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); |
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338 | if (k>=100) return randProbPrimeRounds(k,27); |
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339 | return randProbPrimeRounds(k,40); //number from HAC remark 4.26 (only an estimate) |
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340 | } |
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341 | |
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342 | //return a k-bit probable random prime using n rounds of Miller Rabin (after trial division with small primes) |
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343 | function randProbPrimeRounds(k,n) { |
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344 | var ans, i, divisible, B; |
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345 | B=30000; //B is largest prime to use in trial division |
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346 | ans=int2bigInt(0,k,0); |
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347 | |
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348 | //optimization: try larger and smaller B to find the best limit. |
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349 | |
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350 | if (primes.length==0) |
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351 | primes=findPrimes(30000); //check for divisibility by primes <=30000 |
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352 | |
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353 | if (rpprb.length!=ans.length) |
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354 | rpprb=dup(ans); |
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355 | |
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356 | for (;;) { //keep trying random values for ans until one appears to be prime |
---|
357 | //optimization: pick a random number times L=2*3*5*...*p, plus a |
---|
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 | |
---|