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 | |
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72 | //globals |
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73 | bpe=0; //bits stored per array element |
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74 | mask=0; //AND this with an array element to chop it down to bpe bits |
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75 | radix=mask+1; //equals 2^bpe. A single 1 bit to the left of the last bit of mask. |
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76 | |
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77 | //the digits for converting to different bases |
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78 | digitsStr='0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz_=!@#$%^&*()[]{}|;:,.<>/?`~ \\\'\"+-'; |
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79 | |
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80 | //initialize the global variables |
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81 | for (bpe=0; (1<<(bpe+1)) > (1<<bpe); bpe++); //bpe=number of bits in the mantissa on this platform |
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82 | bpe>>=1; //bpe=number of bits in one element of the array representing the bigInt |
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83 | mask=(1<<bpe)-1; //AND the mask with an integer to get its bpe least significant bits |
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84 | radix=mask+1; //2^bpe. a single 1 bit to the left of the first bit of mask |
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85 | one=int2bigInt(1,1,1); //constant used in powMod_() |
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86 | |
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87 | //the following global variables are scratchpad memory to |
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88 | //reduce dynamic memory allocation in the inner loop |
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89 | t=new Array(0); |
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90 | ss=t; //used in mult_() |
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91 | s0=t; //used in multMod_(), squareMod_() |
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92 | s1=t; //used in powMod_(), multMod_(), squareMod_() |
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93 | s2=t; //used in powMod_(), multMod_() |
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94 | s3=t; //used in powMod_() |
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95 | s4=t; s5=t; //used in mod_() |
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96 | s6=t; //used in bigInt2str() |
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97 | s7=t; //used in powMod_() |
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98 | T=t; //used in GCD_() |
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99 | sa=t; //used in mont_() |
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100 | mr_x1=t; mr_r=t; mr_a=t; //used in millerRabin() |
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101 | 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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102 | 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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103 | |
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104 | 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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105 | 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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106 | |
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107 | rpprb=t; //used in randProbPrimeRounds() (which also uses "primes") |
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108 | |
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109 | //return a copy of x with at least n elements, adding leading zeros if needed |
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110 | function expand(x,n) { |
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111 | var ans=int2bigInt(0,(x.length>n ? x.length : n)*bpe,0); |
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112 | copy_(ans,x); |
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113 | return ans; |
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114 | } |
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115 | //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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116 | function powMod(x,y,n) { |
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117 | var ans=expand(x,n.length); |
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118 | powMod_(ans,bigintTrim(y,2),bigintTrim(n,2),0); //this should work without the bigintTrim, but doesn't |
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119 | return bigintTrim(ans,1); |
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120 | } |
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121 | |
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122 | //return (x-y) for bigInts x and y. Negative answers will be 2s complement |
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123 | function sub(x,y) { |
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124 | var ans=expand(x,(x.length>y.length ? x.length+1 : y.length+1)); |
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125 | sub_(ans,y); |
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126 | return bigintTrim(ans,1); |
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127 | } |
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128 | |
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129 | //return (x+y) for bigInts x and y. |
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130 | function add(x,y) { |
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131 | var ans=expand(x,(x.length>y.length ? x.length+1 : y.length+1)); |
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132 | add_(ans,y); |
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133 | return bigintTrim(ans,1); |
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134 | } |
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135 | //Return the greatest common divisor of bigInts x and y (each with same number of elements). |
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136 | function GCD(x,y) { |
