1//////////////////////////////////////////////////////////////////////////////////////// 2// Big Integer Library v. 5.4 3// Created 2000, last modified 2009 4// Leemon Baird 5// www.leemon.com 6// 7// Version history: 8// v 5.4 3 Oct 2009 9// - added "var i" to greaterShift() so i is not global. (Thanks to Pter Szab for finding that bug) 10// 11// v 5.3 21 Sep 2009 12// - added randProbPrime(k) for probable primes 13// - unrolled loop in mont_ (slightly faster) 14// - millerRabin now takes a bigInt parameter rather than an int 15// 16// v 5.2 15 Sep 2009 17// - fixed capitalization in call to int2bigInt in randBigInt 18// (thanks to Emili Evripidou, Reinhold Behringer, and Samuel Macaleese for finding that bug) 19// 20// v 5.1 8 Oct 2007 21// - renamed inverseModInt_ to inverseModInt since it doesn't change its parameters 22// - added functions GCD and randBigInt, which call GCD_ and randBigInt_ 23// - fixed a bug found by Rob Visser (see comment with his name below) 24// - improved comments 25// 26// This file is public domain. You can use it for any purpose without restriction. 27// I do not guarantee that it is correct, so use it at your own risk. If you use 28// it for something interesting, I'd appreciate hearing about it. If you find 29// any bugs or make any improvements, I'd appreciate hearing about those too. 30// It would also be nice if my name and URL were left in the comments. But none 31// of that is required. 32// 33// This code defines a bigInt library for arbitrary-precision integers. 34// A bigInt is an array of integers storing the value in chunks of bpe bits, 35// little endian (buff[0] is the least significant word). 36// Negative bigInts are stored two's complement. Almost all the functions treat 37// bigInts as nonnegative. The few that view them as two's complement say so 38// in their comments. Some functions assume their parameters have at least one 39// leading zero element. Functions with an underscore at the end of the name put 40// their answer into one of the arrays passed in, and have unpredictable behavior 41// in case of overflow, so the caller must make sure the arrays are big enough to 42// hold the answer. But the average user should never have to call any of the 43// underscored functions. Each important underscored function has a wrapper function 44// of the same name without the underscore that takes care of the details for you. 45// For each underscored function where a parameter is modified, that same variable 46// must not be used as another argument too. So, you cannot square x by doing 47// multMod_(x,x,n). You must use squareMod_(x,n) instead, or do y=dup(x); multMod_(x,y,n). 48// Or simply use the multMod(x,x,n) function without the underscore, where 49// such issues never arise, because non-underscored functions never change 50// their parameters; they always allocate new memory for the answer that is returned. 51// 52// These functions are designed to avoid frequent dynamic memory allocation in the inner loop. 53// For most functions, if it needs a BigInt as a local variable it will actually use 54// a global, and will only allocate to it only when it's not the right size. This ensures 55// that when a function is called repeatedly with same-sized parameters, it only allocates 56// memory on the first call. 57// 58// Note that for cryptographic purposes, the calls to Math.random() must 59// be replaced with calls to a better pseudorandom number generator. 60// 61// In the following, "bigInt" means a bigInt with at least one leading zero element, 62// and "integer" means a nonnegative integer less than radix. In some cases, integer 63// can be negative. Negative bigInts are 2s complement. 64// 65// The following functions do not modify their inputs. 66// Those returning a bigInt, string, or Array will dynamically allocate memory for that value. 67// Those returning a boolean will return the integer 0 (false) or 1 (true). 68// Those returning boolean or int will not allocate memory except possibly on the first 69// time they're called with a given parameter size. 70// 71// bigInt add(x,y) //return (x+y) for bigInts x and y. 72// bigInt addInt(x,n) //return (x+n) where x is a bigInt and n is an integer. 73// string bigInt2str(x,base) //return a string form of bigInt x in a given base, with 2 <= base <= 95 74// int bitSize(x) //return how many bits long the bigInt x is, not counting leading zeros 75// bigInt dup(x) //return a copy of bigInt x 76// boolean equals(x,y) //is the bigInt x equal to the bigint y? 77// boolean equalsInt(x,y) //is bigint x equal to integer y? 78// bigInt expand(x,n) //return a copy of x with at least n elements, adding leading zeros if needed 79// Array findPrimes(n) //return array of all primes less than integer n 80// bigInt GCD(x,y) //return greatest common divisor of bigInts x and y (each with same number of elements). 81// boolean greater(x,y) //is x>y? (x and y are nonnegative bigInts) 82// boolean greaterShift(x,y,shift)//is (x <<(shift*bpe)) > y? 83// bigInt int2bigInt(t,n,m) //return a bigInt equal to integer t, with at least n bits and m array elements 84// bigInt inverseMod(x,n) //return (x**(-1) mod n) for bigInts x and n. If no inverse exists, it returns null 85// int inverseModInt(x,n) //return x**(-1) mod n, for integers x and n. Return 0 if there is no inverse 86// boolean isZero(x) //is the bigInt x equal to zero? 87// boolean millerRabin(x,b) //does one round of Miller-Rabin base integer b say that bigInt x is possibly prime? (b is bigInt, 1<b<x) 88// boolean millerRabinInt(x,b) //does one round of Miller-Rabin base integer b say that bigInt x is possibly prime? (b is int, 1<b<x) 89// bigInt mod(x,n) //return a new bigInt equal to (x mod n) for bigInts x and n. 90// int modInt(x,n) //return x mod n for bigInt x and integer n. 91// bigInt mult(x,y) //return x*y for bigInts x and y. This is faster when y<x. 92// bigInt multMod(x,y,n) //return (x*y mod n) for bigInts x,y,n. For greater speed, let y<x. 93// boolean negative(x) //is bigInt x negative? 