n //c and m should be tuned for a particular machine and value of k, to maximize speed\r\n c=0.1; //c=0.1 in HAC\r\n m=20; //generate this k-bit number by first recursively generating a number that has between k/2 and k-m bits\r\n recLimit=20; //stop recursion when k <=recLimit. Must have recLimit >= 2\r\n\r\n if (s_i2.length!=ans.length) {\r\n s_i2=dup(ans);\r\n s_R =dup(ans);\r\n s_n1=dup(ans);\r\n s_r2=dup(ans);\r\n s_d =dup(ans);\r\n s_x1=dup(ans);\r\n s_x2=dup(ans);\r\n s_b =dup(ans);\r\n s_n =dup(ans);\r\n s_i =dup(ans);\r\n s_rm=dup(ans);\r\n s_q =dup(ans);\r\n s_a =dup(ans);\r\n s_aa=dup(ans);\r\n }\r\n\r\n if (k <= recLimit) { //generate small random primes by trial division up to its square root\r\n pm=(1<<((k+2)>>1))-1; //pm is binary number with all ones, just over sqrt(2^k)\r\n copyInt_(ans,0);\r\n for (dd=1;dd;) {\r\n dd=0;\r\n ans[0]= 1 | (1<<(k-1)) | Math.floor(Math.random()*(1<<k)); //random, k-bit, odd integer, with msb 1\r\n for (j=1;(j<primes.length) && ((primes[j]&pm)==primes[j]);j++) { //trial division by all primes 3...sqrt(2^k)\r\n if (0==(ans[0]%primes[j])) {\r\n dd=1;\r\n break;\r\n }\r\n }\r\n }\r\n carry_(ans);\r\n return;\r\n }\r\n\r\n B=c*k*k; //try small primes up to B (or all the primes[] array if the largest is less than B).\r\n if (k>2*m) //generate this k-bit number by first recursively generating a number that has between k/2 and k-m bits\r\n for (r=1; k-k*r<=m; )\r\n r=pows[Math.floor(Math.random()*512)]; //r=Math.pow(2,Math.random()-1);\r\n else\r\n r=.5;\r\n\r\n //simulation suggests the more complex algorithm using r=.333 is only slightly faster.\r\n\r\n recSize=Math.floor(r*k)+1;\r\n\r\n randTruePrime_(s_q,recSize);\r\n copyInt_(s_i2,0);\r\n s_i2[Math.floor((k-2)/bpe)] |= (1<<((k-2)%bpe)); //s_i2=2^(k-2)\r\n divide_(s_i2,s_q,s_i,s_rm); //s_i=floor((2^(k-1))/(2q))\r\n\r\n z=bitSize(s_i);\r\n\r\n for (;;) {\r\n for (;;) { //generate z-bit numbers until one falls in the range [0,s_i-1]\r\n randBigInt_(s_R,z,0);\r\n if (greater(s_i,s_R))\r\n break;\r\n } //now s_R is in the range [0,s_i-1]\r\n addInt_(s_R,1); //now s_R is in the range [1,s_i]\r\n add_(s_R,s_i); //now s_R is in the range [s_i+1,2*s_i]\r\n\r\n copy_(s_n,s_q);\r\n mult_(s_n,s_R);\r\n multInt_(s_n,2);\r\n addInt_(s_n,1); //s_n=2*s_R*s_q+1\r\n\r\n copy_(s_r2,s_R);\r\n multInt_(s_r2,2); //s_r2=2*s_R\r\n\r\n //check s_n for divisibility by small primes up to B\r\n for (divisible=0,j=0; (j<primes.length) && (primes[j]<B); j++)\r\n if (modInt(s_n,primes[j])==0 && !equalsInt(s_n,primes[j])) {\r\n divisible=1;\r\n break;\r\n }\r\n\r\n if (!divisible) //if it passes small primes check, then try a single Miller-Rabin base 2\r\n if (!millerRabinInt(s_n,2)) //this line represents 75% of the total runtime for randTruePrime_\r\n divisible=1;\r\n\r\n if (!divisible) { //if it passes that test, continue checking s_n\r\n addInt_(s_n,-3);\r\n for (j=s_n.length-1;(s_n[j]==0) && (j>0); j--); //strip leading zeros\r\n for (zz=0,w=s_n[j]; w; (w>>=1),zz++);\r\n zz+=bpe*j; //zz=number of bits in s_n, ignoring leading zeros\r\n for (;;) { //generate z-bit numbers until one falls in the range [0,s_n-1]\r\n randBigInt_(s_a,zz,0);\r\n if (greater(s_n,s_a))\r\n break;\r\n } //now s_a is in the range [0,s_n-1]\r\n addInt_(s_n,3); //now s_a is in the range [0,s_n-4]\r\n addInt_(s_a,2); //now s_a is in the range [2,s_n-2]\r\n copy_(s_b,s_a);\r\n copy_(s_n1,s_n);\r\n addInt_(s_n1,-1);\r\n powMod_(s_b,s_n1,s_n); //s_b=s_a^(s_n-1) modulo