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137 | var xc,yc; |
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138 | xc=dup(x); |
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139 | yc=dup(y); |
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140 | GCD_(xc,yc); |
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141 | return xc; |
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142 | } |
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143 | |
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144 | //set x to the greatest common divisor of bigInts x and y (each with same number of elements). |
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145 | //y is destroyed. |
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146 | function GCD_(x,y) { |
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147 | var i,xp,yp,A,B,C,D,q,sing; |
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148 | if (T.length!=x.length) |
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149 | T=dup(x); |
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150 | |
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151 | sing=1; |
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152 | while (sing) { //while y has nonzero elements other than y[0] |
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153 | sing=0; |
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154 | for (i=1;i<y.length;i++) //check if y has nonzero elements other than 0 |
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155 | if (y[i]) { |
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156 | sing=1; |
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157 | break; |
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158 | } |
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159 | if (!sing) break; //quit when y all zero elements except possibly y[0] |
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160 | |
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161 | for (i=x.length;!x[i] && i>=0;i--); //find most significant element of x |
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162 | xp=x[i]; |
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163 | yp=y[i]; |
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164 | A=1; B=0; C=0; D=1; |
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165 | while ((yp+C) && (yp+D)) { |
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166 | q =Math.floor((xp+A)/(yp+C)); |
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167 | qp=Math.floor((xp+B)/(yp+D)); |
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168 | if (q!=qp) |
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169 | break; |
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170 | 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) |
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171 | t= B-q*D; B=D; D=t; |
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172 | t=xp-q*yp; xp=yp; yp=t; |
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173 | } |
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174 | if (B) { |
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175 | copy_(T,x); |
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176 | linComb_(x,y,A,B); //x=A*x+B*y |
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177 | linComb_(y,T,D,C); //y=D*y+C*T |
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178 | } else { |
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179 | mod_(x,y); |
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180 | copy_(T,x); |
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181 | copy_(x,y); |
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182 | copy_(y,T); |
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183 | } |
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184 | } |
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185 | if (y[0]==0) |
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186 | return; |
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187 | t=modInt(x,y[0]); |
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188 | copyInt_(x,y[0]); |
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189 | y[0]=t; |
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190 | while (y[0]) { |
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191 | x[0]%=y[0]; |
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192 | t=x[0]; x[0]=y[0]; y[0]=t; |
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193 | } |
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194 | } |
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195 | |
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196 | //return x**(-1) mod n, for integers x and n. Return 0 if there is no inverse |
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197 | function inverseModInt(x,n) { |
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198 | var a=1,b=0,t; |
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199 | for (;;) { |
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200 | if (x==1) return a; |
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201 | if (x==0) return 0; |
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202 | b-=a*Math.floor(n/x); |
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203 | n%=x; |
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204 | |
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205 | if (n==1) return b; //to avoid negatives, change this b to n-b, and each -= to += |
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206 | if (n==0) return 0; |
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207 | a-=b*Math.floor(x/n); |
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208 | x%=n; |
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209 | } |
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210 | } |
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211 | |
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212 | //this deprecated function is for backward compatibility only. |
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213 | function inverseModInt_(x,n) { |
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214 | return inverseModInt(x,n); |
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215 | } |
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216 | |