94// bigInt powMod(x,y,n) //return (x**y mod n) where x,y,n are bigInts and ** is exponentiation. 0**0=1. Faster for odd n. 95// bigInt randBigInt(n,s) //return an n-bit random BigInt (n>=1). If s=1, then the most significant of those n bits is set to 1. 96// bigInt randTruePrime(k) //return a new, random, k-bit, true prime bigInt using Maurer's algorithm. 97// bigInt randProbPrime(k) //return a new, random, k-bit, probable prime bigInt (probability it's composite less than 2^-80). 98// bigInt str2bigInt(s,b,n,m) //return a bigInt for number represented in string s in base b with at least n bits and m array elements 99// bigInt sub(x,y) //return (x-y) for bigInts x and y. Negative answers will be 2s complement 100// bigInt trim(x,k) //return a copy of x with exactly k leading zero elements 101// 102//
103// The following functions each have a non-underscored version, which most users should call instead. 104// These functions each write to a single parameter, and the caller is responsible for ensuring the array 105// passed in is large enough to hold the result. 106// 107// void addInt_(x,n) //do x=x+n where x is a bigInt and n is an integer 108// void add_(x,y) //do x=x+y for bigInts x and y 109// void copy_(x,y) //do x=y on bigInts x and y 110// void copyInt_(x,n) //do x=n on bigInt x and integer n 111// void GCD_(x,y) //set x to the greatest common divisor of bigInts x and y, (y is destroyed). (This never overflows its array). 112// boolean inverseMod_(x,n) //do x=x**(-1) mod n, for bigInts x and n. Returns 1 (0) if inverse does (doesn't) exist 113// void mod_(x,n) //do x=x mod n for bigInts x and n. (This never overflows its array). 114// void mult_(x,y) //do x=x*y for bigInts x and y. 115// void multMod_(x,y,n) //do x=x*y mod n for bigInts x,y,n. 116// void powMod_(x,y,n) //do x=x**y mod n, where x,y,n are bigInts (n is odd) and ** is exponentiation. 0**0=1. 117// void randBigInt_(b,n,s) //do b = an n-bit random BigInt. if s=1, then nth bit (most significant bit) is set to 1. n>=1. 118// void randTruePrime_(ans,k) //do ans = a random k-bit true random prime (not just probable prime) with 1 in the msb. 119// void sub_(x,y) //do x=x-y for bigInts x and y. Negative answers will be 2s complement. 120// 121// The following functions do NOT have a non-underscored version. 122// They each write a bigInt result to one or more parameters. The caller is responsible for 123// ensuring the arrays passed in are large enough to hold the results. 124// 125// void addShift_(x,y,ys) //do x=x+(y<<(ys*bpe)) 126// void carry_(x) //do carries and borrows so each element of the bigInt x fits in bpe bits. 127// void divide_(x,y,q,r) //divide x by y giving quotient q and remainder r 128// int divInt_(x,n) //do x=floor(x/n) for bigInt x and integer n, and return the remainder. (This never overflows its array). 129// int eGCD_(x,y,d,a,b) //sets a,b,d to positive bigInts such that d = GCD_(x,y) = a*x-b*y 130// void halve_(x) //do x=floor(|x|/2)*sgn(x) for bigInt x in 2's complement. (This never overflows its array). 131// void leftShift_(x,n) //left shift bigInt x by n bits. n<bpe. 132// void linComb_(x,y,a,b) //do x=a*x+b*y for bigInts x and y and integers a and b 133// void linCombShift_(x,y,b,ys) //do x=x+b*(y<<(ys*bpe)) for bigInts x and y, and integers b and ys 134// void mont_(x,y,n,np) //Montgomery multiplication (see comments where the function is defined) 135// void multInt_(x,n) //do x=x*n where x is a bigInt and n is an integer. 136// void rightShift_(x,n) //right shift bigInt x by n bits. 0 <= n < bpe. (This never overflows its array). 137// void squareMod_(x,n) //do x=x*x mod n for bigInts x,n 138// void 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 180bpe=0; //bits stored per array element 181mask=0; //AND this with an array element to chop it down to bpe bits 182radix=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 185digitsStr='0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz_=!@#$%^&*()[]{}|;:,.<>/?`~ \\\'\"+-'; 186 187//initialize the global variables 188for (bpe=0; (1<<(bpe+1)) > (1<<bpe); bpe++); //bpe=number of bits in the mantissa on this platform 189bpe>>=1; //bpe=number of bits in one element of the array representing the bigInt 190mask=(1<<bpe)-1; //AND the mask with an integer to get its bpe least significant bits 191radix=mask+1; //2^bpe. a single 1 bit to the left of the first bit of mask 192one=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 196t=new Array(0); 197ss=t; //used in mult_() 198s0=t; //used in multMod_(), squareMod_() 199s1=t; //used in powMod_(), multMod_(), squareMod_() 200s2=t; //used in powMod_(), multMod_() 201s3=t; //used in powMod_() 202s4=t; s5=t; //used in mod_() 203s6=t; //used in bigInt2str() 204s7=t; //used in powMod_() 205T=t; //used in GCD_() 206sa=t; //used in mont_() 207mr_x1=t; mr_r=t; mr_a=t; //used in millerRabin() 208eg_v=t; eg_u=t; eg_A=t; eg_B=t; eg_C=t; eg_D=t; //used in eGCD_(), inverseMod_() 209md_q1=t; md_q2=t; md_q3=t; md_r=t; md_r1=t; md_r2=t; md_tt=t; //used in mod_() 210 211primes=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 214rpprb=t; //used in randProbPrimeRounds() (which also uses "primes") 215 216//////////////////////////////////////////////////////////////////////////////////////// 217 218 219//return array of all primes less than integer n 220function findPrimes(n) { 221 var i,s,p,ans; 222 s=new Array(n); 223 for (i=0;i<n;i++) 224 s[i]=0; 225 s[0]=2; 226 p=0; //first p elements of s are primes, the rest are a sieve 227 for(;s[p]<n;) { //s[p] is the pth prime 228 for(i=s[p]*s[p]; i<n; i+=s[p]) //mark multiples of s[p] 229 s[i]=1; 230 p++; 231 s[p]=s[p-1]+1; 232 for(; s[p]<n && s[s[p]]; s[p]++); //find next prime (where s[p]==0) 233 } 234 ans=new Array(p); 235 for(i=0;i<p;i++) 236 ans[i]=s[i]; 237 return ans; 238} 239 240 241//does a single round of Miller-Rabin base b consider x to be a possible prime? 