s_n\r\n addInt_(s_b,-1);\r\n if (isZero(s_b)) {\r\n copy_(s_b,s_a);\r\n powMod_(s_b,s_r2,s_n);\r\n addInt_(s_b,-1);\r\n copy_(s_aa,s_n);\r\n copy_(s_d,s_b);\r\n GCD_(s_d,s_n); //if s_b and s_n are relatively prime, then s_n is a prime\r\n if (equalsInt(s_d,1)) {\r\n copy_(ans,s_aa);\r\n return; //if we've made it this far, then s_n is absolutely guaranteed to be prime\r\n }\r\n }\r\n }\r\n }\r\n}\r\n\r\n//Return an n-bit random BigInt (n>=1). If s=1, then the most significant of those n bits is set to 1.\r\nfunction randBigInt(n,s) {\r\n var a,b;\r\n a=Math.floor((n-1)/bpe)+2; //# array elements to hold the BigInt with a leading 0 element\r\n b=int2bigInt(0,0,a);\r\n randBigInt_(b,n,s);\r\n return b;\r\n}\r\n\r\n//Set b to an n-bit random BigInt. If s=1, then the most significant of those n bits is set to 1.\r\n//Array b must be big enough to hold the result. Must have n>=1\r\nfunction randBigInt_(b,n,s) {\r\n var i,a;\r\n for (i=0;i<b.length;i++)\r\n b[i]=0;\r\n a=Math.floor((n-1)/bpe)+1; //# array elements to hold the BigInt\r\n for (i=0;i<a;i++) {\r\n b[i]=Math.floor(Math.random()*(1<<(bpe-1)));\r\n }\r\n b[a-1] &= (2<<((n-1)%bpe))-1;\r\n if (s==1)\r\n b[a-1] |= (1<<((n-1)%bpe));\r\n}\r\n\r\n//Return the greatest common divisor of bigInts x and y (each with same number of elements).\r\nfunction GCD(x,y) {\r\n var xc,yc;\r\n xc=dup(x);\r\n yc=dup(y);\r\n GCD_(xc,yc);\r\n return xc;\r\n}\r\n\r\n//set x to the greatest common divisor of bigInts x and y (each with same number of elements).\r\n//y is destroyed.\r\nfunction GCD_(x,y) {\r\n var i,xp,yp,A,B,C,D,q,sing;\r\n if (T.length!=x.length)\r\n T=dup(x);\r\n\r\n sing=1;\r\n while (sing) { //while y has nonzero elements other than y[0]\r\n sing=0;\r\n for (i=1;i<y.length;i++) //check if y has nonzero elements other than 0\r\n if (y[i]) {\r\n sing=1;\r\n break;\r\n }\r\n if (!sing) break; //quit when y all zero elements except possibly y[0]\r\n\r\n for (i=x.length;!x[i] && i>=0;i--); //find most significant element of x\r\n xp=x[i];\r\n yp=y[i];\r\n A=1; B=0; C=0; D=1;\r\n while ((yp+C) && (yp+D)) {\r\n q =Math.floor((xp+A)/(yp+C));\r\n qp=Math.floor((xp+B)/(yp+D));\r\n if (q!=qp)\r\n break;\r\n 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)\r\n t= B-q*D; B=D; D=t;\r\n t=xp-q*yp; xp=yp; yp=t;\r\n }\r\n if (B) {\r\n copy_(T,x);\r\n linComb_(x,y,A,B); //x=A*x+B*y\r\n linComb_(y,T,D,C); //y=D*y+C*T\r\n } else {\r\n mod_(x,y);\r\n copy_(T,x);\r\n copy_(x,y);\r\n copy_(y,T);\r\n }\r\n }\r\n if (y[0]==0)\r\n return;\r\n t=modInt(x,y[0]);\r\n copyInt_(x,y[0]);\r\n y[0]=t;\r\n while (y[0]) {\r\n x[0]%=y[0];\r\n t=x[0]; x[0]=y[0]; y[0]=t;\r\n }\r\n}\r\n\r\n//do x=x**(-1) mod n, for bigInts x and n.\r\n//If no inverse exists, it sets x to zero and returns 0, else it returns 1.\r\n//The x array must be at least as large as the n array.\r\nfunction inverseMod_(x,n) {\r\n var k=1+2*Math.max(x.length,n.length);\r\n\r\n if(!(x[0]&1) && !(n[0]&1)) { //if both inputs are even, then inverse doesn't exist\r\n copyInt_(x,0);\r\n return 0;\r\n }\r\n\r\n if (eg_u.length!=k) {\r\n eg_u=new Array(k);\r\n eg_v=new Array(k);\r\n eg_A=new Array(k);\r\n eg_B=new Array(k);\r\n eg_C=new Array(k);\r\n eg_D=new Array(k);\r\n }\r\n\r\n copy_(eg_u,x);\r\n copy_(eg_v,n);\r\n copyInt_(eg_A,1);\r\n copyInt_(eg_B,0);\r\n copyInt_(eg_C,0);\r\n copyInt_(eg_D,1);\r\n for (;;) {\r\n while(!(eg_u[0]&1)) { //while eg_u is even\r\n halve_(eg_u);\r\n if (!