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217 | |
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218 | //Given positive bigInts x and y, change the bigints v, a, and b to positive bigInts such that: |
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219 | // v = GCD_(x,y) = a*x-b*y |
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220 | //The bigInts v, a, b, must have exactly as many elements as the larger of x and y. |
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221 | function eGCD_(x,y,v,a,b) { |
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222 | var g=0; |
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223 | var k=Math.max(x.length,y.length); |
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224 | if (eg_u.length!=k) { |
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225 | eg_u=new Array(k); |
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226 | eg_A=new Array(k); |
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227 | eg_B=new Array(k); |
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228 | eg_C=new Array(k); |
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229 | eg_D=new Array(k); |
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230 | } |
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231 | while(!(x[0]&1) && !(y[0]&1)) { //while x and y both even |
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232 | halve_(x); |
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233 | halve_(y); |
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234 | g++; |
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235 | } |
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236 | copy_(eg_u,x); |
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237 | copy_(v,y); |
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238 | copyInt_(eg_A,1); |
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239 | copyInt_(eg_B,0); |
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240 | copyInt_(eg_C,0); |
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241 | copyInt_(eg_D,1); |
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242 | for (;;) { |
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243 | while(!(eg_u[0]&1)) { //while u is even |
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244 | halve_(eg_u); |
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245 | if (!(eg_A[0]&1) && !(eg_B[0]&1)) { //if A==B==0 mod 2 |
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246 | halve_(eg_A); |
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247 | halve_(eg_B); |
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248 | } else { |
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249 | add_(eg_A,y); halve_(eg_A); |
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250 | sub_(eg_B,x); halve_(eg_B); |
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251 | } |
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252 | } |
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253 | |
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254 | while (!(v[0]&1)) { //while v is even |
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255 | halve_(v); |
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256 | if (!(eg_C[0]&1) && !(eg_D[0]&1)) { //if C==D==0 mod 2 |
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257 | halve_(eg_C); |
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258 | halve_(eg_D); |
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259 | } else { |
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260 | add_(eg_C,y); halve_(eg_C); |
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261 | sub_(eg_D,x); halve_(eg_D); |
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262 | } |
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263 | } |
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264 | |
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265 | if (!greater(v,eg_u)) { //v<=u |
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266 | sub_(eg_u,v); |
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267 | sub_(eg_A,eg_C); |
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268 | sub_(eg_B,eg_D); |
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269 | } else { //v>u |
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270 | sub_(v,eg_u); |
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271 | sub_(eg_C,eg_A); |
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272 | sub_(eg_D,eg_B); |
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273 | } |
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274 | if (equalsInt(eg_u,0)) { |
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275 | if (negative(eg_C)) { //make sure a (C)is nonnegative |
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276 | add_(eg_C,y); |
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277 | sub_(eg_D,x); |
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278 | } |
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279 | multInt_(eg_D,-1); ///make sure b (D) is nonnegative |
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280 | copy_(a,eg_C); |
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281 | copy_(b,eg_D); |
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282 | leftShift_(v,g); |
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283 | return; |
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284 | } |
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285 | } |
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286 | } |
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287 | |
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288 | |
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289 | //is bigInt x negative? |
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290 | function negative(x) { |
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291 | return ((x[x.length-1]>>(bpe-1))&1); |
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292 | } |
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293 | |
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294 | |
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295 | //is (x << (shift*bpe)) > y? |
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296 | //x and y are nonnegative bigInts |
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297 | //shift is a nonnegative integer |
