[2847] | 1 | //////////////////////////////////////////////////////////////////////////////////////// |
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| 2 | // Big Integer Library v. 5.4 |
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| 3 | // Created 2000, last modified 2009 |
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| 4 | // Leemon Baird |
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| 5 | // www.leemon.com |
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| 6 | // |
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| 7 | // Version history: |
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| 8 | // v 5.4 3 Oct 2009 |
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| 9 | // - added "var i" to greaterShift() so i is not global. (Thanks to Pr Szabor finding that bug) |
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| 10 | // |
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| 11 | // v 5.3 21 Sep 2009 |
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| 12 | // - added randProbPrime(k) for probable primes |
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| 13 | // - unrolled loop in mont_ (slightly faster) |
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| 14 | // - millerRabin now takes a bigInt parameter rather than an int |
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| 15 | // |
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| 16 | // v 5.2 15 Sep 2009 |
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| 17 | // - fixed capitalization in call to int2bigInt in randBigInt |
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| 18 | // (thanks to Emili Evripidou, Reinhold Behringer, and Samuel Macaleese for finding that bug) |
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| 19 | // |
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| 20 | // v 5.1 8 Oct 2007 |
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| 21 | // - renamed inverseModInt_ to inverseModInt since it doesn't change its parameters |
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| 22 | // - added functions GCD and randBigInt, which call GCD_ and randBigInt_ |
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| 23 | // - fixed a bug found by Rob Visser (see comment with his name below) |
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| 24 | // - improved comments |
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| 25 | // |
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| 26 | // This file is public domain. You can use it for any purpose without restriction. |
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| 27 | // I do not guarantee that it is correct, so use it at your own risk. If you use |
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| 28 | // it for something interesting, I'd appreciate hearing about it. If you find |
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| 29 | // any bugs or make any improvements, I'd appreciate hearing about those too. |
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| 30 | // It would also be nice if my name and URL were left in the comments. But none |
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| 31 | // of that is required. |
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| 32 | // |
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| 33 | // This code defines a bigInt library for arbitrary-precision integers. |
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| 34 | // A bigInt is an array of integers storing the value in chunks of bpe bits, |
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| 35 | // little endian (buff[0] is the least significant word). |
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| 36 | // Negative bigInts are stored two's complement. Almost all the functions treat |
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| 37 | // bigInts as nonnegative. The few that view them as two's complement say so |
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| 38 | // in their comments. Some functions assume their parameters have at least one |
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| 39 | // leading zero element. Functions with an underscore at the end of the name put |
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| 40 | // their answer into one of the arrays passed in, and have unpredictable behavior |
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| 41 | // in case of overflow, so the caller must make sure the arrays are big enough to |
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| 42 | // hold the answer. But the average user should never have to call any of the |
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| 43 | // underscored functions. Each important underscored function has a wrapper function |
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| 44 | // of the same name without the underscore that takes care of the details for you. |
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| 45 | // For each underscored function where a parameter is modified, that same variable |
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| 46 | // must not be used as another argument too. So, you cannot square x by doing |
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| 47 | // multMod_(x,x,n). You must use squareMod_(x,n) instead, or do y=dup(x); multMod_(x,y,n). |
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| 48 | // Or simply use the multMod(x,x,n) function without the underscore, where |
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| 49 | // such issues never arise, because non-underscored functions never change |
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| 50 | // their parameters; they always allocate new memory for the answer that is returned. |
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| 51 | // |
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| 52 | // These functions are designed to avoid frequent dynamic memory allocation in the inner loop. |
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| 53 | // For most functions, if it needs a BigInt as a local variable it will actually use |
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| 54 | // a global, and will only allocate to it only when it's not the right size. This ensures |
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| 55 | // that when a function is called repeatedly with same-sized parameters, it only allocates |
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| 56 | // memory on the first call. |
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| 57 | // |
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| 58 | // Note that for cryptographic purposes, the calls to Math.random() must |
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| 59 | // be replaced with calls to a better pseudorandom number generator. |
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| 60 | // |
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| 61 | // In the following, "bigInt" means a bigInt with at least one leading zero element, |
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| 62 | // and "integer" means a nonnegative integer less than radix. In some cases, integer |
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| 63 | // can be negative. Negative bigInts are 2s complement. |
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| 64 | // |
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| 65 | // The following functions do not modify their inputs. |
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| 66 | // Those returning a bigInt, string, or Array will dynamically allocate memory for that value. |
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| 67 | // Those returning a boolean will return the integer 0 (false) or 1 (true). |
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| 68 | // Those returning boolean or int will not allocate memory except possibly on the first |
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| 69 | // time they're called with a given parameter size. |
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| 70 | // |
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| 71 | // bigInt add(x,y) //return (x+y) for bigInts x and y. |
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| 72 | // bigInt addInt(x,n) //return (x+n) where x is a bigInt and n is an integer. |
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| 73 | // string bigInt2str(x,base) //return a string form of bigInt x in a given base, with 2 <= base <= 95 |
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| 74 | // int bitSize(x) //return how many bits long the bigInt x is, not counting leading zeros |
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| 75 | // bigInt dup(x) //return a copy of bigInt x |
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| 76 | // boolean equals(x,y) //is the bigInt x equal to the bigint y? |
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| 77 | // boolean equalsInt(x,y) //is bigint x equal to integer y? |
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| 78 | // bigInt expand(x,n) //return a copy of x with at least n elements, adding leading zeros if needed |
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| 79 | // Array findPrimes(n) //return array of all primes less than integer n |
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| 80 | // bigInt GCD(x,y) //return greatest common divisor of bigInts x and y (each with same number of elements). |
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| 81 | // boolean greater(x,y) //is x>y? (x and y are nonnegative bigInts) |
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| 82 | // boolean greaterShift(x,y,shift)//is (x <<(shift*bpe)) > y? |
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| 83 | // bigInt int2bigInt(t,n,m) //return a bigInt equal to integer t, with at least n bits and m array elements |
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| 84 | // bigInt inverseMod(x,n) //return (x**(-1) mod n) for bigInts x and n. If no inverse exists, it returns null |