(eg_A[0]&1) && !(eg_B[0]&1)) { //if eg_A==eg_B==0 mod 2\r\n halve_(eg_A);\r\n halve_(eg_B);\r\n } else {\r\n add_(eg_A,n); halve_(eg_A);\r\n sub_(eg_B,x); halve_(eg_B);\r\n }\r\n }\r\n\r\n while (!(eg_v[0]&1)) { //while eg_v is even\r\n halve_(eg_v);\r\n if (!(eg_C[0]&1) && !(eg_D[0]&1)) { //if eg_C==eg_D==0 mod 2\r\n halve_(eg_C);\r\n halve_(eg_D);\r\n } else {\r\n add_(eg_C,n); halve_(eg_C);\r\n sub_(eg_D,x); halve_(eg_D);\r\n }\r\n }\r\n\r\n if (!greater(eg_v,eg_u)) { //eg_v <= eg_u\r\n sub_(eg_u,eg_v);\r\n sub_(eg_A,eg_C);\r\n sub_(eg_B,eg_D);\r\n } else { //eg_v > eg_u\r\n sub_(eg_v,eg_u);\r\n sub_(eg_C,eg_A);\r\n sub_(eg_D,eg_B);\r\n }\r\n\r\n if (equalsInt(eg_u,0)) {\r\n while (negative(eg_C)) //make sure answer is nonnegative\r\n add_(eg_C,n);\r\n copy_(x,eg_C);\r\n\r\n if (!equalsInt(eg_v,1)) { //if GCD_(x,n)!=1, then there is no inverse\r\n copyInt_(x,0);\r\n return 0;\r\n }\r\n return 1;\r\n }\r\n }\r\n}\r\n\r\n//return x**(-1) mod n, for integers x and n. Return 0 if there is no inverse\r\nfunction inverseModInt(x,n) {\r\n var a=1,b=0,t;\r\n for (;;) {\r\n if (x==1) return a;\r\n if (x==0) return 0;\r\n b-=a*Math.floor(n/x);\r\n n%=x;\r\n\r\n if (n==1) return b; //to avoid negatives, change this b to n-b, and each -= to +=\r\n if (n==0) return 0;\r\n a-=b*Math.floor(x/n);\r\n x%=n;\r\n }\r\n}\r\n\r\n//this deprecated function is for backward compatibility only.\r\nfunction inverseModInt_(x,n) {\r\n return inverseModInt(x,n);\r\n}\r\n\r\n\r\n//Given positive bigInts x and y, change the bigints v, a, and b to positive bigInts such that:\r\n// v = GCD_(x,y) = a*x-b*y\r\n//The bigInts v, a, b, must have exactly as many elements as the larger of x and y.\r\nexport function eGCD_(x,y,v,a,b) {\r\n var g=0;\r\n var k=Math.max(x.length,y.length);\r\n if (eg_u.length!=k) {\r\n eg_u=new Array(k);\r\n eg_A=new Array(k);\r\n eg_B=new Array(k);\r\n eg_C=new Array(k);\r\n eg_D=new Array(k);\r\n }\r\n while(!(x[0]&1) && !(y[0]&1)) { //while x and y both even\r\n halve_(x);\r\n halve_(y);\r\n g++;\r\n }\r\n copy_(eg_u,x);\r\n copy_(v,y);\r\n copyInt_(eg_A,1);\r\n copyInt_(eg_B,0);\r\n copyInt_(eg_C,0);\r\n copyInt_(eg_D,1);\r\n for (;;) {\r\n while(!(eg_u[0]&1)) { //while u is even\r\n halve_(eg_u);\r\n if (!(eg_A[0]&1) && !(eg_B[0]&1)) { //if A==B==0 mod 2\r\n halve_(eg_A);\r\n halve_(eg_B);\r\n } else {\r\n add_(eg_A,y); halve_(eg_A);\r\n sub_(eg_B,x); halve_(eg_B);\r\n }\r\n }\r\n\r\n while (!(v[0]&1)) { //while v is even\r\n halve_(v);\r\n if (!(eg_C[0]&1) && !(eg_D[0]&1)) { //if C==D==0 mod 2\r\n halve_(eg_C);\r\n halve_(eg_D);\r\n } else {\r\n add_(eg_C,y); halve_(eg_C);\r\n sub_(eg_D,x); halve_(eg_D);\r\n }\r\n }\r\n\r\n if (!greater(v,eg_u)) { //v<=u\r\n sub_(eg_u,v);\r\n sub_(eg_A,eg_C);\r\n sub_(eg_B,eg_D);\r\n } else { //v>u\r\n sub_(v,eg_u);\r\n sub_(eg_C,eg_A);\r\n sub_(eg_D,eg_B);\r\n }\r\n if (equalsInt(eg_u,0)) {\r\n while (negative(eg_C)) { //make sure a (C) is nonnegative\r\n add_(eg_C,y);\r\n sub_(eg_D,x);\r\n }\r\n multInt_(eg_D,-1); ///make sure b (D) is nonnegative\r\n copy_(a,eg_C);\r\n copy_(b,eg_D);\r\n leftShift_(v,g);\r\n return;\r\n }\r\n }\r\n}\r\n\r\n\r\n//is bigInt x negative?\r\nfunction negative(x) {\r\n return ((x[x.length-1]>>(bpe-1))&1);\r\n}\r\n\r\n\r\n//is (x << (shift*bpe)) > y?