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1/* 1/*
2 2
3Copyright 2008-2011 Clipperz Srl 3Copyright 2008-2011 Clipperz Srl
4 4
5This file is part of Clipperz's Javascript Crypto Library. 5This file is part of Clipperz Community Edition.
6Javascript Crypto Library provides web developers with an extensive 6Clipperz Community Edition is an online password manager.
7and efficient set of cryptographic functions. The library aims to
8obtain maximum execution speed while preserving modularity and
9reusability.
10For further information about its features and functionalities please 7For further information about its features and functionalities please
11refer to http://www.clipperz.com 8refer to http://www.clipperz.com.
12 9
13* Javascript Crypto Library is free software: you can redistribute 10* Clipperz Community Edition is free software: you can redistribute
14 it and/or modify it under the terms of the GNU Affero General Public 11 it and/or modify it under the terms of the GNU Affero General Public
15 License as published by the Free Software Foundation, either version 12 License as published by the Free Software Foundation, either version
16 3 of the License, or (at your option) any later version. 13 3 of the License, or (at your option) any later version.
17 14
18* Javascript Crypto Library is distributed in the hope that it will 15* Clipperz Community Edition is distributed in the hope that it will
19 be useful, but WITHOUT ANY WARRANTY; without even the implied 16 be useful, but WITHOUT ANY WARRANTY; without even the implied
20 warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. 17 warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.
21 See the GNU Affero General Public License for more details. 18 See the GNU Affero General Public License for more details.
22 19
23* You should have received a copy of the GNU Affero General Public 20* You should have received a copy of the GNU Affero General Public
24 License along with Javascript Crypto Library. If not, see 21 License along with Clipperz Community Edition. If not, see
25 <http://www.gnu.org/licenses/>. 22 <http://www.gnu.org/licenses/>.
26 23
27*/ 24*/
28 25
29if (typeof(Clipperz) == 'undefined') { Clipperz = {}; } 26if (typeof(Clipperz) == 'undefined') { Clipperz = {}; }
30if (typeof(Clipperz.Crypto) == 'undefined') { Clipperz.Crypto = {}; } 27if (typeof(Clipperz.Crypto) == 'undefined') { Clipperz.Crypto = {}; }
31 28
32//############################################################################# 29//#############################################################################
33 //Downloaded on March 05, 2007 from http://www.leemon.com/crypto/BigInt.js 30 //Downloaded on March 05, 2007 from http://www.leemon.com/crypto/BigInt.js
34//############################################################################# 31//#############################################################################
35 32
36 33
37//////////////////////////////////////////////////////////////////////////////////////// 34////////////////////////////////////////////////////////////////////////////////////////
38// Big Integer Library v. 5.0 35// Big Integer Library v. 5.0
39// Created 2000, last modified 2006 36// Created 2000, last modified 2006
40// Leemon Baird 37// Leemon Baird
41// www.leemon.com 38// www.leemon.com
42// 39//
43// This file is public domain. You can use it for any purpose without restriction. 40// This file is public domain. You can use it for any purpose without restriction.
44// I do not guarantee that it is correct, so use it at your own risk. If you use 41// I do not guarantee that it is correct, so use it at your own risk. If you use
45// it for something interesting, I'd appreciate hearing about it. If you find 42// it for something interesting, I'd appreciate hearing about it. If you find
46// any bugs or make any improvements, I'd appreciate hearing about those too. 43// any bugs or make any improvements, I'd appreciate hearing about those too.
47// It would also be nice if my name and address were left in the comments. 44// It would also be nice if my name and address were left in the comments.
48// But none of that is required. 45// But none of that is required.
49// 46//
50// This code defines a bigInt library for arbitrary-precision integers. 47// This code defines a bigInt library for arbitrary-precision integers.
51// A bigInt is an array of integers storing the value in chunks of bpe bits, 48// A bigInt is an array of integers storing the value in chunks of bpe bits,
52// little endian (buff[0] is the least significant word). 49// little endian (buff[0] is the least significant word).
53// Negative bigInts are stored two's complement. 50// Negative bigInts are stored two's complement.
54// Some functions assume their parameters have at least one leading zero element. 51// Some functions assume their parameters have at least one leading zero element.