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| 85 | // int inverseModInt(x,n) //return x**(-1) mod n, for integers x and n. Return 0 if there is no inverse |
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| 86 | // boolean isZero(x) //is the bigInt x equal to zero? |
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| 87 | // 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) |
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| 88 | // 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) |
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| 89 | // bigInt mod(x,n) //return a new bigInt equal to (x mod n) for bigInts x and n. |
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| 90 | // int modInt(x,n) //return x mod n for bigInt x and integer n. |
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| 91 | // bigInt mult(x,y) //return x*y for bigInts x and y. This is faster when y<x. |
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| 92 | // bigInt multMod(x,y,n) //return (x*y mod n) for bigInts x,y,n. For greater speed, let y<x. |
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| 93 | // boolean negative(x) //is bigInt x negative? |
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| 94 | // 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. |
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| 95 | // 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. |
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| 96 | // bigInt randTruePrime(k) //return a new, random, k-bit, true prime bigInt using Maurer's algorithm. |
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| 97 | // bigInt randProbPrime(k) //return a new, random, k-bit, probable prime bigInt (probability it's composite less than 2^-80). |
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| 98 | // 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 |
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| 99 | // bigInt sub(x,y) //return (x-y) for bigInts x and y. Negative answers will be 2s complement |
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| 100 | // bigInt trim(x,k) //return a copy of x with exactly k leading zero elements |
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| 101 | // |
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| 102 | // |
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| 103 | // The following functions each have a non-underscored version, which most users should call instead. |
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| 104 | // These functions each write to a single parameter, and the caller is responsible for ensuring the array |
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| 105 | // passed in is large enough to hold the result. |
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| 106 | // |
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| 107 | // void addInt_(x,n) //do x=x+n where x is a bigInt and n is an integer |
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| 108 | // void add_(x,y) //do x=x+y for bigInts x and y |
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| 109 | // void copy_(x,y) //do x=y on bigInts x and y |
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| 110 | // void copyInt_(x,n) //do x=n on bigInt x and integer n |
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| 111 | // void GCD_(x,y) //set x to the greatest common divisor of bigInts x and y, (y is destroyed). (This never overflows its array). |
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| 112 | // boolean inverseMod_(x,n) //do x=x**(-1) mod n, for bigInts x and n. Returns 1 (0) if inverse does (doesn't) exist |
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| 113 | // void mod_(x,n) //do x=x mod n for bigInts x and n. (This never overflows its array). |
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| 114 | // void mult_(x,y) //do x=x*y for bigInts x and y. |
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| 115 | // void multMod_(x,y,n) //do x=x*y mod n for bigInts x,y,n. |
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| 116 | // 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. |
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| 117 | // 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. |
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| 118 | // void randTruePrime_(ans,k) //do ans = a random k-bit true random prime (not just probable prime) with 1 in the msb. |
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| 119 | // void sub_(x,y) //do x=x-y for bigInts x and y. Negative answers will be 2s complement. |
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| 120 | // |
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| 121 | // The following functions do NOT have a non-underscored version. |
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| 122 | // They each write a bigInt result to one or more parameters. The caller is responsible for |
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| 123 | // ensuring the arrays passed in are large enough to hold the results. |
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| 124 | // |
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| 125 | // void addShift_(x,y,ys) //do x=x+(y<<(ys*bpe)) |
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| 126 | // void carry_(x) //do carries and borrows so each element of the bigInt x fits in bpe bits. |
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| 127 | // void divide_(x,y,q,r) //divide x by y giving quotient q and remainder r |
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| 128 | // int divInt_(x,n) //do x=floor(x/n) for bigInt x and integer n, and return the remainder. (This never overflows its array). |
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| 129 | // int eGCD_(x,y,d,a,b) //sets a,b,d to positive bigInts such that d = GCD_(x,y) = a*x-b*y |
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| 130 | // void halve_(x) //do x=floor(|x|/2)*sgn(x) for bigInt x in 2's complement. (This never overflows its array). |
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| 131 | // void leftShift_(x,n) //left shift bigInt x by n bits. n<bpe. |
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| 132 | // void linComb_(x,y,a,b) //do x=a*x+b*y for bigInts x and y and integers a and b |
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| 133 | // void linCombShift_(x,y,b,ys) //do x=x+b*(y<<(ys*bpe)) for bigInts x and y, and integers b and ys |
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| 134 | // void mont_(x,y,n,np) //Montgomery multiplication (see comments where the function is defined) |
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| 135 | // void multInt_(x,n) //do x=x*n where x is a bigInt and n is an integer. |
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| 136 | // void rightShift_(x,n) //right shift bigInt x by n bits. 0 <= n < bpe. (This never overflows its array). |
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| 137 | // void squareMod_(x,n) //do x=x*x mod n for bigInts x,n |
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| 138 | // void subShift_(x,y,ys) //do x=x-(y<<(ys*bpe)). Negative answers will be 2s complement. |
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| 139 | // |
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| 140 | // The following functions are based on algorithms from the _Handbook of Applied Cryptography_ |
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| 141 | // powMod_() = algorithm 14.94, Montgomery exponentiation |
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| 142 | // eGCD_,inverseMod_() = algorithm 14.61, Binary extended GCD_ |
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| 143 | // GCD_() = algorothm 14.57, Lehmer's algorithm |
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| 144 | // mont_() = algorithm 14.36, Montgomery multiplication |
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| 145 | // divide_() = algorithm 14.20 Multiple-precision division |
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| 146 | // squareMod_() = algorithm 14.16 Multiple-precision squaring |
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| 147 | // randTruePrime_() = algorithm 4.62, Maurer's algorithm |
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| 148 | // millerRabin() = algorithm 4.24, Miller-Rabin algorithm |
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| 149 | // |
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| 150 | // Profiling shows: |
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| 151 | // randTruePrime_() spends: |
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| 152 | // 10% of its time in calls to powMod_() |
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| 153 | // 85% of its time in calls to millerRabin() |
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| 154 | // millerRabin() spends: |
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| 155 | // 99% of its time in calls to powMod_() (always with a base of 2) |
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| 156 | // powMod_() spends: |
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| 157 | // 94% of its time in calls to mont_() (almost always with x==y) |
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| 158 | // |
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| 159 | // This suggests there are several ways to speed up this library slightly: |
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| 160 | // - convert powMod_ to use a Montgomery form of k-ary window (or maybe a Montgomery form of sliding window) |
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| 161 | // -- this should especially focus on being fast when raising 2 to a power mod n |
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| 162 | // - convert randTruePrime_() to use a minimum r of 1/3 instead of 1/2 with the appropriate change to the test |
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| 163 | // - tune the parameters in randTruePrime_(), including c, m, and recLimit |