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298 | function greaterShift(x,y,shift) { |
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299 | var i, kx=x.length, ky=y.length; |
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300 | k=((kx+shift)<ky) ? (kx+shift) : ky; |
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301 | for (i=ky-1-shift; i<kx && i>=0; i++) |
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302 | if (x[i]>0) |
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303 | return 1; //if there are nonzeros in x to the left of the first column of y, then x is bigger |
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304 | for (i=kx-1+shift; i<ky; i++) |
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305 | if (y[i]>0) |
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306 | return 0; //if there are nonzeros in y to the left of the first column of x, then x is not bigger |
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307 | for (i=k-1; i>=shift; i--) |
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308 | if (x[i-shift]>y[i]) return 1; |
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309 | else if (x[i-shift]<y[i]) return 0; |
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310 | return 0; |
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311 | } |
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312 | |
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313 | //is x > y? (x and y both nonnegative) |
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314 | function greater(x,y) { |
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315 | var i; |
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316 | var k=(x.length<y.length) ? x.length : y.length; |
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317 | |
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318 | for (i=x.length;i<y.length;i++) |
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319 | if (y[i]) |
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320 | return 0; //y has more digits |
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321 | |
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322 | for (i=y.length;i<x.length;i++) |
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323 | if (x[i]) |
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324 | return 1; //x has more digits |
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325 | |
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326 | for (i=k-1;i>=0;i--) |
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327 | if (x[i]>y[i]) |
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328 | return 1; |
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329 | else if (x[i]<y[i]) |
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330 | return 0; |
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331 | return 0; |
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332 | } |
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333 | |
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334 | //divide x by y giving quotient q and remainder r. (q=floor(x/y), r=x mod y). All 4 are bigints. |
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335 | //x must have at least one leading zero element. |
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336 | //y must be nonzero. |
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337 | //q and r must be arrays that are exactly the same length as x. (Or q can have more). |
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338 | //Must have x.length >= y.length >= 2. |
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339 | function divide_(x,y,q,r) { |
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340 | var kx, ky; |
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341 | var i,j,y1,y2,c,a,b; |
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342 | copy_(r,x); |
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343 | for (ky=y.length;y[ky-1]==0;ky--); //ky is number of elements in y, not including leading zeros |
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344 | |
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345 | //normalize: ensure the most significant element of y has its highest bit set |
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346 | b=y[ky-1]; |
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347 | for (a=0; b; a++) |
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348 | b>>=1; |
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349 | a=bpe-a; //a is how many bits to shift so that the high order bit of y is leftmost in its array element |
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350 | leftShift_(y,a); //multiply both by 1<<a now, then divide both by that at the end |
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351 | leftShift_(r,a); |
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352 | |
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353 | //Rob Visser discovered a bug: the following line was originally just before the normalization. |
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354 | for (kx=r.length;r[kx-1]==0 && kx>ky;kx--); //kx is number of elements in normalized x, not including leading zeros |
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355 | |
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356 | copyInt_(q,0); // q=0 |
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357 | while (!greaterShift(y,r,kx-ky)) { // while (leftShift_(y,kx-ky) <= r) { |
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358 | subShift_(r,y,kx-ky); // r=r-leftShift_(y,kx-ky) |
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359 | q[kx-ky]++; // q[kx-ky]++; |
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360 | } // } |
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361 | |
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362 | for (i=kx-1; i>=ky; i--) { |
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363 | if (r[i]==y[ky-1]) |
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364 | q[i-ky]=mask; |
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365 | else |
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366 | q[i-ky]=Math.floor((r[i]*radix+r[i-1])/y[ky-1]); |
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367 | |