55// Functions with an underscore at the end of the name have unpredictable behavior in case of overflow, 52// Functions with an underscore at the end of the name have unpredictable behavior in case of overflow,
56// so the caller must make sure overflow won't happen. 53// so the caller must make sure overflow won't happen.
57// For each function where a parameter is modified, that same 54// For each function where a parameter is modified, that same
58// variable must not be used as another argument too. 55// variable must not be used as another argument too.
59// So, you cannot square x by doing multMod_(x,x,n). 56// So, you cannot square x by doing multMod_(x,x,n).
60// You must use squareMod_(x,n) instead, or do y=dup(x); multMod_(x,y,n). 57// You must use squareMod_(x,n) instead, or do y=dup(x); multMod_(x,y,n).
61// 58//
62// These functions are designed to avoid frequent dynamic memory allocation in the inner loop. 59// These functions are designed to avoid frequent dynamic memory allocation in the inner loop.
63// For most functions, if it needs a BigInt as a local variable it will actually use 60// For most functions, if it needs a BigInt as a local variable it will actually use
64// a global, and will only allocate to it when it's not the right size. This ensures 61// a global, and will only allocate to it when it's not the right size. This ensures
65// that when a function is called repeatedly with same-sized parameters, it only allocates 62// that when a function is called repeatedly with same-sized parameters, it only allocates
66// memory on the first call. 63// memory on the first call.
67// 64//
68// Note that for cryptographic purposes, the calls to Math.random() must 65// Note that for cryptographic purposes, the calls to Math.random() must
69// be replaced with calls to a better pseudorandom number generator. 66// be replaced with calls to a better pseudorandom number generator.
70// 67//
71// In the following, "bigInt" means a bigInt with at least one leading zero element, 68// In the following, "bigInt" means a bigInt with at least one leading zero element,
72// and "integer" means a nonnegative integer less than radix. In some cases, integer 69// and "integer" means a nonnegative integer less than radix. In some cases, integer
73// can be negative. Negative bigInts are 2s complement. 70// can be negative. Negative bigInts are 2s complement.
74// 71//
75// The following functions do not modify their inputs, but dynamically allocate memory every time they are called: 72// The following functions do not modify their inputs, but dynamically allocate memory every time they are called:
76// 73//
77// function bigInt2str(x,base) //convert a bigInt into a string in a given base, from base 2 up to base 95 74// function bigInt2str(x,base) //convert a bigInt into a string in a given base, from base 2 up to base 95
78// function dup(x) //returns a copy of bigInt x 75// function dup(x) //returns a copy of bigInt x
79// function findPrimes(n) //return array of all primes less than integer n 76// function findPrimes(n) //return array of all primes less than integer n
80// function int2bigInt(t,n,m) //convert integer t to a bigInt with at least n bits and m array elements 77// function int2bigInt(t,n,m) //convert integer t to a bigInt with at least n bits and m array elements
81// function int2bigInt(s,b,n,m) //convert string s in base b to a bigInt with at least n bits and m array elements 78// function int2bigInt(s,b,n,m) //convert string s in base b to a bigInt with at least n bits and m array elements
82// function trim(x,k) //return a copy of x with exactly k leading zero elements 79// function trim(x,k) //return a copy of x with exactly k leading zero elements
83// 80//
84// The following functions do not modify their inputs, so there is never a problem with the result being too big: 81// The following functions do not modify their inputs, so there is never a problem with the result being too big:
85// 82//
86// function bitSize(x) //returns how many bits long the bigInt x is, not counting leading zeros 83// function bitSize(x) //returns how many bits long the bigInt x is, not counting leading zeros
87// function equals(x,y) //is the bigInt x equal to the bigint y? 84// function equals(x,y) //is the bigInt x equal to the bigint y?
88// function equalsInt(x,y) //is bigint x equal to integer y? 85// function equalsInt(x,y) //is bigint x equal to integer y?
89// function greater(x,y) //is x>y? (x and y are nonnegative bigInts) 86// function greater(x,y) //is x>y? (x and y are nonnegative bigInts)
90// function greaterShift(x,y,shift)//is (x <<(shift*bpe)) > y? 87// function greaterShift(x,y,shift)//is (x <<(shift*bpe)) > y?