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| 164 | // - speed up the single loop in mont_() that takes 95% of the runtime, perhaps by reducing checking |
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| 165 | // within the loop when all the parameters are the same length. |
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| 166 | // |
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| 167 | // There are several ideas that look like they wouldn't help much at all: |
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| 168 | // - replacing trial division in randTruePrime_() with a sieve (that speeds up something taking almost no time anyway) |
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| 169 | // - 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) |
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| 170 | // - speeding up mont_(x,y,n,np) when x==y by doing a non-modular, non-Montgomery square |
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| 171 | // followed by a Montgomery reduction. The intermediate answer will be twice as long as x, so that |
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| 172 | // method would be slower. This is unfortunate because the code currently spends almost all of its time |
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| 173 | // doing mont_(x,x,...), both for randTruePrime_() and powMod_(). A faster method for Montgomery squaring |
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| 174 | // would have a large impact on the speed of randTruePrime_() and powMod_(). HAC has a couple of poorly-worded |
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| 175 | // sentences that seem to imply it's faster to do a non-modular square followed by a single |
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| 176 | // Montgomery reduction, but that's obviously wrong. |
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| 177 | //////////////////////////////////////////////////////////////////////////////////////// |
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| 178 | |
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| 179 | //globals |
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| 180 | bpe=0; //bits stored per array element |
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| 181 | mask=0; //AND this with an array element to chop it down to bpe bits |
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| 182 | radix=mask+1; //equals 2^bpe. A single 1 bit to the left of the last bit of mask. |
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| 183 | |
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| 184 | //the digits for converting to different bases |
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| 185 | digitsStr='0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz_=!@#$%^&*()[]{}|;:,.<>/?`~ \\\'\"+-'; |
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| 186 | |
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| 187 | //initialize the global variables |
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| 188 | for (bpe=0; (1<<(bpe+1)) > (1<<bpe); bpe++); //bpe=number of bits in the mantissa on this platform |
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| 189 | bpe>>=1; //bpe=number of bits in one element of the array representing the bigInt |
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| 190 | mask=(1<<bpe)-1; //AND the mask with an integer to get its bpe least significant bits |
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| 191 | radix=mask+1; //2^bpe. a single 1 bit to the left of the first bit of mask |
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| 192 | one=int2bigInt(1,1,1); //constant used in powMod_() |
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| 193 | |
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| 194 | //the following global variables are scratchpad memory to |
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| 195 | //reduce dynamic memory allocation in the inner loop |
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| 196 | t=new Array(0); |
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| 197 | ss=t; //used in mult_() |
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| 198 | s0=t; //used in multMod_(), squareMod_() |
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| 199 | s1=t; //used in powMod_(), multMod_(), squareMod_() |
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| 200 | s2=t; //used in powMod_(), multMod_() |
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| 201 | s3=t; //used in powMod_() |
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| 202 | s4=t; s5=t; //used in mod_() |
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| 203 | s6=t; //used in bigInt2str() |
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| 204 | s7=t; //used in powMod_() |
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| 205 | T=t; //used in GCD_() |
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| 206 | sa=t; //used in mont_() |
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| 207 | mr_x1=t; mr_r=t; mr_a=t; //used in millerRabin() |
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| 208 | eg_v=t; eg_u=t; eg_A=t; eg_B=t; eg_C=t; eg_D=t; //used in eGCD_(), inverseMod_() |
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| 209 | md_q1=t; md_q2=t; md_q3=t; md_r=t; md_r1=t; md_r2=t; md_tt=t; //used in mod_() |
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| 210 | |
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| 211 | primes=t; pows=t; s_i=t; s_i2=t; s_R=t; s_rm=t; s_q=t; s_n1=t; |
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| 212 | 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_() |
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| 213 | |
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| 214 | rpprb=t; //used in randProbPrimeRounds() (which also uses "primes") |
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| 215 | |
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| 216 | //////////////////////////////////////////////////////////////////////////////////////// |
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| 217 | |
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| 218 | |
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| 219 | //return array of all primes less than integer n |
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| 220 | function findPrimes(n) { |
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| 221 | var i,s,p,ans; |
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| 222 | s=new Array(n); |
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| 223 | for (i=0;i<n;i++) |
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| 224 | s[i]=0; |
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| 225 | s[0]=2; |
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| 226 | p=0; //first p elements of s are primes, the rest are a sieve |
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| 227 | for(;s[p]<n;) { //s[p] is the pth prime |
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| 228 | for(i=s[p]*s[p]; i<n; i+=s[p]) //mark multiples of s[p] |
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| 229 | s[i]=1; |
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| 230 | p++; |
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| 231 | s[p]=s[p-1]+1; |
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| 232 | for(; s[p]<n && s[s[p]]; s[p]++); //find next prime (where s[p]==0) |
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| 233 | } |
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| 234 | ans=new Array(p); |
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| 235 | for(i=0;i<p;i++) |
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| 236 | ans[i]=s[i]; |
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| 237 | return ans; |
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| 238 | } |
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| 239 | |
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| 240 | |
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| 241 | //does a single round of Miller-Rabin base b consider x to be a possible prime? |
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| 242 | //x is a bigInt, and b is an integer, with b<x |
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| 243 | function millerRabinInt(x,b) { |
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| 244 | if (mr_x1.length!=x.length) { |
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| 245 | mr_x1=dup(x); |
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| 246 | mr_r=dup(x); |
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| 247 | mr_a=dup(x); |
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| 248 | } |
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| 249 | |
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| 250 | copyInt_(mr_a,b); |
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| 251 | return millerRabin(x,mr_a); |
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| 252 | } |
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| 253 | |
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| 254 | //does a single round of Miller-Rabin base b consider x to be a possible prime? |
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| 255 | //x and b are bigInts with b<x |
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| 256 | function millerRabin(x,b) { |
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| 257 | var i,j,k,s; |
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| 258 | |
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| 259 | if (mr_x1.length!=x.length) { |
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| 260 | mr_x1=dup(x); |
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| 261 | mr_r=dup(x); |
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| 262 | mr_a=dup(x); |
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| 263 | } |
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| 264 | |
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| 265 | copy_(mr_a,b); |
