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| 1 | /* | ||
| 2 | |||
| 3 | Copyright 2008-2013 Clipperz Srl | ||
| 4 | |||
| 5 | This file is part of Clipperz, the online password manager. | ||
| 6 | For further information about its features and functionalities please | ||
| 7 | refer to http://www.clipperz.com. | ||
| 8 | |||
| 9 | * Clipperz is free software: you can redistribute it and/or modify it | ||
| 10 | under the terms of the GNU Affero General Public License as published | ||
| 11 | by the Free Software Foundation, either version 3 of the License, or | ||
| 12 | (at your option) any later version. | ||
| 13 | |||
| 14 | * Clipperz is distributed in the hope that it will be useful, but | ||
| 15 | WITHOUT ANY WARRANTY; without even the implied warranty of | ||
| 16 | MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. | ||
| 17 | See the GNU Affero General Public License for more details. | ||
| 18 | |||
| 19 | * You should have received a copy of the GNU Affero General Public | ||
| 20 | License along with Clipperz. If not, see http://www.gnu.org/licenses/. | ||
| 21 | |||
| 22 | */ | ||
| 23 | |||
| 24 | if (typeof(Clipperz) == 'undefined') { Clipperz = {}; } | ||
| 25 | if (typeof(Clipperz.Crypto) == 'undefined') { Clipperz.Crypto = {}; } | ||
| 26 | |||
| 27 | //############################################################################# | ||
| 28 | //Downloaded on March 05, 2007 from http://www.leemon.com/crypto/BigInt.js | ||
| 29 | //############################################################################# | ||
| 30 | |||
| 31 | |||
| 32 | //////////////////////////////////////////////////////////////////////////////////////// | ||
| 33 | // Big Integer Library v. 5.0 | ||
| 34 | // Created 2000, last modified 2006 | ||
| 35 | // Leemon Baird | ||
| 36 | // www.leemon.com | ||
| 37 | // | ||
| 38 | // This file is public domain. You can use it for any purpose without restriction. | ||
| 39 | // I do not guarantee that it is correct, so use it at your own risk. If you use | ||
| 40 | // it for something interesting, I'd appreciate hearing about it. If you find | ||
| 41 | // any bugs or make any improvements, I'd appreciate hearing about those too. | ||
| 42 | // It would also be nice if my name and address were left in the comments. | ||
| 43 | // But none of that is required. | ||
| 44 | // | ||
| 45 | // This code defines a bigInt library for arbitrary-precision integers. | ||
| 46 | // A bigInt is an array of integers storing the value in chunks of bpe bits, | ||
| 47 | // little endian (buff[0] is the least significant word). | ||
| 48 | // Negative bigInts are stored two's complement. | ||
| 49 | // Some functions assume their parameters have at least one leading zero element. | ||
| 50 | // Functions with an underscore at the end of the name have unpredictable behavior in case of overflow, | ||
| 51 | // so the caller must make sure overflow won't happen. | ||
| 52 | // For each function where a parameter is modified, that same | ||
| 53 | // variable must not be used as another argument too. | ||
| 54 | // So, you cannot square x by doing multMod_(x,x,n). | ||
| 55 | // You must use squareMod_(x,n) instead, or do y=dup(x); multMod_(x,y,n). | ||
| 56 | // | ||
| 57 | // These functions are designed to avoid frequent dynamic memory allocation in the inner loop. | ||
| 58 | // For most functions, if it needs a BigInt as a local variable it will actually use | ||
| 59 | // a global, and will only allocate to it when it's not the right size. This ensures | ||
| 60 | // that when a function is called repeatedly with same-sized parameters, it only allocates | ||
| 61 | // memory on the first call. | ||
| 62 | // | ||
| 63 | // Note that for cryptographic purposes, the calls to Math.random() must | ||
| 64 | // be replaced with calls to a better pseudorandom number generator. | ||
| 65 | // | ||
| 66 | // In the following, "bigInt" means a bigInt with at least one leading zero element, | ||
| 67 | // and "integer" means a nonnegative integer less than radix. In some cases, integer | ||
| 68 | // can be negative. Negative bigInts are 2s complement. | ||
| 69 | // | ||
| 70 | // The following functions do not modify their inputs, but dynamically allocate memory every time they are called: | ||
| 71 | // | ||
| 72 | // function bigInt2str(x,base) //convert a bigInt into a string in a given base, from base 2 up to base 95 | ||
| 73 | // function dup(x) //returns a copy of bigInt x | ||
| 74 | // function findPrimes(n) //return array of all primes less than integer n | ||
| 75 | // function int2bigInt(t,n,m) //convert integer t to a bigInt with at least n bits and m array elements | ||
| 76 | // function int2bigInt(s,b,n,m) //convert string s in base b to a bigInt with at least n bits and m array elements | ||
| 77 | // function trim(x,k) //return a copy of x with exactly k leading zero elements | ||
| 78 | // | ||
| 79 | // The following functions do not modify their inputs, so there is never a problem with the result being too big: | ||
| 80 | // | ||
| 81 | // function bitSize(x) //returns how many bits long the bigInt x is, not counting leading zeros | ||
| 82 | // function equals(x,y) //is the bigInt x equal to the bigint y? | ||
| 83 | // function equalsInt(x,y) //is bigint x equal to integer y? | ||
| 84 | // function greater(x,y) //is x>y? (x and y are nonnegative bigInts) | ||
| 85 | // function greaterShift(x,y,shift)//is (x <<(shift*bpe)) > y? | ||
| 86 | // function isZero(x) //is the bigInt x equal to zero? | ||
| 87 | // function millerRabin(x,b) //does one round of Miller-Rabin base integer b say that bigInt x is possibly prime (as opposed to definitely composite)? | ||
| 88 | // function modInt(x,n) //return x mod n for bigInt x and integer n. | ||
| 89 | // function negative(x) //is bigInt x negative? | ||
| 90 | // | ||
| 91 | // The following functions do not modify their inputs, but allocate memory and call functions with underscores | ||
| 92 | // | ||
| 93 | // function add(x,y) //return (x+y) for bigInts x and y. | ||
| 94 | // function addInt(x,n) //return (x+n) where x is a bigInt and n is an integer. | ||
| 95 | // function expand(x,n) //return a copy of x with at least n elements, adding leading zeros if needed | ||
| 96 | // function inverseMod(x,n) //return (x**(-1) mod n) for bigInts x and n. If no inverse exists, it returns null | ||
| 97 | // function mod(x,n) //return a new bigInt equal to (x mod n) for bigInts x and n. | ||
| 98 | // function mult(x,y) //return x*y for bigInts x and y. This is faster when y<x. | ||
| 99 | // function multMod(x,y,n) //return (x*y mod n) for bigInts x,y,n. For greater speed, let y<x. | ||
| 100 | // function 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. | ||
| 101 | // function randTruePrime(k) //return a new, random, k-bit, true prime using Maurer's algorithm. | ||
| 102 | // function sub(x,y) //return (x-y) for bigInts x and y. Negative answers will be 2s complement | ||
| 103 | // | ||
| 104 | // The following functions write a bigInt result to one of the parameters, but | ||
| 105 | // the result is never bigger than the original, so there can't be overflow problems: | ||
| 106 | // | ||
| 107 | // function divInt_(x,n) //do x=floor(x/n) for bigInt x and integer n, and return the remainder | ||
| 108 | // function GCD_(x,y) //set x to the greatest common divisor of bigInts x and y, (y is destroyed). | ||
| 109 | // function halve_(x) //do x=floor(|x|/2)*sgn(x) for bigInt x in 2's complement | ||
| 110 | // function mod_(x,n) //do x=x mod n for bigInts x and n. | ||
| 111 | // function rightShift_(x,n) //right shift bigInt x by n bits. 0 <= n < bpe. | ||
| 112 | // | ||
| 113 | // The following functions write a bigInt result to one of the parameters. The caller is responsible for | ||
| 114 | // ensuring it is large enough to hold the result. | ||
| 115 | // | ||
| 116 | // function addInt_(x,n) //do x=x+n where x is a bigInt and n is an integer | ||
| 117 | // function add_(x,y) //do x=x+y for bigInts x and y | ||
| 118 | // function addShift_(x,y,ys) //do x=x+(y<<(ys*bpe)) | ||
| 119 | // function copy_(x,y) //do x=y on bigInts x and y | ||
| 120 | // function copyInt_(x,n) //do x=n on bigInt x and integer n | ||
| 121 | // function carry_(x) //do carries and borrows so each element of the bigInt x fits in bpe bits. | ||
| 122 | // function divide_(x,y,q,r) //divide_ x by y giving quotient q and remainder r | ||
| 123 | // function eGCD_(x,y,d,a,b) //sets a,b,d to positive big integers such that d = GCD_(x,y) = a*x-b*y | ||
| 124 | // function inverseMod_(x,n) //do x=x**(-1) mod n, for bigInts x and n. Returns 1 (0) if inverse does (doesn't) exist | ||