91// function isZero(x) //is the bigInt x equal to zero? 88// function isZero(x) //is the bigInt x equal to zero?
92// 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)? 89// 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)?
93// function modInt(x,n) //return x mod n for bigInt x and integer n. 90// function modInt(x,n) //return x mod n for bigInt x and integer n.
94// function negative(x) //is bigInt x negative? 91// function negative(x) //is bigInt x negative?
95// 92//
96// The following functions do not modify their inputs, but allocate memory and call functions with underscores 93// The following functions do not modify their inputs, but allocate memory and call functions with underscores
97// 94//
98// function add(x,y) //return (x+y) for bigInts x and y. 95// function add(x,y) //return (x+y) for bigInts x and y.
99// function addInt(x,n) //return (x+n) where x is a bigInt and n is an integer. 96// function addInt(x,n) //return (x+n) where x is a bigInt and n is an integer.
100// function expand(x,n) //return a copy of x with at least n elements, adding leading zeros if needed 97// function expand(x,n) //return a copy of x with at least n elements, adding leading zeros if needed
101// function inverseMod(x,n) //return (x**(-1) mod n) for bigInts x and n. If no inverse exists, it returns null 98// function inverseMod(x,n) //return (x**(-1) mod n) for bigInts x and n. If no inverse exists, it returns null
102// function mod(x,n) //return a new bigInt equal to (x mod n) for bigInts x and n. 99// function mod(x,n) //return a new bigInt equal to (x mod n) for bigInts x and n.
103// function mult(x,y) //return x*y for bigInts x and y. This is faster when y<x. 100// function mult(x,y) //return x*y for bigInts x and y. This is faster when y<x.
104// function multMod(x,y,n) //return (x*y mod n) for bigInts x,y,n. For greater speed, let y<x. 101// function multMod(x,y,n) //return (x*y mod n) for bigInts x,y,n. For greater speed, let y<x.
105// 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. 102// 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.
106// function randTruePrime(k) //return a new, random, k-bit, true prime using Maurer's algorithm. 103// function randTruePrime(k) //return a new, random, k-bit, true prime using Maurer's algorithm.
107// function sub(x,y) //return (x-y) for bigInts x and y. Negative answers will be 2s complement 104// function sub(x,y) //return (x-y) for bigInts x and y. Negative answers will be 2s complement
108// 105//
109// The following functions write a bigInt result to one of the parameters, but 106// The following functions write a bigInt result to one of the parameters, but
110// the result is never bigger than the original, so there can't be overflow problems: 107// the result is never bigger than the original, so there can't be overflow problems:
111// 108//
112// function divInt_(x,n) //do x=floor(x/n) for bigInt x and integer n, and return the remainder 109// function divInt_(x,n) //do x=floor(x/n) for bigInt x and integer n, and return the remainder
113// function GCD_(x,y) //set x to the greatest common divisor of bigInts x and y, (y is destroyed). 110// function GCD_(x,y) //set x to the greatest common divisor of bigInts x and y, (y is destroyed).
114// function halve_(x) //do x=floor(|x|/2)*sgn(x) for bigInt x in 2's complement 111// function halve_(x) //do x=floor(|x|/2)*sgn(x) for bigInt x in 2's complement
115// function mod_(x,n) //do x=x mod n for bigInts x and n. 112// function mod_(x,n) //do x=x mod n for bigInts x and n.
116// function rightShift_(x,n) //right shift bigInt x by n bits. 0 <= n < bpe. 113// function rightShift_(x,n) //right shift bigInt x by n bits. 0 <= n < bpe.
117// 114//
118// The following functions write a bigInt result to one of the parameters. The caller is responsible for 115// The following functions write a bigInt result to one of the parameters. The caller is responsible for
119// ensuring it is large enough to hold the result. 116// ensuring it is large enough to hold the result.