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| 266 | copy_(mr_r,x); |
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| 267 | copy_(mr_x1,x); |
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| 268 | |
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| 269 | addInt_(mr_r,-1); |
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| 270 | addInt_(mr_x1,-1); |
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| 271 | |
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| 272 | //s=the highest power of two that divides mr_r |
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| 273 | k=0; |
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| 274 | for (i=0;i<mr_r.length;i++) |
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| 275 | for (j=1;j<mask;j<<=1) |
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| 276 | if (x[i] & j) { |
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| 277 | s=(k<mr_r.length+bpe ? k : 0); |
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| 278 | i=mr_r.length; |
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| 279 | j=mask; |
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| 280 | } else |
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| 281 | k++; |
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| 282 | |
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| 283 | if (s) |
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| 284 | rightShift_(mr_r,s); |
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| 285 | |
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| 286 | powMod_(mr_a,mr_r,x); |
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| 287 | |
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| 288 | if (!equalsInt(mr_a,1) && !equals(mr_a,mr_x1)) { |
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| 289 | j=1; |
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| 290 | while (j<=s-1 && !equals(mr_a,mr_x1)) { |
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| 291 | squareMod_(mr_a,x); |
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| 292 | if (equalsInt(mr_a,1)) { |
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| 293 | return 0; |
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| 294 | } |
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| 295 | j++; |
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| 296 | } |
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| 297 | if (!equals(mr_a,mr_x1)) { |
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| 298 | return 0; |
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| 299 | } |
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| 300 | } |
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| 301 | return 1; |
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| 302 | } |
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| 303 | |
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| 304 | //returns how many bits long the bigInt is, not counting leading zeros. |
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| 305 | function bitSize(x) { |
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| 306 | var j,z,w; |
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| 307 | for (j=x.length-1; (x[j]==0) && (j>0); j--); |
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| 308 | for (z=0,w=x[j]; w; (w>>=1),z++); |
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| 309 | z+=bpe*j; |
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| 310 | return z; |
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| 311 | } |
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| 312 | |
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| 313 | //return a copy of x with at least n elements, adding leading zeros if needed |
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| 314 | function expand(x,n) { |
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| 315 | var ans=int2bigInt(0,(x.length>n ? x.length : n)*bpe,0); |
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| 316 | copy_(ans,x); |
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| 317 | return ans; |
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| 318 | } |
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| 319 | |
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| 320 | //return a k-bit true random prime using Maurer's algorithm. |
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| 321 | function randTruePrime(k) { |
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| 322 | var ans=int2bigInt(0,k,0); |
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| 323 | randTruePrime_(ans,k); |
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| 324 | return trim(ans,1); |
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| 325 | } |
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| 326 | |
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| 327 | //return a k-bit random probable prime with probability of error < 2^-80 |
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| 328 | function randProbPrime(k) { |
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| 329 | if (k>=600) return randProbPrimeRounds(k,2); //numbers from HAC table 4.3 |
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| 330 | if (k>=550) return randProbPrimeRounds(k,4); |
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| 331 | if (k>=500) return randProbPrimeRounds(k,5); |
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| 332 | if (k>=400) return randProbPrimeRounds(k,6); |
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| 333 | if (k>=350) return randProbPrimeRounds(k,7); |
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| 334 | if (k>=300) return randProbPrimeRounds(k,9); |
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| 335 | if (k>=250) return randProbPrimeRounds(k,12); //numbers from HAC table 4.4 |
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| 336 | if (k>=200) return randProbPrimeRounds(k,15); |
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| 337 | if (k>=150) return randProbPrimeRounds(k,18); |
---|
| 338 | if (k>=100) return randProbPrimeRounds(k,27); |
---|
| 339 | return randProbPrimeRounds(k,40); //number from HAC remark 4.26 (only an estimate) |
---|
| 340 | } |
---|
| 341 | |
---|
| 342 | //return a k-bit probable random prime using n rounds of Miller Rabin (after trial division with small primes) |
---|
| 343 | function randProbPrimeRounds(k,n) { |
---|
| 344 | var ans, i, divisible, B; |
---|
| 345 | B=30000; //B is largest prime to use in trial division |
---|
| 346 | ans=int2bigInt(0,k,0); |
---|
| 347 | |
---|
| 348 | //optimization: try larger and smaller B to find the best limit. |
---|
| 349 | |
---|
| 350 | if (primes.length==0) |
---|
| 351 | primes=findPrimes(30000); //check for divisibility by primes <=30000 |
---|
| 352 | |
---|
| 353 | if (rpprb.length!=ans.length) |
---|
| 354 | rpprb=dup(ans); |
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| 355 | |
---|
| 356 | for (;;) { //keep trying random values for ans until one appears to be prime |
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| 357 | //optimization: pick a random number times L=2*3*5*...*p, plus a |
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| 358 | // random element of the list of all numbers in [0,L) not divisible by any prime up to p. |
---|
| 359 | // This can reduce the amount of random number generation. |
---|
| 360 | |
---|
| 361 | randBigInt_(ans,k,0); //ans = a random odd number to check |
---|
| 362 | ans[0] |= 1; |
---|
| 363 | divisible=0; |
---|
| 364 | |
---|
| 365 | //check ans for divisibility by small primes up to B |
---|
| 366 | for (i=0; (i<primes.length) && (primes[i]<=B); i++) |
---|
| 367 | if (modInt(ans,primes[i])==0 && !equalsInt(ans,primes[i])) { |
---|
| 368 | divisible=1; |
---|
| 369 | break; |
---|
| 370 | } |
---|
| 371 | |
---|
| 372 | //optimization: change millerRabin so the base can be bigger than the number being checked, then eliminate the while here. |
---|
| 373 | |
---|
| 374 | //do n rounds of Miller Rabin, with random bases less than ans |
---|
| 375 | for (i=0; i<n && !divisible; i++) { |
---|
| 376 | randBigInt_(rpprb,k,0); |
---|
| 377 | while(!greater(ans,rpprb)) //pick a random rpprb that's < ans |
---|
| 378 | randBigInt_(rpprb,k,0); |
---|
| 379 | if (!millerRabin(ans,rpprb)) |
---|
| 380 | divisible=1; |
---|
| 381 | } |
---|
| 382 | |
---|
| 383 | if(!divisible) |
---|
| 384 | return ans; |
---|
| 385 | } |
---|
| 386 | } |
---|
| 387 | |
---|
| 388 | //return a new bigInt equal to (x mod n) for bigInts x and n. |
---|
| 389 | function mod(x,n) { |
---|
| 390 | var ans=dup(x); |
---|
| 391 | mod_(ans,n); |
---|
| 392 | return trim(ans,1); |
---|
| 393 | } |
---|
| 394 | |
---|
| 395 | //return (x+n) where x is a bigInt and n is an integer. |
---|
| 396 | function addInt(x,n) { |
---|
| 397 | var ans=expand(x,x.length+1); |
---|
| 398 | addInt_(ans,n); |
---|
| 399 | return trim(ans,1); |
---|
| 400 | } |
---|
| 401 | |
---|
| 402 | //return x*y for bigInts x and y. This is faster when y<x. |
---|
| 403 | function mult(x,y) { |
---|
| 404 | var ans=expand(x,x.length+y.length); |
---|
| 405 | mult_(ans,y); |
---|
| 406 | return trim(ans,1); |
---|
| 407 | } |
---|
| 408 | |
---|
| 409 | //return (x**y mod n) where x,y,n are bigInts and ** is exponentiation. 0**0=1. Faster for odd n. |
---|
| 410 | function powMod(x,y,n) { |
---|
| 411 | var ans=expand(x,n.length); |
---|
| 412 | powMod_(ans,trim(y,2),trim(n,2),0); //this should work without the trim, but doesn't |
---|
| 413 | return trim(ans,1); |
---|
| 414 | } |
---|
| 415 | |
---|
| 416 | //return (x-y) for bigInts x and y. Negative answers will be 2s complement |
---|
| 417 | function sub(x,y) { |
---|
| 418 | var ans=expand(x,(x.length>y.length ? x.length+1 : y.length+1)); |
---|
| 419 | sub_(ans,y); |
---|
| 420 | return trim(ans,1); |
---|
| 421 | } |
---|
| 422 | |
---|
| 423 | //return (x+y) for bigInts x and y. |
---|
| 424 | function add(x,y) { |
---|
| 425 | var ans=expand(x,(x.length>y.length ? x.length+1 : y.length+1)); |
---|
| 426 | add_(ans,y); |