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368 | //The following for(;;) loop is equivalent to the commented while loop, |
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369 | //except that the uncommented version avoids overflow. |
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370 | //The commented loop comes from HAC, which assumes r[-1]==y[-1]==0 |
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371 | // while (q[i-ky]*(y[ky-1]*radix+y[ky-2]) > r[i]*radix*radix+r[i-1]*radix+r[i-2]) |
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372 | // q[i-ky]--; |
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373 | for (;;) { |
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374 | y2=(ky>1 ? y[ky-2] : 0)*q[i-ky]; |
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375 | c=y2>>bpe; |
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376 | y2=y2 & mask; |
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377 | y1=c+q[i-ky]*y[ky-1]; |
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378 | c=y1>>bpe; |
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379 | y1=y1 & mask; |
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380 | |
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381 | if (c==r[i] ? y1==r[i-1] ? y2>(i>1 ? r[i-2] : 0) : y1>r[i-1] : c>r[i]) |
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382 | q[i-ky]--; |
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383 | else |
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384 | break; |
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385 | } |
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386 | |
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387 | linCombShift_(r,y,-q[i-ky],i-ky); //r=r-q[i-ky]*leftShift_(y,i-ky) |
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388 | if (negative(r)) { |
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389 | addShift_(r,y,i-ky); //r=r+leftShift_(y,i-ky) |
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390 | q[i-ky]--; |
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391 | } |
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392 | } |
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393 | |
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394 | rightShift_(y,a); //undo the normalization step |
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395 | rightShift_(r,a); //undo the normalization step |
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396 | } |
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397 | |
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398 | //do carries and borrows so each element of the bigInt x fits in bpe bits. |
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399 | function carry_(x) { |
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400 | var i,k,c,b; |
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401 | k=x.length; |
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402 | c=0; |
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403 | for (i=0;i<k;i++) { |
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404 | c+=x[i]; |
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405 | b=0; |
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406 | if (c<0) { |
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407 | b=-(c>>bpe); |
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408 | c+=b*radix; |
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409 | } |
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410 | x[i]=c & mask; |
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411 | c=(c>>bpe)-b; |
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412 | } |
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413 | } |
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414 | |
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415 | //return x mod n for bigInt x and integer n. |
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416 | function modInt(x,n) { |
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417 | var i,c=0; |
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418 | for (i=x.length-1; i>=0; i--) |
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419 | c=(c*radix+x[i])%n; |
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420 | return c; |
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421 | } |
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422 | |
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423 | //convert the integer t into a bigInt with at least the given number of bits. |
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424 | //the returned array stores the bigInt in bpe-bit chunks, little endian (buff[0] is least significant word) |
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425 | //Pad the array with leading zeros so that it has at least minSize elements. |
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426 | //There will always be at least one leading 0 element. |
---|
427 | function int2bigInt(t,bits,minSize) { |
---|
428 | var i,k; |
---|
429 | k=Math.ceil(bits/bpe)+1; |
---|
430 | k=minSize>k ? minSize : k; |
---|
431 | buff=new Array(k); |
---|
432 | copyInt_(buff,t); |
---|
433 | return buff; |
---|
434 | } |
---|
435 | |
---|
436 | //return the bigInt given a string representation in a given base. |
---|
437 | //Pad the array with leading zeros so that it has at least minSize elements. |
---|
438 | //If base=-1, then it reads in a space-separated list of array elements in decimal. |
---|
439 | //The array will always have at least one leading zero, unless base=-1. |
---|
440 | function str2bigInt(s,base,minSize) { |
---|
441 | var d, i, j, x, y, kk; |
---|
442 | var k=s.length; |
---|
443 | if (base==-1) { //comma-separated list of array elements in decimal |
---|
444 | x=new Array(0); |
---|
445 | for (;;) { |
---|
446 | y=new Array(x.length+1); |
---|
447 | for (i=0;i<x.length;i++) |
---|
448 | y[i+1]=x[i]; |
---|
449 | y[0]=parseInt(s,10); |
---|
450 | x=y; |
---|
451 | d=s.indexOf(',',0); |
---|
452 | if (d<1) |
---|
453 | break; |
---|
454 | s=s.substring(d+1); |
---|
455 | if (s.length==0) |
---|
456 | break; |
---|
457 | } |
---|
458 | if (x.length<minSize) { |
---|
459 | y=new Array(minSize); |
---|
460 | copy_(y,x); |
---|
461 | return y; |
---|
462 | } |
---|
463 | return x; |
---|
464 | } |
---|
465 | |
---|
466 | x=int2bigInt(0,base*k,0); |
---|