| 125 | // function inverseModInt_(x,n) //return x**(-1) mod n, for integers x and n. Return 0 if there is no inverse | ||
| 126 | // function leftShift_(x,n) //left shift bigInt x by n bits. n<bpe. | ||
| 127 | // function linComb_(x,y,a,b) //do x=a*x+b*y for bigInts x and y and integers a and b | ||
| 128 | // function linCombShift_(x,y,b,ys) //do x=x+b*(y<<(ys*bpe)) for bigInts x and y, and integers b and ys | ||
| 129 | // function mont_(x,y,n,np) //Montgomery multiplication (see comments where the function is defined) | ||
| 130 | // function mult_(x,y) //do x=x*y for bigInts x and y. | ||
| 131 | // function multInt_(x,n) //do x=x*n where x is a bigInt and n is an integer. | ||
| 132 | // function multMod_(x,y,n) //do x=x*y mod n for bigInts x,y,n. | ||
| 133 | // function 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. | ||
| 134 | // function 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. | ||
| 135 | // function randTruePrime_(ans,k) //do ans = a random k-bit true random prime (not just probable prime) with 1 in the msb. | ||
| 136 | // function squareMod_(x,n) //do x=x*x mod n for bigInts x,n | ||
| 137 | // function sub_(x,y) //do x=x-y for bigInts x and y. Negative answers will be 2s complement. | ||
| 138 | // function subShift_(x,y,ys) //do x=x-(y<<(ys*bpe)). Negative answers will be 2s complement. | ||
| 139 | // | ||
| 140 | // The following functions are based on algorithms from the _Handbook of Applied Cryptography_ | ||
| 141 | // powMod_() = algorithm 14.94, Montgomery exponentiation | ||
| 142 | // eGCD_,inverseMod_() = algorithm 14.61, Binary extended GCD_ | ||
| 143 | // GCD_() = algorothm 14.57, Lehmer's algorithm | ||
| 144 | // mont_() = algorithm 14.36, Montgomery multiplication | ||
| 145 | // divide_() = algorithm 14.20 Multiple-precision division | ||
| 146 | // squareMod_() = algorithm 14.16 Multiple-precision squaring | ||
| 147 | // randTruePrime_() = algorithm 4.62, Maurer's algorithm | ||
| 148 | // millerRabin() = algorithm 4.24, Miller-Rabin algorithm | ||
| 149 | // | ||
| 150 | // Profiling shows: | ||
| 151 | // randTruePrime_() spends: | ||
| 152 | // 10% of its time in calls to powMod_() | ||
| 153 | // 85% of its time in calls to millerRabin() | ||
| 154 | // millerRabin() spends: | ||
| 155 | // 99% of its time in calls to powMod_() (always with a base of 2) | ||
| 156 | // powMod_() spends: | ||
| 157 | // 94% of its time in calls to mont_() (almost always with x==y) | ||
| 158 | // | ||
| 159 | // This suggests there are several ways to speed up this library slightly: | ||
| 160 | // - convert powMod_ to use a Montgomery form of k-ary window (or maybe a Montgomery form of sliding window) | ||
| 161 | // -- this should especially focus on being fast when raising 2 to a power mod n | ||
| 162 | // - convert randTruePrime_() to use a minimum r of 1/3 instead of 1/2 with the appropriate change to the test | ||
| 163 | // - tune the parameters in randTruePrime_(), including c, m, and recLimit | ||
| 164 | // - speed up the single loop in mont_() that takes 95% of the runtime, perhaps by reducing checking | ||
| 165 | // within the loop when all the parameters are the same length. | ||
| 166 | // | ||
| 167 | // There are several ideas that look like they wouldn't help much at all: | ||
| 168 | // - replacing trial division in randTruePrime_() with a sieve (that speeds up something taking almost no time anyway) | ||
| 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) | ||
| 170 | // - speeding up mont_(x,y,n,np) when x==y by doing a non-modular, non-Montgomery square | ||
| 171 | // followed by a Montgomery reduction. The intermediate answer will be twice as long as x, so that | ||
| 172 | // method would be slower. This is unfortunate because the code currently spends almost all of its time | ||
| 173 | // doing mont_(x,x,...), both for randTruePrime_() and powMod_(). A faster method for Montgomery squaring | ||
| 174 | // would have a large impact on the speed of randTruePrime_() and powMod_(). HAC has a couple of poorly-worded | ||
| 175 | // sentences that seem to imply it's faster to do a non-modular square followed by a single | ||
| 176 | // Montgomery reduction, but that's obviously wrong. | ||
| 177 | //////////////////////////////////////////////////////////////////////////////////////// | ||
| 178 | |||
| 179 | //globals | ||
| 180 | bpe=0; //bits stored per array element | ||
| 181 | mask=0; //AND this with an array element to chop it down to bpe bits | ||
| 182 | radix=mask+1; //equals 2^bpe. A single 1 bit to the left of the last bit of mask. | ||
| 183 | |||
| 184 | //the digits for converting to different bases | ||
| 185 | digitsStr='0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz_=!@#$%^&*()[]{}|;:,.<>/?`~ \\\'\"+-'; | ||
| 186 | |||
| 187 | //initialize the global variables | ||
| 188 | for (bpe=0; (1<<(bpe+1)) > (1<<bpe); bpe++); //bpe=number of bits in the mantissa on this platform | ||
| 189 | bpe>>=1; //bpe=number of bits in one element of the array representing the bigInt | ||
| 190 | mask=(1<<bpe)-1; //AND the mask with an integer to get its bpe least significant bits | ||
| 191 | radix=mask+1; //2^bpe. a single 1 bit to the left of the first bit of mask | ||
| 192 | one=int2bigInt(1,1,1); //constant used in powMod_() | ||
| 193 | |||
| 194 | //the following global variables are scratchpad memory to | ||
| 195 | //reduce dynamic memory allocation in the inner loop | ||
| 196 | t=new Array(0); | ||
| 197 | ss=t; //used in mult_() | ||
| 198 | s0=t; //used in multMod_(), squareMod_() | ||
| 199 | s1=t; //used in powMod_(), multMod_(), squareMod_() | ||
| 200 | s2=t; //used in powMod_(), multMod_() | ||
| 201 | s3=t; //used in powMod_() | ||
| 202 | s4=t; s5=t; //used in mod_() | ||
| 203 | s6=t; //used in bigInt2str() | ||
| 204 | s7=t; //used in powMod_() | ||
| 205 | T=t; //used in GCD_() | ||
| 206 | sa=t; //used in mont_() | ||
| 207 | mr_x1=t; mr_r=t; mr_a=t; //used in millerRabin() | ||
| 208 | eg_v=t; eg_u=t; eg_A=t; eg_B=t; eg_C=t; eg_D=t; //used in eGCD_(), inverseMod_() | ||
| 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_() | ||
| 210 | |||
| 211 | primes=t; pows=t; s_i=t; s_i2=t; s_R=t; s_rm=t; s_q=t; s_n1=t; | ||
| 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_() | ||
| 213 | |||
| 214 | //////////////////////////////////////////////////////////////////////////////////////// | ||
| 215 | |||
| 216 | //return array of all primes less than integer n | ||
| 217 | function findPrimes(n) { | ||
| 218 | var i,s,p,ans; | ||
| 219 | s=new Array(n); | ||
| 220 | for (i=0;i<n;i++) | ||
| 221 | s[i]=0; | ||
| 222 | s[0]=2; | ||
| 223 | p=0; //first p elements of s are primes, the rest are a sieve | ||
| 224 | for(;s[p]<n;) { //s[p] is the pth prime | ||
| 225 | for(i=s[p]*s[p]; i<n; i+=s[p]) //mark multiples of s[p] | ||
| 226 | s[i]=1; | ||
| 227 | p++; | ||
| 228 | s[p]=s[p-1]+1; | ||
| 229 | for(; s[p]<n && s[s[p]]; s[p]++); //find next prime (where s[p]==0) | ||
| 230 | } | ||
| 231 | ans=new Array(p); | ||
| 232 | for(i=0;i<p;i++) | ||
| 233 | ans[i]=s[i]; | ||
| 234 | return ans; | ||
| 235 | } | ||
| 236 | |||
| 237 | //does a single round of Miller-Rabin base b consider x to be a possible prime? | ||
| 238 | //x is a bigInt, and b is an integer | ||
| 239 | function millerRabin(x,b) { | ||
| 240 | var i,j,k,s; | ||
| 241 | |||
| 242 | if (mr_x1.length!=x.length) { | ||
| 243 | mr_x1=dup(x); | ||
| 244 | mr_r=dup(x); | ||
| 245 | mr_a=dup(x); | ||
| 246 | } | ||
| 247 | |||
| 248 | copyInt_(mr_a,b); | ||
| 249 | copy_(mr_r,x); | ||
| 250 | copy_(mr_x1,x); | ||
| 251 | |||
| 252 | addInt_(mr_r,-1); | ||
| 253 | addInt_(mr_x1,-1); | ||
| 254 | |||
| 255 | //s=the highest power of two that divides mr_r | ||
| 256 | k=0; | ||
| 257 | for (i=0;i<mr_r.length;i++) | ||
| 258 | for (j=1;j<mask;j<<=1) | ||
| 259 | if (x[i] & j) { | ||
| 260 | s=(k<mr_r.length+bpe ? k : 0); | ||
| 261 | i=mr_r.length; | ||
| 262 | j=mask; | ||
| 263 | } else | ||
| 264 | k++; | ||
| 265 | |||
| 266 | if (s) | ||
| 267 | rightShift_(mr_r,s); | ||
| 268 | |||
| 269 | powMod_(mr_a,mr_r,x); | ||
| 270 | |||
| 271 | if (!equalsInt(mr_a,1) && !equals(mr_a,mr_x1)) { | ||
| 272 | j=1; | ||
| 273 | while (j<=s-1 && !equals(mr_a,mr_x1)) { | ||
| 274 | squareMod_(mr_a,x); | ||
| 275 | if (equalsInt(mr_a,1)) { | ||
| 276 | return 0; | ||
| 277 | } | ||
| 278 | j++; | ||
| 279 | } | ||
| 280 | if (!equals(mr_a,mr_x1)) { | ||
| 281 | return 0; | ||
| 282 | } | ||
| 283 | } | ||
| 284 | return 1; | ||
| 285 | } | ||
| 286 | |||
| 287 | //returns how many bits long the bigInt is, not counting leading zeros. | ||