242//x is a bigInt, and b is an integer, with b<x 243function millerRabinInt(x,b) { 244 if (mr_x1.length!=x.length) { 245 mr_x1=dup(x); 246 mr_r=dup(x); 247 mr_a=dup(x); 248 } 249 250 copyInt_(mr_a,b); 251 return millerRabin(x,mr_a); 252} 253 254//does a single round of Miller-Rabin base b consider x to be a possible prime? 255//x and b are bigInts with b<x 256function millerRabin(x,b) { 257 var i,j,k,s; 258 259 if (mr_x1.length!=x.length) { 260 mr_x1=dup(x); 261 mr_r=dup(x); 262 mr_a=dup(x); 263 } 264 265 copy_(mr_a,b); 266 copy_(mr_r,x); 267 copy_(mr_x1,x); 268 269 addInt_(mr_r,-1); 270 addInt_(mr_x1,-1); 271 272 //s=the highest power of two that divides mr_r 273 k=0; 274 for (i=0;i<mr_r.length;i++) 275 for (j=1;j<mask;j<<=1) 276 if (x[i] & j) { 277 s=(k<mr_r.length+bpe ? k : 0); 278 i=mr_r.length; 279 j=mask; 280 } else 281 k++; 282 283 if (s) 284 rightShift_(mr_r,s); 285 286 powMod_(mr_a,mr_r,x); 287 288 if (!equalsInt(mr_a,1) && !equals(mr_a,mr_x1)) { 289 j=1; 290 while (j<=s-1 && !equals(mr_a,mr_x1)) { 291 squareMod_(mr_a,x); 292 if (equalsInt(mr_a,1)) { 293 return 0; 294 } 295 j++; 296 } 297 if (!equals(mr_a,mr_x1)) { 298 return 0; 299 } 300 } 301 return 1; 302} 303 304//returns how many bits long the bigInt is, not counting leading zeros. 305function bitSize(x) { 306 var j,z,w; 307 for (j=x.length-1; (x[j]==0) && (j>0); j--); 308 for (z=0,w=x[j]; w; (w>>=1),z++); 309 z+=bpe*j; 310 return z; 311} 312 313//return a copy of x with at least n elements, adding leading zeros if needed 314function expand(x,n) { 315 var ans=int2bigInt(0,(x.length>n ? x.length : n)*bpe,0); 316 copy_(ans,x); 317 return ans; 318} 319 320//return a k-bit true random prime using Maurer's algorithm. 321function randTruePrime(k) { 322 var ans=int2bigInt(0,k,0); 323 randTruePrime_(ans,k); 324 return trim(ans,1); 325} 326 327//return a k-bit random probable prime with probability of error < 2^-80 328function randProbPrime(k) { 329 if (k>=600) return randProbPrimeRounds(k,2); //numbers from HAC table 4.3 330 if (k>=550) return randProbPrimeRounds(k,4); 331 if (k>=500) return randProbPrimeRounds(k,5); 332 if (k>=400) return randProbPrimeRounds(k,6); 333 if (k>=350) return randProbPrimeRounds(k,7); 334 if (k>=300) return randProbPrimeRounds(k,9); 335 if (k>=250) return randProbPrimeRounds(k,12); //numbers from HAC table 4.4 336 if (k>=200) return randProbPrimeRounds(k,15); 337 if (k>=150) return randProbPrimeRounds(k,18); 338 if (k>=100) return randProbPrimeRounds(k,27); 339 return randProbPrimeRounds(k,40); //number from HAC remark 4.26 (only an estimate) 340} 341 342//return a k-bit probable random prime using n rounds of Miller Rabin (after trial division
342with small primes) 343function randProbPrimeRounds(k,n) { 344 var ans, i, divisible, B; 345 B=30000; //B is largest prime to use in trial division 346 ans=int2bigInt(0,k,0); 347 348 //optimization: try larger and smaller B to find the best limit. 349 350 if (primes.length==0) 351 primes=findPrimes(30000); //check for divisibility by primes <=30000 352 353 if (rpprb.length!=ans.length) 354 rpprb=dup(ans); 355 356 for (;;) { //keep trying random values for ans until one appears to be prime 357 //optimization: pick a random number times L=2*3*5*...*p, plus a 358 // random element of the list of all numbers in [0,L) not divisible by any prime up to p. 359 // This can reduce the amount of random number generation. 360 361 randBigInt_(ans,k,0); //ans = a random odd number to check 362 ans[0] |= 1; 363 divisible=0; 364 365 //check ans for divisibility by small primes up to B 366 for (i=0; (i<primes.length) && (primes[i]<=B); i++) 367 if (modInt(ans,primes[i])==0 && !equalsInt(ans,primes[i])) { 368 divisible=1; 369 break; 370 } 371 372 //optimization: change millerRabin so the base can be bigger than the number being checked, then eliminate the while here. 373 374 //do n rounds of Miller Rabin, with random bases less than ans 375 for (i=0; i<n && !divisible; i++) { 376 randBigInt_(rpprb,k,0); 377 while(!greater(ans,rpprb)) //pick a random rpprb that's < ans 378 randBigInt_(rpprb,k,0); 379 if (!millerRabin(ans,rpprb)) 380 divisible=1; 381 } 382 383 if(!divisible) 384 return ans; 385 } 386} 387 388//return a new bigInt equal to (x mod n) for bigInts x and n. 389function mod(x,n) { 390 var ans=dup(x); 391 mod_(ans,n); 392 return trim(ans,1); 393} 394 395//return (x+n) where x is a bigInt and n is an integer. 396function addInt(x,n) { 397 var ans=expand(x,x.length+1); 398 addInt_(ans,n); 399 return trim(ans,1); 400} 401 402//return x*y for bigInts x and y. This is faster when y<x. 403function mult(x,y) { 404 var ans=expand(x,x.length+y.length); 405 mult_(ans,y); 406 return trim(ans,1); 407} 408 409//return (x**y mod n) where x,y,n are bigInts and ** is exponentiation. 