467 | for (i=0;i<k;i++) { |
---|
468 | d=digitsStr.indexOf(s.substring(i,i+1),0); |
---|
469 | if (base<=36 && d>=36) //convert lowercase to uppercase if base<=36 |
---|
470 | d-=26; |
---|
471 | if (d>=base || d<0) { //stop at first illegal character |
---|
472 | break; |
---|
473 | } |
---|
474 | multInt_(x,base); |
---|
475 | addInt_(x,d); |
---|
476 | } |
---|
477 | |
---|
478 | for (k=x.length;k>0 && !x[k-1];k--); //strip off leading zeros |
---|
479 | k=minSize>k+1 ? minSize : k+1; |
---|
480 | y=new Array(k); |
---|
481 | kk=k<x.length ? k : x.length; |
---|
482 | for (i=0;i<kk;i++) |
---|
483 | y[i]=x[i]; |
---|
484 | for (;i<k;i++) |
---|
485 | y[i]=0; |
---|
486 | return y; |
---|
487 | } |
---|
488 | |
---|
489 | //is bigint x equal to integer y? |
---|
490 | //y must have less than bpe bits |
---|
491 | function equalsInt(x,y) { |
---|
492 | var i; |
---|
493 | if (x[0]!=y) |
---|
494 | return 0; |
---|
495 | for (i=1;i<x.length;i++) |
---|
496 | if (x[i]) |
---|
497 | return 0; |
---|
498 | return 1; |
---|
499 | } |
---|
500 | |
---|
501 | //are bigints x and y equal? |
---|
502 | //this works even if x and y are different lengths and have arbitrarily many leading zeros |
---|
503 | function equals(x,y) { |
---|
504 | var i; |
---|
505 | var k=x.length<y.length ? x.length : y.length; |
---|
506 | for (i=0;i<k;i++) |
---|
507 | if (x[i]!=y[i]) |
---|
508 | return 0; |
---|
509 | if (x.length>y.length) { |
---|
510 | for (;i<x.length;i++) |
---|
511 | if (x[i]) |
---|
512 | return 0; |
---|
513 | } else { |
---|
514 | for (;i<y.length;i++) |
---|
515 | if (y[i]) |
---|
516 | return 0; |
---|
517 | } |
---|
518 | return 1; |
---|
519 | } |
---|
520 | |
---|
521 | //is the bigInt x equal to zero? |
---|
522 | function isZero(x) { |
---|
523 | var i; |
---|
524 | for (i=0;i<x.length;i++) |
---|
525 | if (x[i]) |
---|
526 | return 0; |
---|
527 | return 1; |
---|
528 | } |
---|
529 | |
---|
530 | //convert a bigInt into a string in a given base, from base 2 up to base 95. |
---|
531 | //Base -1 prints the contents of the array representing the number. |
---|
532 | function bigInt2str(x,base) { |
---|
533 | var i,t,s=""; |
---|
534 | |
---|
535 | if (s6.length!=x.length) |
---|
536 | s6=dup(x); |
---|
537 | else |
---|
538 | copy_(s6,x); |
---|
539 | |
---|
540 | if (base==-1) { //return the list of array contents |
---|
541 | for (i=x.length-1;i>0;i--) |
---|
542 | s+=x[i]+','; |
---|
543 | s+=x[0]; |
---|
544 | } |
---|
545 | else { //return it in the given base |
---|
546 | while (!isZero(s6)) { |
---|
547 | t=divInt_(s6,base); //t=s6 % base; s6=floor(s6/base); |
---|
548 | s=digitsStr.substring(t,t+1)+s; |
---|
549 | } |
---|
550 | } |
---|
551 | if (s.length==0) |
---|
552 | s="0"; |
---|
553 | return s; |
---|
554 | } |
---|
555 | |
---|
556 | //returns a duplicate of bigInt x |
---|
557 | function dup(x) { |
---|
558 | var i; |
---|
559 | buff=new Array(x.length); |
---|
560 | copy_(buff,x); |
---|
561 | return buff; |
---|
562 | } |
---|
563 | |
---|
564 | //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). |
---|
565 | function copy_(x,y) { |
---|
566 | var i; |
---|
567 | var k=x.length<y.length ? x.length : y.length; |
---|
568 | for (i=0;i<k;i++) |
---|
569 | x[i]=y[i]; |
---|
570 | for (i=k;i<x.length;i++) |
---|
571 | x[i]=0; |
---|
572 | } |
---|
573 | |
---|
574 | //do x=y on bigInt x and integer y. |
---|
575 | function copyInt_(x,n) { |
---|
576 | var i,c; |
---|
577 | for (c=n,i=0;i<x.length;i++) { |
---|
578 | x[i]=c & mask; |
---|
579 | c>>=bpe; |
---|
580 | } |
---|
581 | } |
---|
582 | |
---|
583 | //do x=x+n where x is a bigInt and n is an integer. |
---|
584 | //x must be large enough to hold the result. |
---|
585 | function addInt_(x,n) { |
---|
586 | var i,k,c,b; |
---|
587 | x[0]+=n; |
---|
588 | k=x.length; |
---|
589 | c=0; |
---|
590 | for (i=0;i<k;i++) { |
---|
591 | c+=x[i]; |
---|
592 | b=0; |
---|
593 | if (c<0) { |
---|
594 | b=-(c>>bpe); |
---|
595 | c+=b*radix; |
---|
596 | } |
---|
597 | x[i]=c & mask; |
---|
598 | c=(c>>bpe)-b; |
---|
599 | if (!c) return; //stop carrying as soon as the carry is zero |
---|
600 | } |
---|
601 | } |
---|
602 | |
---|
603 | //right shift bigInt x by n bits. 0 <= n < bpe. |
---|
604 | function rightShift_(x,n) { |
---|
605 | var i; |
---|
606 | var k=Math.floor(n/bpe); |
---|
607 | if (k) { |
---|
608 | for (i=0;i<x.length-k;i++) //right shift x by k elements |
---|
609 | x[i]=x[i+k]; |
---|
610 | for (;i<x.length;i++) |
---|
611 | x[i]=0; |
---|
612 | n%=bpe; |
---|
613 | } |
---|
614 | for (i=0;i<x.length-1;i++) { |
---|
615 | x[i]=mask & ((x[i+1]<<(bpe-n)) | (x[i]>>n)); |
---|
616 | } |
---|
617 | x[i]>>=n; |
---|
618 | } |
---|
619 | |
---|
620 | //do x=floor(|x|/2)*sgn(x) for bigInt x in 2's complement |
---|
621 | function halve_(x) { |
---|
622 | var i; |
---|
623 | for (i=0;i<x.length-1;i++) { |
---|
624 | x[i]=mask & ((x[i+1]<<(bpe-1)) | (x[i]>>1)); |
---|
625 | } |
---|
626 | x[i]=(x[i]>>1) | (x[i] & (radix>>1)); //most significant bit stays the same |
---|
627 | } |
---|
628 | |
---|
629 | //left shift bigInt x by n bits. |
---|
630 | function leftShift_(x,n) { |
---|
631 | var i; |
---|
632 | var k=Math.floor(n/bpe); |
---|
633 | if (k) { |
---|
634 | for (i=x.length; i>=k; i--) //left shift x by k elements |
---|
635 | x[i]=x[i-k]; |
---|
636 | for (;i>=0;i--) |
---|
637 | x[i]=0; |
---|
638 | n%=bpe; |
---|
639 | } |
---|
640 | if (!n) |
---|
641 | return; |
---|
642 | for (i=x.length-1;i>0;i--) { |
---|
643 | x[i]=mask & ((x[i]<<n) | (x[i-1]>>(bpe-n))); |
---|
644 | } |
---|
645 | x[i]=mask & (x[i]<<n); |
---|
646 | } |
---|
647 | |
---|
648 | //do x=x*n where x is a bigInt and n is an integer. |
---|
649 | //x must be large enough to hold the result. |
---|
650 | function multInt_(x,n) { |
---|
651 | var i,k,c,b; |
---|
652 | if (!n) |
---|
653 | return; |
---|
654 | k=x.length; |
---|
655 | c=0; |
---|
656 | for (i=0;i<k;i++) { |
---|
657 | c+=x[i]*n; |
---|
658 | b=0; |
---|
659 | if (c<0) { |
---|
660 | b=-(c>>bpe); |
---|
661 | c+=b*radix; |
---|
662 | } |
---|
663 | x[i]=c & mask; |
---|
664 | c=(c>>bpe)-b; |
---|
665 | } |
---|
666 | } |
---|
667 | |
---|
668 | //do x=floor(x/n) for bigInt x and integer n, and return the remainder |
---|
669 | function divInt_(x,n) { |
---|
670 | var i,r=0,s; |
---|
671 | for (i=x.length-1;i>=0;i--) { |
---|
672 | s=r*radix+x[i]; |
---|
673 | x[i]=Math.floor(s/n); |
---|
674 | r=s%n; |
---|
675 | } |