| 288 | function bitSize(x) { | ||
| 289 | var j,z,w; | ||
| 290 | for (j=x.length-1; (x[j]==0) && (j>0); j--); | ||
| 291 | for (z=0,w=x[j]; w; (w>>=1),z++); | ||
| 292 | z+=bpe*j; | ||
| 293 | return z; | ||
| 294 | } | ||
| 295 | |||
| 296 | //return a copy of x with at least n elements, adding leading zeros if needed | ||
| 297 | function expand(x,n) { | ||
| 298 | var ans=int2bigInt(0,(x.length>n ? x.length : n)*bpe,0); | ||
| 299 | copy_(ans,x); | ||
| 300 | return ans; | ||
| 301 | } | ||
| 302 | |||
| 303 | //return a k-bit true random prime using Maurer's algorithm. | ||
| 304 | function randTruePrime(k) { | ||
| 305 | var ans=int2bigInt(0,k,0); | ||
| 306 | randTruePrime_(ans,k); | ||
| 307 | return trim(ans,1); | ||
| 308 | } | ||
| 309 | |||
| 310 | //return a new bigInt equal to (x mod n) for bigInts x and n. | ||
| 311 | function mod(x,n) { | ||
| 312 | var ans=dup(x); | ||
| 313 | mod_(ans,n); | ||
| 314 | return trim(ans,1); | ||
| 315 | } | ||
| 316 | |||
| 317 | //return (x+n) where x is a bigInt and n is an integer. | ||
| 318 | function addInt(x,n) { | ||
| 319 | var ans=expand(x,x.length+1); | ||
| 320 | addInt_(ans,n); | ||
| 321 | return trim(ans,1); | ||
| 322 | } | ||
| 323 | |||
| 324 | //return x*y for bigInts x and y. This is faster when y<x. | ||
| 325 | function mult(x,y) { | ||
| 326 | var ans=expand(x,x.length+y.length); | ||
| 327 | mult_(ans,y); | ||
| 328 | return trim(ans,1); | ||
| 329 | } | ||
| 330 | |||
| 331 | //return (x**y mod n) where x,y,n are bigInts and ** is exponentiation. 0**0=1. Faster for odd n. | ||
| 332 | function powMod(x,y,n) { | ||
| 333 | var ans=expand(x,n.length); | ||
| 334 | powMod_(ans,trim(y,2),trim(n,2),0); //this should work without the trim, but doesn't | ||
| 335 | return trim(ans,1); | ||
| 336 | } | ||
| 337 | |||
| 338 | //return (x-y) for bigInts x and y. Negative answers will be 2s complement | ||
| 339 | function sub(x,y) { | ||
| 340 | var ans=expand(x,(x.length>y.length ? x.length+1 : y.length+1)); | ||
| 341 | sub_(ans,y); | ||
| 342 | return trim(ans,1); | ||
| 343 | } | ||
| 344 | |||
| 345 | //return (x+y) for bigInts x and y. | ||
| 346 | function add(x,y) { | ||
| 347 | var ans=expand(x,(x.length>y.length ? x.length+1 : y.length+1)); | ||
| 348 | add_(ans,y); | ||
| 349 | return trim(ans,1); | ||
| 350 | } | ||
| 351 | |||
| 352 | //return (x**(-1) mod n) for bigInts x and n. If no inverse exists, it returns null | ||
| 353 | function inverseMod(x,n) { | ||
| 354 | var ans=expand(x,n.length); | ||
| 355 | var s; | ||
| 356 | s=inverseMod_(ans,n); | ||
| 357 | return s ? trim(ans,1) : null; | ||
| 358 | } | ||
| 359 | |||
| 360 | //return (x*y mod n) for bigInts x,y,n. For greater speed, let y<x. | ||
| 361 | function multMod(x,y,n) { | ||
| 362 | var ans=expand(x,n.length); | ||
| 363 | multMod_(ans,y,n); | ||
| 364 | return trim(ans,1); | ||
| 365 | } | ||
| 366 | |||
| 367 | //generate a k-bit true random prime using Maurer's algorithm, | ||
| 368 | //and put it into ans. The bigInt ans must be large enough to hold it. | ||
| 369 | function randTruePrime_(ans,k) { | ||
| 370 | var c,m,pm,dd,j,r,B,divisible,z,zz,recSize; | ||
| 371 | |||
| 372 | if (primes.length==0) | ||
| 373 | primes=findPrimes(30000); //check for divisibility by primes <=30000 | ||
| 374 | |||
| 375 | if (pows.length==0) { | ||
| 376 | pows=new Array(512); | ||
| 377 | for (j=0;j<512;j++) { | ||
| 378 | pows[j]=Math.pow(2,j/511.-1.); | ||
| 379 | } | ||
| 380 | } | ||
| 381 | |||
| 382 | //c and m should be tuned for a particular machine and value of k, to maximize speed | ||
| 383 | //this was: c=primes[primes.length-1]/k/k; //check using all the small primes. (c=0.1 in HAC) | ||
| 384 | c=0.1; | ||
| 385 | m=20; //generate this k-bit number by first recursively generating a number that has between k/2 and k-m bits | ||
| 386 | recLimit=20; /*must be at least 2 (was 29)*/ //stop recursion when k <=recLimit | ||
| 387 | |||
| 388 | if (s_i2.length!=ans.length) { | ||
| 389 | s_i2=dup(ans); | ||
| 390 | s_R =dup(ans); | ||
| 391 | s_n1=dup(ans); | ||
| 392 | s_r2=dup(ans); | ||
| 393 | s_d =dup(ans); | ||
| 394 | s_x1=dup(ans); | ||
| 395 | s_x2=dup(ans); | ||
| 396 | s_b =dup(ans); | ||
| 397 | s_n =dup(ans); | ||
| 398 | s_i =dup(ans); | ||
| 399 | s_rm=dup(ans); | ||
| 400 | s_q =dup(ans); | ||
| 401 | s_a =dup(ans); | ||
| 402 | s_aa=dup(ans); | ||
| 403 | } | ||
| 404 | |||
| 405 | if (k <= recLimit) { //generate small random primes by trial division up to its square root | ||
| 406 | pm=(1<<((k+2)>>1))-1; //pm is binary number with all ones, just over sqrt(2^k) | ||
| 407 | copyInt_(ans,0); | ||
| 408 | for (dd=1;dd;) { | ||
| 409 | dd=0; | ||
| 410 | ans[0]= 1 | (1<<(k-1)) | Math.floor(Math.random()*(1<<k)); //random, k-bit, odd integer, with msb 1 | ||
| 411 | for (j=1;(j<primes.length) && ((primes[j]&pm)==primes[j]);j++) { //trial division by all primes 3...sqrt(2^k) | ||
| 412 | if (0==(ans[0]%primes[j])) { | ||
| 413 | dd=1; | ||
| 414 | break; | ||
| 415 | } | ||
| 416 | } | ||
| 417 | } | ||
| 418 | carry_(ans); | ||
| 419 | return; | ||
| 420 | } | ||
| 421 | |||
| 422 | B=c*k*k; //try small primes up to B (or all the primes[] array if the largest is less than B). | ||
| 423 | if (k>2*m) //generate this k-bit number by first recursively generating a number that has between k/2 and k-m bits | ||
| 424 | for (r=1; k-k*r<=m; ) | ||
| 425 | r=pows[Math.floor(Math.random()*512)]; //r=Math.pow(2,Math.random()-1); | ||
| 426 | else | ||
| 427 | r=.5; | ||
| 428 | |||
| 429 | //simulation suggests the more complex algorithm using r=.333 is only slightly faster. | ||
| 430 | |||
| 431 | recSize=Math.floor(r*k)+1; | ||
| 432 | |||
| 433 | randTruePrime_(s_q,recSize); | ||
| 434 | copyInt_(s_i2,0); | ||
| 435 | s_i2[Math.floor((k-2)/bpe)] |= (1<<((k-2)%bpe)); //s_i2=2^(k-2) | ||
| 436 | divide_(s_i2,s_q,s_i,s_rm); //s_i=floor((2^(k-1))/(2q)) | ||
| 437 | |||
| 438 | z=bitSize(s_i); | ||
| 439 | |||
| 440 | for (;;) { | ||
| 441 | for (;;) { //generate z-bit numbers until one falls in the range [0,s_i-1] | ||
| 442 | randBigInt_(s_R,z,0); | ||
| 443 | if (greater(s_i,s_R)) | ||
| 444 | break; | ||
| 445 | } //now s_R is in the range [0,s_i-1] | ||
| 446 | addInt_(s_R,1); //now s_R is in the range [1,s_i] | ||
| 447 | add_(s_R,s_i); //now s_R is in the range [s_i+1,2*s_i] | ||
| 448 | |||
| 449 | copy_(s_n,s_q); | ||
| 450 | mult_(s_n,s_R); | ||
| 451 | multInt_(s_n,2); | ||
| 452 | addInt_(s_n,1); //s_n=2*s_R*s_q+1 | ||
| 453 | |||
| 454 | copy_(s_r2,s_R); | ||
| 455 | multInt_(s_r2,2); //s_r2=2*s_R | ||
| 456 | |||
| 457 | //check s_n for divisibility by small primes up to B | ||
| 458 | for (divisible=0,j=0; (j<primes.length) && (primes[j]<B); j++) | ||
| 459 | if (modInt(s_n,primes[j])==0) { | ||
| 460 | divisible=1; | ||
| 461 | break; | ||
| 462 | } | ||
| 463 | |||
| 464 | if (!divisible) //if it passes small primes check, then try a single Miller-Rabin base 2 | ||
| 465 | if (!millerRabin(s_n,2)) //this line represents 75% of the total runtime for randTruePrime_ | ||
| 466 | divisible=1; | ||
| 467 | |||
| 468 | if (!divisible) { //if it passes that test, continue checking s_n | ||
| 469 | addInt_(s_n,-3); | ||
| 470 | for (j=s_n.length-1;(s_n[j]==0) && (j>0); j--); //strip leading zeros | ||
| 471 | for (zz=0,w=s_n[j]; w; (w>>=1),zz++); | ||
| 472 | zz+=bpe*j; //zz=number of bits in s_n, ignoring leading zeros | ||
| 473 | for (;;) { //generate z-bit numbers until one falls in the range [0,s_n-1] | ||
| 474 | randBigInt_(s_a,zz,0); | ||
| 475 | if (greater(s_n,s_a)) | ||
| 476 | break; | ||
| 477 | } //now s_a is in the range [0,s_n-1] | ||
| 478 | addInt_(s_n,3); //now s_a is in the range [0,s_n-4] | ||
| 479 | addInt_(s_a,2); //now s_a is in the range [2,s_n-2] | ||
| 480 | copy_(s_b,s_a); | ||
| 481 | copy_(s_n1,s_n); | ||
| 482 | addInt_(s_n1,-1); | ||
| 483 | powMod_(s_b,s_n1,s_n); //s_b=s_a^(s_n-1) modulo s_n | ||
| 484 | addInt_(s_b,-1); | ||
| 485 | if (isZero(s_b)) { | ||
| 486 | copy_(s_b,s_a); | ||
| 487 | powMod_(s_b,s_r2,s_n); | ||
| 488 | addInt_(s_b,-1); | ||
| 489 | copy_(s_aa,s_n); | ||
| 490 | copy_(s_d,s_b); | ||
| 491 | GCD_(s_d,s_n); //if s_b and s_n are relatively prime, then s_n is a prime | ||
| 492 | if (equalsInt(s_d,1)) { | ||
| 493 | copy_(ans,s_aa); | ||
| 494 | return; //if we've made it this far, then s_n is absolutely guaranteed to be prime | ||
| 495 | } | ||
| 496 | } | ||
| 497 | } | ||
| 498 | } | ||
| 499 | } | ||
| 500 | |||
| 501 | //set b to an n-bit random BigInt. If s=1, then nth bit (most significant bit) is set to 1. | ||
| 502 | //array b must be big enough to hold the result. Must have n>=1 | ||