---|
| 427 | return trim(ans,1); |
---|
| 428 | } |
---|
| 429 | |
---|
| 430 | //return (x**(-1) mod n) for bigInts x and n. If no inverse exists, it returns null |
---|
| 431 | function inverseMod(x,n) { |
---|
| 432 | var ans=expand(x,n.length); |
---|
| 433 | var s; |
---|
| 434 | s=inverseMod_(ans,n); |
---|
| 435 | return s ? trim(ans,1) : null; |
---|
| 436 | } |
---|
| 437 | |
---|
| 438 | //return (x*y mod n) for bigInts x,y,n. For greater speed, let y<x. |
---|
| 439 | function multMod(x,y,n) { |
---|
| 440 | var ans=expand(x,n.length); |
---|
| 441 | multMod_(ans,y,n); |
---|
| 442 | return trim(ans,1); |
---|
| 443 | } |
---|
| 444 | |
---|
| 445 | //generate a k-bit true random prime using Maurer's algorithm, |
---|
| 446 | //and put it into ans. The bigInt ans must be large enough to hold it. |
---|
| 447 | function randTruePrime_(ans,k) { |
---|
| 448 | var c,m,pm,dd,j,r,B,divisible,z,zz,recSize; |
---|
| 449 | |
---|
| 450 | if (primes.length==0) |
---|
| 451 | primes=findPrimes(30000); //check for divisibility by primes <=30000 |
---|
| 452 | |
---|
| 453 | if (pows.length==0) { |
---|
| 454 | pows=new Array(512); |
---|
| 455 | for (j=0;j<512;j++) { |
---|
| 456 | pows[j]=Math.pow(2,j/511.-1.); |
---|
| 457 | } |
---|
| 458 | } |
---|
| 459 | |
---|
| 460 | //c and m should be tuned for a particular machine and value of k, to maximize speed |
---|
| 461 | c=0.1; //c=0.1 in HAC |
---|
| 462 | m=20; //generate this k-bit number by first recursively generating a number that has between k/2 and k-m bits |
---|
| 463 | recLimit=20; //stop recursion when k <=recLimit. Must have recLimit >= 2 |
---|
| 464 | |
---|
| 465 | if (s_i2.length!=ans.length) { |
---|
| 466 | s_i2=dup(ans); |
---|
| 467 | s_R =dup(ans); |
---|
| 468 | s_n1=dup(ans); |
---|
| 469 | s_r2=dup(ans); |
---|
| 470 | s_d =dup(ans); |
---|
| 471 | s_x1=dup(ans); |
---|
| 472 | s_x2=dup(ans); |
---|
| 473 | s_b =dup(ans); |
---|
| 474 | s_n =dup(ans); |
---|
| 475 | s_i =dup(ans); |
---|
| 476 | s_rm=dup(ans); |
---|
| 477 | s_q =dup(ans); |
---|
| 478 | s_a =dup(ans); |
---|
| 479 | s_aa=dup(ans); |
---|
| 480 | } |
---|
| 481 | |
---|
| 482 | if (k <= recLimit) { //generate small random primes by trial division up to its square root |
---|
| 483 | pm=(1<<((k+2)>>1))-1; //pm is binary number with all ones, just over sqrt(2^k) |
---|
| 484 | copyInt_(ans,0); |
---|
| 485 | for (dd=1;dd;) { |
---|
| 486 | dd=0; |
---|
| 487 | ans[0]= 1 | (1<<(k-1)) | Math.floor(Math.random()*(1<<k)); //random, k-bit, odd integer, with msb 1 |
---|
| 488 | for (j=1;(j<primes.length) && ((primes[j]&pm)==primes[j]);j++) { //trial division by all primes 3...sqrt(2^k) |
---|
| 489 | if (0==(ans[0]%primes[j])) { |
---|
| 490 | dd=1; |
---|
| 491 | break; |
---|
| 492 | } |
---|
| 493 | } |
---|
| 494 | } |
---|
| 495 | carry_(ans); |
---|
| 496 | return; |
---|
| 497 | } |
---|
| 498 | |
---|
| 499 | B=c*k*k; //try small primes up to B (or all the primes[] array if the largest is less than B). |
---|
| 500 | if (k>2*m) //generate this k-bit number by first recursively generating a number that has between k/2 and k-m bits |
---|
| 501 | for (r=1; k-k*r<=m; ) |
---|
| 502 | r=pows[Math.floor(Math.random()*512)]; //r=Math.pow(2,Math.random()-1); |
---|
| 503 | else |
---|
| 504 | r=.5; |
---|
| 505 | |
---|
| 506 | //simulation suggests the more complex algorithm using r=.333 is only slightly faster. |
---|
| 507 | |
---|
| 508 | recSize=Math.floor(r*k)+1; |
---|
| 509 | |
---|
| 510 | randTruePrime_(s_q,recSize); |
---|
| 511 | copyInt_(s_i2,0); |
---|
| 512 | s_i2[Math.floor((k-2)/bpe)] |= (1<<((k-2)%bpe)); //s_i2=2^(k-2) |
---|
| 513 | divide_(s_i2,s_q,s_i,s_rm); //s_i=floor((2^(k-1))/(2q)) |
---|
| 514 | |
---|
| 515 | z=bitSize(s_i); |
---|
| 516 | |
---|
| 517 | for (;;) { |
---|
| 518 | for (;;) { //generate z-bit numbers until one falls in the range [0,s_i-1] |
---|
| 519 | randBigInt_(s_R,z,0); |
---|
| 520 | if (greater(s_i,s_R)) |
---|
| 521 | break; |
---|
| 522 | } //now s_R is in the range [0,s_i-1] |
---|
| 523 | addInt_(s_R,1); //now s_R is in the range [1,s_i] |
---|
| 524 | add_(s_R,s_i); //now s_R is in the range [s_i+1,2*s_i] |
---|
| 525 | |
---|
| 526 | copy_(s_n,s_q); |
---|
| 527 | mult_(s_n,s_R); |
---|
| 528 | multInt_(s_n,2); |
---|
| 529 | addInt_(s_n,1); //s_n=2*s_R*s_q+1 |
---|
| 530 | |
---|
| 531 | copy_(s_r2,s_R); |
---|
| 532 | multInt_(s_r2,2); //s_r2=2*s_R |
---|
| 533 | |
---|
| 534 | //check s_n for divisibility by small primes up to B |
---|
| 535 | for (divisible=0,j=0; (j<primes.length) && (primes[j]<B); j++) |
---|
| 536 | if (modInt(s_n,primes[j])==0 && !equalsInt(s_n,primes[j])) { |
---|
| 537 | divisible=1; |
---|
| 538 | break; |
---|
| 539 | } |
---|
| 540 | |
---|
| 541 | if (!divisible) //if it passes small primes check, then try a single Miller-Rabin base 2 |
---|
| 542 | if (!millerRabinInt(s_n,2)) //this line represents 75% of the total runtime for randTruePrime_ |
---|
| 543 | divisible=1; |
---|
| 544 | |
---|
| 545 | if (!divisible) { //if it passes that test, continue checking s_n |
---|
| 546 | addInt_(s_n,-3); |
---|
| 547 | for (j=s_n.length-1;(s_n[j]==0) && (j>0); j--); //strip leading zeros |
---|
| 548 | for (zz=0,w=s_n[j]; w; (w>>=1),zz++); |
---|
| 549 | zz+=bpe*j; //zz=number of bits in s_n, ignoring leading zeros |
---|
| 550 | for (;;) { //generate z-bit numbers until one falls in the range [0,s_n-1] |
---|
| 551 | randBigInt_(s_a,zz,0); |
---|
| 552 | if (greater(s_n,s_a)) |
---|
| 553 | break; |
---|
| 554 | } //now s_a is in the range [0,s_n-1] |
---|
| 555 | addInt_(s_n,3); //now s_a is in the range [0,s_n-4] |
---|
| 556 | addInt_(s_a,2); //now s_a is in the range [2,s_n-2] |
---|
| 557 | copy_(s_b,s_a); |
---|
| 558 | copy_(s_n1,s_n); |
---|
| 559 | addInt_(s_n1,-1); |
---|
| 560 | powMod_(s_b,s_n1,s_n); //s_b=s_a^(s_n-1) modulo s_n |
---|
| 561 | addInt_(s_b,-1); |
---|
| 562 | if (isZero(s_b)) { |
---|
| 563 | copy_(s_b,s_a); |
---|
| 564 | powMod_(s_b,s_r2,s_n); |
---|
| 565 | addInt_(s_b,-1); |
---|
| 566 | copy_(s_aa,s_n); |
---|
| 567 | copy_(s_d,s_b); |
---|
| 568 | GCD_(s_d,s_n); //if s_b and s_n are relatively prime, then s_n is a prime |
---|
| 569 | if (equalsInt(s_d,1)) { |
---|
| 570 | copy_(ans,s_aa); |
---|
| 571 | return; //if we've made it this far, then s_n is absolutely guaranteed to be prime |
---|
| 572 | } |
---|
| 573 | } |
---|
| 574 | } |
---|
| 575 | } |
---|
| 576 | } |
---|
| 577 | |
---|
| 578 | //Return an n-bit random BigInt (n>=1). If s=1, then the most significant of those n bits is set to 1. |
---|
| 579 | function randBigInt(n,s) { |
---|
| 580 | var a,b; |
---|
| 581 | a=Math.floor((n-1)/bpe)+2; //# array elements to hold the BigInt with a leading 0 element |
---|
| 582 | b=int2bigInt(0,0,a); |
---|
| 583 | randBigInt_(b,n,s); |
---|
| 584 | return b; |
---|
| 585 | } |
---|
| 586 | |
---|
| 587 | //Set b to an n-bit random BigInt. If s=1, then the most significant of those n bits is set to 1. |
---|
| 588 | //Array b must be big enough to hold the result. Must have n>=1 |
---|
| 589 | function randBigInt_(b,n,s) { |
---|
| 590 | var i,a; |
---|
| 591 | for (i=0;i<b.length;i++) |
---|
| 592 | b[i]=0; |
---|
| 593 | a=Math.floor((n-1)/bpe)+1; //# array elements to hold the BigInt |
---|
| 594 | for (i=0;i<a;i++) { |
---|
| 595 | b[i]=Math.floor(Math.random()*(1<<(bpe-1))); |
---|
| 596 | } |
---|
| 597 | b[a-1] &= (2<<((n-1)%bpe))-1; |
---|
| 598 | if (s==1) |
---|
| 599 | b[a-1] |= (1<<((n-1)%bpe)); |
---|
| 600 | } |
---|
| 601 | |
---|
| 602 | //Return the greatest common divisor of bigInts x and y (each with same number of elements). |
---|
| 603 | function GCD(x,y) { |
---|
| 604 | var xc,yc; |
---|
| 605 | xc=dup(x); |
---|
| 606 | yc=dup(y); |
---|
| 607 | GCD_(xc,yc); |
---|
| 608 | return xc; |
---|
| 609 | } |
---|
| 610 | |
---|
| 611 | //set x to the greatest common divisor of bigInts x and y (each with same number of elements). |
---|
| 612 | //y is destroyed. |
---|
| 613 | function GCD_(x,y) { |
---|
| 614 | var i,xp,yp,A,B,C,D,q,sing; |
---|
| 615 | if (T.length!=x.length) |
---|
| 616 | T=dup(x); |
---|
| 617 | |
---|
| 618 | sing=1; |
---|
| 619 | while (sing) { //while y has nonzero elements other than y[0] |
---|
| 620 | sing=0; |
---|
| 621 | for (i=1;i<y.length;i++) //check if y has nonzero elements other than 0 |
---|
| 622 | if (y[i]) { |
---|
| 623 | sing=1; |
---|
| 624 | break; |
---|
| 625 | } |
---|
| 626 | if (!sing) break; //quit when y all zero elements except possibly y[0] |
---|
| 627 | |
---|
| 628 | for (i=x.length;!x[i] && i>=0;i--); //find most significant element of x |
---|
| 629 | xp=x[i]; |
---|
| 630 | yp=y[i]; |
---|
| 631 | A=1; B=0; C=0; D=1; |
---|
| 632 | while ((yp+C) && (yp+D)) { |
---|
| 633 | q =Math.floor((xp+A)/(yp+C)); |
---|
| 634 | qp=Math.floor((xp+B)/(yp+D)); |
---|
| 635 | if (q!=qp) |
---|
| 636 | break; |
---|
| 637 | 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) |
---|
| 638 | t= B-q*D; B=D; D=t; |
---|
| 639 | t=xp-q*yp; xp=yp; yp=t; |
---|
| 640 | } |
---|
| 641 | if (B) { |
---|
| 642 | copy_(T,x); |
---|
| 643 | linComb_(x,y,A,B); //x=A*x+B*y |
---|
| 644 | linComb_(y,T,D,C); //y=D*y+C*T |
---|
| 645 | } else { |
---|
| 646 | mod_(x,y); |
---|
| 647 | copy_(T,x); |
---|
| 648 | copy_(x,y); |
---|
| 649 | copy_(y,T); |
---|
| 650 | } |
---|
| 651 | } |
---|
| 652 | if (y[0]==0) |
---|
| 653 | return; |
---|
| 654 | t=modInt(x,y[0]); |
---|
| 655 | copyInt_(x,y[0]); |
---|
| 656 | y[0]=t; |
---|
| 657 | while (y[0]) { |
---|
| 658 | x[0]%=y[0]; |
---|
| 659 | t=x[0]; x[0]=y[0]; y[0]=t; |
---|
| 660 | } |
---|
| 661 | } |
---|
| 662 | |
---|
| 663 | //do x=x**(-1) mod n, for bigInts x and n. |
---|
| 664 | //If no inverse exists, it sets x to zero and returns 0, else it returns 1. |
---|
| 665 | //The x array must be at least as large as the n array. |
---|
| 666 | function inverseMod_(x,n) { |
---|
| 667 | var k=1+2*Math.max(x.length,n.length); |
---|
| 668 | |
---|
| 669 | if(!(x[0]&1) && !(n[0]&1)) { //if both inputs are even, then inverse doesn't exist |
---|
| 670 | copyInt_(x,0); |
---|
| 671 | return 0; |
---|
| 672 | } |
---|
| 673 | |
---|
| 674 | if (eg_u.length!=k) { |
---|
| 675 | eg_u=new Array(k); |
