1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
|
/*
* Number Theory Functions
* (C) 1999-2007 Jack Lloyd
*
* Botan is released under the Simplified BSD License (see license.txt)
*/
#ifndef BOTAN_NUMBER_THEORY_H__
#define BOTAN_NUMBER_THEORY_H__
#include <botan/bigint.h>
#include <botan/pow_mod.h>
#include <botan/rng.h>
namespace Botan {
/**
* Fused multiply-add
* @param a an integer
* @param b an integer
* @param c an integer
* @return (a*b)+c
*/
BigInt BOTAN_DLL mul_add(const BigInt& a,
const BigInt& b,
const BigInt& c);
/**
* Fused subtract-multiply
* @param a an integer
* @param b an integer
* @param c an integer
* @return (a-b)*c
*/
BigInt BOTAN_DLL sub_mul(const BigInt& a,
const BigInt& b,
const BigInt& c);
/**
* Fused multiply-subtract
* @param a an integer
* @param b an integer
* @param c an integer
* @return (a*b)-c
*/
BigInt BOTAN_DLL mul_sub(const BigInt& a,
const BigInt& b,
const BigInt& c);
/**
* Return the absolute value
* @param n an integer
* @return absolute value of n
*/
inline BigInt abs(const BigInt& n) { return n.abs(); }
/**
* Compute the greatest common divisor
* @param x a positive integer
* @param y a positive integer
* @return gcd(x,y)
*/
BigInt BOTAN_DLL gcd(const BigInt& x, const BigInt& y);
/**
* Least common multiple
* @param x a positive integer
* @param y a positive integer
* @return z, smallest integer such that z % x == 0 and z % y == 0
*/
BigInt BOTAN_DLL lcm(const BigInt& x, const BigInt& y);
/**
* @param x an integer
* @return (x*x)
*/
BigInt BOTAN_DLL square(const BigInt& x);
/**
* Modular inversion
* @param x a positive integer
* @param modulus a positive integer
* @return y st (x*y) % modulus == 1 or 0 if no such value
* Not const time
*/
BigInt BOTAN_DLL inverse_mod(const BigInt& x,
const BigInt& modulus);
/**
* Const time modular inversion
* Requires the modulus be odd
*/
BigInt BOTAN_DLL ct_inverse_mod_odd_modulus(const BigInt& n, const BigInt& mod);
/**
* Return a^-1 * 2^k mod b
* Returns k, between n and 2n
* Not const time
*/
size_t BOTAN_DLL almost_montgomery_inverse(BigInt& result,
const BigInt& a,
const BigInt& b);
/**
* Call almost_montgomery_inverse and correct the result to a^-1 mod b
*/
BigInt BOTAN_DLL normalized_montgomery_inverse(const BigInt& a, const BigInt& b);
/**
* Compute the Jacobi symbol. If n is prime, this is equivalent
* to the Legendre symbol.
* @see http://mathworld.wolfram.com/JacobiSymbol.html
*
* @param a is a non-negative integer
* @param n is an odd integer > 1
* @return (n / m)
*/
s32bit BOTAN_DLL jacobi(const BigInt& a,
const BigInt& n);
/**
* Modular exponentation
* @param b an integer base
* @param x a positive exponent
* @param m a positive modulus
* @return (b^x) % m
*/
BigInt BOTAN_DLL power_mod(const BigInt& b,
const BigInt& x,
const BigInt& m);
/**
* Compute the square root of x modulo a prime using the
* Shanks-Tonnelli algorithm
*
* @param x the input
* @param p the prime
* @return y such that (y*y)%p == x, or -1 if no such integer
*/
BigInt BOTAN_DLL ressol(const BigInt& x, const BigInt& p);
/*
* Compute -input^-1 mod 2^MP_WORD_BITS. Returns zero if input
* is even. If input is odd, input and 2^n are relatively prime
* and an inverse exists.
*/
word BOTAN_DLL monty_inverse(word input);
/**
* @param x a positive integer
* @return count of the zero bits in x, or, equivalently, the largest
* value of n such that 2^n divides x evenly. Returns zero if
* n is less than or equal to zero.
*/
size_t BOTAN_DLL low_zero_bits(const BigInt& x);
/**
* Check for primality
* @param n a positive integer to test for primality
* @param rng a random number generator
* @param prob chance of false positive is bounded by 1/2**prob
* @param is_random true if n was randomly chosen by us
* @return true if all primality tests passed, otherwise false
*/
bool BOTAN_DLL is_prime(const BigInt& n,
RandomNumberGenerator& rng,
size_t prob = 56,
bool is_random = false);
inline bool quick_check_prime(const BigInt& n, RandomNumberGenerator& rng)
{ return is_prime(n, rng, 32); }
inline bool check_prime(const BigInt& n, RandomNumberGenerator& rng)
{ return is_prime(n, rng, 56); }
inline bool verify_prime(const BigInt& n, RandomNumberGenerator& rng)
{ return is_prime(n, rng, 80); }
/**
* Randomly generate a prime
* @param rng a random number generator
* @param bits how large the resulting prime should be in bits
* @param coprime a positive integer the result should be coprime to
* @param equiv a non-negative number that the result should be
equivalent to modulo equiv_mod
* @param equiv_mod the modulus equiv should be checked against
* @return random prime with the specified criteria
*/
BigInt BOTAN_DLL random_prime(RandomNumberGenerator& rng,
size_t bits, const BigInt& coprime = 1,
size_t equiv = 1, size_t equiv_mod = 2);
/**
* Return a 'safe' prime, of the form p=2*q+1 with q prime
* @param rng a random number generator
* @param bits is how long the resulting prime should be
* @return prime randomly chosen from safe primes of length bits
*/
BigInt BOTAN_DLL random_safe_prime(RandomNumberGenerator& rng,
size_t bits);
/**
* Generate DSA parameters using the FIPS 186 kosherizer
* @param rng a random number generator
* @param p_out where the prime p will be stored
* @param q_out where the prime q will be stored
* @param pbits how long p will be in bits
* @param qbits how long q will be in bits
* @return random seed used to generate this parameter set
*/
std::vector<byte> BOTAN_DLL
generate_dsa_primes(RandomNumberGenerator& rng,
BigInt& p_out, BigInt& q_out,
size_t pbits, size_t qbits);
/**
* Generate DSA parameters using the FIPS 186 kosherizer
* @param rng a random number generator
* @param p_out where the prime p will be stored
* @param q_out where the prime q will be stored
* @param pbits how long p will be in bits
* @param qbits how long q will be in bits
* @param seed the seed used to generate the parameters
* @return true if seed generated a valid DSA parameter set, otherwise
false. p_out and q_out are only valid if true was returned.
*/
bool BOTAN_DLL
generate_dsa_primes(RandomNumberGenerator& rng,
BigInt& p_out, BigInt& q_out,
size_t pbits, size_t qbits,
const std::vector<byte>& seed);
/**
* The size of the PRIMES[] array
*/
const size_t PRIME_TABLE_SIZE = 6541;
/**
* A const array of all primes less than 65535
*/
extern const u16bit BOTAN_DLL PRIMES[];
}
#endif
|