| 503 | function randBigInt_(b,n,s) { | ||
| 504 | var i,a; | ||
| 505 | for (i=0;i<b.length;i++) | ||
| 506 | b[i]=0; | ||
| 507 | a=Math.floor((n-1)/bpe)+1; //# array elements to hold the BigInt | ||
| 508 | for (i=0;i<a;i++) { | ||
| 509 | b[i]=Math.floor(Math.random()*(1<<(bpe-1))); | ||
| 510 | } | ||
| 511 | b[a-1] &= (2<<((n-1)%bpe))-1; | ||
| 512 | if (s) | ||
| 513 | b[a-1] |= (1<<((n-1)%bpe)); | ||
| 514 | } | ||
| 515 | |||
| 516 | //set x to the greatest common divisor of x and y. | ||
| 517 | //x,y are bigInts with the same number of elements. y is destroyed. | ||
| 518 | function GCD_(x,y) { | ||
| 519 | var i,xp,yp,A,B,C,D,q,sing; | ||
| 520 | if (T.length!=x.length) | ||
| 521 | T=dup(x); | ||
| 522 | |||
| 523 | sing=1; | ||
| 524 | while (sing) { //while y has nonzero elements other than y[0] | ||
| 525 | sing=0; | ||
| 526 | for (i=1;i<y.length;i++) //check if y has nonzero elements other than 0 | ||
| 527 | if (y[i]) { | ||
| 528 | sing=1; | ||
| 529 | break; | ||
| 530 | } | ||
| 531 | if (!sing) break; //quit when y all zero elements except possibly y[0] | ||
| 532 | |||
| 533 | for (i=x.length;!x[i] && i>=0;i--); //find most significant element of x | ||
| 534 | xp=x[i]; | ||
| 535 | yp=y[i]; | ||
| 536 | A=1; B=0; C=0; D=1; | ||
| 537 | while ((yp+C) && (yp+D)) { | ||
| 538 | q =Math.floor((xp+A)/(yp+C)); | ||
| 539 | qp=Math.floor((xp+B)/(yp+D)); | ||
| 540 | if (q!=qp) | ||
| 541 | break; | ||
| 542 | 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) | ||
| 543 | t= B-q*D; B=D; D=t; | ||
| 544 | t=xp-q*yp; xp=yp; yp=t; | ||
| 545 | } | ||
| 546 | if (B) { | ||
| 547 | copy_(T,x); | ||
| 548 | linComb_(x,y,A,B); //x=A*x+B*y | ||
| 549 | linComb_(y,T,D,C); //y=D*y+C*T | ||
| 550 | } else { | ||
| 551 | mod_(x,y); | ||
| 552 | copy_(T,x); | ||
| 553 | copy_(x,y); | ||
| 554 | copy_(y,T); | ||
| 555 | } | ||
| 556 | } | ||
| 557 | if (y[0]==0) | ||
| 558 | return; | ||
| 559 | t=modInt(x,y[0]); | ||
| 560 | copyInt_(x,y[0]); | ||
| 561 | y[0]=t; | ||
| 562 | while (y[0]) { | ||
| 563 | x[0]%=y[0]; | ||
| 564 | t=x[0]; x[0]=y[0]; y[0]=t; | ||
| 565 | } | ||
| 566 | } | ||
| 567 | |||
| 568 | //do x=x**(-1) mod n, for bigInts x and n. | ||
| 569 | //If no inverse exists, it sets x to zero and returns 0, else it returns 1. | ||
| 570 | //The x array must be at least as large as the n array. | ||
| 571 | function inverseMod_(x,n) { | ||
| 572 | var k=1+2*Math.max(x.length,n.length); | ||
| 573 | |||
| 574 | if(!(x[0]&1) && !(n[0]&1)) { //if both inputs are even, then inverse doesn't exist | ||
| 575 | copyInt_(x,0); | ||
| 576 | return 0; | ||
| 577 | } | ||
| 578 | |||
| 579 | if (eg_u.length!=k) { | ||
| 580 | eg_u=new Array(k); | ||
| 581 | eg_v=new Array(k); | ||
| 582 | eg_A=new Array(k); | ||
| 583 | eg_B=new Array(k); | ||
| 584 | eg_C=new Array(k); | ||
| 585 | eg_D=new Array(k); | ||
| 586 | } | ||
| 587 | |||
| 588 | copy_(eg_u,x); | ||
| 589 | copy_(eg_v,n); | ||
| 590 | copyInt_(eg_A,1); | ||
| 591 | copyInt_(eg_B,0); | ||
| 592 | copyInt_(eg_C,0); | ||
| 593 | copyInt_(eg_D,1); | ||
| 594 | for (;;) { | ||
| 595 | while(!(eg_u[0]&1)) { //while eg_u is even | ||
| 596 | halve_(eg_u); | ||
| 597 | if (!(eg_A[0]&1) && !(eg_B[0]&1)) { //if eg_A==eg_B==0 mod 2 | ||
| 598 | halve_(eg_A); | ||
| 599 | halve_(eg_B); | ||
| 600 | } else { | ||
| 601 | add_(eg_A,n); halve_(eg_A); | ||
| 602 | sub_(eg_B,x); halve_(eg_B); | ||
| 603 | } | ||
| 604 | } | ||
| 605 | |||
| 606 | while (!(eg_v[0]&1)) { //while eg_v is even | ||
| 607 | halve_(eg_v); | ||
| 608 | if (!(eg_C[0]&1) && !(eg_D[0]&1)) { //if eg_C==eg_D==0 mod 2 | ||
| 609 | halve_(eg_C); | ||
| 610 | halve_(eg_D); | ||
| 611 | } else { | ||
| 612 | add_(eg_C,n); halve_(eg_C); | ||
| 613 | sub_(eg_D,x); halve_(eg_D); | ||
| 614 | } | ||
| 615 | } | ||
| 616 | |||
| 617 | if (!greater(eg_v,eg_u)) { //eg_v <= eg_u | ||
| 618 | sub_(eg_u,eg_v); | ||
| 619 | sub_(eg_A,eg_C); | ||
| 620 | sub_(eg_B,eg_D); | ||
| 621 | } else { //eg_v > eg_u | ||
| 622 | sub_(eg_v,eg_u); | ||
| 623 | sub_(eg_C,eg_A); | ||
| 624 | sub_(eg_D,eg_B); | ||
| 625 | } | ||
| 626 | |||
| 627 | if (equalsInt(eg_u,0)) { | ||
| 628 | if (negative(eg_C)) //make sure answer is nonnegative | ||
| 629 | add_(eg_C,n); | ||
| 630 | copy_(x,eg_C); | ||
| 631 | |||
| 632 | if (!equalsInt(eg_v,1)) { //if GCD_(x,n)!=1, then there is no inverse | ||
| 633 | copyInt_(x,0); | ||
| 634 | return 0; | ||
| 635 | } | ||
| 636 | return 1; | ||
| 637 | } | ||
| 638 | } | ||
| 639 | } | ||
| 640 | |||
| 641 | //return x**(-1) mod n, for integers x and n. Return 0 if there is no inverse | ||
| 642 | function inverseModInt_(x,n) { | ||
| 643 | var a=1,b=0,t; | ||
| 644 | for (;;) { | ||
| 645 | if (x==1) return a; | ||
| 646 | if (x==0) return 0; | ||
| 647 | b-=a*Math.floor(n/x); | ||
| 648 | n%=x; | ||
| 649 | |||
| 650 | if (n==1) return b; //to avoid negatives, change this b to n-b, and each -= to += | ||
| 651 | if (n==0) return 0; | ||
| 652 | a-=b*Math.floor(x/n); | ||
| 653 | x%=n; | ||
| 654 | } | ||
| 655 | } | ||
| 656 | |||
| 657 | //Given positive bigInts x and y, change the bigints v, a, and b to positive bigInts such that: | ||
| 658 | // v = GCD_(x,y) = a*x-b*y | ||
| 659 | //The bigInts v, a, b, must have exactly as many elements as the larger of x and y. | ||
| 660 | function eGCD_(x,y,v,a,b) { | ||
| 661 | var g=0; | ||
| 662 | var k=Math.max(x.length,y.length); | ||
| 663 | if (eg_u.length!=k) { | ||
| 664 | eg_u=new Array(k); | ||
| 665 | eg_A=new Array(k); | ||
| 666 | eg_B=new Array(k); | ||
| 667 | eg_C=new Array(k); | ||
| 668 | eg_D=new Array(k); | ||
| 669 | } | ||
| 670 | while(!(x[0]&1) && !(y[0]&1)) { //while x and y both even | ||
| 671 | halve_(x); | ||
| 672 | halve_(y); | ||
| 673 | g++; | ||
| 674 | } | ||
| 675 | copy_(eg_u,x); | ||
| 676 | copy_(v,y); | ||
| 677 | copyInt_(eg_A,1); | ||
| 678 | copyInt_(eg_B,0); | ||
| 679 | copyInt_(eg_C,0); | ||
| 680 | copyInt_(eg_D,1); | ||
| 681 | for (;;) { | ||
| 682 | while(!(eg_u[0]&1)) { //while u is even | ||
| 683 | halve_(eg_u); | ||
| 684 | if (!(eg_A[0]&1) && !(eg_B[0]&1)) { //if A==B==0 mod 2 | ||
| 685 | halve_(eg_A); | ||
| 686 | halve_(eg_B); | ||
| 687 | } else { | ||
| 688 | add_(eg_A,y); halve_(eg_A); | ||
| 689 | sub_(eg_B,x); halve_(eg_B); | ||
| 690 | } | ||
| 691 | } | ||
| 692 | |||
| 693 | while (!(v[0]&1)) { //while v is even | ||
| 694 | halve_(v); | ||
| 695 | if (!(eg_C[0]&1) && !(eg_D[0]&1)) { //if C==D==0 mod 2 | ||
| 696 | halve_(eg_C); | ||
| 697 | halve_(eg_D); | ||
| 698 | } else { | ||
| 699 | add_(eg_C,y); halve_(eg_C); | ||
| 700 | sub_(eg_D,x); halve_(eg_D); | ||
| 701 | } | ||
| 702 | } | ||
| 703 | |||
| 704 | if (!greater(v,eg_u)) { //v<=u | ||
| 705 | sub_(eg_u,v); | ||
| 706 | sub_(eg_A,eg_C); | ||
| 707 | sub_(eg_B,eg_D); | ||
| 708 | } else { //v>u | ||
| 709 | sub_(v,eg_u); | ||
| 710 | sub_(eg_C,eg_A); | ||
| 711 | sub_(eg_D,eg_B); | ||
| 712 | } | ||
| 713 | if (equalsInt(eg_u,0)) { | ||
| 714 | if (negative(eg_C)) { //make sure a (C)is nonnegative | ||
| 715 | add_(eg_C,y); | ||
| 716 | sub_(eg_D,x); | ||
| 717 | } | ||
| 718 | multInt_(eg_D,-1); ///make sure b (D) is nonnegative | ||
| 719 | copy_(a,eg_C); | ||
| 720 | copy_(b,eg_D); | ||
| 721 | leftShift_(v,g); | ||
| 722 | return; | ||
| 723 | } | ||
| 724 | } | ||
| 725 | } | ||
| 726 | |||
| 727 | |||
| 728 | //is bigInt x negative? | ||
| 729 | function negative(x) { | ||
| 730 | return ((x[x.length-1]>>(bpe-1))&1); | ||
| 731 | } | ||
| 732 | |||
| 733 | |||
| 734 | //is (x << (shift*bpe)) > y? | ||
| 735 | //x and y are nonnegative bigInts | ||
| 736 | //shift is a nonnegative integer | ||
| 737 | function greaterShift(x,y,shift) { | ||
| 738 | var kx=x.length, ky=y.length; | ||
| 739 | k=((kx+shift)<ky) ? (kx+shift) : ky; | ||
| 740 | for (i=ky-1-shift; i<kx && i>=0; i++) | ||
| 741 | if (x[i]>0) | ||
| 742 | return 1; //if there are nonzeros in x to the left of the first column of y, then x is bigger | ||
| 743 | for (i=kx-1+shift; i<ky; i++) | ||
| 744 | if (y[i]>0) | ||
| 745 | return 0; //if there are nonzeros in y to the left of the first column of x, then x is not bigger | ||
| 746 | for (i=k-1; i>=shift; i--) | ||
| 747 | if (x[i-shift]>y[i]) return 1; | ||
| 748 | else if (x[i-shift]<y[i]) return 0; | ||
| 749 | return 0; | ||
| 750 | } | ||
| 751 | |||
| 752 | //is x > y? (x and y both nonnegative) | ||
| 753 | function greater(x,y) { | ||
| 754 | var i; | ||
| 755 | var k=(x.length<y.length) ? x.length : y.length; | ||
| 756 | |||
| 757 | for (i=x.length;i<y.length;i++) | ||
| 758 | if (y[i]) | ||
| 759 | return 0; //y has more digits | ||
| 760 | |||
| 761 | for (i=y.length;i<x.length;i++) | ||