\r\n//x and y are nonnegative bigInts\r\n//shift is a nonnegative integer\r\nfunction greaterShift(x,y,shift) {\r\n var i, kx=x.length, ky=y.length;\r\n k=((kx+shift)<ky) ? (kx+shift) : ky;\r\n for (i=ky-1-shift; i<kx && i>=0; i++)\r\n if (x[i]>0)\r\n return 1; //if there are nonzeros in x to the left of the first column of y, then x is bigger\r\n for (i=kx-1+shift; i<ky; i++)\r\n if (y[i]>0)\r\n return 0; //if there are nonzeros in y to the left of the first column of x, then x is not bigger\r\n for (i=k-1; i>=shift; i--)\r\n if (x[i-shift]>y[i]) return 1;\r\n else if (x[i-shift]<y[i]) return 0;\r\n return 0;\r\n}\r\n\r\n//is x > y? (x and y both nonnegative)\r\nexport function greater(x,y) {\r\n var i;\r\n var k=(x.length<y.length) ? x.length : y.length;\r\n\r\n for (i=x.length;i<y.length;i++)\r\n if (y[i])\r\n return 0; //y has more digits\r\n\r\n for (i=y.length;i<x.length;i++)\r\n if (x[i])\r\n return 1; //x has more digits\r\n\r\n for (i=k-1;i>=0;i--)\r\n if (x[i]>y[i])\r\n return 1;\r\n else if (x[i]<y[i])\r\n return 0;\r\n return 0;\r\n}\r\n\r\n//divide x by y giving quotient q and remainder r. (q=floor(x/y), r=x mod y). All 4 are bigints.\r\n//x must have at least one leading zero element.\r\n//y must be nonzero.\r\n//q and r must be arrays that are exactly the same length as x. (Or q can have more).\r\n//Must have x.length >= y.length >= 2.\r\nexport function divide_(x,y,q,r) {\r\n var kx, ky;\r\n var i,j,y1,y2,c,a,b;\r\n copy_(r,x);\r\n for (ky=y.length;y[ky-1]==0;ky--); //ky is number of elements in y, not including leading zeros\r\n\r\n //normalize: ensure the most significant element of y has its highest bit set\r\n b=y[ky-1];\r\n for (a=0; b; a++)\r\n b>>=1;\r\n a=bpe-a; //a is how many bits to shift so that the high order bit of y is leftmost in its array element\r\n leftShift_(y,a); //multiply both by 1<<a now, then divide both by that at the end\r\n leftShift_(r,a);\r\n\r\n //Rob Visser discovered a bug: the following line was originally just before the normalization.\r\n for (kx=r.length;r[kx-1]==0 && kx>ky;kx--); //kx is number of elements in normalized x, not including leading zeros\r\n\r\n copyInt_(q,0); // q=0\r\n while (!greaterShift(y,r,kx-ky)) { // while (leftShift_(y,kx-ky) <= r) {\r\n subShift_(r,y,kx-ky); // r=r-leftShift_(y,kx-ky)\r\n q[kx-ky]++; // q[kx-ky]++;\r\n } // }\r\n\r\n for (i=kx-1; i>=ky; i--) {\r\n if (r[i]==y[ky-1])\r\n q[i-ky]=mask;\r\n else\r\n q[i-ky]=Math.floor((r[i]*radix+r[i-1])/y[ky-1]);\r\n\r\n //The following for(;;) loop is equivalent to the commented while loop,\r\n //except that the uncommented version avoids overflow.\r\n //The commented loop comes from HAC, which assumes r[-1]==y[-1]==0\r\n // while (q[i-ky]*(y[ky-1]*radix+y[ky-2]) > r[i]*radix*radix+r[i-1]*radix+r[i-2])\r\n // q[i-ky]--;\r\n for (;;) {\r\n y2=(ky>1 ? y[ky-2] : 0)*q[i-ky];\r\n c=y2>>bpe;\r\n y2=y2 & mask;\r\n y1=c+q[i-ky]*y[ky-1];\r\n c=y1>>bpe;\r\n y1=y1 & mask;\r\n\r\n if (c==r[i] ? y1==r[i-1] ? y2>(i>1 ? r[i-2] : 0) : y1>r[i-1] : c>r[i])\r\n q[i-ky]--;\r\n else\r\n break;\r\n }\r\n\r\n linCombShift_(r,y,-q[i-ky],i-ky); //r=r-q[i-ky]*leftShift_(y,i-ky)\r\n if (negative(r)) {\r\n addShift_(r,y,i-ky); //r=r+leftShift_(y,i-ky)\r\n q[i-ky]--;\r\n }\r\n }\r\n\r\n rightShift_(y,a); //undo the normalization step\r\n rightShift_(r,a); //undo the normalization step\r\n}\r\n\r\n//do carries and borrows so each element of the bigInt x fits in bpe bits.\r\nfunction carry_(x) {\r\n var i,k,c,b;\r\n k=x.length;\r\n c=0;\r\n for (i=0;i<k;i++) {\r\n c+=x[i];\r\n b=0;\r\n if (c<0) {\r\n b=-(c>>bpe);\r\n c+=b*radix;\r\n }\r\n x[i]=c & mask;\r\n c=(c>>bpe)-b;\r\n }\r\n}\r\n\r\n//return x mod n for bigInt x and integer n.