---|
676 | return r; |
---|
677 | } |
---|
678 | |
---|
679 | //do the linear combination x=a*x+b*y for bigInts x and y, and integers a and b. |
---|
680 | //x must be large enough to hold the answer. |
---|
681 | function linComb_(x,y,a,b) { |
---|
682 | var i,c,k,kk; |
---|
683 | k=x.length<y.length ? x.length : y.length; |
---|
684 | kk=x.length; |
---|
685 | for (c=0,i=0;i<k;i++) { |
---|
686 | c+=a*x[i]+b*y[i]; |
---|
687 | x[i]=c & mask; |
---|
688 | c>>=bpe; |
---|
689 | } |
---|
690 | for (i=k;i<kk;i++) { |
---|
691 | c+=a*x[i]; |
---|
692 | x[i]=c & mask; |
---|
693 | c>>=bpe; |
---|
694 | } |
---|
695 | } |
---|
696 | |
---|
697 | //do the linear combination x=a*x+b*(y<<(ys*bpe)) for bigInts x and y, and integers a, b and ys. |
---|
698 | //x must be large enough to hold the answer. |
---|
699 | function linCombShift_(x,y,b,ys) { |
---|
700 | var i,c,k,kk; |
---|
701 | k=x.length<ys+y.length ? x.length : ys+y.length; |
---|
702 | kk=x.length; |
---|
703 | for (c=0,i=ys;i<k;i++) { |
---|
704 | c+=x[i]+b*y[i-ys]; |
---|
705 | x[i]=c & mask; |
---|
706 | c>>=bpe; |
---|
707 | } |
---|
708 | for (i=k;c && i<kk;i++) { |
---|
709 | c+=x[i]; |
---|
710 | x[i]=c & mask; |
---|
711 | c>>=bpe; |
---|
712 | } |
---|
713 | } |
---|
714 | |
---|
715 | //do x=x+(y<<(ys*bpe)) for bigInts x and y, and integers a,b and ys. |
---|
716 | //x must be large enough to hold the answer. |
---|
717 | function addShift_(x,y,ys) { |
---|
718 | var i,c,k,kk; |
---|
719 | k=x.length<ys+y.length ? x.length : ys+y.length; |
---|
720 | kk=x.length; |
---|
721 | for (c=0,i=ys;i<k;i++) { |
---|
722 | c+=x[i]+y[i-ys]; |
---|
723 | x[i]=c & mask; |
---|
724 | c>>=bpe; |
---|
725 | } |
---|
726 | for (i=k;c && i<kk;i++) { |
---|
727 | c+=x[i]; |
---|
728 | x[i]=c & mask; |
---|
729 | c>>=bpe; |
---|
730 | } |
---|
731 | } |
---|
732 | |
---|
733 | //do x=x-(y<<(ys*bpe)) for bigInts x and y, and integers a,b and ys. |
---|
734 | //x must be large enough to hold the answer. |
---|
735 | function subShift_(x,y,ys) { |
---|
736 | var i,c,k,kk; |
---|
737 | k=x.length<ys+y.length ? x.length : ys+y.length; |
---|
738 | kk=x.length; |
---|
739 | for (c=0,i=ys;i<k;i++) { |
---|
740 | c+=x[i]-y[i-ys]; |
---|
741 | x[i]=c & mask; |
---|
742 | c>>=bpe; |
---|
743 | } |
---|
744 | for (i=k;c && i<kk;i++) { |
---|
745 | c+=x[i]; |
---|
746 | x[i]=c & mask; |
---|
747 | c>>=bpe; |
---|
748 | } |
---|
749 | } |
---|
750 | |
---|
751 | //do x=x-y for bigInts x and y. |
---|
752 | //x must be large enough to hold the answer. |
---|
753 | //negative answers will be 2s complement |
---|
754 | function sub_(x,y) { |
---|
755 | var i,c,k,kk; |
---|
756 | k=x.length<y.length ? x.length : y.length; |
---|
757 | for (c=0,i=0;i<k;i++) { |
---|
758 | c+=x[i]-y[i]; |
---|
759 | x[i]=c & mask; |
---|
760 | c>>=bpe; |
---|
761 | } |
---|
762 | for (i=k;c && i<x.length;i++) { |
---|
763 | c+=x[i]; |
---|
764 | x[i]=c & mask; |
---|
765 | c>>=bpe; |
---|
766 | } |
---|
767 | } |
---|
768 | |
---|
769 | //do x=x+y for bigInts x and y. |
---|
770 | //x must be large enough to hold the answer. |
---|
771 | function add_(x,y) { |
---|
772 | var i,c,k,kk; |
---|
773 | k=x.length<y.length ? x.length : y.length; |
---|
774 | for (c=0,i=0;i<k;i++) { |
---|
775 | c+=x[i]+y[i]; |
---|
776 | x[i]=c & mask; |
---|
777 | c>>=bpe; |
---|
778 | } |
---|
779 | for (i=k;c && i<x.length;i++) { |
---|
780 | c+=x[i]; |
---|
781 | x[i]=c & mask; |
---|
782 | c>>=bpe; |
---|
783 | } |
---|
784 | } |
---|
785 | |
---|
786 | //do x=x*y for bigInts x and y. This is faster when y<x. |
---|
787 | function mult_(x,y) { |
---|
788 | var i; |
---|
789 | if (ss.length!=2*x.length) |
---|
790 | ss=new Array(2*x.length); |
---|
791 | copyInt_(ss,0); |
---|
792 | for (i=0;i<y.length;i++) |
---|
793 | if (y[i]) |
---|
794 | linCombShift_(ss,x,y[i],i); //ss=1*ss+y[i]*(x<<(i*bpe)) |
---|
795 | copy_(x,ss); |
---|
796 | } |
---|
797 | |
---|
798 | //do x=x mod n for bigInts x and n. |
---|
799 | function mod_(x,n) { |
---|
800 | if (s4.length!=x.length) |
---|
801 | s4=dup(x); |
---|
802 | else |
---|
803 | copy_(s4,x); |
---|
804 | if (s5.length!=x.length) |
---|
805 | s5=dup(x); |
---|
806 | divide_(s4,n,s5,x); //x = remainder of s4 / n |
---|
807 | } |
---|
808 | |
---|
809 | //do x=x*y mod n for bigInts x,y,n. |
---|
810 | //for greater speed, let y<x. |
---|
811 | function multMod_(x,y,n) { |
---|
812 | var i; |
---|
813 | if (s0.length!=2*x.length) |
---|
814 | s0=new Array(2*x.length); |
---|
815 | copyInt_(s0,0); |
---|
816 | for (i=0;i<y.length;i++) |
---|
817 | if (y[i]) |
---|
818 | linCombShift_(s0,x,y[i],i); //s0=1*s0+y[i]*(x<<(i*bpe)) |
---|
819 | mod_(s0,n); |
---|
820 | copy_(x,s0); |
---|
821 | } |
---|
822 | |
---|
823 | //do x=x*x mod n for bigInts x,n. |
---|
824 | function squareMod_(x,n) { |
---|
825 | var i,j,d,c,kx,kn,k; |
---|
826 | for (kx=x.length; kx>0 && !x[kx-1]; kx--); //ignore leading zeros in x |
---|
827 | 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 |
---|
828 | if (s0.length!=k) |
---|
829 | s0=new Array(k); |
---|
830 | copyInt_(s0,0); |
---|
831 | for (i=0;i<kx;i++) { |
---|
832 | c=s0[2*i]+x[i]*x[i]; |
---|
833 | s0[2*i]=c & mask; |
---|
834 | c>>=bpe; |
---|
835 | for (j=i+1;j<kx;j++) { |
---|
836 | c=s0[i+j]+2*x[i]*x[j]+c; |
---|
837 | s0[i+j]=(c & mask); |
---|
838 | c>>=bpe; |
---|
839 | } |
---|
840 | s0[i+kx]=c; |
---|
841 | } |
---|
842 | mod_(s0,n); |
---|
843 | copy_(x,s0); |
---|
844 | } |
---|
845 | |
---|
846 | //return x with exactly k leading zero elements |
---|
847 | function bigintTrim(x,k) { |
---|
848 | var i,y; |
---|
849 | for (i=x.length; i>0 && !x[i-1]; i--); |
---|
850 | y=new Array(i+k); |
---|
851 | copy_(y,x); |
---|
852 | return y; |
---|
853 | } |
---|
854 | |
---|
855 | //do x=x**y mod n, where x,y,n are bigInts and ** is exponentiation. 0**0=1. |
---|
856 | //this is faster when n is odd. x usually needs to have as many elements as n. |
---|
857 | function powMod_(x,y,n) { |
---|
858 | var k1,k2,kn,np; |
---|
859 | if(s7.length!=n.length) |
---|
860 | s7=dup(n); |
---|
861 | |
---|
862 | //for even modulus, use a simple square-and-multiply algorithm, |
---|
863 | //rather than using the more complex Montgomery algorithm. |
---|
864 | if ((n[0]&1)==0) { |
---|
865 | copy_(s7,x); |
---|
866 | copyInt_(x,1); |
---|
867 | while(!equalsInt(y,0)) { |
---|
868 | if (y[0]&1) |
---|
869 | multMod_(x,s7,n); |
---|
870 | divInt_(y,2); |
---|