120// 117//
121// function addInt_(x,n) //do x=x+n where x is a bigInt and n is an integer 118// function addInt_(x,n) //do x=x+n where x is a bigInt and n is an integer
122// function add_(x,y) //do x=x+y for bigInts x and y 119// function add_(x,y) //do x=x+y for bigInts x and y
123// function addShift_(x,y,ys) //do x=x+(y<<(ys*bpe)) 120// function addShift_(x,y,ys) //do x=x+(y<<(ys*bpe))
124// function copy_(x,y) //do x=y on bigInts x and y 121// function copy_(x,y) //do x=y on bigInts x and y
125// function copyInt_(x,n) //do x=n on bigInt x and integer n 122// function copyInt_(x,n) //do x=n on bigInt x and integer n
126// function carry_(x) //do carries and borrows so each element of the bigInt x fits in bpe bits. 123// function carry_(x) //do carries and borrows so each element of the bigInt x fits in bpe bits.
127// function divide_(x,y,q,r) //divide_ x by y giving quotient q and remainder r 124// function divide_(x,y,q,r) //divide_ x by y giving quotient q and remainder r
128// 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 125// 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
129// function inverseMod_(x,n) //do x=x**(-1) mod n, for bigInts x and n. Returns 1 (0) if inverse does (doesn't) exist 126// function inverseMod_(x,n) //do x=x**(-1) mod n, for bigInts x and n. Returns 1 (0) if inverse does (doesn't) exist
130// function inverseModInt_(x,n) //return x**(-1) mod n, for integers x and n. Return 0 if there is no inverse 127// function inverseModInt_(x,n) //return x**(-1) mod n, for integers x and n. Return 0 if there is no inverse
131// function leftShift_(x,n) //left shift bigInt x by n bits. n<bpe. 128// function leftShift_(x,n) //left shift bigInt x by n bits. n<bpe.
132// function linComb_(x,y,a,b) //do x=a*x+b*y for bigInts x and y and integers a and b 129// function linComb_(x,y,a,b) //do x=a*x+b*y for bigInts x and y and integers a and b
133// function linCombShift_(x,y,b,ys) //do x=x+b*(y<<(ys*bpe)) for bigInts x and y, and integers b and ys 130// function linCombShift_(x,y,b,ys) //do x=x+b*(y<<(ys*bpe)) for bigInts x and y, and integers b and ys
134// function mont_(x,y,n,np) //Montgomery multiplication (see comments where the function is defined) 131// function mont_(x,y,n,np) //Montgomery multiplication (see comments where the function is defined)
135// function mult_(x,y) //do x=x*y for bigInts x and y. 132// function mult_(x,y) //do x=x*y for bigInts x and y.
136// function multInt_(x,n) //do x=x*n where x is a bigInt and n is an integer. 133// function multInt_(x,n) //do x=x*n where x is a bigInt and n is an integer.
137// function multMod_(x,y,n) //do x=x*y mod n for bigInts x,y,n. 134// function multMod_(x,y,n) //do x=x*y mod n for bigInts x,y,n.
138// 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. 135// 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.
139// 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. 136// 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.
140// function randTruePrime_(ans,k) //do ans = a random k-bit true random prime (not just probable prime) with 1 in the msb. 137// function randTruePrime_(ans,k) //do ans = a random k-bit true random prime (not just probable prime) with 1 in the msb.
141// function squareMod_(x,n) //do x=x*x mod n for bigInts x,n 138// function squareMod_(x,n) //do x=x*x mod n for bigInts x,n
142// function sub_(x,y) //do x=x-y for bigInts x and y. Negative answers will be 2s complement. 139// function sub_(x,y) //do x=x-y for bigInts x and y. Negative answers will be 2s complement.
143// function subShift_(x,y,ys) //do x=x-(y<<(ys*bpe)). Negative answers will be 2s complement. 140// function subShift_(x,y,ys) //do x=x-(y<<(ys*bpe)). Negative answers will be 2s complement.