---|
| 676 | eg_v=new Array(k); |
---|
| 677 | eg_A=new Array(k); |
---|
| 678 | eg_B=new Array(k); |
---|
| 679 | eg_C=new Array(k); |
---|
| 680 | eg_D=new Array(k); |
---|
| 681 | } |
---|
| 682 | |
---|
| 683 | copy_(eg_u,x); |
---|
| 684 | copy_(eg_v,n); |
---|
| 685 | copyInt_(eg_A,1); |
---|
| 686 | copyInt_(eg_B,0); |
---|
| 687 | copyInt_(eg_C,0); |
---|
| 688 | copyInt_(eg_D,1); |
---|
| 689 | for (;;) { |
---|
| 690 | while(!(eg_u[0]&1)) { //while eg_u is even |
---|
| 691 | halve_(eg_u); |
---|
| 692 | if (!(eg_A[0]&1) && !(eg_B[0]&1)) { //if eg_A==eg_B==0 mod 2 |
---|
| 693 | halve_(eg_A); |
---|
| 694 | halve_(eg_B); |
---|
| 695 | } else { |
---|
| 696 | add_(eg_A,n); halve_(eg_A); |
---|
| 697 | sub_(eg_B,x); halve_(eg_B); |
---|
| 698 | } |
---|
| 699 | } |
---|
| 700 | |
---|
| 701 | while (!(eg_v[0]&1)) { //while eg_v is even |
---|
| 702 | halve_(eg_v); |
---|
| 703 | if (!(eg_C[0]&1) && !(eg_D[0]&1)) { //if eg_C==eg_D==0 mod 2 |
---|
| 704 | halve_(eg_C); |
---|
| 705 | halve_(eg_D); |
---|
| 706 | } else { |
---|
| 707 | add_(eg_C,n); halve_(eg_C); |
---|
| 708 | sub_(eg_D,x); halve_(eg_D); |
---|
| 709 | } |
---|
| 710 | } |
---|
| 711 | |
---|
| 712 | if (!greater(eg_v,eg_u)) { //eg_v <= eg_u |
---|
| 713 | sub_(eg_u,eg_v); |
---|
| 714 | sub_(eg_A,eg_C); |
---|
| 715 | sub_(eg_B,eg_D); |
---|
| 716 | } else { //eg_v > eg_u |
---|
| 717 | sub_(eg_v,eg_u); |
---|
| 718 | sub_(eg_C,eg_A); |
---|
| 719 | sub_(eg_D,eg_B); |
---|
| 720 | } |
---|
| 721 | |
---|
| 722 | if (equalsInt(eg_u,0)) { |
---|
| 723 | if (negative(eg_C)) //make sure answer is nonnegative |
---|
| 724 | add_(eg_C,n); |
---|
| 725 | copy_(x,eg_C); |
---|
| 726 | |
---|
| 727 | if (!equalsInt(eg_v,1)) { //if GCD_(x,n)!=1, then there is no inverse |
---|
| 728 | copyInt_(x,0); |
---|
| 729 | return 0; |
---|
| 730 | } |
---|
| 731 | return 1; |
---|
| 732 | } |
---|
| 733 | } |
---|
| 734 | } |
---|
| 735 | |
---|
| 736 | //return x**(-1) mod n, for integers x and n. Return 0 if there is no inverse |
---|
| 737 | function inverseModInt(x,n) { |
---|
| 738 | var a=1,b=0,t; |
---|
| 739 | for (;;) { |
---|
| 740 | if (x==1) return a; |
---|
| 741 | if (x==0) return 0; |
---|
| 742 | b-=a*Math.floor(n/x); |
---|
| 743 | n%=x; |
---|
| 744 | |
---|
| 745 | if (n==1) return b; //to avoid negatives, change this b to n-b, and each -= to += |
---|
| 746 | if (n==0) return 0; |
---|
| 747 | a-=b*Math.floor(x/n); |
---|
| 748 | x%=n; |
---|
| 749 | } |
---|
| 750 | } |
---|
| 751 | |
---|
| 752 | //this deprecated function is for backward compatibility only. |
---|
| 753 | function inverseModInt_(x,n) { |
---|
| 754 | return inverseModInt(x,n); |
---|
| 755 | } |
---|
| 756 | |
---|
| 757 | |
---|
| 758 | //Given positive bigInts x and y, change the bigints v, a, and b to positive bigInts such that: |
---|
| 759 | // v = GCD_(x,y) = a*x-b*y |
---|
| 760 | //The bigInts v, a, b, must have exactly as many elements as the larger of x and y. |
---|
| 761 | function eGCD_(x,y,v,a,b) { |
---|
| 762 | var g=0; |
---|
| 763 | var k=Math.max(x.length,y.length); |
---|
| 764 | if (eg_u.length!=k) { |
---|
| 765 | eg_u=new Array(k); |
---|
| 766 | eg_A=new Array(k); |
---|
| 767 | eg_B=new Array(k); |
---|
| 768 | eg_C=new Array(k); |
---|
| 769 | eg_D=new Array(k); |
---|
| 770 | } |
---|
| 771 | while(!(x[0]&1) && !(y[0]&1)) { //while x and y both even |
---|
| 772 | halve_(x); |
---|
| 773 | halve_(y); |
---|
| 774 | g++; |
---|
| 775 | } |
---|
| 776 | copy_(eg_u,x); |
---|
| 777 | copy_(v,y); |
---|
| 778 | copyInt_(eg_A,1); |
---|
| 779 | copyInt_(eg_B,0); |
---|
| 780 | copyInt_(eg_C,0); |
---|
| 781 | copyInt_(eg_D,1); |
---|
| 782 | for (;;) { |
---|
| 783 | while(!(eg_u[0]&1)) { //while u is even |
---|
| 784 | halve_(eg_u); |
---|
| 785 | if (!(eg_A[0]&1) && !(eg_B[0]&1)) { //if A==B==0 mod 2 |
---|
| 786 | halve_(eg_A); |
---|
| 787 | halve_(eg_B); |
---|
| 788 | } else { |
---|
| 789 | add_(eg_A,y); halve_(eg_A); |
---|
| 790 | sub_(eg_B,x); halve_(eg_B); |
---|
| 791 | } |
---|
| 792 | } |
---|
| 793 | |
---|
| 794 | while (!(v[0]&1)) { //while v is even |
---|
| 795 | halve_(v); |
---|
| 796 | if (!(eg_C[0]&1) && !(eg_D[0]&1)) { //if C==D==0 mod 2 |
---|
| 797 | halve_(eg_C); |
---|
| 798 | halve_(eg_D); |
---|
| 799 | } else { |
---|
| 800 | add_(eg_C,y); halve_(eg_C); |
---|
| 801 | sub_(eg_D,x); halve_(eg_D); |
---|
| 802 | } |
---|
| 803 | } |
---|
| 804 | |
---|
| 805 | if (!greater(v,eg_u)) { //v<=u |
---|
| 806 | sub_(eg_u,v); |
---|
| 807 | sub_(eg_A,eg_C); |
---|
| 808 | sub_(eg_B,eg_D); |
---|
| 809 | } else { //v>u |
---|
| 810 | sub_(v,eg_u); |
---|
| 811 | sub_(eg_C,eg_A); |
---|
| 812 | sub_(eg_D,eg_B); |
---|
| 813 | } |
---|
| 814 | if (equalsInt(eg_u,0)) { |
---|
| 815 | if (negative(eg_C)) { //make sure a (C)is nonnegative |
---|
| 816 | add_(eg_C,y); |
---|
| 817 | sub_(eg_D,x); |
---|
| 818 | } |
---|
| 819 | multInt_(eg_D,-1); ///make sure b (D) is nonnegative |
---|
| 820 | copy_(a,eg_C); |
---|
| 821 | copy_(b,eg_D); |
---|
| 822 | leftShift_(v,g); |
---|
| 823 | return; |
---|
| 824 | } |
---|
| 825 | } |
---|
| 826 | } |
---|
| 827 | |
---|
| 828 | |
---|
| 829 | //is bigInt x negative? |
---|
| 830 | function negative(x) { |
---|
| 831 | return ((x[x.length-1]>>(bpe-1))&1); |
---|
| 832 | } |
---|
| 833 | |
---|
| 834 | |
---|
| 835 | //is (x << (shift*bpe)) > y? |
---|
| 836 | //x and y are nonnegative bigInts |
---|
| 837 | //shift is a nonnegative integer |
---|
| 838 | function greaterShift(x,y,shift) { |
---|
| 839 | var i, kx=x.length, ky=y.length; |
---|
| 840 | k=((kx+shift)<ky) ? (kx+shift) : ky; |
---|
| 841 | for (i=ky-1-shift; i<kx && i>=0; i++) |
---|
| 842 | if (x[i]>0) |
---|
| 843 | return 1; //if there are nonzeros in x to the left of the first column of y, then x is bigger |
---|
| 844 | for (i=kx-1+shift; i<ky; i++) |
---|
| 845 | if (y[i]>0) |
---|
| 846 | return 0; //if there are nonzeros in y to the left of the first column of x, then x is not bigger |
---|
| 847 | for (i=k-1; i>=shift; i--) |
---|
| 848 | if (x[i-shift]>y[i]) return 1; |
---|
| 849 | else if (x[i-shift]<y[i]) return 0; |
---|
| 850 | return 0; |
---|
| 851 | } |
---|
| 852 | |
---|
| 853 | //is x > y? (x and y both nonnegative) |
---|
| 854 | function greater(x,y) { |
---|
| 855 | var i; |
---|
| 856 | var k=(x.length<y.length) ? x.length : y.length; |
---|
| 857 | |
---|
| 858 | for (i=x.length;i<y.length;i++) |
---|
| 859 | if (y[i]) |
---|
| 860 | return 0; //y has more digits |
---|
| 861 | |
---|
| 862 | for (i=y.length;i<x.length;i++) |
---|
| 863 | if (x[i]) |
---|
| 864 | return 1; //x has more digits |
---|
| 865 | |
---|
| 866 | for (i=k-1;i>=0;i--) |
---|
| 867 | if (x[i]>y[i]) |
---|
| 868 | return 1; |
---|
| 869 | else if (x[i]<y[i]) |
---|
| 870 | return 0; |
---|
| 871 | return 0; |
---|
| 872 | } |
---|
| 873 | |
---|
| 874 | //divide x by y giving quotient q and remainder r. (q=floor(x/y), r=x mod y). All 4 are bigints. |
---|
| 875 | //x must have at least one leading zero element. |
---|
| 876 | //y must be nonzero. |
---|
| 877 | //q and r must be arrays that are exactly the same length as x. (Or q can have more). |
---|
| 878 | //Must have x.length >= y.length >= 2. |
---|
| 879 | function divide_(x,y,q,r) { |
---|
| 880 | var kx, ky; |
---|
| 881 | var i,j,y1,y2,c,a,b; |
---|
| 882 | copy_(r,x); |
---|
| 883 | for (ky=y.length;y[ky-1]==0;ky--); //ky is number of elements in y, not including leading zeros |
---|
| 884 | |
---|
| 885 | //normalize: ensure the most significant element of y has its highest bit set |
---|
| 886 | b=y[ky-1]; |
---|
| 887 | for (a=0; b; a++) |
---|
| 888 | b>>=1; |
---|
| 889 | a=bpe-a; //a is how many bits to shift so that the high order bit of y is leftmost in its array element |
---|
| 890 | leftShift_(y,a); //multiply both by 1<<a now, then divide both by that at the end |
---|
| 891 | leftShift_(r,a); |
---|
| 892 | |
---|
| 893 | //Rob Visser discovered a bug: the following line was originally just before the normalization. |
---|
| 894 | for (kx=r.length;r[kx-1]==0 && kx>ky;kx--); //kx is number of elements in normalized x, not including leading zeros |
---|
| 895 | |
---|
| 896 | copyInt_(q,0); // q=0 |
---|
| 897 | while (!greaterShift(y,r,kx-ky)) { // while (leftShift_(y,kx-ky) <= r) { |
---|
| 898 | subShift_(r,y,kx-ky); // r=r-leftShift_(y,kx-ky) |
---|
| 899 | q[kx-ky]++; // q[kx-ky]++; |
---|
| 900 | } // } |
---|
| 901 | |
---|
| 902 | for (i=kx-1; i>=ky; i--) { |
---|
| 903 | if (r[i]==y[ky-1]) |
---|
| 904 | q[i-ky]=mask; |
---|
| 905 | else |
---|
| 906 | q[i-ky]=Math.floor((r[i]*radix+r[i-1])/y[ky-1]); |
---|
| 907 | |
---|
| 908 | //The following for(;;) loop is equivalent to the commented while loop, |
---|
| 909 | //except that the uncommented version avoids overflow. |
---|
| 910 | //The commented loop comes from HAC, which assumes r[-1]==y[-1]==0 |
---|
| 911 | // while (q[i-ky]*(y[ky-1]*radix+y[ky-2]) > r[i]*radix*radix+r[i-1]*radix+r[i-2]) |
---|
| 912 | // q[i-ky]--; |
---|
| 913 | for (;;) { |
---|
| 914 | y2=(ky>1 ? y[ky-2] : 0)*q[i-ky]; |
---|
| 915 | c=y2>>bpe; |
---|
| 916 | y2=y2 & mask; |
---|
| 917 | y1=c+q[i-ky]*y[ky-1]; |
---|
| 918 | c=y1>>bpe; |
---|
| 919 | y1=y1 & mask; |
---|
| 920 | |
---|
| 921 | if (c==r[i] ? y1==r[i-1] ? y2>(i>1 ? r[i-2] : 0) : y1>r[i-1] : c>r[i]) |
---|
| 922 | q[i-ky]--; |
---|
| 923 | else |
---|
| 924 | break; |
---|
| 925 | } |
---|
| 926 | |
---|
| 927 | linCombShift_(r,y,-q[i-ky],i-ky); //r=r-q[i-ky]*leftShift_(y,i-ky) |
---|
| 928 | if (negative(r)) { |
---|
| 929 | addShift_(r,y,i-ky); //r=r+leftShift_(y,i-ky) |
---|
| 930 | q[i-ky]--; |
---|
| 931 | } |
---|
| 932 | } |
---|
| 933 | |
---|
| 934 | rightShift_(y,a); //undo the normalization step |
---|
| 935 | rightShift_(r,a); //undo the normalization step |
---|
| 936 | } |
---|
| 937 | |
---|
| 938 | //do carries and borrows so each element of the bigInt x fits in bpe bits. |
---|
| 939 | function carry_(x) { |
---|
| 940 | var i,k,c,b; |
---|
| 941 | k=x.length; |
---|
| 942 | c=0; |
---|
| 943 | for (i=0;i<k;i++) { |
---|
| 944 | c+=x[i]; |
---|
| 945 | b=0; |
---|
| 946 | if (c<0) { |
---|
| 947 | b=-(c>>bpe); |
---|
| 948 | c+=b*radix; |
---|