0**0=1. Faster for odd n. 410function powMod(x,y,n) { 411 var ans=expand(x,n.length); 412 powMod_(ans,trim(y,2),trim(n,2),0); //this should work without the trim, but doesn't 413 return trim(ans,1); 414} 415 416//return (x-y) for bigInts x and y. Negative answers will be 2s complement 417function sub(x,y) { 418 var ans=expand(x,(x.length>y.length ? x.length+1 : y.length+1)); 419 sub_(ans,y); 420 return trim(ans,1); 421} 422 423//return (x+y) for bigInts x and y. 424function add(x,y) { 425 var ans=expand(x,(x.length>y.length ? x.length+1 : y.length+1)); 426 add_(ans,y); 427 return trim(ans,1); 428} 429 430//return (x**(-1) mod n) for bigInts x and n. If no inverse exists, it returns null 431function inverseMod(x,n) { 432 var ans=expand(x,n.length); 433 var s; 434 s=inverseMod_(ans,n); 435 return s ? trim(ans,1) : null; 436} 437 438//return (x*y mod n) for bigInts x,y,n. For greater speed, let y<x. 439function multMod(x,y,n) { 440 var ans=expand(x,n.length); 441 multMod_(ans,y,n); 442 return trim(ans,1); 443} 444 445//generate a k-bit true random prime using Maurer's algorithm, 446//and put it into ans. The bigInt ans must be large enough to hold it. 447function randTruePrime_(ans,k) { 448 var c,m,pm,dd,j,r,B,divisible,z,zz,recSize; 449 450 if (primes.length==0) 451 primes=findPrimes(30000); //check for divisibility by primes <=30000 452 453 if (pows.length==0) { 454 pows=new Array(512); 455 for (j=0;j<512;j++) { 456 pows[j]=Math.pow(2,j/511.-1.); 457 } 458 } 459 460 //c and m should be tuned for a particular machine and value of k, to maximize speed 461 c=0.1; //c=0.1 in HAC 462 m=20; //generate this k-bit number by first recursively generating a number that has between k/2 and k-m bits 463 recLimit=20; //stop recursion when k <=recLimit. Must have recLimit >= 2 464 465 if (s_i2.length!=ans.length) { 466 s_i2=dup(ans); 467 s_R =dup(ans); 468 s_n1=dup(ans); 469 s_r2=dup(ans); 470 s_d =dup(ans); 471 s_x1=dup(ans); 472 s_x2=dup(ans); 473 s_b =dup(ans); 474 s_n =dup(ans); 475 s_i =dup(ans); 476 s_rm=dup(ans); 477 s_q =dup(ans); 478 s_a =dup(ans); 479 s_aa=dup(ans); 480 } 481 482 if (k <= recLimit) { //generate small random primes by trial division up to its square root 483 pm=(1<<((k+2)>>1))-1; //pm is binary number with all ones, just over sqrt(2^k) 484 copyInt_(ans,0); 485 for (dd=1;dd;) { 486 dd=0; 487 ans[0]= 1 | (1<<(k-1)) | Math.floor(Math.random()*(1<<k)); //random, k-bit, odd integer, with msb 1 488 for (j=1;(j<primes.length) && ((primes[j]&pm)==primes[j]);j++) { //trial division by all primes 3...sqrt(2^k) 489 if (0==(ans[0]%primes[j])) { 490 dd=1; 491 break; 492 } 493 } 494 } 495 carry_(ans); 496 return; 497 } 498 499 B=c*k*k; //try small primes up to B (or all the primes[] array if the largest is less than B). 500 if (k>2*m) //generate this k-bit number by first recursively generating a number that has between k/2 and k-m bits 501 for (r=1; k-k*r<=m; ) 502 r=pows[Math.floor(Math.random()*512)]; //r=Math.pow(2,Math.random()-1); 503 else 504 r=.5; 505 506 //simulation suggests the more complex algorithm using r=.333 is only slightly faster. 507 508 recSize=Math.floor(r*k)+1; 509 510 randTruePrime_(s_q,recSize); 511 copyInt_(s_i2,0); 512 s_i2[Math.floor((k-2)/bpe)] |= (1<<((k-2)%bpe)); //s_i2=2^(k-2) 513 divide_(s_i2,s_q,s_i,s_rm); //s_i=floor((2^(k-1))/(2q)) 514 515 z=bitSize(s_i); 516 517 for (;;) { 518 for (;;) { //generate z-bit numbers until one falls in the range [0,s_i-1] 519 randBigInt_(s_R,z,0); 520 if (greater(s_i,s_R)) 521 break; 522 } //now s_R is in the range [0,s_i-1] 523 addInt_(s_R,1); //now s_R is in the range [1,s_i] 524 add_(s_R,s_i); //now s_R is in the range [s_i+1,2*s_i] 525 526 copy_(s_n,s_q); 527 mult_(s_n,s_R); 528 multInt_(s_n,2); 529 addInt_(s_n,1); //s_n=2*s_R*s_q+1 530 531 copy_(s_r2,s_R); 532 multInt_(s_r2,2); //s_r2=2*s_R 533 534 //check s_n for divisibility by small primes up to B 535 for (divisible=0,j=0; (j<primes.length) && (primes[j]<B); j++) 536 if (modInt(s_n,primes[j])==0 && !equalsInt(s_n,primes[j])) { 537 divisible=1; 538 break; 539 } 540 541 if (!divisible) //if it passes small primes check, then try a single Miller-Rabin base 2 542 if (!millerRabinInt(s_n,2)) //this line represents 75% of the total runtime for randTruePrime_ 543 divisible=1; 544 545 if (!divisible) { //if it passes that test, continue checking s_n 546 addInt_(s_n,-3); 547 for (j=s_n.length-1;(s_n[j]==0) && (j>0); j--); //strip leading zeros 548 for (zz=0,w=s_n[j]; w; (w>>=1),zz++); 549 zz+=bpe*j; //zz=number of bits in s_n, ignoring leading zeros 550 for (;;) { //generate z-bit numbers until one falls in the range [0,s_n-1] 551 randBigInt_(s_a,zz,0); 552 if (greater(s_n,s_a)) 553 break; 554 } //now s_a is in the range [0,s_n-1] 555 addInt_(s_n,3); //now s_a is in the range [0,s_n-4] 556 addInt_(s_a,2); //now s_a is in the range [2,s_n-2] 557 copy_(s_b,s_a); 558 copy_(s_n1,s_n); 559 addInt_(s_n1,-1); 560 powMod_(s_b,s_n1,s_n); //s_b=s_a^(s_n-1) modulo s_n 561 addInt_(s_b,-1); 562 if (isZero(s_b)) { 563 copy_(s_b,s_a); 564 powMod_(s_b,s_r2,s_n); 565 addInt_(s_b,-1); 566 copy_(s_aa,s_n); 567 copy_(s_d,s_b); 568 GCD_(s_d,s_n); //if s_b and s_n are relatively prime, then s_n is a prime 569 if (equalsInt(s_d,1)) { 570 copy_(ans,s_aa); 571 return; //if we've made it this far, then s_n is absolutely guaranteed to be prime 572 } 573 } 574 } 575 } 576} 577 578//Return an n-bit random BigInt (n>=1). If s=1, then the most significant of those n bits is set to 1. 579function randBigInt(n,s) { 580 var a,b; 581 a=Math.floor((n-1)/bpe)+2; //# array elements to hold the BigInt with a leading 0 element 582 b=int2bigInt(0,0,a); 583 randBigInt_(b,n,s); 584 return b; 585} 586 587//Set b to an n-bit random BigInt. If s=1, then the most significant of those n bits is set to 1. 588//Array b must be big enough to hold the result. Must have n>=1 589function randBigInt_(b,n,s) { 590 var i,a; 591 for (i=0;i<b.length;i++) 592 b[i]=0; 593 a=Math.floor((n-1)/bpe)+1; //# array elements to hold the BigInt 594 for (i=0;i<a;i++) { 595 b[i]=Math.floor(Math.random()*(1<<(bpe-1))); 596 } 597 b[a-1] &= (2<<((n-1)%bpe))-1; 598 if (s==1) 599 b[a-1] |= (1<<((n-1)%bpe)); 600} 601 602//Return the greatest common divisor of bigInts x and y (each with same number of elements). 603function GCD(x,y) { 604 var xc,yc; 605 xc=dup(x); 606 yc=dup(y); 607 GCD_(xc,yc); 608 return xc; 609} 610 611//set x to the greatest common divisor of bigInts x and y (each with same number of elements). 612//y is destroyed. 613function GCD_(x,y) { 614 var i,xp,yp,A,B,C,D,q,sing; 615 if (T.length!