\r\nfunction modInt(x,n) {\r\n var i,c=0;\r\n for (i=x.length-1; i>=0; i--)\r\n c=(c*radix+x[i])%n;\r\n return c;\r\n}\r\n\r\n//convert the integer t into a bigInt with at least the given number of bits.\r\n//the returned array stores the bigInt in bpe-bit chunks, little endian (buff[0] is least significant word)\r\n//Pad the array with leading zeros so that it has at least minSize elements.\r\n//There will always be at least one leading 0 element.\r\nfunction int2bigInt(t,bits,minSize) {\r\n var i,k;\r\n k=Math.ceil(bits/bpe)+1;\r\n k=minSize>k ? minSize : k;\r\n var buff=new Array(k);\r\n copyInt_(buff,t);\r\n return buff;\r\n}\r\n\r\n//return the bigInt given a string representation in a given base.\r\n//Pad the array with leading zeros so that it has at least minSize elements.\r\n//If base=-1, then it reads in a space-separated list of array elements in decimal.\r\n//The array will always have at least one leading zero, unless base=-1.\r\nexport function str2bigInt(s,base,minSize) {\r\n var d, i, j, x, y, kk;\r\n var k=s.length;\r\n if (base==-1) { //comma-separated list of array elements in decimal\r\n x=new Array(0);\r\n for (;;) {\r\n y=new Array(x.length+1);\r\n for (i=0;i<x.length;i++)\r\n y[i+1]=x[i];\r\n y[0]=parseInt(s,10);\r\n x=y;\r\n d=s.indexOf(',',0);\r\n if (d<1)\r\n break;\r\n s=s.substring(d+1);\r\n if (s.length==0)\r\n break;\r\n }\r\n if (x.length<minSize) {\r\n y=new Array(minSize);\r\n copy_(y,x);\r\n return y;\r\n }\r\n return x;\r\n }\r\n\r\n x=int2bigInt(0,base*k,0);\r\n for (i=0;i<k;i++) {\r\n d=digitsStr.indexOf(s.substring(i,i+1),0);\r\n if (base<=36 && d>=36) //convert lowercase to uppercase if base<=36\r\n d-=26;\r\n if (d>=base || d<0) { //stop at first illegal character\r\n break;\r\n }\r\n multInt_(x,base);\r\n addInt_(x,d);\r\n }\r\n\r\n for (k=x.length;k>0 && !x[k-1];k--); //strip off leading zeros\r\n k=minSize>k+1 ? minSize : k+1;\r\n y=new Array(k);\r\n kk=k<x.length ? k : x.length;\r\n for (i=0;i<kk;i++)\r\n y[i]=x[i];\r\n for (;i<k;i++)\r\n y[i]=0;\r\n return y;\r\n}\r\n\r\n//is bigint x equal to integer y?\r\n//y must have less than bpe bits\r\nexport function equalsInt(x,y) {\r\n var i;\r\n if (x[0]!=y)\r\n return 0;\r\n for (i=1;i<x.length;i++)\r\n if (x[i])\r\n return 0;\r\n return 1;\r\n}\r\n\r\n//are bigints x and y equal?\r\n//this works even if x and y are different lengths and have arbitrarily many leading zeros\r\nfunction equals(x,y) {\r\n var i;\r\n var k=x.length<y.length ? x.length : y.length;\r\n for (i=0;i<k;i++)\r\n if (x[i]!=y[i])\r\n return 0;\r\n if (x.length>y.length) {\r\n for (;i<x.length;i++)\r\n if (x[i])\r\n return 0;\r\n } else {\r\n for (;i<y.length;i++)\r\n if (y[i])\r\n return 0;\r\n }\r\n return 1;\r\n}\r\n\r\n//is the bigInt x equal to zero?\r\nexport function isZero(x) {\r\n var i;\r\n for (i=0;i<x.length;i++)\r\n if (x[i])\r\n return 0;\r\n return 1;\r\n}\r\n\r\n//convert a bigInt into a string in a given base, from base 2 up to base 95.\r\n//Base -1 prints the contents of the array representing the number.\r\nexport function bigInt2str(x,base) {\r\n var i,t,s=\"\";\r\n\r\n if (s6.length!