| 949 | } |
---|
| 950 | x[i]=c & mask; |
---|
| 951 | c=(c>>bpe)-b; |
---|
| 952 | } |
---|
| 953 | } |
---|
| 954 | |
---|
| 955 | //return x mod n for bigInt x and integer n. |
---|
| 956 | function modInt(x,n) { |
---|
| 957 | var i,c=0; |
---|
| 958 | for (i=x.length-1; i>=0; i--) |
---|
| 959 | c=(c*radix+x[i])%n; |
---|
| 960 | return c; |
---|
| 961 | } |
---|
| 962 | |
---|
| 963 | //convert the integer t into a bigInt with at least the given number of bits. |
---|
| 964 | //the returned array stores the bigInt in bpe-bit chunks, little endian (buff[0] is least significant word) |
---|
| 965 | //Pad the array with leading zeros so that it has at least minSize elements. |
---|
| 966 | //There will always be at least one leading 0 element. |
---|
| 967 | function int2bigInt(t,bits,minSize) { |
---|
| 968 | var i,k; |
---|
| 969 | k=Math.ceil(bits/bpe)+1; |
---|
| 970 | k=minSize>k ? minSize : k; |
---|
| 971 | buff=new Array(k); |
---|
| 972 | copyInt_(buff,t); |
---|
| 973 | return buff; |
---|
| 974 | } |
---|
| 975 | |
---|
| 976 | //return the bigInt given a string representation in a given base. |
---|
| 977 | //Pad the array with leading zeros so that it has at least minSize elements. |
---|
| 978 | //If base=-1, then it reads in a space-separated list of array elements in decimal. |
---|
| 979 | //The array will always have at least one leading zero, unless base=-1. |
---|
| 980 | function str2bigInt(s,base,minSize) { |
---|
| 981 | var d, i, j, x, y, kk; |
---|
| 982 | var k=s.length; |
---|
| 983 | if (base==-1) { //comma-separated list of array elements in decimal |
---|
| 984 | x=new Array(0); |
---|
| 985 | for (;;) { |
---|
| 986 | y=new Array(x.length+1); |
---|
| 987 | for (i=0;i<x.length;i++) |
---|
| 988 | y[i+1]=x[i]; |
---|
| 989 | y[0]=parseInt(s,10); |
---|
| 990 | x=y; |
---|
| 991 | d=s.indexOf(',',0); |
---|
| 992 | if (d<1) |
---|
| 993 | break; |
---|
| 994 | s=s.substring(d+1); |
---|
| 995 | if (s.length==0) |
---|
| 996 | break; |
---|
| 997 | } |
---|
| 998 | if (x.length<minSize) { |
---|
| 999 | y=new Array(minSize); |
---|
| 1000 | copy_(y,x); |
---|
| 1001 | return y; |
---|
| 1002 | } |
---|
| 1003 | return x; |
---|
| 1004 | } |
---|
| 1005 | |
---|
| 1006 | x=int2bigInt(0,base*k,0); |
---|
| 1007 | for (i=0;i<k;i++) { |
---|
| 1008 | d=digitsStr.indexOf(s.substring(i,i+1),0); |
---|
| 1009 | if (base<=36 && d>=36) //convert lowercase to uppercase if base<=36 |
---|
| 1010 | d-=26; |
---|
| 1011 | if (d>=base || d<0) { //stop at first illegal character |
---|
| 1012 | break; |
---|
| 1013 | } |
---|
| 1014 | multInt_(x,base); |
---|
| 1015 | addInt_(x,d); |
---|
| 1016 | } |
---|
| 1017 | |
---|
| 1018 | for (k=x.length;k>0 && !x[k-1];k--); //strip off leading zeros |
---|
| 1019 | k=minSize>k+1 ? minSize : k+1; |
---|
| 1020 | y=new Array(k); |
---|
| 1021 | kk=k<x.length ? k : x.length; |
---|
| 1022 | for (i=0;i<kk;i++) |
---|
| 1023 | y[i]=x[i]; |
---|
| 1024 | for (;i<k;i++) |
---|
| 1025 | y[i]=0; |
---|
| 1026 | return y; |
---|
| 1027 | } |
---|
| 1028 | |
---|
| 1029 | //is bigint x equal to integer y? |
---|
| 1030 | //y must have less than bpe bits |
---|
| 1031 | function equalsInt(x,y) { |
---|
| 1032 | var i; |
---|
| 1033 | if (x[0]!=y) |
---|
| 1034 | return 0; |
---|
| 1035 | for (i=1;i<x.length;i++) |
---|
| 1036 | if (x[i]) |
---|
| 1037 | return 0; |
---|
| 1038 | return 1; |
---|
| 1039 | } |
---|
| 1040 | |
---|
| 1041 | //are bigints x and y equal? |
---|
| 1042 | //this works even if x and y are different lengths and have arbitrarily many leading zeros |
---|
| 1043 | function equals(x,y) { |
---|
| 1044 | var i; |
---|
| 1045 | var k=x.length<y.length ? x.length : y.length; |
---|
| 1046 | for (i=0;i<k;i++) |
---|
| 1047 | if (x[i]!=y[i]) |
---|
| 1048 | return 0; |
---|
| 1049 | if (x.length>y.length) { |
---|
| 1050 | for (;i<x.length;i++) |
---|
| 1051 | if (x[i]) |
---|
| 1052 | return 0; |
---|
| 1053 | } else { |
---|
| 1054 | for (;i<y.length;i++) |
---|
| 1055 | if (y[i]) |
---|
| 1056 | return 0; |
---|
| 1057 | } |
---|
| 1058 | return 1; |
---|
| 1059 | } |
---|
| 1060 | |
---|
| 1061 | //is the bigInt x equal to zero? |
---|
| 1062 | function isZero(x) { |
---|
| 1063 | var i; |
---|
| 1064 | for (i=0;i<x.length;i++) |
---|
| 1065 | if (x[i]) |
---|
| 1066 | return 0; |
---|
| 1067 | return 1; |
---|
| 1068 | } |
---|
| 1069 | |
---|
| 1070 | //convert a bigInt into a string in a given base, from base 2 up to base 95. |
---|
| 1071 | //Base -1 prints the contents of the array representing the number. |
---|
| 1072 | function bigInt2str(x,base) { |
---|
| 1073 | var i,t,s=""; |
---|
| 1074 | |
---|
| 1075 | if (s6.length!=x.length) |
---|
| 1076 | s6=dup(x); |
---|
| 1077 | else |
---|
| 1078 | copy_(s6,x); |
---|
| 1079 | |
---|
| 1080 | if (base==-1) { //return the list of array contents |
---|
| 1081 | for (i=x.length-1;i>0;i--) |
---|
| 1082 | s+=x[i]+','; |
---|
| 1083 | s+=x[0]; |
---|
| 1084 | } |
---|
| 1085 | else { //return it in the given base |
---|
| 1086 | while (!isZero(s6)) { |
---|
| 1087 | t=divInt_(s6,base); //t=s6 % base; s6=floor(s6/base); |
---|
| 1088 | s=digitsStr.substring(t,t+1)+s; |
---|
| 1089 | } |
---|
| 1090 | } |
---|
| 1091 | if (s.length==0) |
---|
| 1092 | s="0"; |
---|
| 1093 | return s; |
---|
| 1094 | } |
---|
| 1095 | |
---|
| 1096 | //returns a duplicate of bigInt x |
---|
| 1097 | function dup(x) { |
---|
| 1098 | var i; |
---|
| 1099 | buff=new Array(x.length); |
---|
| 1100 | copy_(buff,x); |
---|
| 1101 | return buff; |
---|
| 1102 | } |
---|
| 1103 | |
---|
| 1104 | //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). |
---|
| 1105 | function copy_(x,y) { |
---|
| 1106 | var i; |
---|
| 1107 | var k=x.length<y.length ? x.length : y.length; |
---|
| 1108 | for (i=0;i<k;i++) |
---|
| 1109 | x[i]=y[i]; |
---|
| 1110 | for (i=k;i<x.length;i++) |
---|
| 1111 | x[i]=0; |
---|
| 1112 | } |
---|
| 1113 | |
---|
| 1114 | //do x=y on bigInt x and integer y. |
---|
| 1115 | function copyInt_(x,n) { |
---|
| 1116 | var i,c; |
---|
| 1117 | for (c=n,i=0;i<x.length;i++) { |
---|
| 1118 | x[i]=c & mask; |
---|
| 1119 | c>>=bpe; |
---|
| 1120 | } |
---|
| 1121 | } |
---|
| 1122 | |
---|
| 1123 | //do x=x+n where x is a bigInt and n is an integer. |
---|
| 1124 | //x must be large enough to hold the result. |
---|
| 1125 | function addInt_(x,n) { |
---|
| 1126 | var i,k,c,b; |
---|
| 1127 | x[0]+=n; |
---|
| 1128 | k=x.length; |
---|
| 1129 | c=0; |
---|
| 1130 | for (i=0;i<k;i++) { |
---|
| 1131 | c+=x[i]; |
---|
| 1132 | b=0; |
---|
| 1133 | if (c<0) { |
---|
| 1134 | b=-(c>>bpe); |
---|
| 1135 | c+=b*radix; |
---|
| 1136 | } |
---|
| 1137 | x[i]=c & mask; |
---|
| 1138 | c=(c>>bpe)-b; |
---|
| 1139 | if (!c) return; //stop carrying as soon as the carry is zero |
---|
| 1140 | } |
---|
| 1141 | } |
---|
| 1142 | |
---|
| 1143 | //right shift bigInt x by n bits. 0 <= n < bpe. |
---|
| 1144 | function rightShift_(x,n) { |
---|
| 1145 | var i; |
---|
| 1146 | var k=Math.floor(n/bpe); |
---|
| 1147 | if (k) { |
---|
| 1148 | for (i=0;i<x.length-k;i++) //right shift x by k elements |
---|
| 1149 | x[i]=x[i+k]; |
---|
| 1150 | for (;i<x.length;i++) |
---|
| 1151 | x[i]=0; |
---|
| 1152 | n%=bpe; |
---|
| 1153 | } |
---|
| 1154 | for (i=0;i<x.length-1;i++) { |
---|
| 1155 | x[i]=mask & ((x[i+1]<<(bpe-n)) | (x[i]>>n)); |
---|
| 1156 | } |
---|
| 1157 | x[i]>>=n; |
---|
| 1158 | } |
---|
| 1159 | |
---|
| 1160 | //do x=floor(|x|/2)*sgn(x) for bigInt x in 2's complement |
---|
| 1161 | function halve_(x) { |
---|
| 1162 | var i; |
---|
| 1163 | for (i=0;i<x.length-1;i++) { |
---|
| 1164 | x[i]=mask & ((x[i+1]<<(bpe-1)) | (x[i]>>1)); |
---|
| 1165 | } |
---|
| 1166 | x[i]=(x[i]>>1) | (x[i] & (radix>>1)); //most significant bit stays the same |
---|
| 1167 | } |
---|
| 1168 | |
---|
| 1169 | //left shift bigInt x by n bits. |
---|
| 1170 | function leftShift_(x,n) { |
---|
| 1171 | var i; |
---|
| 1172 | var k=Math.floor(n/bpe); |
---|
| 1173 | if (k) { |
---|
| 1174 | for (i=x.length; i>=k; i--) //left shift x by k elements |
---|
| 1175 | x[i]=x[i-k]; |
---|
| 1176 | for (;i>=0;i--) |
---|
| 1177 | x[i]=0; |
---|
| 1178 | n%=bpe; |
---|
| 1179 | } |
---|
| 1180 | if (!n) |
---|
| 1181 | return; |
---|
| 1182 | for (i=x.length-1;i>0;i--) { |
---|
| 1183 | x[i]=mask & ((x[i]<<n) | (x[i-1]>>(bpe-n))); |
---|
| 1184 | } |
---|
| 1185 | x[i]=mask & (x[i]<<n); |
---|
| 1186 | } |
---|
| 1187 | |
---|
| 1188 | //do x=x*n where x is a bigInt and n is an integer. |
---|
| 1189 | //x must be large enough to hold the result. |
---|
| 1190 | function multInt_(x,n) { |
---|
| 1191 | var i,k,c,b; |
---|
| 1192 | if (!n) |
---|
| 1193 | return; |
---|
| 1194 | k=x.length; |
---|
| 1195 | c=0; |
---|
| 1196 | for (i=0;i<k;i++) { |
---|
| 1197 | c+=x[i]*n; |
---|
| 1198 | b=0; |
---|
| 1199 | if (c<0) { |
---|
| 1200 | b=-(c>>bpe); |
---|
| 1201 | c+=b*radix; |
---|
| 1202 | } |
---|
| 1203 | x[i]=c & mask; |
---|
| 1204 | c=(c>>bpe)-b; |
---|
| 1205 | } |
---|
| 1206 | } |
---|
| 1207 | |
---|
| 1208 | //do x=floor(x/n) for bigInt x and integer n, and return the remainder |
---|
| 1209 | function divInt_(x,n) { |
---|
| 1210 | var i,r=0,s; |
---|
| 1211 | for (i=x.length-1;i>=0;i--) { |
---|
| 1212 | s=r*radix+x[i]; |
---|
| 1213 | x[i]=Math.floor(s/n); |
---|
| 1214 | r=s%n; |
---|
| 1215 | } |
---|
| 1216 | return r; |
---|
| 1217 | } |
---|
| 1218 | |
---|
| 1219 | //do the linear combination x=a*x+b*y for bigInts x and y, and integers a and b. |
---|
| 1220 | //x must be large enough to hold the answer. |
---|
| 1221 | function linComb_(x,y,a,b) { |
---|
| 1222 | var i,c,k,kk; |
---|
| 1223 | k=x.length<y.length ? x.length : y.length; |
---|
| 1224 | kk=x.length; |
---|
| 1225 | for (c=0,i=0;i<k;i++) { |
---|
| 1226 | c+=a*x[i]+b*y[i]; |
---|
| 1227 | x[i]=c & mask; |
---|
| 1228 | c>>=bpe; |
---|
| 1229 | } |
---|
| 1230 | for (i=k;i<kk;i++) { |
---|
| 1231 | c+=a*x[i]; |
---|
| 1232 | x[i]=c & mask; |
---|
| 1233 | c>>=bpe; |
---|
| 1234 | } |
---|
| 1235 | } |
---|
| 1236 | |
---|
| 1237 | //do the linear combination x=a*x+b*(y<<(ys*bpe)) for bigInts x and y, and integers a, b and ys. |
---|
| 1238 | //x must be large enough to hold the answer. |