| 762 | if (x[i]) | ||
| 763 | return 1; //x has more digits | ||
| 764 | |||
| 765 | for (i=k-1;i>=0;i--) | ||
| 766 | if (x[i]>y[i]) | ||
| 767 | return 1; | ||
| 768 | else if (x[i]<y[i]) | ||
| 769 | return 0; | ||
| 770 | return 0; | ||
| 771 | } | ||
| 772 | |||
| 773 | //divide_ x by y giving quotient q and remainder r. (q=floor(x/y), r=x mod y). All 4 are bigints. | ||
| 774 | //x must have at least one leading zero element. | ||
| 775 | //y must be nonzero. | ||
| 776 | //q and r must be arrays that are exactly the same length as x. | ||
| 777 | //the x array must have at least as many elements as y. | ||
| 778 | function divide_(x,y,q,r) { | ||
| 779 | var kx, ky; | ||
| 780 | var i,j,y1,y2,c,a,b; | ||
| 781 | copy_(r,x); | ||
| 782 | for (ky=y.length;y[ky-1]==0;ky--); //kx,ky is number of elements in x,y, not including leading zeros | ||
| 783 | for (kx=r.length;r[kx-1]==0 && kx>ky;kx--); | ||
| 784 | |||
| 785 | //normalize: ensure the most significant element of y has its highest bit set | ||
| 786 | b=y[ky-1]; | ||
| 787 | for (a=0; b; a++) | ||
| 788 | b>>=1; | ||
| 789 | a=bpe-a; //a is how many bits to shift so that the high order bit of y is leftmost in its array element | ||
| 790 | leftShift_(y,a); //multiply both by 1<<a now, then divide_ both by that at the end | ||
| 791 | leftShift_(r,a); | ||
| 792 | |||
| 793 | copyInt_(q,0); // q=0 | ||
| 794 | while (!greaterShift(y,r,kx-ky)) { // while (leftShift_(y,kx-ky) <= r) { | ||
| 795 | subShift_(r,y,kx-ky); // r=r-leftShift_(y,kx-ky) | ||
| 796 | q[kx-ky]++; // q[kx-ky]++; | ||
| 797 | } // } | ||
| 798 | |||
| 799 | for (i=kx-1; i>=ky; i--) { | ||
| 800 | if (r[i]==y[ky-1]) | ||
| 801 | q[i-ky]=mask; | ||
| 802 | else | ||
| 803 | q[i-ky]=Math.floor((r[i]*radix+r[i-1])/y[ky-1]); | ||
| 804 | |||
| 805 | //The following for(;;) loop is equivalent to the commented while loop, | ||
| 806 | //except that the uncommented version avoids overflow. | ||
| 807 | //The commented loop comes from HAC, which assumes r[-1]==y[-1]==0 | ||
| 808 | // while (q[i-ky]*(y[ky-1]*radix+y[ky-2]) > r[i]*radix*radix+r[i-1]*radix+r[i-2]) | ||
| 809 | // q[i-ky]--; | ||
| 810 | for (;;) { | ||
| 811 | y2=(ky>1 ? y[ky-2] : 0)*q[i-ky]; | ||
| 812 | c=y2>>bpe; | ||
| 813 | y2=y2 & mask; | ||
| 814 | y1=c+q[i-ky]*y[ky-1]; | ||
| 815 | c=y1>>bpe; | ||
| 816 | y1=y1 & mask; | ||
| 817 | |||
| 818 | if (c==r[i] ? y1==r[i-1] ? y2>(i>1 ? r[i-2] : 0) : y1>r[i-1] : c>r[i]) | ||
| 819 | q[i-ky]--; | ||
| 820 | else | ||
| 821 | break; | ||
| 822 | } | ||
| 823 | |||
| 824 | linCombShift_(r,y,-q[i-ky],i-ky); //r=r-q[i-ky]*leftShift_(y,i-ky) | ||
| 825 | if (negative(r)) { | ||
| 826 | addShift_(r,y,i-ky); //r=r+leftShift_(y,i-ky) | ||
| 827 | q[i-ky]--; | ||
| 828 | } | ||
| 829 | } | ||
| 830 | |||
| 831 | rightShift_(y,a); //undo the normalization step | ||
| 832 | rightShift_(r,a); //undo the normalization step | ||
| 833 | } | ||
| 834 | |||
| 835 | //do carries and borrows so each element of the bigInt x fits in bpe bits. | ||
| 836 | function carry_(x) { | ||
| 837 | var i,k,c,b; | ||
| 838 | k=x.length; | ||
| 839 | c=0; | ||
| 840 | for (i=0;i<k;i++) { | ||
| 841 | c+=x[i]; | ||
| 842 | b=0; | ||
| 843 | if (c<0) { | ||
| 844 | b=-(c>>bpe); | ||
| 845 | c+=b*radix; | ||
| 846 | } | ||
| 847 | x[i]=c & mask; | ||
| 848 | c=(c>>bpe)-b; | ||
| 849 | } | ||
| 850 | } | ||
| 851 | |||
| 852 | //return x mod n for bigInt x and integer n. | ||
| 853 | function modInt(x,n) { | ||
| 854 | var i,c=0; | ||
| 855 | for (i=x.length-1; i>=0; i--) | ||
| 856 | c=(c*radix+x[i])%n; | ||
| 857 | return c; | ||
| 858 | } | ||
| 859 | |||
| 860 | //convert the integer t into a bigInt with at least the given number of bits. | ||
| 861 | //the returned array stores the bigInt in bpe-bit chunks, little endian (buff[0] is least significant word) | ||
| 862 | //Pad the array with leading zeros so that it has at least minSize elements. | ||
| 863 | //There will always be at least one leading 0 element. | ||
| 864 | function int2bigInt(t,bits,minSize) { | ||
| 865 | var i,k; | ||
| 866 | k=Math.ceil(bits/bpe)+1; | ||
| 867 | k=minSize>k ? minSize : k; | ||
| 868 | buff=new Array(k); | ||
| 869 | copyInt_(buff,t); | ||
| 870 | return buff; | ||
| 871 | } | ||
| 872 | |||
| 873 | //return the bigInt given a string representation in a given base. | ||
| 874 | //Pad the array with leading zeros so that it has at least minSize elements. | ||
| 875 | //If base=-1, then it reads in a space-separated list of array elements in decimal. | ||
| 876 | //The array will always have at least one leading zero, unless base=-1. | ||
| 877 | function str2bigInt(s,base,minSize) { | ||
| 878 | var d, i, j, x, y, kk; | ||
| 879 | var k=s.length; | ||
| 880 | if (base==-1) { //comma-separated list of array elements in decimal | ||
| 881 | x=new Array(0); | ||
| 882 | for (;;) { | ||
| 883 | y=new Array(x.length+1); | ||
| 884 | for (i=0;i<x.length;i++) | ||
| 885 | y[i+1]=x[i]; | ||
| 886 | y[0]=parseInt(s,10); | ||
| 887 | x=y; | ||
| 888 | d=s.indexOf(',',0); | ||
| 889 | if (d<1) | ||
| 890 | break; | ||
| 891 | s=s.substring(d+1); | ||
| 892 | if (s.length==0) | ||
| 893 | break; | ||
| 894 | } | ||
| 895 | if (x.length<minSize) { | ||
| 896 | y=new Array(minSize); | ||
| 897 | copy_(y,x); | ||
| 898 | return y; | ||
| 899 | } | ||
| 900 | return x; | ||
| 901 | } | ||
| 902 | |||
| 903 | x=int2bigInt(0,base*k,0); | ||
| 904 | for (i=0;i<k;i++) { | ||
| 905 | d=digitsStr.indexOf(s.substring(i,i+1),0); | ||
| 906 | if (base<=36 && d>=36) //convert lowercase to uppercase if base<=36 | ||
| 907 | d-=26; | ||
| 908 | if (d<base && d>=0) { //ignore illegal characters | ||
| 909 | multInt_(x,base); | ||
| 910 | addInt_(x,d); | ||
| 911 | } | ||
| 912 | } | ||
| 913 | |||
| 914 | for (k=x.length;k>0 && !x[k-1];k--); //strip off leading zeros | ||
| 915 | k=minSize>k+1 ? minSize : k+1; | ||
| 916 | y=new Array(k); | ||
| 917 | kk=k<x.length ? k : x.length; | ||
| 918 | for (i=0;i<kk;i++) | ||
| 919 | y[i]=x[i]; | ||
| 920 | for (;i<k;i++) | ||
| 921 | y[i]=0; | ||
| 922 | return y; | ||
| 923 | } | ||
| 924 | |||
| 925 | //is bigint x equal to integer y? | ||
| 926 | //y must have less than bpe bits | ||
| 927 | function equalsInt(x,y) { | ||
| 928 | var i; | ||
| 929 | if (x[0]!=y) | ||
| 930 | return 0; | ||
| 931 | for (i=1;i<x.length;i++) | ||
| 932 | if (x[i]) | ||
| 933 | return 0; | ||
| 934 | return 1; | ||
| 935 | } | ||
| 936 | |||
| 937 | //are bigints x and y equal? | ||
| 938 | //this works even if x and y are different lengths and have arbitrarily many leading zeros | ||
| 939 | function equals(x,y) { | ||
| 940 | var i; | ||
| 941 | var k=x.length<y.length ? x.length : y.length; | ||
| 942 | for (i=0;i<k;i++) | ||
| 943 | if (x[i]!=y[i]) | ||
| 944 | return 0; | ||
| 945 | if (x.length>y.length) { | ||
| 946 | for (;i<x.length;i++) | ||
| 947 | if (x[i]) | ||
| 948 | return 0; | ||
| 949 | } else { | ||
| 950 | for (;i<y.length;i++) | ||
| 951 | if (y[i]) | ||
| 952 | return 0; | ||
| 953 | } | ||
| 954 | return 1; | ||
| 955 | } | ||
| 956 | |||
| 957 | //is the bigInt x equal to zero? | ||
| 958 | function isZero(x) { | ||
| 959 | var i; | ||
| 960 | for (i=0;i<x.length;i++) | ||
| 961 | if (x[i]) | ||
| 962 | return 0; | ||
| 963 | return 1; | ||
| 964 | } | ||
| 965 | |||
| 966 | //convert a bigInt into a string in a given base, from base 2 up to base 95. | ||
| 967 | //Base -1 prints the contents of the array representing the number. | ||
| 968 | function bigInt2str(x,base) { | ||
| 969 | var i,t,s=""; | ||
| 970 | |||
| 971 | if (s6.length!=x.length) | ||
| 972 | s6=dup(x); | ||
| 973 | else | ||
| 974 | copy_(s6,x); | ||
| 975 | |||
| 976 | if (base==-1) { //return the list of array contents | ||
| 977 | for (i=x.length-1;i>0;i--) | ||
| 978 | s+=x[i]+','; | ||
| 979 | s+=x[0]; | ||
| 980 | } | ||
| 981 | else { //return it in the given base | ||
| 982 | while (!isZero(s6)) { | ||
| 983 | t=divInt_(s6,base); //t=s6 % base; s6=floor(s6/base); | ||
| 984 | s=digitsStr.substring(t,t+1)+s; | ||
| 985 | } | ||
| 986 | } | ||
| 987 | if (s.length==0) | ||
| 988 | s="0"; | ||
| 989 | return s; | ||
| 990 | } | ||
| 991 | |||
| 992 | //returns a duplicate of bigInt x | ||
| 993 | function dup(x) { | ||
| 994 | var i; | ||
| 995 | buff=new Array(x.length); | ||
| 996 | copy_(buff,x); | ||
| 997 | return buff; | ||
| 998 | } | ||
| 999 | |||
| 1000 | //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). | ||
| 1001 | function copy_(x,y) { | ||
| 1002 | var i; | ||
| 1003 | var k=x.length<y.length ? x.length : y.length; | ||
| 1004 | for (i=0;i<k;i++) | ||
| 1005 | x[i]=y[i]; | ||
| 1006 | for (i=k;i<x.length;i++) | ||