=x.length) 616 T=dup(x); 617 618 sing=1; 619 while (sing) { //while y has nonzero elements other than y[0] 620 sing=0; 621 for (i=1;i<y.length;i++) //check if y has nonzero elements other than 0 622 if (y[i]) { 623 sing=1; 624 break; 625 } 626 if (!sing) break; //quit when y all zero elements except possibly y[0] 627 628 for (i=x.length;!x[i] && i>=0;i--); //find most significant element of x 629 xp=x[i]; 630 yp=y[i]; 631 A=1; B=0; C=0; D=1; 632 while ((yp+C) && (yp+D)) { 633 q =Math.floor((xp+A)/(yp+C)); 634 qp=Math.floor((xp+B)/(yp+D)); 635 if (q!=qp) 636 break; 637 t= A-q*C; A=C; C=t; // do (A,B,xp, C,D,yp) = (C,D,yp, A,B,xp) - q*(0,0,0, C,D,yp) 638 t= B-q*D; B=D; D=t; 639 t=xp-q*yp; xp=yp; yp=t; 640 } 641 if (B) { 642 copy_(T,x); 643 linComb_(x,y,A,B); //x=A*x+B*y 644 linComb_(y,T,D,C); //y=D*y+C*T 645 } else { 646 mod_(x,y); 647 copy_(T,x); 648 copy_(x,y); 649 copy_(y,T); 650 } 651 } 652 if (y[0]==0) 653 return; 654 t=modInt(x,y[0]); 655 copyInt_(x,y[0]); 656 y[0]=t; 657 while (y[0]) { 658 x[0]%=y[0]; 659 t=x[0]; x[0]=y[0]; y[0]=t; 660 } 661} 662 663//do x=x**(-1) mod n, for bigInts x and n. 664//If no inverse exists, it sets x to zero and returns 0, else it returns 1. 665//The x array must be at least as large as the n array. 666function inverseMod_(x,n) { 667 var k=1+2*Math.max(x.length,n.length); 668 669 if(!(x[0]&1) && !(n[0]&1)) { //if both inputs are even, then inverse doesn't exist 670 copyInt_(x,0); 671 return 0; 672 } 673 674 if (eg_u.length!=k) { 675 eg_u=new Array(k); 676 eg_v=new Array(k); 677 eg_A=new Array(k); 678 eg_B=new Array(k); 679 eg_C=new Array(k); 680 eg_D=new Array(k); 681 } 682 683 copy_(eg_u,x); 684 copy_(eg_v,n); 685 copyInt_(eg_A,1); 686 copyInt_(eg_B,0); 687 copyInt_(eg_C,0); 688 copyInt_(eg_D,1); 689 for (;;) { 690 while(!(eg_u[0]&1)) { //while eg_u is even 691 halve_(eg_u); 692 if (!(eg_A[0]&1) && !(eg_B[0]&1)) { //if eg_A==eg_B==0 mod 2 693 halve_(eg_A); 694 halve_(eg_B); 695 } else { 696 add_(eg_A,n); halve_(eg_A); 697 sub_(eg_B,x); halve_(eg_B); 698 } 699 } 700 701 while (!(eg_v[0]&1)) { //while eg_v is even 702 halve_(eg_v); 703 if (!(eg_C[0]&1) && !(eg_D[0]&1)) { //if eg_C==eg_D==0 mod 2 704 halve_(eg_C); 705 halve_(eg_D); 706 } else { 707 add_(eg_C,n); halve_(eg_C); 708 sub_(eg_D,x); halve_(eg_D); 709 } 710 } 711 712 if (!greater(eg_v,eg_u)) { //eg_v <= eg_u 713 sub_(eg_u,eg_v); 714 sub_(eg_A,eg_C); 715 sub_(eg_B,eg_D); 716 } else { //eg_v > eg_u 717 sub_(eg_v,eg_u); 718 sub_(eg_C,eg_A); 719 sub_(eg_D,eg_B); 720 } 721 722 if (equalsInt(eg_u,0)) { 723 if (negative(eg_C)) //make sure answer is nonnegative 724 add_(eg_C,n); 725 copy_(x,eg_C); 726 727 if (!equalsInt(eg_v,1)) { //if GCD_(x,n)!=1, then there is no inverse 728 copyInt_(x,0); 729 return 0; 730 } 731 return 1; 732 } 733 } 734} 735 736//return x**(-1) mod n, for integers x and n. Return 0 if there is no inverse 737function inverseModInt(x,n) { 738 var a=1,b=0,t; 739 for (;;) { 740 if (x==1) return a; 741 if (x==0) return 0; 742 b-=a*Math.floor(n/x); 743 n%=x; 744 745 if (n==1) return b; //to avoid negatives, change this b to n-b, and each -= to += 746 if (n==0) return 0; 747 a-=b*Math.floor(x/n); 748 x%=n; 749 } 750} 751 752//this deprecated function is for backward compatibility only. 753function inverseModInt_(x,n) { 754 return inverseModInt(x,n); 755} 756 757 758//Given positive bigInts x and y, change the bigints v, a, and b to positive bigInts such that: 759// v = GCD_(x,y) = a*x-b*y 760//The bigInts v, a, b, must have exactly as many elements as the larger of x and y. 761function eGCD_(x,y,v,a,b) { 762 var g=0; 763 var k=Math.max(x.length,y.length); 764 if (eg_u.length!=k) { 765 eg_u=new Array(k); 766 eg_A=new Array(k); 767 eg_B=new Array(k); 768 eg_C=new Array(k); 769 eg_D=new Array(k); 770 } 771 while(!(x[0]&1) && !(y[0]&1)) { //while x and y both even 772 halve_(x); 773 halve_(y); 774 g++; 775 } 776 copy_(eg_u,x); 777 copy_(v,y); 778 copyInt_(eg_A,1); 779 copyInt_(eg_B,0); 780 copyInt_(eg_C,0); 781 copyInt_(eg_D,1); 782 for (;;) { 783 while(!(eg_u[0]&1)) { //while u is even 784 halve_(eg_u); 785 if (!(eg_A[0]&1) && !(eg_B[0]&1)) { //if A==B==0 mod 2 786 halve_(eg_A); 787 halve_(eg_B); 788 } else { 789 add_(eg_A,y); halve_(eg_A); 790 sub_(eg_B,x); halve_(eg_B); 791 } 792 } 793 794 while (!(v[0]&1)) { //while v is even 795 halve_(v); 796 if (!(eg_C[0]&1) && !(eg_D[0]&1)) { //if C==D==0 mod 2 797 halve_(eg_C); 798 halve_(eg_D); 799 } else { 800 add_(eg_C,y); halve_(eg_C); 801 sub_(eg_D,x); halve_(eg_D); 802 } 803 } 804 805 if (!greater(v,eg_u)) { //v<=u 806 sub_(eg_u,v); 807 sub_(eg_A,eg_C); 808 sub_(eg_B,eg_D); 809 } else { //v>u 810 sub_(v,eg_u); 811 sub_(eg_C,eg_A); 812 sub_(eg_D,eg_B); 813 } 814 if (equalsInt(eg_u,0)) { 815 if (negative(eg_C)) { //make sure a (C)is nonnegative 816 add_(eg_C,y); 817 sub_(eg_D,x); 818 } 819 multInt_(eg_D,-1); ///make sure b (D) is nonnegative 820 copy_(a,eg_C); 821 copy_(b,eg_D); 822 leftShift_(v,g); 823 return; 824 } 825 } 826} 827 828 829//is bigInt x negative? 830function negative(x) { 831 return ((x[x.length-1]>>(bpe-1))&1); 832} 833 834 835//is (x << (shift*bpe)) > y? 836//x and y are nonnegative bigInts