144// 141//
145// The following functions are based on algorithms from the _Handbook of Applied Cryptography_ 142// The following functions are based on algorithms from the _Handbook of Applied Cryptography_
146// powMod_() = algorithm 14.94, Montgomery exponentiation 143// powMod_() = algorithm 14.94, Montgomery exponentiation
147// eGCD_,inverseMod_() = algorithm 14.61, Binary extended GCD_ 144// eGCD_,inverseMod_() = algorithm 14.61, Binary extended GCD_
148// GCD_() = algorothm 14.57, Lehmer's algorithm 145// GCD_() = algorothm 14.57, Lehmer's algorithm
149// mont_() = algorithm 14.36, Montgomery multiplication 146// mont_() = algorithm 14.36, Montgomery multiplication
150// divide_() = algorithm 14.20 Multiple-precision division 147// divide_() = algorithm 14.20 Multiple-precision division
151// squareMod_() = algorithm 14.16 Multiple-precision squaring 148// squareMod_() = algorithm 14.16 Multiple-precision squaring
152// randTruePrime_() = algorithm 4.62, Maurer's algorithm 149// randTruePrime_() = algorithm 4.62, Maurer's algorithm
153// millerRabin() = algorithm 4.24, Miller-Rabin algorithm 150// millerRabin() = algorithm 4.24, Miller-Rabin algorithm
154// 151//
155// Profiling shows: 152// Profiling shows:
156// randTruePrime_() spends: 153// randTruePrime_() spends:
157// 10% of its time in calls to powMod_() 154// 10% of its time in calls to powMod_()
158// 85% of its time in calls to millerRabin() 155// 85% of its time in calls to millerRabin()
159// millerRabin() spends: 156// millerRabin() spends:
160// 99% of its time in calls to powMod_() (always with a base of 2) 157// 99% of its time in calls to powMod_() (always with a base of 2)
161// powMod_() spends: 158// powMod_() spends:
162// 94% of its time in calls to mont_() (almost always with x==y) 159// 94% of its time in calls to mont_() (almost always with x==y)
163// 160//
164// This suggests there are several ways to speed up this library slightly: 161// This suggests there are several ways to speed up this library slightly:
165// - convert powMod_ to use a Montgomery form of k-ary window (or maybe a Montgomery form of sliding window) 162// - convert powMod_ to use a Montgomery form of k-ary window (or maybe a Montgomery form of sliding window)
166// -- this should especially focus on being fast when raising 2 to a power mod n 163// -- this should especially focus on being fast when raising 2 to a power mod n
167// - convert randTruePrime_() to use a minimum r of 1/3 instead of 1/2 with the appropriate change to the test 164// - convert randTruePrime_() to use a minimum r of 1/3 instead of 1/2 with the appropriate change to the test
168// - tune the parameters in randTruePrime_(), including c, m, and recLimit 165// - tune the parameters in randTruePrime_(), including c, m, and recLimit
169// - speed up the single loop in mont_() that takes 95% of the runtime, perhaps by reducing checking 166// - speed up the single loop in mont_() that takes 95% of the runtime, perhaps by reducing checking
170// within the loop when all the parameters are the same length. 167// within the loop when all the parameters are the same length.
171// 168//
172// There are several ideas that look like they wouldn't help much at all: 169// There are several ideas that look like they wouldn't help much at all:
173// - replacing trial division in randTruePrime_() with a sieve (that speeds up something taking almost no time anyway) 170// - replacing trial division in randTruePrime_() with a sieve (that speeds up something taking almost no time anyway)
174// - 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) 171// - 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)
175// - speeding up mont_(x,y,n,np) when x==y by doing a non-modular, non-Montgomery square 172// - speeding up mont_(x,y,n,np) when x==y by doing a non-modular, non-Montgomery square
176// followed by a Montgomery reduction. The intermediate answer will be twice as long as x, so that 173// followed by a Montgomery reduction. The intermediate answer will be twice as long as x, so that
177// method would be slower. This is unfortunate because the code currently spends almost all of its time 174// method would be slower. This is unfortunate because the code currently spends almost all of its time
178// doing mont_(x,x,...), both for randTruePrime_() and powMod_(). A faster method for Montgomery squaring 175// doing mont_(x,x,...), both for randTruePrime_() and powMod_(). A faster method for Montgomery squaring
179// would have a large impact on the speed of randTruePrime_() and powMod_(). HAC has a couple of poorly-worded 176// would have a large impact on the speed of randTruePrime_() and powMod_(). HAC has a couple of poorly-worded
180// sentences that seem to imply it's faster to do a non-modular square followed by a single 177// sentences that seem to imply it's faster to do a non-modular square followed by a single
181// Montgomery reduction, but that's obviously wrong. 178// Montgomery reduction, but that's obviously wrong.