| 1007 | x[i]=0; | ||
| 1008 | } | ||
| 1009 | |||
| 1010 | //do x=y on bigInt x and integer y. | ||
| 1011 | function copyInt_(x,n) { | ||
| 1012 | var i,c; | ||
| 1013 | for (c=n,i=0;i<x.length;i++) { | ||
| 1014 | x[i]=c & mask; | ||
| 1015 | c>>=bpe; | ||
| 1016 | } | ||
| 1017 | } | ||
| 1018 | |||
| 1019 | //do x=x+n where x is a bigInt and n is an integer. | ||
| 1020 | //x must be large enough to hold the result. | ||
| 1021 | function addInt_(x,n) { | ||
| 1022 | var i,k,c,b; | ||
| 1023 | x[0]+=n; | ||
| 1024 | k=x.length; | ||
| 1025 | c=0; | ||
| 1026 | for (i=0;i<k;i++) { | ||
| 1027 | c+=x[i]; | ||
| 1028 | b=0; | ||
| 1029 | if (c<0) { | ||
| 1030 | b=-(c>>bpe); | ||
| 1031 | c+=b*radix; | ||
| 1032 | } | ||
| 1033 | x[i]=c & mask; | ||
| 1034 | c=(c>>bpe)-b; | ||
| 1035 | if (!c) return; //stop carrying as soon as the carry_ is zero | ||
| 1036 | } | ||
| 1037 | } | ||
| 1038 | |||
| 1039 | //right shift bigInt x by n bits. 0 <= n < bpe. | ||
| 1040 | function rightShift_(x,n) { | ||
| 1041 | var i; | ||
| 1042 | var k=Math.floor(n/bpe); | ||
| 1043 | if (k) { | ||
| 1044 | for (i=0;i<x.length-k;i++) //right shift x by k elements | ||
| 1045 | x[i]=x[i+k]; | ||
| 1046 | for (;i<x.length;i++) | ||
| 1047 | x[i]=0; | ||
| 1048 | n%=bpe; | ||
| 1049 | } | ||
| 1050 | for (i=0;i<x.length-1;i++) { | ||
| 1051 | x[i]=mask & ((x[i+1]<<(bpe-n)) | (x[i]>>n)); | ||
| 1052 | } | ||
| 1053 | x[i]>>=n; | ||
| 1054 | } | ||
| 1055 | |||
| 1056 | //do x=floor(|x|/2)*sgn(x) for bigInt x in 2's complement | ||
| 1057 | function halve_(x) { | ||
| 1058 | var i; | ||
| 1059 | for (i=0;i<x.length-1;i++) { | ||
| 1060 | x[i]=mask & ((x[i+1]<<(bpe-1)) | (x[i]>>1)); | ||
| 1061 | } | ||
| 1062 | x[i]=(x[i]>>1) | (x[i] & (radix>>1)); //most significant bit stays the same | ||
| 1063 | } | ||
| 1064 | |||
| 1065 | //left shift bigInt x by n bits. | ||
| 1066 | function leftShift_(x,n) { | ||
| 1067 | var i; | ||
| 1068 | var k=Math.floor(n/bpe); | ||
| 1069 | if (k) { | ||
| 1070 | for (i=x.length; i>=k; i--) //left shift x by k elements | ||
| 1071 | x[i]=x[i-k]; | ||
| 1072 | for (;i>=0;i--) | ||
| 1073 | x[i]=0; | ||
| 1074 | n%=bpe; | ||
| 1075 | } | ||
| 1076 | if (!n) | ||
| 1077 | return; | ||
| 1078 | for (i=x.length-1;i>0;i--) { | ||
| 1079 | x[i]=mask & ((x[i]<<n) | (x[i-1]>>(bpe-n))); | ||
| 1080 | } | ||
| 1081 | x[i]=mask & (x[i]<<n); | ||
| 1082 | } | ||
| 1083 | |||
| 1084 | //do x=x*n where x is a bigInt and n is an integer. | ||
| 1085 | //x must be large enough to hold the result. | ||
| 1086 | function multInt_(x,n) { | ||
| 1087 | var i,k,c,b; | ||
| 1088 | if (!n) | ||
| 1089 | return; | ||
| 1090 | k=x.length; | ||
| 1091 | c=0; | ||
| 1092 | for (i=0;i<k;i++) { | ||
| 1093 | c+=x[i]*n; | ||
| 1094 | b=0; | ||
| 1095 | if (c<0) { | ||
| 1096 | b=-(c>>bpe); | ||
| 1097 | c+=b*radix; | ||
| 1098 | } | ||
| 1099 | x[i]=c & mask; | ||
| 1100 | c=(c>>bpe)-b; | ||
| 1101 | } | ||
| 1102 | } | ||
| 1103 | |||
| 1104 | //do x=floor(x/n) for bigInt x and integer n, and return the remainder | ||
| 1105 | function divInt_(x,n) { | ||
| 1106 | var i,r=0,s; | ||
| 1107 | for (i=x.length-1;i>=0;i--) { | ||
| 1108 | s=r*radix+x[i]; | ||
| 1109 | x[i]=Math.floor(s/n); | ||
| 1110 | r=s%n; | ||
| 1111 | } | ||
| 1112 | return r; | ||
| 1113 | } | ||
| 1114 | |||
| 1115 | //do the linear combination x=a*x+b*y for bigInts x and y, and integers a and b. | ||
| 1116 | //x must be large enough to hold the answer. | ||
| 1117 | function linComb_(x,y,a,b) { | ||
| 1118 | var i,c,k,kk; | ||
| 1119 | k=x.length<y.length ? x.length : y.length; | ||
| 1120 | kk=x.length; | ||
| 1121 | for (c=0,i=0;i<k;i++) { | ||
| 1122 | c+=a*x[i]+b*y[i]; | ||
| 1123 | x[i]=c & mask; | ||
| 1124 | c>>=bpe; | ||
| 1125 | } | ||
| 1126 | for (i=k;i<kk;i++) { | ||
| 1127 | c+=a*x[i]; | ||
| 1128 | x[i]=c & mask; | ||
| 1129 | c>>=bpe; | ||
| 1130 | } | ||
| 1131 | } | ||
| 1132 | |||
| 1133 | //do the linear combination x=a*x+b*(y<<(ys*bpe)) for bigInts x and y, and integers a, b and ys. | ||
| 1134 | //x must be large enough to hold the answer. | ||
| 1135 | function linCombShift_(x,y,b,ys) { | ||
| 1136 | var i,c,k,kk; | ||
| 1137 | k=x.length<ys+y.length ? x.length : ys+y.length; | ||
| 1138 | kk=x.length; | ||
| 1139 | for (c=0,i=ys;i<k;i++) { | ||
| 1140 | c+=x[i]+b*y[i-ys]; | ||
| 1141 | x[i]=c & mask; | ||
| 1142 | c>>=bpe; | ||
| 1143 | } | ||
| 1144 | for (i=k;c && i<kk;i++) { | ||
| 1145 | c+=x[i]; | ||
| 1146 | x[i]=c & mask; | ||
| 1147 | c>>=bpe; | ||
| 1148 | } | ||
| 1149 | } | ||
| 1150 | |||
| 1151 | //do x=x+(y<<(ys*bpe)) for bigInts x and y, and integers a,b and ys. | ||
| 1152 | //x must be large enough to hold the answer. | ||
| 1153 | function addShift_(x,y,ys) { | ||
| 1154 | var i,c,k,kk; | ||
| 1155 | k=x.length<ys+y.length ? x.length : ys+y.length; | ||
| 1156 | kk=x.length; | ||
| 1157 | for (c=0,i=ys;i<k;i++) { | ||
| 1158 | c+=x[i]+y[i-ys]; | ||
| 1159 | x[i]=c & mask; | ||
| 1160 | c>>=bpe; | ||
| 1161 | } | ||
| 1162 | for (i=k;c && i<kk;i++) { | ||
| 1163 | c+=x[i]; | ||
| 1164 | x[i]=c & mask; | ||
| 1165 | c>>=bpe; | ||
| 1166 | } | ||
| 1167 | } | ||
| 1168 | |||
| 1169 | //do x=x-(y<<(ys*bpe)) for bigInts x and y, and integers a,b and ys. | ||
| 1170 | //x must be large enough to hold the answer. | ||
| 1171 | function subShift_(x,y,ys) { | ||
| 1172 | var i,c,k,kk; | ||
| 1173 | k=x.length<ys+y.length ? x.length : ys+y.length; | ||
| 1174 | kk=x.length; | ||
| 1175 | for (c=0,i=ys;i<k;i++) { | ||
| 1176 | c+=x[i]-y[i-ys]; | ||
| 1177 | x[i]=c & mask; | ||
| 1178 | c>>=bpe; | ||
| 1179 | } | ||
| 1180 | for (i=k;c && i<kk;i++) { | ||
| 1181 | c+=x[i]; | ||
| 1182 | x[i]=c & mask; | ||
| 1183 | c>>=bpe; | ||
| 1184 | } | ||
| 1185 | } | ||
| 1186 | |||
| 1187 | //do x=x-y for bigInts x and y. | ||
| 1188 | //x must be large enough to hold the answer. | ||
| 1189 | //negative answers will be 2s complement | ||
| 1190 | function sub_(x,y) { | ||
| 1191 | var i,c,k,kk; | ||
| 1192 | k=x.length<y.length ? x.length : y.length; | ||
| 1193 | for (c=0,i=0;i<k;i++) { | ||
| 1194 | c+=x[i]-y[i]; | ||
| 1195 | x[i]=c & mask; | ||
| 1196 | c>>=bpe; | ||
| 1197 | } | ||
| 1198 | for (i=k;c && i<x.length;i++) { | ||
| 1199 | c+=x[i]; | ||
| 1200 | x[i]=c & mask; | ||
| 1201 | c>>=bpe; | ||
| 1202 | } | ||
| 1203 | } | ||
| 1204 | |||
| 1205 | //do x=x+y for bigInts x and y. | ||
| 1206 | //x must be large enough to hold the answer. | ||
| 1207 | function add_(x,y) { | ||
| 1208 | var i,c,k,kk; | ||
| 1209 | k=x.length<y.length ? x.length : y.length; | ||
| 1210 | for (c=0,i=0;i<k;i++) { | ||
| 1211 | c+=x[i]+y[i]; | ||
| 1212 | x[i]=c & mask; | ||
| 1213 | c>>=bpe; | ||
| 1214 | } | ||
| 1215 | for (i=k;c && i<x.length;i++) { | ||
| 1216 | c+=x[i]; | ||
| 1217 | x[i]=c & mask; | ||
| 1218 | c>>=bpe; | ||
| 1219 | } | ||
| 1220 | } | ||
| 1221 | |||
| 1222 | //do x=x*y for bigInts x and y. This is faster when y<x. | ||
| 1223 | function mult_(x,y) { | ||
| 1224 | var i; | ||
| 1225 | if (ss.length!=2*x.length) | ||
| 1226 | ss=new Array(2*x.length); | ||
| 1227 | copyInt_(ss,0); | ||
| 1228 | for (i=0;i<y.length;i++) | ||
| 1229 | if (y[i]) | ||
| 1230 | linCombShift_(ss,x,y[i],i); //ss=1*ss+y[i]*(x<<(i*bpe)) | ||
| 1231 | copy_(x,ss); | ||
| 1232 | } | ||
| 1233 | |||
| 1234 | //do x=x mod n for bigInts x and n. | ||
| 1235 | function mod_(x,n) { | ||
| 1236 | if (s4.length!=x.length) | ||
| 1237 | s4=dup(x); | ||
| 1238 | else | ||
| 1239 | copy_(s4,x); | ||
| 1240 | if (s5.length!=x.length) | ||
| 1241 | s5=dup(x); | ||
| 1242 | divide_(s4,n,s5,x); //x = remainder of s4 / n | ||
| 1243 | } | ||
| 1244 | |||
| 1245 | //do x=x*y mod n for bigInts x,y,n. | ||
| 1246 | //for greater speed, let y<x. | ||
| 1247 | function multMod_(x,y,n) { | ||
| 1248 | var i; | ||
| 1249 | if (s0.length!=2*x.length) | ||
| 1250 | s0=new Array(2*x.length); | ||
| 1251 | copyInt_(s0,0); | ||
| 1252 | for (i=0;i<y.length;i++) | ||
| 1253 | if (y[i]) | ||
| 1254 | linCombShift_(s0,x,y[i],i); //s0=1*s0+y[i]*(x<<(i*bpe)) | ||
| 1255 | mod_(s0,n); | ||
| 1256 | copy_(x,s0); | ||
| 1257 | } | ||
| 1258 | |||
| 1259 | //do x=x*x mod n for bigInts x,n. | ||
| 1260 | function squareMod_(x,n) { | ||
| 1261 | var i,j,d,c,kx,kn,k; | ||
| 1262 | for (kx=x.length; kx>0 && !x[kx-1]; kx--); //ignore leading zeros in x | ||
| 1263 | 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 | ||
| 1264 | if (s0.length!=k) | ||
| 1265 | s0=new Array(k); | ||
| 1266 | copyInt_(s0,0); | ||
| 1267 | for (i=0;i<kx;i++) { | ||
| 1268 | c=s0[2*i]+x[i]*x[i]; | ||
| 1269 | s0[2*i]=c & mask; | ||
| 1270 | c>>=bpe; | ||
| 1271 | for (j=i+1;j<kx;j++) { | ||
| 1272 | c=s0[i+j]+2*x[i]*x[j]+c; | ||