=x.length)\r\n s6=dup(x);\r\n else\r\n copy_(s6,x);\r\n\r\n if (base==-1) { //return the list of array contents\r\n for (i=x.length-1;i>0;i--)\r\n s+=x[i]+',';\r\n s+=x[0];\r\n }\r\n else { //return it in the given base\r\n while (!isZero(s6)) {\r\n t=divInt_(s6,base); //t=s6 % base; s6=floor(s6/base);\r\n s=digitsStr.substring(t,t+1)+s;\r\n }\r\n }\r\n if (s.length==0)\r\n s=\"0\";\r\n return s;\r\n}\r\n\r\n//returns a duplicate of bigInt x\r\nfunction dup(x) {\r\n var i;\r\n buff=new Array(x.length);\r\n copy_(buff,x);\r\n return buff;\r\n}\r\n\r\n//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).\r\nexport function copy_(x,y) {\r\n var i;\r\n var k=x.length<y.length ? x.length : y.length;\r\n for (i=0;i<k;i++)\r\n x[i]=y[i];\r\n for (i=k;i<x.length;i++)\r\n x[i]=0;\r\n}\r\n\r\n//do x=y on bigInt x and integer y.\r\nexport function copyInt_(x,n) {\r\n var i,c;\r\n for (c=n,i=0;i<x.length;i++) {\r\n x[i]=c & mask;\r\n c>>=bpe;\r\n }\r\n}\r\n\r\n//do x=x+n where x is a bigInt and n is an integer.\r\n//x must be large enough to hold the result.\r\nfunction addInt_(x,n) {\r\n var i,k,c,b;\r\n x[0]+=n;\r\n k=x.length;\r\n c=0;\r\n for (i=0;i<k;i++) {\r\n c+=x[i];\r\n b=0;\r\n if (c<0) {\r\n b=-(c>>bpe);\r\n c+=b*radix;\r\n }\r\n x[i]=c & mask;\r\n c=(c>>bpe)-b;\r\n if (!c) return; //stop carrying as soon as the carry is zero\r\n }\r\n}\r\n\r\n//right shift bigInt x by n bits. 0 <= n < bpe.\r\nexport function rightShift_(x,n) {\r\n var i;\r\n var k=Math.floor(n/bpe);\r\n if (k) {\r\n for (i=0;i<x.length-k;i++) //right shift x by k elements\r\n x[i]=x[i+k];\r\n for (;i<x.length;i++)\r\n x[i]=0;\r\n n%=bpe;\r\n }\r\n for (i=0;i<x.length-1;i++) {\r\n x[i]=mask & ((x[i+1]<<(bpe-n)) | (x[i]>>n));\r\n }\r\n x[i]>>=n;\r\n}\r\n\r\n//do x=floor(|x|/2)*sgn(x) for bigInt x in 2's complement\r\nfunction halve_(x) {\r\n var i;\r\n for (i=0;i<x.length-1;i++) {\r\n x[i]=mask & ((x[i+1]<<(bpe-1)) | (x[i]>>1));\r\n }\r\n x[i]=(x[i]>>1) | (x[i] & (radix>>1)); //most significant bit stays the same\r\n}\r\n\r\n//left shift bigInt x by n bits.\r\nfunction leftShift_(x,n) {\r\n var i;\r\n var k=Math.floor(n/bpe);\r\n if (k) {\r\n for (i=x.length; i>=k; i--) //left shift x by k elements\r\n x[i]=x[i-k];\r\n for (;i>=0;i--)\r\n x[i]=0;\r\n n%=bpe;\r\n }\r\n if (!n)\r\n return;\r\n for (i=x.length-1;i>0;i--) {\r\n x[i]=mask & ((x[i]<<n) | (x[i-1]>>(bpe-n)));\r\n }\r\n x[i]=mask & (x[i]<<n);\r\n}\r\n\r\n//do x=x*n where x is a bigInt and n is an integer.\r\n//x must be large enough to hold the result.\r\nfunction multInt_(x,n) {\r\n var i,k,c,b;\r\n if (!n)\r\n return;\r\n k=x.length;\r\n c=0;\r\n for (i=0;i<k;i++) {\r\n c+=x[i]*n;\r\n b=0;\r\n if (c<0) {\r\n b=-(c>>bpe);\r\n c+=b*radix;\r\n }\r\n x[i]=c & mask;\r\n c=(c>>bpe)-b;\r\n }\r\n}\r\n\r\n//do x=floor(x/n) for bigInt x and integer n, and return the remainder\r\nfunction divInt_(x,n) {\r\n var i,r=0,s;\r\n for (i=x.length-1;i>=0;i--) {\r\n s=r*radix+x[i];\r\n x[i]=Math.floor(s/n);\r\n r=s%n;\r\n }\r\n return r;\r\n}\r\n\r\n//do the linear combination x=a*x+b*y for bigInts x and y, and integers a and b.\r\n//x must be large enough to hold the answer.\r\nfunction linComb_(x,y,a,b) {\r\n var i,c,k,kk;\r\n k=x.length<y.length ? x.length : y.length;\r\n kk=x.length;\r\n for (c=0,i=0;i<k;i++) {\r\n c+=a*x[i]+b*y[i];\r\n x[i]=c & mask;\r\n c>>=bpe;\r\n }\r\n for (i=k;i<kk;i++) {\r\n c+=a*x[i];\r\n x[i]=c & mask;\r\n c>>=bpe;\r\n }\r\n}\r\n\r\n//do the linear combination x=a*x+b*(y<<(ys*bpe)) for bigInts x and y, and integers a, b and ys.\r\n//x must be large enough to hold the answer.