---|
| 1239 | function linCombShift_(x,y,b,ys) { |
---|
| 1240 | var i,c,k,kk; |
---|
| 1241 | k=x.length<ys+y.length ? x.length : ys+y.length; |
---|
| 1242 | kk=x.length; |
---|
| 1243 | for (c=0,i=ys;i<k;i++) { |
---|
| 1244 | c+=x[i]+b*y[i-ys]; |
---|
| 1245 | x[i]=c & mask; |
---|
| 1246 | c>>=bpe; |
---|
| 1247 | } |
---|
| 1248 | for (i=k;c && i<kk;i++) { |
---|
| 1249 | c+=x[i]; |
---|
| 1250 | x[i]=c & mask; |
---|
| 1251 | c>>=bpe; |
---|
| 1252 | } |
---|
| 1253 | } |
---|
| 1254 | |
---|
| 1255 | //do x=x+(y<<(ys*bpe)) for bigInts x and y, and integers a,b and ys. |
---|
| 1256 | //x must be large enough to hold the answer. |
---|
| 1257 | function addShift_(x,y,ys) { |
---|
| 1258 | var i,c,k,kk; |
---|
| 1259 | k=x.length<ys+y.length ? x.length : ys+y.length; |
---|
| 1260 | kk=x.length; |
---|
| 1261 | for (c=0,i=ys;i<k;i++) { |
---|
| 1262 | c+=x[i]+y[i-ys]; |
---|
| 1263 | x[i]=c & mask; |
---|
| 1264 | c>>=bpe; |
---|
| 1265 | } |
---|
| 1266 | for (i=k;c && i<kk;i++) { |
---|
| 1267 | c+=x[i]; |
---|
| 1268 | x[i]=c & mask; |
---|
| 1269 | c>>=bpe; |
---|
| 1270 | } |
---|
| 1271 | } |
---|
| 1272 | |
---|
| 1273 | //do x=x-(y<<(ys*bpe)) for bigInts x and y, and integers a,b and ys. |
---|
| 1274 | //x must be large enough to hold the answer. |
---|
| 1275 | function subShift_(x,y,ys) { |
---|
| 1276 | var i,c,k,kk; |
---|
| 1277 | k=x.length<ys+y.length ? x.length : ys+y.length; |
---|
| 1278 | kk=x.length; |
---|
| 1279 | for (c=0,i=ys;i<k;i++) { |
---|
| 1280 | c+=x[i]-y[i-ys]; |
---|
| 1281 | x[i]=c & mask; |
---|
| 1282 | c>>=bpe; |
---|
| 1283 | } |
---|
| 1284 | for (i=k;c && i<kk;i++) { |
---|
| 1285 | c+=x[i]; |
---|
| 1286 | x[i]=c & mask; |
---|
| 1287 | c>>=bpe; |
---|
| 1288 | } |
---|
| 1289 | } |
---|
| 1290 | |
---|
| 1291 | //do x=x-y for bigInts x and y. |
---|
| 1292 | //x must be large enough to hold the answer. |
---|
| 1293 | //negative answers will be 2s complement |
---|
| 1294 | function sub_(x,y) { |
---|
| 1295 | var i,c,k,kk; |
---|
| 1296 | k=x.length<y.length ? x.length : y.length; |
---|
| 1297 | for (c=0,i=0;i<k;i++) { |
---|
| 1298 | c+=x[i]-y[i]; |
---|
| 1299 | x[i]=c & mask; |
---|
| 1300 | c>>=bpe; |
---|
| 1301 | } |
---|
| 1302 | for (i=k;c && i<x.length;i++) { |
---|
| 1303 | c+=x[i]; |
---|
| 1304 | x[i]=c & mask; |
---|
| 1305 | c>>=bpe; |
---|
| 1306 | } |
---|
| 1307 | } |
---|
| 1308 | |
---|
| 1309 | //do x=x+y for bigInts x and y. |
---|
| 1310 | //x must be large enough to hold the answer. |
---|
| 1311 | function add_(x,y) { |
---|
| 1312 | var i,c,k,kk; |
---|
| 1313 | k=x.length<y.length ? x.length : y.length; |
---|
| 1314 | for (c=0,i=0;i<k;i++) { |
---|
| 1315 | c+=x[i]+y[i]; |
---|
| 1316 | x[i]=c & mask; |
---|
| 1317 | c>>=bpe; |
---|
| 1318 | } |
---|
| 1319 | for (i=k;c && i<x.length;i++) { |
---|
| 1320 | c+=x[i]; |
---|
| 1321 | x[i]=c & mask; |
---|
| 1322 | c>>=bpe; |
---|
| 1323 | } |
---|
| 1324 | } |
---|
| 1325 | |
---|
| 1326 | //do x=x*y for bigInts x and y. This is faster when y<x. |
---|
| 1327 | function mult_(x,y) { |
---|
| 1328 | var i; |
---|
| 1329 | if (ss.length!=2*x.length) |
---|
| 1330 | ss=new Array(2*x.length); |
---|
| 1331 | copyInt_(ss,0); |
---|
| 1332 | for (i=0;i<y.length;i++) |
---|
| 1333 | if (y[i]) |
---|
| 1334 | linCombShift_(ss,x,y[i],i); //ss=1*ss+y[i]*(x<<(i*bpe)) |
---|
| 1335 | copy_(x,ss); |
---|
| 1336 | } |
---|
| 1337 | |
---|
| 1338 | //do x=x mod n for bigInts x and n. |
---|
| 1339 | function mod_(x,n) { |
---|
| 1340 | if (s4.length!=x.length) |
---|
| 1341 | s4=dup(x); |
---|
| 1342 | else |
---|
| 1343 | copy_(s4,x); |
---|
| 1344 | if (s5.length!=x.length) |
---|
| 1345 | s5=dup(x); |
---|
| 1346 | divide_(s4,n,s5,x); //x = remainder of s4 / n |
---|
| 1347 | } |
---|
| 1348 | |
---|
| 1349 | //do x=x*y mod n for bigInts x,y,n. |
---|
| 1350 | //for greater speed, let y<x. |
---|
| 1351 | function multMod_(x,y,n) { |
---|
| 1352 | var i; |
---|
| 1353 | if (s0.length!=2*x.length) |
---|
| 1354 | s0=new Array(2*x.length); |
---|
| 1355 | copyInt_(s0,0); |
---|
| 1356 | for (i=0;i<y.length;i++) |
---|
| 1357 | if (y[i]) |
---|
| 1358 | linCombShift_(s0,x,y[i],i); //s0=1*s0+y[i]*(x<<(i*bpe)) |
---|
| 1359 | mod_(s0,n); |
---|
| 1360 | copy_(x,s0); |
---|
| 1361 | } |
---|
| 1362 | |
---|
| 1363 | //do x=x*x mod n for bigInts x,n. |
---|
| 1364 | function squareMod_(x,n) { |
---|
| 1365 | var i,j,d,c,kx,kn,k; |
---|
| 1366 | for (kx=x.length; kx>0 && !x[kx-1]; kx--); //ignore leading zeros in x |
---|
| 1367 | 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 |
---|
| 1368 | if (s0.length!=k) |
---|
| 1369 | s0=new Array(k); |
---|
| 1370 | copyInt_(s0,0); |
---|
| 1371 | for (i=0;i<kx;i++) { |
---|
| 1372 | c=s0[2*i]+x[i]*x[i]; |
---|
| 1373 | s0[2*i]=c & mask; |
---|
| 1374 | c>>=bpe; |
---|
| 1375 | for (j=i+1;j<kx;j++) { |
---|
| 1376 | c=s0[i+j]+2*x[i]*x[j]+c; |
---|
| 1377 | s0[i+j]=(c & mask); |
---|
| 1378 | c>>=bpe; |
---|
| 1379 | } |
---|
| 1380 | s0[i+kx]=c; |
---|
| 1381 | } |
---|
| 1382 | mod_(s0,n); |
---|
| 1383 | copy_(x,s0); |
---|
| 1384 | } |
---|
| 1385 | |
---|
| 1386 | //return x with exactly k leading zero elements |
---|
| 1387 | function trim(x,k) { |
---|
| 1388 | var i,y; |
---|
| 1389 | for (i=x.length; i>0 && !x[i-1]; i--); |
---|
| 1390 | y=new Array(i+k); |
---|
| 1391 | copy_(y,x); |
---|
| 1392 | return y; |
---|
| 1393 | } |
---|
| 1394 | |
---|
| 1395 | //do x=x**y mod n, where x,y,n are bigInts and ** is exponentiation. 0**0=1. |
---|
| 1396 | //this is faster when n is odd. x usually needs to have as many elements as n. |
---|
| 1397 | function powMod_(x,y,n) { |
---|
| 1398 | var k1,k2,kn,np; |
---|
| 1399 | if(s7.length!=n.length) |
---|
| 1400 | s7=dup(n); |
---|
| 1401 | |
---|
| 1402 | //for even modulus, use a simple square-and-multiply algorithm, |
---|
| 1403 | //rather than using the more complex Montgomery algorithm. |
---|
| 1404 | if ((n[0]&1)==0) { |
---|
| 1405 | copy_(s7,x); |
---|
| 1406 | copyInt_(x,1); |
---|
| 1407 | while(!equalsInt(y,0)) { |
---|
| 1408 | if (y[0]&1) |
---|
| 1409 | multMod_(x,s7,n); |
---|
| 1410 | divInt_(y,2); |
---|
| 1411 | squareMod_(s7,n); |
---|
| 1412 | } |
---|
| 1413 | return; |
---|
| 1414 | } |
---|
| 1415 | |
---|
| 1416 | //calculate np from n for the Montgomery multiplications |
---|
| 1417 | copyInt_(s7,0); |
---|
| 1418 | for (kn=n.length;kn>0 && !n[kn-1];kn--); |
---|
| 1419 | np=radix-inverseModInt(modInt(n,radix),radix); |
---|
| 1420 | s7[kn]=1; |
---|
| 1421 | multMod_(x ,s7,n); // x = x * 2**(kn*bp) mod n |
---|
| 1422 | |
---|
| 1423 | if (s3.length!=x.length) |
---|
| 1424 | s3=dup(x); |
---|
| 1425 | else |
---|
| 1426 | copy_(s3,x); |
---|
| 1427 | |
---|
| 1428 | for (k1=y.length-1;k1>0 & !y[k1]; k1--); //k1=first nonzero element of y |
---|
| 1429 | if (y[k1]==0) { //anything to the 0th power is 1 |
---|
| 1430 | copyInt_(x,1); |
---|
| 1431 | return; |
---|
| 1432 | } |
---|
| 1433 | for (k2=1<<(bpe-1);k2 && !(y[k1] & k2); k2>>=1); //k2=position of first 1 bit in y[k1] |
---|
| 1434 | for (;;) { |
---|
| 1435 | if (!(k2>>=1)) { //look at next bit of y |
---|
| 1436 | k1--; |
---|
| 1437 | if (k1<0) { |
---|
| 1438 | mont_(x,one,n,np); |
---|
| 1439 | return; |
---|
| 1440 | } |
---|
| 1441 | k2=1<<(bpe-1); |
---|
| 1442 | } |
---|
| 1443 | mont_(x,x,n,np); |
---|
| 1444 | |
---|
| 1445 | if (k2 & y[k1]) //if next bit is a 1 |
---|
| 1446 | mont_(x,s3,n,np); |
---|
| 1447 | } |
---|
| 1448 | } |
---|
| 1449 | |
---|
| 1450 | |
---|
| 1451 | //do x=x*y*Ri mod n for bigInts x,y,n, |
---|
| 1452 | // where Ri = 2**(-kn*bpe) mod n, and kn is the |
---|
| 1453 | // number of elements in the n array, not |
---|
| 1454 | // counting leading zeros. |
---|
| 1455 | //x array must have at least as many elemnts as the n array |
---|
| 1456 | //It's OK if x and y are the same variable. |
---|
| 1457 | //must have: |
---|
| 1458 | // x,y < n |
---|
| 1459 | // n is odd |
---|
| 1460 | // np = -(n^(-1)) mod radix |
---|
| 1461 | function mont_(x,y,n,np) { |
---|
| 1462 | var i,j,c,ui,t,ks; |
---|
| 1463 | var kn=n.length; |
---|
| 1464 | var ky=y.length; |
---|
| 1465 | |
---|
| 1466 | if (sa.length!=kn) |
---|
| 1467 | sa=new Array(kn); |
---|
| 1468 | |
---|
| 1469 | copyInt_(sa,0); |
---|
| 1470 | |
---|
| 1471 | for (;kn>0 && n[kn-1]==0;kn--); //ignore leading zeros of n |
---|
| 1472 | for (;ky>0 && y[ky-1]==0;ky--); //ignore leading zeros of y |
---|
| 1473 | ks=sa.length-1; //sa will never have more than this many nonzero elements. |
---|
| 1474 | |
---|
| 1475 | //the following loop consumes 95% of the runtime for randTruePrime_() and powMod_() for large numbers |
---|
| 1476 | for (i=0; i<kn; i++) { |
---|
| 1477 | t=sa[0]+x[i]*y[0]; |
---|
| 1478 | ui=((t & mask) * np) & mask; //the inner "& mask" was needed on Safari (but not MSIE) at one time |
---|
| 1479 | c=(t+ui*n[0]) >> bpe; |
---|
| 1480 | t=x[i]; |
---|
| 1481 | |
---|
| 1482 | //do sa=(sa+x[i]*y+ui*n)/b where b=2**bpe. Loop is unrolled 5-fold for speed |
---|
| 1483 | j=1; |
---|
| 1484 | for (;j<ky-4;) { c+=sa[j]+ui*n[j]+t*y[j]; sa[j-1]=c & mask; c>>=bpe; j++; |
---|
| 1485 | c+=sa[j]+ui*n[j]+t*y[j]; sa[j-1]=c & mask; c>>=bpe; j++; |
---|
| 1486 | c+=sa[j]+ui*n[j]+t*y[j]; sa[j-1]=c & mask; c>>=bpe; j++; |
---|
| 1487 | c+=sa[j]+ui*n[j]+t*y[j]; sa[j-1]=c & mask; c>>=bpe; j++; |
---|
| 1488 | c+=sa[j]+ui*n[j]+t*y[j]; sa[j-1]=c & mask; c>>=bpe; j++; } |
---|
| 1489 | for (;j<ky;) { c+=sa[j]+ui*n[j]+t*y[j]; sa[j-1]=c & mask; c>>=bpe; j++; } |
---|
| 1490 | for (;j<kn-4;) { c+=sa[j]+ui*n[j]; sa[j-1]=c & mask; c>>=bpe; j++; |
---|
| 1491 | c+=sa[j]+ui*n[j]; sa[j-1]=c & mask; c>>=bpe; j++; |
---|
| 1492 | c+=sa[j]+ui*n[j]; sa[j-1]=c & mask; c>>=bpe; j++; |
---|
| 1493 | c+=sa[j]+ui*n[j]; sa[j-1]=c & mask; c>>=bpe; j++; |
---|
| 1494 | c+=sa[j]+ui*n[j]; sa[j-1]=c & mask; c>>=bpe; j++; } |
---|
| 1495 | for (;j<kn;) { c+=sa[j]+ui*n[j]; sa[j-1]=c & mask; c>>=bpe; j++; } |
---|
| 1496 | for (;j<ks;) { c+=sa[j]; sa[j-1]=c & mask; c>>=bpe; j++; } |
---|
| 1497 | sa[j-1]=c & mask; |
---|
| 1498 | } |
---|
| 1499 | |
---|
| 1500 | if (!greater(n,sa)) |
---|
| 1501 | sub_(sa,n); |
---|
| 1502 | copy_(x,sa); |
---|
| 1503 | } |
---|
| 1504 | |
---|
| 1505 | |
---|