| 1273 | s0[i+j]=(c & mask); | ||
| 1274 | c>>=bpe; | ||
| 1275 | } | ||
| 1276 | s0[i+kx]=c; | ||
| 1277 | } | ||
| 1278 | mod_(s0,n); | ||
| 1279 | copy_(x,s0); | ||
| 1280 | } | ||
| 1281 | |||
| 1282 | //return x with exactly k leading zero elements | ||
| 1283 | function trim(x,k) { | ||
| 1284 | var i,y; | ||
| 1285 | for (i=x.length; i>0 && !x[i-1]; i--); | ||
| 1286 | y=new Array(i+k); | ||
| 1287 | copy_(y,x); | ||
| 1288 | return y; | ||
| 1289 | } | ||
| 1290 | |||
| 1291 | //do x=x**y mod n, where x,y,n are bigInts and ** is exponentiation. 0**0=1. | ||
| 1292 | //this is faster when n is odd. x usually needs to have as many elements as n. | ||
| 1293 | function powMod_(x,y,n) { | ||
| 1294 | var k1,k2,kn,np; | ||
| 1295 | if(s7.length!=n.length) | ||
| 1296 | s7=dup(n); | ||
| 1297 | |||
| 1298 | //for even modulus, use a simple square-and-multiply algorithm, | ||
| 1299 | //rather than using the more complex Montgomery algorithm. | ||
| 1300 | if ((n[0]&1)==0) { | ||
| 1301 | copy_(s7,x); | ||
| 1302 | copyInt_(x,1); | ||
| 1303 | while(!equalsInt(y,0)) { | ||
| 1304 | if (y[0]&1) | ||
| 1305 | multMod_(x,s7,n); | ||
| 1306 | divInt_(y,2); | ||
| 1307 | squareMod_(s7,n); | ||
| 1308 | } | ||
| 1309 | return; | ||
| 1310 | } | ||
| 1311 | |||
| 1312 | //calculate np from n for the Montgomery multiplications | ||
| 1313 | copyInt_(s7,0); | ||
| 1314 | for (kn=n.length;kn>0 && !n[kn-1];kn--); | ||
| 1315 | np=radix-inverseModInt_(modInt(n,radix),radix); | ||
| 1316 | s7[kn]=1; | ||
| 1317 | multMod_(x ,s7,n); // x = x * 2**(kn*bp) mod n | ||
| 1318 | |||
| 1319 | if (s3.length!=x.length) | ||
| 1320 | s3=dup(x); | ||
| 1321 | else | ||
| 1322 | copy_(s3,x); | ||
| 1323 | |||
| 1324 | for (k1=y.length-1;k1>0 & !y[k1]; k1--); //k1=first nonzero element of y | ||
| 1325 | if (y[k1]==0) { //anything to the 0th power is 1 | ||
| 1326 | copyInt_(x,1); | ||
| 1327 | return; | ||
| 1328 | } | ||
| 1329 | for (k2=1<<(bpe-1);k2 && !(y[k1] & k2); k2>>=1); //k2=position of first 1 bit in y[k1] | ||
| 1330 | for (;;) { | ||
| 1331 | if (!(k2>>=1)) { //look at next bit of y | ||
| 1332 | k1--; | ||
| 1333 | if (k1<0) { | ||
| 1334 | mont_(x,one,n,np); | ||
| 1335 | return; | ||
| 1336 | } | ||
| 1337 | k2=1<<(bpe-1); | ||
| 1338 | } | ||
| 1339 | mont_(x,x,n,np); | ||
| 1340 | |||
| 1341 | if (k2 & y[k1]) //if next bit is a 1 | ||
| 1342 | mont_(x,s3,n,np); | ||
| 1343 | } | ||
| 1344 | } | ||
| 1345 | |||
| 1346 | //do x=x*y*Ri mod n for bigInts x,y,n, | ||
| 1347 | // where Ri = 2**(-kn*bpe) mod n, and kn is the | ||
| 1348 | // number of elements in the n array, not | ||
| 1349 | // counting leading zeros. | ||
| 1350 | //x must be large enough to hold the answer. | ||
| 1351 | //It's OK if x and y are the same variable. | ||
| 1352 | //must have: | ||
| 1353 | // x,y < n | ||
| 1354 | // n is odd | ||
| 1355 | // np = -(n^(-1)) mod radix | ||
| 1356 | function mont_(x,y,n,np) { | ||
| 1357 | var i,j,c,ui,t; | ||
| 1358 | var kn=n.length; | ||
| 1359 | var ky=y.length; | ||
| 1360 | |||
| 1361 | if (sa.length!=kn) | ||
| 1362 | sa=new Array(kn); | ||
| 1363 | |||
| 1364 | for (;kn>0 && n[kn-1]==0;kn--); //ignore leading zeros of n | ||
| 1365 | //this function sometimes gives wrong answers when the next line is uncommented | ||
| 1366 | //for (;ky>0 && y[ky-1]==0;ky--); //ignore leading zeros of y | ||
| 1367 | |||
| 1368 | copyInt_(sa,0); | ||
| 1369 | |||
| 1370 | //the following loop consumes 95% of the runtime for randTruePrime_() and powMod_() for large keys | ||
| 1371 | for (i=0; i<kn; i++) { | ||
| 1372 | t=sa[0]+x[i]*y[0]; | ||
| 1373 | ui=((t & mask) * np) & mask; //the inner "& mask" is needed on Macintosh MSIE, but not windows MSIE | ||
| 1374 | c=(t+ui*n[0]) >> bpe; | ||
| 1375 | t=x[i]; | ||
| 1376 | |||
| 1377 | //do sa=(sa+x[i]*y+ui*n)/b where b=2**bpe | ||
| 1378 | for (j=1;j<ky;j++) { | ||
| 1379 | c+=sa[j]+t*y[j]+ui*n[j]; | ||
| 1380 | sa[j-1]=c & mask; | ||
| 1381 | c>>=bpe; | ||
| 1382 | } | ||
| 1383 | for (;j<kn;j++) { | ||
| 1384 | c+=sa[j]+ui*n[j]; | ||
| 1385 | sa[j-1]=c & mask; | ||
| 1386 | c>>=bpe; | ||
| 1387 | } | ||
| 1388 | sa[j-1]=c & mask; | ||
| 1389 | } | ||
| 1390 | |||
| 1391 | if (!greater(n,sa)) | ||
| 1392 | sub_(sa,n); | ||
| 1393 | copy_(x,sa); | ||
| 1394 | } | ||
| 1395 | |||
| 1396 | |||
| 1397 | |||
| 1398 | |||
| 1399 | //############################################################################# | ||
| 1400 | //############################################################################# | ||
| 1401 | //############################################################################# | ||
| 1402 | //############################################################################# | ||
| 1403 | //############################################################################# | ||
| 1404 | //############################################################################# | ||
| 1405 | //############################################################################# | ||
| 1406 | |||
| 1407 | |||
| 1408 | |||
| 1409 | |||
| 1410 | |||
| 1411 | //############################################################################# | ||
| 1412 | |||
| 1413 | Clipperz.Crypto.BigInt = function (aValue, aBase) { | ||
| 1414 | varbase; | ||
| 1415 | varvalue; | ||
| 1416 | |||
| 1417 | if (typeof(aValue) == 'object') { | ||
| 1418 | this._internalValue = aValue; | ||
| 1419 | } else { | ||
| 1420 | if (typeof(aValue) == 'undefined') { | ||
| 1421 | value = "0"; | ||
| 1422 | } else { | ||
| 1423 | value = aValue + ""; | ||
| 1424 | } | ||
| 1425 | |||
| 1426 | if (typeof(aBase) == 'undefined') { | ||
| 1427 | base = 10; | ||
| 1428 | } else { | ||
| 1429 | base = aBase; | ||
| 1430 | } | ||
| 1431 | |||
| 1432 | this._internalValue = str2bigInt(value, base, 1, 1); | ||
| 1433 | } | ||
| 1434 | |||
| 1435 | return this; | ||
| 1436 | } | ||
| 1437 | |||
| 1438 | //============================================================================= | ||
| 1439 | |||
| 1440 | MochiKit.Base.update(Clipperz.Crypto.BigInt.prototype, { | ||
| 1441 | |||
| 1442 | 'clone': function() { | ||
| 1443 | return new Clipperz.Crypto.BigInt(this.internalValue()); | ||
| 1444 | }, | ||
| 1445 | |||
| 1446 | //------------------------------------------------------------------------- | ||
| 1447 | |||
| 1448 | 'internalValue': function () { | ||
| 1449 | return this._internalValue; | ||
| 1450 | }, | ||
| 1451 | |||
| 1452 | //------------------------------------------------------------------------- | ||
| 1453 | |||
| 1454 | 'isBigInt': true, | ||
| 1455 | |||
| 1456 | //------------------------------------------------------------------------- | ||
| 1457 | |||
| 1458 | 'toString': function(aBase) { | ||
| 1459 | return this.asString(aBase); | ||
| 1460 | }, | ||
| 1461 | |||
| 1462 | //------------------------------------------------------------------------- | ||
| 1463 | |||
| 1464 | 'asString': function (aBase, minimumLength) { | ||
| 1465 | varresult; | ||
| 1466 | varbase; | ||
| 1467 | |||
| 1468 | if (typeof(aBase) == 'undefined') { | ||
| 1469 | base = 10; | ||
| 1470 | } else { | ||
| 1471 | base = aBase; | ||
| 1472 | } | ||
| 1473 | |||
| 1474 | result = bigInt2str(this.internalValue(), base).toLowerCase(); | ||
| 1475 | |||
| 1476 | if ((typeof(minimumLength) != 'undefined') && (result.length < minimumLength)) { | ||
| 1477 | var i, c; | ||
| 1478 | c = (minimumLength - result.length); | ||
| 1479 | for (i=0; i<c; i++) { | ||
| 1480 | result = '0' + result; | ||
| 1481 | } | ||
| 1482 | } | ||
| 1483 | |||
| 1484 | return result; | ||
| 1485 | }, | ||
| 1486 | |||
| 1487 | //------------------------------------------------------------------------- | ||
| 1488 | |||
| 1489 | 'asByteArray': function() { | ||
| 1490 | return new Clipperz.ByteArray("0x" + this.asString(16), 16); | ||
| 1491 | }, | ||
| 1492 | |||
| 1493 | //------------------------------------------------------------------------- | ||
| 1494 | |||
| 1495 | 'equals': function (aValue) { | ||
| 1496 | var result; | ||
| 1497 | |||
| 1498 | if (aValue.isBigInt) { | ||
| 1499 | result = equals(this.internalValue(), aValue.internalValue()); | ||
| 1500 | } else if (typeof(aValue) == "number") { | ||
| 1501 | result = equalsInt(this.internalValue(), aValue); | ||
| 1502 | } else { | ||
| 1503 | throw Clipperz.Crypt.BigInt.exception.UnknownType; | ||
| 1504 | } | ||
| 1505 | |||
| 1506 | return result; | ||
| 1507 | }, | ||
| 1508 | |||
| 1509 | //------------------------------------------------------------------------- | ||
| 1510 | |||
| 1511 | 'compare': function(aValue) { | ||
| 1512 | /* | ||
| 1513 | var result; | ||
| 1514 | var thisAsString; | ||