\r\nfunction linCombShift_(x,y,b,ys) {\r\n var i,c,k,kk;\r\n k=x.length<ys+y.length ? x.length : ys+y.length;\r\n kk=x.length;\r\n for (c=0,i=ys;i<k;i++) {\r\n c+=x[i]+b*y[i-ys];\r\n x[i]=c & mask;\r\n c>>=bpe;\r\n }\r\n for (i=k;c && i<kk;i++) {\r\n c+=x[i];\r\n x[i]=c & mask;\r\n c>>=bpe;\r\n }\r\n}\r\n\r\n//do x=x+(y<<(ys*bpe)) for bigInts x and y, and integers a,b and ys.\r\n//x must be large enough to hold the answer.\r\nfunction addShift_(x,y,ys) {\r\n var i,c,k,kk;\r\n k=x.length<ys+y.length ? x.length : ys+y.length;\r\n kk=x.length;\r\n for (c=0,i=ys;i<k;i++) {\r\n c+=x[i]+y[i-ys];\r\n x[i]=c & mask;\r\n c>>=bpe;\r\n }\r\n for (i=k;c && i<kk;i++) {\r\n c+=x[i];\r\n x[i]=c & mask;\r\n c>>=bpe;\r\n }\r\n}\r\n\r\n//do x=x-(y<<(ys*bpe)) for bigInts x and y, and integers a,b and ys.\r\n//x must be large enough to hold the answer.\r\nfunction subShift_(x,y,ys) {\r\n var i,c,k,kk;\r\n k=x.length<ys+y.length ? x.length : ys+y.length;\r\n kk=x.length;\r\n for (c=0,i=ys;i<k;i++) {\r\n c+=x[i]-y[i-ys];\r\n x[i]=c & mask;\r\n c>>=bpe;\r\n }\r\n for (i=k;c && i<kk;i++) {\r\n c+=x[i];\r\n x[i]=c & mask;\r\n c>>=bpe;\r\n }\r\n}\r\n\r\n//do x=x-y for bigInts x and y.\r\n//x must be large enough to hold the answer.\r\n//negative answers will be 2s complement\r\nexport function sub_(x,y) {\r\n var i,c,k,kk;\r\n k=x.length<y.length ? x.length : y.length;\r\n for (c=0,i=0;i<k;i++) {\r\n c+=x[i]-y[i];\r\n x[i]=c & mask;\r\n c>>=bpe;\r\n }\r\n for (i=k;c && i<x.length;i++) {\r\n c+=x[i];\r\n x[i]=c & mask;\r\n c>>=bpe;\r\n }\r\n}\r\n\r\n//do x=x+y for bigInts x and y.\r\n//x must be large enough to hold the answer.\r\nexport function add_(x,y) {\r\n var i,c,k,kk;\r\n k=x.length<y.length ? x.length : y.length;\r\n for (c=0,i=0;i<k;i++) {\r\n c+=x[i]+y[i];\r\n x[i]=c & mask;\r\n c>>=bpe;\r\n }\r\n for (i=k;c && i<x.length;i++) {\r\n c+=x[i];\r\n x[i]=c & mask;\r\n c>>=bpe;\r\n }\r\n}\r\n\r\n//do x=x*y for bigInts x and y. This is faster when y<x.\r\nfunction mult_(x,y) {\r\n var i;\r\n if (ss.length!=2*x.length)\r\n ss=new Array(2*x.length);\r\n copyInt_(ss,0);\r\n for (i=0;i<y.length;i++)\r\n if (y[i])\r\n linCombShift_(ss,x,y[i],i); //ss=1*ss+y[i]*(x<<(i*bpe))\r\n copy_(x,ss);\r\n}\r\n\r\n//do x=x mod n for bigInts x and n.\r\nfunction mod_(x,n) {\r\n if (s4.length!=x.length)\r\n s4=dup(x);\r\n else\r\n copy_(s4,x);\r\n if (s5.length!=x.length)\r\n s5=dup(x);\r\n divide_(s4,n,s5,x); //x = remainder of s4 / n\r\n}\r\n\r\n//do x=x*y mod n for bigInts x,y,n.\r\n//for greater speed, let y<x.\r\nfunction multMod_(x,y,n) {\r\n var i;\r\n if (s0.length!=2*x.length)\r\n s0=new Array(2*x.length);\r\n copyInt_(s0,0);\r\n for (i=0;i<y.length;i++)\r\n if (y[i])\r\n linCombShift_(s0,x,y[i],i); //s0=1*s0+y[i]*(x<<(i*bpe))\r\n mod_(s0,n);\r\n copy_(x,s0);\r\n}\r\n\r\n//do x=x*x mod n for bigInts x,n.\r\nfunction squareMod_(x,n) {\r\n var i,j,d,c,kx,kn,k;\r\n for (kx=x.length; kx>0 && !x[kx-1]; kx--); //ignore leading zeros in x\r\n 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\r\n if (s0.length!=k)\r\n s0=new Array(k);\r\n copyInt_(s0,0);\r\n for (i=0;i<kx;i++) {\r\n c=s0[2*i]+x[i]*x[i];\r\n s0[2*i]=c & mask;\r\n c>>=bpe;\r\n for (j=i+1;j<kx;j++) {\r\n c=s0[i+j]+2*x[i]*x[j]+c;\r\n s0[i+j]=(c & mask);\r\n c>>=bpe;\r\n }\r\n s0[i+kx]=c;\r\n }\r\n mod_(s0,n);\r\n copy_(x,s0);\r\n}\r\n\r\n//return x with exactly k leading zero elements\r\nfunction trim(x,k) {\r\n var i,y;\r\n for (i=x.length; i>0 && !x[i-1]; i--);\r\n y=new Array(i+k);\r\n copy_(y,x);\r\n return y;\r\n}\r\n\r\n//do x=x**y mod n, where x,y,n are bigInts and ** is exponentiation. 