837//shift is a nonnegative integer 838function greaterShift(x,y,shift) { 839 var i, kx=x.length, ky=y.length; 840 k=((kx+shift)<ky) ? (kx+shift) : ky; 841 for (i=ky-1-shift; i<kx && i>=0; i++) 842 if (x[i]>0) 843 return 1; //if there are nonzeros in x to the left of the first column of y, then x is bigger 844 for (i=kx-1+shift; i<ky; i++) 845 if (y[i]>0) 846 return 0; //if there are nonzeros in y to the left of the first column of x, then x is not bigger 847 for (i=k-1; i>=shift; i--) 848 if (x[i-shift]>y[i]) return 1; 849 else if (x[i-shift]<y[i]) return 0; 850 return 0; 851} 852 853//is x > y? (x and y both nonnegative) 854function greater(x,y) { 855 var i; 856 var k=(x.length<y.length) ? x.length : y.length; 857 858 for (i=x.length;i<y.length;i++) 859 if (y[i]) 860 return 0; //y has more digits 861 862 for (i=y.length;i<x.length;i++) 863 if (x[i]) 864 return 1; //x has more digits 865 866 for (i=k-1;i>=0;i--) 867 if (x[i]>y[i]) 868 return 1; 869 else if (x[i]<y[i]) 870 return 0; 871 return 0; 872} 873 874//divide x by y giving quotient q and remainder r. (q=floor(x/y), r=x mod y). All 4 are bigints. 875//x must have at least one leading zero element. 876//y must be nonzero. 877//q and r must be arrays that are exactly the same length as x. (Or q can have more). 878//Must have x.length >= y.length >= 2. 879function divide_(x,y,q,r) { 880 var kx, ky; 881 var i,j,y1,y2,c,a,b; 882 copy_(r,x); 883 for (ky=y.length;y[ky-1]==0;ky--); //ky is number of elements in y, not including leading zeros 884 885 //normalize: ensure the most significant element of y has its highest bit set 886 b=y[ky-1]; 887 for (a=0; b; a++) 888 b>>=1; 889 a=bpe-a; //a is how many bits to shift so that the high order bit of y is leftmost in its array element 890 leftShift_(y,a); //multiply both by 1<<a now, then divide both by that at the end 891 leftShift_(r,a); 892 893 //Rob Visser discovered a bug: the following line was originally just before the normalization. 894 for (kx=r.length;r[kx-1]==0 && kx>ky;kx--); //kx is number of elements in normalized x, not including leading zeros 895 896 copyInt_(q,0); // q=0 897 while (!greaterShift(y,r,kx-ky)) { // while (leftShift_(y,kx-ky) <= r) { 898 subShift_(r,y,kx-ky); // r=r-leftShift_(y,kx-ky) 899 q[kx-ky]++; // q[kx-ky]++; 900 } // } 901 902 for (i=kx-1; i>=ky; i--) { 903 if (r[i]==y[ky-1]) 904 q[i-ky]=mask; 905 else 906 q[i-ky]=Math.floor((r[i]*radix+r[i-1])/y[ky-1]); 907 908 //The following for(;;) loop is equivalent to the commented while loop, 909 //except that the uncommented version avoids overflow. 910 //The commented loop comes from HAC, which assumes r[-1]==y[-1]==0 911 // while (q[i-ky]*(y[ky-1]*radix+y[ky-2]) > r[i]*radix*radix+r[i-1]*radix+r[i-2]) 912 // q[i-ky]--; 913 for (;;) { 914 y2=(ky>1 ? y[ky-2] : 0)*q[i-ky]; 915 c=y2>>bpe; 916 y2=y2 & mask; 917 y1=c+q[i-ky]*y[ky-1]; 918 c=y1>>bpe; 919 y1=y1 & mask; 920 921 if (c==r[i] ? y1==r[i-1] ? y2>(i>1 ? r[i-2] : 0) : y1>r[i-1] : c>r[i]) 922 q[i-ky]--; 923 else 924 break; 925 } 926 927 linCombShift_(r,y,-q[i-ky],i-ky); //r=r-q[i-ky]*leftShift_(y,i-ky) 928 if (negative(r)) { 929 addShift_(r,y,i-ky); //r=r+leftShift_(y,i-ky) 930 q[i-ky]--; 931 } 932 } 933 934 rightShift_(y,a); //undo the normalization step 935 rightShift_(r,a); //undo the normalization step 936} 937 938//do carries and borrows so each element of the bigInt x fits in bpe bits. 939function carry_(x) { 940 var i,k,c,b; 941 k=x.length; 942 c=0; 943 for (i=0;i<k;i++) { 944 c+=x[i]; 945 b=0; 946 if (c<0) { 947 b=-(c>>bpe); 948 c+=b*radix; 949 } 950 x[i]=c & mask; 951 c=(c>>bpe)-b; 952 } 953} 954 955//return x mod n for bigInt x and integer n. 956function modInt(x,n) { 957 var i,c=0; 958 for (i=x.length-1; i>=0; i--) 959 c=(c*radix+x[i])%n; 960 return c; 961} 962 963//convert the integer t into a bigInt with at least the given number of bits. 964//the returned array stores the bigInt in bpe-bit chunks, little endian (buff[0] is least significant word) 965//Pad the array with leading zeros so that it has at least minSize elements. 966//There will always be at least one leading 0 element. 967function int2bigInt(t,bits,minSize) { 968 var i,k; 969 k=Math.ceil(bits/bpe)+1; 970 k=minSize>k ? minSize : k; 971 buff=new Array(k); 972 copyInt_(buff,t); 973 return buff; 974} 975 976//return the bigInt given a string representation in a given base. 977//Pad the array with leading zeros so that it has at least minSize elements.
978//If base=-1, then it reads in a space-separated list of array elements in decimal. 979//The array will always have at least one leading zero, unless base=-1. 980function str2bigInt(s,base,minSize) { 981 var d, i, j, x, y, kk; 982 var k=s.length; 983 if (base==-1) { //comma-separated list of array elements in decimal 984 x=new Array(0); 985 for (;;) { 986 y=new Array(x.length+1); 987 for (i=0;i<x.length;i++) 988 y[i+1]=x[i]; 989 y[0]=parseInt(s,10); 990 x=y; 991 d=s.indexOf(',',0); 992 if (d<1) 993 break; 994 s=s.substring(d+1); 995 if (s.length==0) 996 break; 997 } 998 if (x.length<minSize) { 999 y=new Array(minSize); 1000 copy_(y,x); 1001 return y; 1002 } 1003 return x; 1004 } 1005 1006 x=int2bigInt(0,base*k,0); 1007 for (i=0;i<k;i++) { 1008 d=digitsStr.indexOf(s.substring(i,i+1),0); 1009 if (base<=36 && d>=36) //convert lowercase to uppercase if base<=36 1010 d-=26; 1011 if (d>=base || d<0) { //stop at first illegal character 1012 break; 1013 } 1014 multInt_(x,base); 1015 addInt_(x,d); 1016 } 1017 1018 for (k=x.length;k>0 && !x[k-1];k--); //strip off leading zeros 1019 k=minSize>k+1 ? minSize : k+1; 1020 y=new Array(k); 1021 kk=k<x.length ? k : x.length; 1022 for (i=0;i<kk;i++) 1023 y[i]=x[i]; 1024 for (;i<k;i++) 1025 y[i]=0; 1026 return y; 1027} 1028 1029//is bigint x equal to integer y? 1030//y must have less than bpe bits 1031function equalsInt(x,y) { 1032 var i; 1033 if (x[0]!=y) 1034 return 0; 1035 for (i=1;i<x.length;i++) 1036 if (x[i]) 1037 return 0; 1038 return 1; 1039} 1040 1041//are bigints x and y equal? 1042//this works even if x and y are different lengths and have arbitrarily many leading zeros 1043function equals(x,y) { 1044 var i; 1045 var k=x.length<y.length ? x.length : y.length; 1046 for (i=0;i<k;i++) 1047 if (x[i]!=y[i]) 1048 return 0; 1049 if (x.length>y.length) { 1050 for (;i<x.length;i++) 1051 if (x[i]) 1052 return 0; 1053 } else { 1054 for (;i<y.length;i++) 1055 if (y[i]) 1056 return 0; 1057 } 1058 return 1; 1059} 1060 1061//is the bigInt x equal to zero? 1062function isZero(x) { 1063 var i; 1064 for (i=0;i<x.length;i++) 1065 if (x[i]) 1066 return 0; 1067 return 1; 1068} 1069 1070//convert a bigInt into a string in a given base, from base 2 up to base 95. 