| 1515 | var aValueAsString; | ||
| 1516 | |||
| 1517 | thisAsString = this.asString(10); | ||
| 1518 | aValueAsString = aValue.asString(10); | ||
| 1519 | |||
| 1520 | result = MochiKit.Base.compare(thisAsString.length, aValueAsString.length); | ||
| 1521 | if (result == 0) { | ||
| 1522 | result = MochiKit.Base.compare(thisAsString, aValueAsString); | ||
| 1523 | } | ||
| 1524 | |||
| 1525 | return result; | ||
| 1526 | */ | ||
| 1527 | var result; | ||
| 1528 | |||
| 1529 | if (equals(this.internalValue(), aValue.internalValue())) { | ||
| 1530 | result = 0; | ||
| 1531 | } else if (greater(this.internalValue(), aValue.internalValue())) { | ||
| 1532 | result = 1; | ||
| 1533 | } else { | ||
| 1534 | result = -1; | ||
| 1535 | } | ||
| 1536 | |||
| 1537 | return result; | ||
| 1538 | }, | ||
| 1539 | |||
| 1540 | //------------------------------------------------------------------------- | ||
| 1541 | |||
| 1542 | 'add': function (aValue) { | ||
| 1543 | var result; | ||
| 1544 | |||
| 1545 | if (aValue.isBigInt) { | ||
| 1546 | result = add(this.internalValue(), aValue.internalValue()); | ||
| 1547 | } else { | ||
| 1548 | result = addInt(this.internalValue(), aValue); | ||
| 1549 | } | ||
| 1550 | |||
| 1551 | return new Clipperz.Crypto.BigInt(result); | ||
| 1552 | }, | ||
| 1553 | |||
| 1554 | //------------------------------------------------------------------------- | ||
| 1555 | |||
| 1556 | 'subtract': function (aValue) { | ||
| 1557 | var result; | ||
| 1558 | var value; | ||
| 1559 | |||
| 1560 | if (aValue.isBigInt) { | ||
| 1561 | value = aValue; | ||
| 1562 | } else { | ||
| 1563 | value = new Clipperz.Crypto.BigInt(aValue); | ||
| 1564 | } | ||
| 1565 | |||
| 1566 | result = sub(this.internalValue(), value.internalValue()); | ||
| 1567 | |||
| 1568 | return new Clipperz.Crypto.BigInt(result); | ||
| 1569 | }, | ||
| 1570 | |||
| 1571 | //------------------------------------------------------------------------- | ||
| 1572 | |||
| 1573 | 'multiply': function (aValue, aModule) { | ||
| 1574 | var result; | ||
| 1575 | var value; | ||
| 1576 | |||
| 1577 | if (aValue.isBigInt) { | ||
| 1578 | value = aValue; | ||
| 1579 | } else { | ||
| 1580 | value = new Clipperz.Crypto.BigInt(aValue); | ||
| 1581 | } | ||
| 1582 | |||
| 1583 | if (typeof(aModule) == 'undefined') { | ||
| 1584 | result = mult(this.internalValue(), value.internalValue()); | ||
| 1585 | } else { | ||
| 1586 | if (greater(this.internalValue(), value.internalValue())) { | ||
| 1587 | result = multMod(this.internalValue(), value.internalValue(), aModule); | ||
| 1588 | } else { | ||
| 1589 | result = multMod(value.internalValue(), this.internalValue(), aModule); | ||
| 1590 | } | ||
| 1591 | } | ||
| 1592 | |||
| 1593 | return new Clipperz.Crypto.BigInt(result); | ||
| 1594 | }, | ||
| 1595 | |||
| 1596 | //------------------------------------------------------------------------- | ||
| 1597 | |||
| 1598 | 'module': function (aModule) { | ||
| 1599 | varresult; | ||
| 1600 | var module; | ||
| 1601 | |||
| 1602 | if (aModule.isBigInt) { | ||
| 1603 | module = aModule; | ||
| 1604 | } else { | ||
| 1605 | module = new Clipperz.Crypto.BigInt(aModule); | ||
| 1606 | } | ||
| 1607 | |||
| 1608 | result = mod(this.internalValue(), module.internalValue()); | ||
| 1609 | |||
| 1610 | return new Clipperz.Crypto.BigInt(result); | ||
| 1611 | }, | ||
| 1612 | |||
| 1613 | //------------------------------------------------------------------------- | ||
| 1614 | |||
| 1615 | 'powerModule': function(aValue, aModule) { | ||
| 1616 | varresult; | ||
| 1617 | varvalue; | ||
| 1618 | var module; | ||
| 1619 | |||
| 1620 | if (aValue.isBigInt) { | ||
| 1621 | value = aValue; | ||
| 1622 | } else { | ||
| 1623 | value = new Clipperz.Crypto.BigInt(aValue); | ||
| 1624 | } | ||
| 1625 | |||
| 1626 | if (aModule.isBigInt) { | ||
| 1627 | module = aModule; | ||
| 1628 | } else { | ||
| 1629 | module = new Clipperz.Crypto.BigInt(aModule); | ||
| 1630 | } | ||
| 1631 | |||
| 1632 | if (aValue == -1) { | ||
| 1633 | result = inverseMod(this.internalValue(), module.internalValue()); | ||
| 1634 | } else { | ||
| 1635 | result = powMod(this.internalValue(), value.internalValue(), module.internalValue()); | ||
| 1636 | } | ||
| 1637 | |||
| 1638 | return new Clipperz.Crypto.BigInt(result); | ||
| 1639 | }, | ||
| 1640 | |||
| 1641 | //------------------------------------------------------------------------- | ||
| 1642 | |||
| 1643 | 'xor': function(aValue) { | ||
| 1644 | var result; | ||
| 1645 | varthisByteArray; | ||
| 1646 | var aValueByteArray; | ||
| 1647 | var xorArray; | ||
| 1648 | |||
| 1649 | thisByteArray = new Clipperz.ByteArray("0x" + this.asString(16), 16); | ||
| 1650 | aValueByteArray = new Clipperz.ByteArray("0x" + aValue.asString(16), 16); | ||
| 1651 | xorArray = thisByteArray.xorMergeWithBlock(aValueByteArray, 'right'); | ||
| 1652 | result = new Clipperz.Crypto.BigInt(xorArray.toHexString(), 16); | ||
| 1653 | |||
| 1654 | return result; | ||
| 1655 | }, | ||
| 1656 | |||
| 1657 | //------------------------------------------------------------------------- | ||
| 1658 | |||
| 1659 | 'shiftLeft': function(aNumberOfBitsToShift) { | ||
| 1660 | var result; | ||
| 1661 | var internalResult; | ||
| 1662 | var wholeByteToShift; | ||
| 1663 | var bitsLeftToShift; | ||
| 1664 | |||
| 1665 | wholeByteToShift = Math.floor(aNumberOfBitsToShift / 8); | ||
| 1666 | bitsLeftToShift = aNumberOfBitsToShift % 8; | ||
| 1667 | |||
| 1668 | if (wholeByteToShift == 0) { | ||
| 1669 | internalResult = this.internalValue(); | ||
| 1670 | } else { | ||
| 1671 | var hexValue; | ||
| 1672 | var i,c; | ||
| 1673 | |||
| 1674 | hexValue = this.asString(16); | ||
| 1675 | c = wholeByteToShift; | ||
| 1676 | for (i=0; i<c; i++) { | ||
| 1677 | hexValue += "00"; | ||
| 1678 | } | ||
| 1679 | internalResult = str2bigInt(hexValue, 16, 1, 1); | ||
| 1680 | } | ||
| 1681 | |||
| 1682 | if (bitsLeftToShift > 0) { | ||
| 1683 | leftShift_(internalResult, bitsLeftToShift); | ||
| 1684 | } | ||
| 1685 | result = new Clipperz.Crypto.BigInt(internalResult); | ||
| 1686 | |||
| 1687 | return result; | ||
| 1688 | }, | ||
| 1689 | |||
| 1690 | //------------------------------------------------------------------------- | ||
| 1691 | |||
| 1692 | 'bitSize': function() { | ||
| 1693 | return bitSize(this.internalValue()); | ||
| 1694 | }, | ||
| 1695 | |||
| 1696 | //------------------------------------------------------------------------- | ||
| 1697 | |||
| 1698 | 'isBitSet': function(aBitPosition) { | ||
| 1699 | var result; | ||
| 1700 | |||
| 1701 | if (this.asByteArray().bitAtIndex(aBitPosition) == 0) { | ||
| 1702 | result = false; | ||
| 1703 | } else { | ||
| 1704 | result = true; | ||
| 1705 | }; | ||
| 1706 | |||
| 1707 | return result; | ||
| 1708 | }, | ||
| 1709 | |||
| 1710 | //------------------------------------------------------------------------- | ||
| 1711 | __syntaxFix__: "syntax fix" | ||
| 1712 | |||
| 1713 | }); | ||
| 1714 | |||
| 1715 | //############################################################################# | ||
| 1716 | |||
| 1717 | Clipperz.Crypto.BigInt.randomPrime = function(aBitSize) { | ||
| 1718 | return new Clipperz.Crypto.BigInt(randTruePrime(aBitSize)); | ||
| 1719 | } | ||
| 1720 | |||
| 1721 | //############################################################################# | ||
| 1722 | //############################################################################# | ||
| 1723 | |||
| 1724 | Clipperz.Crypto.BigInt.ZERO = new Clipperz.Crypto.BigInt(0); | ||
| 1725 | |||
| 1726 | //############################################################################# | ||
| 1727 | |||
| 1728 | Clipperz.Crypto.BigInt.equals = function(a, b) { | ||
| 1729 | return a.equals(b); | ||
| 1730 | } | ||
| 1731 | |||
| 1732 | Clipperz.Crypto.BigInt.add = function(a, b) { | ||
| 1733 | return a.add(b); | ||
| 1734 | } | ||
| 1735 | |||
| 1736 | Clipperz.Crypto.BigInt.subtract = function(a, b) { | ||
| 1737 | return a.subtract(b); | ||
| 1738 | } | ||
| 1739 | |||
| 1740 | Clipperz.Crypto.BigInt.multiply = function(a, b, module) { | ||
| 1741 | return a.multiply(b, module); | ||
| 1742 | } | ||
| 1743 | |||
| 1744 | Clipperz.Crypto.BigInt.module = function(a, module) { | ||
| 1745 | return a.module(module); | ||
| 1746 | } | ||
| 1747 | |||
| 1748 | Clipperz.Crypto.BigInt.powerModule = function(a, b, module) { | ||
| 1749 | return a.powerModule(b, module); | ||
| 1750 | } | ||
| 1751 | |||
| 1752 | Clipperz.Crypto.BigInt.exception = { | ||
| 1753 | UnknownType: new MochiKit.Base.NamedError("Clipperz.Crypto.BigInt.exception.UnknownType") | ||
| 1754 | } | ||