871 | squareMod_(s7,n); |
---|
872 | } |
---|
873 | return; |
---|
874 | } |
---|
875 | |
---|
876 | //calculate np from n for the Montgomery multiplications |
---|
877 | copyInt_(s7,0); |
---|
878 | for (kn=n.length;kn>0 && !n[kn-1];kn--); |
---|
879 | np=radix-inverseModInt(modInt(n,radix),radix); |
---|
880 | s7[kn]=1; |
---|
881 | multMod_(x ,s7,n); // x = x * 2**(kn*bp) mod n |
---|
882 | |
---|
883 | if (s3.length!=x.length) |
---|
884 | s3=dup(x); |
---|
885 | else |
---|
886 | copy_(s3,x); |
---|
887 | |
---|
888 | for (k1=y.length-1;k1>0 & !y[k1]; k1--); //k1=first nonzero element of y |
---|
889 | if (y[k1]==0) { //anything to the 0th power is 1 |
---|
890 | copyInt_(x,1); |
---|
891 | return; |
---|
892 | } |
---|
893 | for (k2=1<<(bpe-1);k2 && !(y[k1] & k2); k2>>=1); //k2=position of first 1 bit in y[k1] |
---|
894 | for (;;) { |
---|
895 | if (!(k2>>=1)) { //look at next bit of y |
---|
896 | k1--; |
---|
897 | if (k1<0) { |
---|
898 | mont_(x,one,n,np); |
---|
899 | return; |
---|
900 | } |
---|
901 | k2=1<<(bpe-1); |
---|
902 | } |
---|
903 | mont_(x,x,n,np); |
---|
904 | |
---|
905 | if (k2 & y[k1]) //if next bit is a 1 |
---|
906 | mont_(x,s3,n,np); |
---|
907 | } |
---|
908 | } |
---|
909 | |
---|
910 | |
---|
911 | //do x=x*y*Ri mod n for bigInts x,y,n, |
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912 | // where Ri = 2**(-kn*bpe) mod n, and kn is the |
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913 | // number of elements in the n array, not |
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914 | // counting leading zeros. |
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915 | //x array must have at least as many elemnts as the n array |
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916 | //It's OK if x and y are the same variable. |
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917 | //must have: |
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918 | // x,y < n |
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919 | // n is odd |
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920 | // np = -(n^(-1)) mod radix |
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921 | function mont_(x,y,n,np) { |
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922 | var i,j,c,ui,t,ks; |
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923 | var kn=n.length; |
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924 | var ky=y.length; |
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925 | |
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926 | if (sa.length!=kn) |
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927 | sa=new Array(kn); |
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928 | |
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929 | copyInt_(sa,0); |
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930 | |
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931 | for (;kn>0 && n[kn-1]==0;kn--); //ignore leading zeros of n |
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932 | for (;ky>0 && y[ky-1]==0;ky--); //ignore leading zeros of y |
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933 | ks=sa.length-1; //sa will never have more than this many nonzero elements. |
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934 | |
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935 | //the following loop consumes 95% of the runtime for randTruePrime_() and powMod_() for large numbers |
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936 | for (i=0; i<kn; i++) { |
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937 | t=sa[0]+x[i]*y[0]; |
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938 | ui=((t & mask) * np) & mask; //the inner "& mask" was needed on Safari (but not MSIE) at one time |
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939 | c=(t+ui*n[0]) >> bpe; |
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940 | t=x[i]; |
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941 | |
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942 | //do sa=(sa+x[i]*y+ui*n)/b where b=2**bpe. Loop is unrolled 5-fold for speed |
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943 | j=1; |
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944 | for (;j<ky-4;) { c+=sa[j]+ui*n[j]+t*y[j]; sa[j-1]=c & mask; c>>=bpe; j++; |
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945 | c+=sa[j]+ui*n[j]+t*y[j]; sa[j-1]=c & mask; c>>=bpe; j++; |
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946 | c+=sa[j]+ui*n[j]+t*y[j]; sa[j-1]=c & mask; c>>=bpe; j++; |
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947 | c+=sa[j]+ui*n[j]+t*y[j]; sa[j-1]=c & mask; c>>=bpe; j++; |
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948 | c+=sa[j]+ui*n[j]+t*y[j]; sa[j-1]=c & mask; c>>=bpe; j++; } |
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949 | for (;j<ky;) { c+=sa[j]+ui*n[j]+t*y[j]; sa[j-1]=c & mask; c>>=bpe; j++; } |
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950 | for (;j<kn-4;) { c+=sa[j]+ui*n[j]; sa[j-1]=c & mask; c>>=bpe; j++; |
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951 | c+=sa[j]+ui*n[j]; sa[j-1]=c & mask; c>>=bpe; j++; |
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952 | c+=sa[j]+ui*n[j]; sa[j-1]=c & mask; c>>=bpe; j++; |
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953 | c+=sa[j]+ui*n[j]; sa[j-1]=c & mask; c>>=bpe; j++; |
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954 | c+=sa[j]+ui*n[j]; sa[j-1]=c & mask; c>>=bpe; j++; } |
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955 | for (;j<kn;) { c+=sa[j]+ui*n[j]; sa[j-1]=c & mask; c>>=bpe; j++; } |
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956 | for (;j<ks;) { c+=sa[j]; sa[j-1]=c & mask; c>>=bpe; j++; } |
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957 | sa[j-1]=c & mask; |
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958 | } |
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959 | |
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960 | if (!greater(n,sa)) |
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961 | sub_(sa,n); |
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962 | copy_(x,sa); |
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963 | } |
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964 | |
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965 | |
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