1071//Base -1 prints the contents of the array representing the number. 1072function bigInt2str(x,base) { 1073 var i,t,s=""; 1074 1075 if (s6.length!=x.length) 1076 s6=dup(x); 1077 else 1078 copy_(s6,x); 1079 1080 if (base==-1) { //return the list of array contents 1081 for (i=x.length-1;i>0;i--) 1082 s+=x[i]+','; 1083 s+=x[0]; 1084 } 1085 else { //return it in the given base 1086 while (!isZero(s6)) { 1087 t=divInt_(s6,base); //t=s6 % base; s6=floor(s6/base); 1088 s=digitsStr.substring(t,t+1)+s; 1089 } 1090 } 1091 if (s.length==0) 1092 s="0"; 1093 return s; 1094} 1095 1096//returns a duplicate of bigInt x 1097function dup(x) { 1098 var i; 1099 buff=new Array(x.length); 1100 copy_(buff,x); 1101 return buff; 1102} 1103 1104//do x=y on bigInts x and y. x must be an array at least as big as y (not counting the leading zeros in y). 1105function copy_(x,y) { 1106 var i; 1107 var k=x.length<y.length ? x.length : y.length; 1108 for (i=0;i<k;i++) 1109 x[i]=y[i]; 1110 for (i=k;i<x.length;i++) 1111 x[i]=0; 1112} 1113 1114//do x=y on bigInt x and integer y. 1115function copyInt_(x,n) { 1116 var i,c; 1117 for (c=n,i=0;i<x.length;i++) { 1118 x[i]=c & mask; 1119 c>>=bpe; 1120 } 1121} 1122 1123//do x=x+n where x is a bigInt and n is an integer. 1124//x must be large enough to hold the result. 1125function addInt_(x,n) { 1126 var i,k,c,b; 1127 x[0]+=n; 1128 k=x.length; 1129 c=0; 1130 for (i=0;i<k;i++) { 1131 c+=x[i]; 1132 b=0; 1133 if (c<0) { 1134 b=-(c>>bpe); 1135 c+=b*radix; 1136 } 1137 x[i]=c & mask; 1138 c=(c>>bpe)-b; 1139 if (!c) return; //stop carrying as soon as the carry is zero 1140 } 1141} 1142 1143//right shift bigInt x by n bits. 0 <= n < bpe. 1144function rightShift_(x,n) { 1145 var i; 1146 var k=Math.floor(n/bpe); 1147 if (k) { 1148 for (i=0;i<x.length-k;i++) //right shift x by k elements 1149 x[i]=x[i+k]; 1150 for (;i<x.length;i++) 1151 x[i]=0; 1152 n%=bpe; 1153 } 1154 for (i=0;i<x.length-1;i++) { 1155 x[i]=mask & ((x[i+1]<<(bpe-n)) | (x[i]>>n)); 1156 } 1157 x[i]>>=n; 1158} 1159 1160//do x=floor(|x|/2)*sgn(x) for bigInt x in 2's complement 1161function halve_(x) { 1162 var i; 1163 for (i=0;i<x.length-1;i++) { 1164 x[i]=mask & ((x[i+1]<<(bpe-1)) | (x[i]>>1)); 1165 } 1166 x[i]=(x[i]>>1) | (x[i] & (radix>>1)); //most significant bit stays the same 1167} 1168 1169//left shift bigInt x by n bits. 1170function leftShift_(x,n) { 1171 var i; 1172 var k=Math.floor(n/bpe); 1173 if (k) { 1174 for (i=x.length; i>=k; i--) //left shift x by k elements 1175 x[i]=x[i-k]; 1176 for (;i>=0;i--) 1177 x[i]=0; 1178 n%=bpe; 1179 } 1180 if (!n) 1181 return; 1182 for (i=x.length-1;i>0;i--) { 1183 x[i]=mask & ((x[i]<<n) | (x[i-1]>>(bpe-n))); 1184 } 1185 x[i]=mask & (x[i]<<n); 1186} 1187 1188//do x=x*n where x is a bigInt and n is an integer. 1189//x must be large enough to hold the result. 1190function multInt_(x,n) { 1191 var i,k,c,b; 1192 if (!n) 1193 return; 1194 k=x.length; 1195 c=0; 1196 for (i=0;i<k;i++) { 1197 c+=x[i]*n; 1198 b=0; 1199 if (c<0) { 1200 b=-(c>>bpe); 1201 c+=b*radix; 1202 } 1203 x[i]=c & mask; 1204 c=(c>>bpe)-b; 1205 } 1206} 1207 1208//do x=floor(x/n) for bigInt x and integer n, and return the remainder 1209function divInt_(x,n) { 1210 var i,r=0,s; 1211 for (i=x.length-1;i>=0;i--) { 1212 s=r*radix+x[i]; 1213 x[i]=Math.floor(s/n); 1214 r=s%n; 1215 } 1216 return r; 1217} 1218 1219//do the linear combination x=a*x+b*y for bigInts x and y, and integers a and b. 1220//x must be large enough to hold the answer. 1221function linComb_(x,y,a,b) { 1222 var i,c,k,kk; 1223 k=x.length<y.length ? x.length : y.length; 1224 kk=x.length; 1225 for (c=0,i=0;i<k;i++) { 1226 c+=a*x[i]+b*y[i]; 1227 x[i]=c & mask; 1228 c>>=bpe; 1229 } 1230 for (i=k;i<kk;i++) { 1231 c+=a*x[i]; 1232 x[i]=c & mask; 1233 c>>=bpe; 1234 } 1235} 1236 1237//do the linear combination x=a*x+b*(y<<(ys*bpe)) for bigInts x and y, and integers a, b and ys. 1238//x must be large enough to hold the answer. 1239function linCombShift_(x,y,b,ys) { 1240 var i,c,k,kk; 1241 k=x.length<ys+y.length ? x.length : ys+y.length; 1242 kk=x.length; 1243 for (c=0,i=ys;i<k;i++) { 1244 c+=x[i]+b*y[i-ys]; 1245 x[i]=c & mask; 1246 c>>=bpe; 1247 } 1248 for (i=k;c && i<kk;i++) { 1249 c+=x[i]; 1250 x[i]=c & mask; 1251 c>>=bpe; 1252 } 1253} 1254 1255//do x=x+(y<<(ys*bpe)) for bigInts x and y, and integers a,b and ys. 1256//x must be large enough to hold the answer. 1257function addShift_(x,y,ys) { 1258 var i,c,k,kk; 1259 k=x.length<ys+y.length ? x.length : ys+y.length; 1260 kk=x.length; 1261 for (c=0,i=ys;i<k;i++) { 1262 c+=x[i]+y[i-ys]; 1263 x[i]=c & mask; 1264 c>>=bpe; 1265 } 1266 for (i=k;c && i<kk;i++) { 1267 c+=x[i]; 1268 x[i]=c & mask; 1269 c>>=bpe; 1270 } 1271} 1272 1273//do x=x-(y<<(ys*bpe)) for bigInts x and y, and integers a,b and ys. 1274//x must be large enough to hold the answer. 1275function subShift_(x,y,ys) { 1276 var i,c,k,kk; 1277 k=x.length<ys+y.length ? x.length : ys+y.length; 1278 kk=x.length; 1279 for (c=0,i=ys;i<k;i++) { 1280 c+=x[i]-y[i-ys]; 1281 x[i]=c & mask; 1282 c>>=bpe; 1283 } 1284 for (i=k;c && i<kk;i++) { 1285 c+=x[i]; 1286 x[i]=c & mask; 1287 c>>=bpe; 1288 } 1289} 1290 1291//do x=x-y for bigInts x and y. 1292//x must be large enough to hold the answer. 1293//negative answers will be 2s complement 1294function sub_(x,y) { 1295 var i,c,k,kk; 1296 k=x.length<y.length ? x.length : y.length; 1297 for (c=0,i=0;i<k;i++) { 1298 c+=x[i]-y[i]; 1299 x[i]=c & mask; 1300 c>>=bpe; 1301 } 1302 for (i=k;c && i<x.length;i++) { 1303 c+=x[i]; 1304 x[i]=c & mask; 1305 c>>=bpe; 1306 } 1307} 1308 1309//do x=x+y for bigInts x and y. 1310//x must be large enough to hold the answer. 