0**0=1.\r\n//this is faster when n is odd. x usually needs to have as many elements as n.\r\nfunction powMod_(x,y,n) {\r\n var k1,k2,kn,np;\r\n if(s7.length!=n.length)\r\n s7=dup(n);\r\n\r\n //for even modulus, use a simple square-and-multiply algorithm,\r\n //rather than using the more complex Montgomery algorithm.\r\n if ((n[0]&1)==0) {\r\n copy_(s7,x);\r\n copyInt_(x,1);\r\n while(!equalsInt(y,0)) {\r\n if (y[0]&1)\r\n multMod_(x,s7,n);\r\n divInt_(y,2);\r\n squareMod_(s7,n);\r\n }\r\n return;\r\n }\r\n\r\n //calculate np from n for the Montgomery multiplications\r\n copyInt_(s7,0);\r\n for (kn=n.length;kn>0 && !n[kn-1];kn--);\r\n np=radix-inverseModInt(modInt(n,radix),radix);\r\n s7[kn]=1;\r\n multMod_(x ,s7,n); // x = x * 2**(kn*bp) mod n\r\n\r\n if (s3.length!=x.length)\r\n s3=dup(x);\r\n else\r\n copy_(s3,x);\r\n\r\n for (k1=y.length-1;k1>0 & !y[k1]; k1--); //k1=first nonzero element of y\r\n if (y[k1]==0) { //anything to the 0th power is 1\r\n copyInt_(x,1);\r\n return;\r\n }\r\n for (k2=1<<(bpe-1);k2 && !(y[k1] & k2); k2>>=1); //k2=position of first 1 bit in y[k1]\r\n for (;;) {\r\n if (!(k2>>=1)) { //look at next bit of y\r\n k1--;\r\n if (k1<0) {\r\n mont_(x,one,n,np);\r\n return;\r\n }\r\n k2=1<<(bpe-1);\r\n }\r\n mont_(x,x,n,np);\r\n\r\n if (k2 & y[k1]) //if next bit is a 1\r\n mont_(x,s3,n,np);\r\n }\r\n}\r\n\r\n\r\n//do x=x*y*Ri mod n for bigInts x,y,n,\r\n// where Ri = 2**(-kn*bpe) mod n, and kn is the\r\n// number of elements in the n array, not\r\n// counting leading zeros.\r\n//x array must have at least as many elemnts as the n array\r\n//It's OK if x and y are the same variable.\r\n//must have:\r\n// x,y < n\r\n// n is odd\r\n// np = -(n^(-1)) mod radix\r\nfunction mont_(x,y,n,np) {\r\n var i,j,c,ui,t,ks;\r\n var kn=n.length;\r\n var ky=y.length;\r\n\r\n if (sa.length!=kn)\r\n sa=new Array(kn);\r\n\r\n copyInt_(sa,0);\r\n\r\n for (;kn>0 && n[kn-1]==0;kn--); //ignore leading zeros of n\r\n for (;ky>0 && y[ky-1]==0;ky--); //ignore leading zeros of y\r\n ks=sa.length-1; //sa will never have more than this many nonzero elements.\r\n\r\n //the following loop consumes 95% of the runtime for randTruePrime_() and powMod_() for large numbers\r\n for (i=0; i<kn; i++) {\r\n t=sa[0]+x[i]*y[0];\r\n ui=((t & mask) * np) & mask; //the inner \"& mask\" was needed on Safari (but not MSIE) at one time\r\n c=(t+ui*n[0]) >> bpe;\r\n t=x[i];\r\n\r\n //do sa=(sa+x[i]*y+ui*n)/b where b=2**bpe. Loop is unrolled 5-fold for speed\r\n j=1;\r\n for (;j<ky-4;) { c+=sa[j]+ui*n[j]+t*y[j]; sa[j-1]=c & mask; c>>=bpe; j++;\r\n c+=sa[j]+ui*n[j]+t*y[j]; sa[j-1]=c & mask; c>>=bpe; j++;\r\n c+=sa[j]+ui*n[j]+t*y[j]; sa[j-1]=c & mask; c>>=bpe; j++;\r\n c+=sa[j]+ui*n[j]+t*y[j]; sa[j-1]=c & mask; c>>=bpe; j++;\r\n c+=sa[j]+ui*n[j]+t*y[j]; sa[j-1]=c & mask; c>>=bpe; j++; }\r\n for (;j<ky;) { c+=sa[j]+ui*n[j]+t*y[j]; sa[j-1]=c & mask; c>>=bpe; j++; }\r\n for (;j<kn-4;) { c+=sa[j]+ui*n[j]; sa[j-1]=c & mask; c>>=bpe; j++;\r\n c+=sa[j]+ui*n[j]; sa[j-1]=c & mask; c>>=bpe; j++;\r\n c+=sa[j]+ui*n[j]; sa[j-1]=c & mask; c>>=bpe; j++;\r\n c+=sa[j]+ui*n[j]; sa[j-1]=c & mask; c>>=bpe; j++;\r\n c+=sa[j]+ui*n[j]; sa[j-1]=c & mask; c>>=bpe; j++; }\r\n for (;j<kn;) { c+=sa[j]+ui*n[j]; sa[j-1]=c & mask; c>>=bpe; j++; }\r\n for (;j<ks;) { c+=sa[j]; sa[j-1]=c & mask; c>>=bpe; j++; }\r\n sa[j-1]=c & mask;\r\n }\r\n\r\n if (!greater(n,sa))\r\n sub_(sa,n);\r\n copy_(x,sa);\r\n}\r\n\r\n"]} |