182//////////////////////////////////////////////////////////////////////////////////////// 179////////////////////////////////////////////////////////////////////////////////////////
183 180
184//globals 181//globals
185bpe=0; //bits stored per array element 182bpe=0; //bits stored per array element
186mask=0; //AND this with an array element to chop it down to bpe bits 183mask=0; //AND this with an array element to chop it down to bpe bits
187radix=mask+1; //equals 2^bpe. A single 1 bit to the left of the last bit of mask. 184radix=mask+1; //equals 2^bpe. A single 1 bit to the left of the last bit of mask.
188 185
189//the digits for converting to different bases 186//the digits for converting to different bases
190digitsStr='0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz_=!@#$%^&*()[]{}|;:,.<>/?`~ \\\'\"+-'; 187digitsStr='0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz_=!@#$%^&*()[]{}|;:,.<>/?`~ \\\'\"+-';
191 188
192//initialize the global variables 189//initialize the global variables
193for (bpe=0; (1<<(bpe+1)) > (1<<bpe); bpe++); //bpe=number of bits in the mantissa on this platform 190for (bpe=0; (1<<(bpe+1)) > (1<<bpe); bpe++); //bpe=number of bits in the mantissa on this platform
194bpe>>=1; //bpe=number of bits in one element of the array representing the bigInt 191bpe>>=1; //bpe=number of bits in one element of the array representing the bigInt
195mask=(1<<bpe)-1; //AND the mask with an integer to get its bpe least significant bits 192mask=(1<<bpe)-1; //AND the mask with an integer to get its bpe least significant bits
196radix=mask+1; //2^bpe. a single 1 bit to the left of the first bit of mask 193radix=mask+1; //2^bpe. a single 1 bit to the left of the first bit of mask
197one=int2bigInt(1,1,1); //constant used in powMod_() 194one=int2bigInt(1,1,1); //constant used in powMod_()
198 195
199//the following global variables are scratchpad memory to 196//the following global variables are scratchpad memory to
200//reduce dynamic memory allocation in the inner loop 197//reduce dynamic memory allocation in the inner loop
201t=new Array(0); 198t=new Array(0);
202ss=t; //used in mult_() 199ss=t; //used in mult_()
203s0=t; //used in multMod_(), squareMod_() 200s0=t; //used in multMod_(), squareMod_()
204s1=t; //used in powMod_(), multMod_(), squareMod_() 201s1=t; //used in powMod_(), multMod_(), squareMod_()
205s2=t; //used in powMod_(), multMod_() 202s2=t; //used in powMod_(), multMod_()
206s3=t; //used in powMod_() 203s3=t; //used in powMod_()
207s4=t; s5=t; //used in mod_() 204s4=t; s5=t; //used in mod_()
208s6=t; //used in bigInt2str() 205s6=t; //used in bigInt2str()
209s7=t; //used in powMod_() 206s7=t; //used in powMod_()
210T=t; //used in GCD_() 207T=t; //used in GCD_()
211sa=t; //used in mont_() 208sa=t; //used in mont_()
212mr_x1=t; mr_r=t; mr_a=t; //used in millerRabin() 209mr_x1=t; mr_r=t; mr_a=t; //used in millerRabin()
213eg_v=t; eg_u=t; eg_A=t; eg_B=t; eg_C=t; eg_D=t; //used in eGCD_(), inverseMod_() 210eg_v=t; eg_u=t; eg_A=t; eg_B=t; eg_C=t; eg_D=t; //used in eGCD_(), inverseMod_()
214md_q1=t; md_q2=t; md_q3=t; md_r=t; md_r1=t; md_r2=t; md_tt=t; //used in mod_() 211md_q1=t; md_q2=t; md_q3=t; md_r=t; md_r1=t; md_r2=t; md_tt=t; //used in mod_()
215 212
216primes=t; pows=t; s_i=t; s_i2=t; s_R=t; s_rm=t; s_q=t; s_n1=t; 213primes=t; pows=t; s_i=t; s_i2=t; s_R=t; s_rm=t; s_q=t; s_n1=t;