1311function add_(x,y) { 1312 var i,c,k,kk; 1313 k=x.length<y.length ? x.length : y.length; 1314 for (c=0,i=0;i<k;i++) { 1315 c+=x[i]+y[i]; 1316 x[i]=c & mask; 1317 c>>=bpe; 1318 } 1319 for (i=k;c && i<x.length;i++) { 1320 c+=x[i]; 1321 x[i]=c & mask; 1322 c>>=bpe; 1323 } 1324} 1325 1326//do x=x*y for bigInts x and y. This is faster when y<x. 1327function mult_(x,y) { 1328 var i; 1329 if (ss.length!=2*x.length) 1330 ss=new Array(2*x.length); 1331 copyInt_(ss,0); 1332 for (i=0;i<y.length;i++) 1333 if (y[i]) 1334 linCombShift_(ss,x,y[i],i); //ss=1*ss+y[i]*(x<<(i*bpe)) 1335 copy_(x,ss); 1336} 1337 1338//do x=x mod n for bigInts x and n. 1339function mod_(x,n) { 1340 if (s4.length!=x.length) 1341 s4=dup(x); 1342 else 1343 copy_(s4,x); 1344 if (s5.length!=x.length) 1345 s5=dup(x); 1346 divide_(s4,n,s5,x); //x = remainder of s4 / n 1347} 1348 1349//do x=x*y mod n for bigInts x,y,n. 1350//for greater speed, let y<x. 1351function multMod_(x,y,n) { 1352 var i; 1353 if (s0.length!=2*x.length) 1354 s0=new Array(2*x.length); 1355 copyInt_(s0,0); 1356 for (i=0;i<y.length;i++) 1357 if (y[i]) 1358 linCombShift_(s0,x,y[i],i); //s0=1*s0+y[i]*(x<<(i*bpe)) 1359 mod_(s0,n); 1360 copy_(x,s0); 1361} 1362 1363//do x=x*x mod n for bigInts x,n. 1364function squareMod_(x,n) { 1365 var i,j,d,c,kx,kn,k; 1366 for (kx=x.length; kx>0 && !x[kx-1]; kx--); //ignore leading zeros in x 1367 k=kx>n.length ? 2*kx : 2*n.length; //k=# elements in the product, which is twice the elements in the larger of x and n 1368 if (s0.length!=k) 1369 s0=new Array(k); 1370 copyInt_(s0,0); 1371 for (i=0;i<kx;i++) { 1372 c=s0[2*i]+x[i]*x[i]; 1373 s0[2*i]=c & mask; 1374 c>>=bpe; 1375 for (j=i+1;j<kx;j++) { 1376 c=s0[i+j]+2*x[i]*x[j]+c; 1377 s0[i+j]=(c & mask); 1378 c>>=bpe; 1379 } 1380 s0[i+kx]=c; 1381 } 1382 mod_(s0,n); 1383 copy_(x,s0); 1384} 1385 1386//return x with exactly k leading zero elements 1387function trim(x,k) { 1388 var i,y; 1389 for (i=x.length; i>0 && !x[i-1]; i--); 1390 y=new Array(i+k); 1391 copy_(y,x); 1392 return y; 1393} 1394 1395//do x=x**y mod n, where x,y,n are bigInts and ** is exponentiation. 0**0=1. 1396//this is faster when n is odd. x usually needs to have as many elements as n. 1397function powMod_(x,y,n) { 1398 var k1,k2,kn,np; 1399 if(s7.length!=n.length) 1400 s7=dup(n); 1401 1402 //for even modulus, use a simple square-and-multiply algorithm, 1403 //rather than using the more complex Montgomery algorithm. 1404 if ((n[0]&1)==0) { 1405 copy_(s7,x); 1406 copyInt_(x,1); 1407 while(!equalsInt(y,0)) { 1408 if (y[0]&1) 1409 multMod_(x,s7,n); 1410 divInt_(y,2); 1411 squareMod_(s7,n); 1412 } 1413 return; 1414 } 1415 1416 //calculate np from n for the Montgomery multiplications 1417 copyInt_(s7,0); 1418 for (kn=n.length;kn>0 && !n[kn-1];kn--); 1419 np=radix-inverseModInt(modInt(n,radix),radix); 1420 s7[kn]=1; 1421 multMod_(x ,s7,n); // x = x * 2**(kn*bp) mod n 1422 1423 if (s3.length!=x.length) 1424 s3=dup(x); 1425 else 1426 copy_(s3,x); 1427 1428 for (k1=y.length-1;k1>0 & !y[k1]; k1--); //k1=first nonzero element of y 1429 if (y[k1]==0) { //anything to the 0th power is 1 1430 copyInt_(x,1); 1431 return; 1432 } 1433 for (k2=1<<(bpe-1);k2 && !(y[k1] & k2); k2>>=1); //k2=position of first 1 bit in y[k1] 1434 for (;;) { 1435 if (!(k2>>=1)) { //look at next bit of y 1436 k1--; 1437 if (k1<0) { 1438 mont_(x,one,n,np); 1439 return; 1440 } 1441 k2=1<<(bpe-1); 1442 } 1443 mont_(x,x,n,np); 1444 1445 if (k2 & y[k1]) //if next bit is a 1 1446 mont_(x,s3,n,np); 1447 } 1448} 1449 1450 1451//do x=x*y*Ri mod n for bigInts x,y,n, 1452// where Ri = 2**(-kn*bpe) mod n, and kn is the 1453// number of elements in the n array, not 1454// counting leading zeros. 1455//x array must have at least as many elemnts as the n array 1456//It's OK if x and y are the same variable. 1457//must have: 1458// x,y < n 1459// n is odd 1460// np = -(n^(-1)) mod radix 1461function mont_(x,y,n,np) { 1462 var i,j,c,ui,t,ks; 1463 var kn=n.length; 1464 var ky=y.length; 1465 1466 if (sa.length!=kn) 1467 sa=new Array(kn); 1468 1469 copyInt_(sa,0); 1470 1471 for (;kn>0 && n[kn-1]==0;kn--); //ignore leading zeros of n 1472 for (;ky>0 && y[ky-1]==0;ky--); //ignore leading zeros of y 1473 ks=sa.length-1; //sa will never have more than this many nonzero elements. 1474 1475 //the following loop consumes 95% of the runtime for randTruePrime_() and powMod_() for large numbers 1476 for (i=0; i<kn; i++) { 1477 t=sa[0]+x[i]*y[0]; 1478 ui=((t & mask) * np) & mask; //the inner "& mask" was needed on Safari (but not MSIE) at one time 1479 c=(t+ui*n[0]) >> bpe; 1480 t=x[i]; 1481 1482 //do sa=(sa+x[i]*y+ui*n)/b where b=2**bpe. Loop is unrolled 5-fold for speed 1483 j=1; 1484 for (;j<ky-4;) { c+=sa[j]+ui*n[j]+t*y[j]; sa[j-1]=c & mask; c>>=bpe; j++; 1485 c+=sa[j]+ui*n[j]+t*y[j]; sa[j-1]=c & mask; c>>=bpe; j++; 1486 c+=sa[j]+ui*n[j]+t*y[j]; sa[j-1]=c & mask; c>>=bpe; j++; 1487 c+=sa[j]+ui*n[j]+t*y[j]; sa[j-1]=c & mask; c>>=bpe; j++; 1488 c+=sa[j]+ui*n[j]+t*y[j]; sa[j-1]=c & mask; c>>=bpe; j++; } 1489 for (;j<ky;) { c+=sa[j]+ui*n[j]+t*y[j]; sa[j-1]=c & mask; c>>=bpe; j++; } 1490 for (;j<kn-4;) { c+=sa[j]+ui*n[j]; sa[j-1]=c & mask; c>>=bpe; j++; 1491 c+=sa[j]+ui*n[j]; sa[j-1]=c & mask; c>>=bpe; j++; 1492 c+=sa[j]+ui*n[j]; sa[j-1]=c & mask; c>>=bpe; j++; 1493 c+=sa[j]+ui*n[j]; sa[j-1]=c & mask; c>>=bpe; j++; 1494 c+=sa[j]+ui*n[j]; sa[j-1]=c & mask; c>>=bpe; j++; } 1495 for (;j<kn;) { c+=sa[j]+ui*n[j]; sa[j-1]=c & mask; c>>=bpe; j++; } 1496 for (;j<ks;) { c+=sa[j]; sa[j-1]=c & mask; c>>=bpe; j++; } 1497 sa[j-1]=c & mask; 1498 } 1499 1500 if (!greater(n,sa)) 1501 sub_(sa,n); 1502 copy_(x,sa); 1503} 1504 1505
Line numbers count LF bytes from the start of the resource, as the search results do. Vendor segments are library code the classifier recognised; they are stored but not indexed. Bytes are shown as Latin1 characters, one per byte.