numthry.h

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/*
* Number Theory Functions
* (C) 1999-2007,2018 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>

namespace Botan {

class RandomNumberGenerator;

/**
* Fused multiply-add
* @param a an integer
* @param b an integer
* @param c an integer
* @return (a*b)+c
*/
BigInt BOTAN_PUBLIC_API(2,0) 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_PUBLIC_API(2,0) 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_PUBLIC_API(2,0) 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_PUBLIC_API(2,0) 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_PUBLIC_API(2,0) lcm(const BigInt& x, const BigInt& y);

/**
* @param x an integer
* @return (x*x)
*/
BigInt BOTAN_PUBLIC_API(2,0) 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_PUBLIC_API(2,0) inverse_mod(const BigInt& x,
                                         const BigInt& modulus);

/**
* Modular inversion using extended binary Euclidian algorithm
* @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_PUBLIC_API(2,5) inverse_euclid(const BigInt& x,
                                            const BigInt& modulus);

/**
* Const time modular inversion
* Requires the modulus be odd
*/
BigInt BOTAN_PUBLIC_API(2,0) 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_PUBLIC_API(2,0) 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_PUBLIC_API(2,0) 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)
*/
int32_t BOTAN_PUBLIC_API(2,0) 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_PUBLIC_API(2,0) 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_PUBLIC_API(2,0) ressol(const BigInt& x, const BigInt& p);

/*
* Compute -input^-1 mod 2^MP_WORD_BITS. Throws an exception if input
* is even. If input is odd, then input and 2^n are relatively prime
* and an inverse exists.
*/
word BOTAN_PUBLIC_API(2,0) 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_PUBLIC_API(2,0) 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_PUBLIC_API(2,0) is_prime(const BigInt& n,
                                    RandomNumberGenerator& rng,
                                    size_t prob = 64,
                                    bool is_random = false);

/**
* Test if the positive integer x is a perfect square ie if there
* exists some positive integer y st y*y == x
* See FIPS 186-4 sec C.4
* @return 0 if the integer is not a perfect square, otherwise
*         returns the positive y st y*y == x
*/
BigInt BOTAN_PUBLIC_API(2,8) is_perfect_square(const BigInt& x);

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 suitable for discrete logarithm parameters
* @param rng a random number generator
* @param bits how large the resulting prime should be in bits
* @param coprime a positive integer that (prime - 1) 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
* @param prob use test so false positive is bounded by 1/2**prob
* @return random prime with the specified criteria
*/
BigInt BOTAN_PUBLIC_API(2,0) random_prime(RandomNumberGenerator& rng,
                                          size_t bits,
                                          const BigInt& coprime = 0,
                                          size_t equiv = 1,
                                          size_t equiv_mod = 2,
                                          size_t prob = 128);

/**
* Generate a prime suitable for RSA p/q
* @param keygen_rng a random number generator
* @param prime_test_rng a random number generator
* @param bits how large the resulting prime should be in bits (must be >= 512)
* @param coprime a positive integer that (prime - 1) should be coprime to
* @param prob use test so false positive is bounded by 1/2**prob
* @return random prime with the specified criteria
*/
BigInt BOTAN_PUBLIC_API(2,7) generate_rsa_prime(RandomNumberGenerator& keygen_rng,
                                                RandomNumberGenerator& prime_test_rng,
                                                size_t bits,
                                                const BigInt& coprime,
                                                size_t prob = 128);

/**
* 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_PUBLIC_API(2,0) 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<uint8_t> BOTAN_PUBLIC_API(2,0)
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
* @param offset optional offset from seed to start searching at
* @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_PUBLIC_API(2,0)
generate_dsa_primes(RandomNumberGenerator& rng,
                    BigInt& p_out, BigInt& q_out,
                    size_t pbits, size_t qbits,
                    const std::vector<uint8_t>& seed,
                    size_t offset = 0);

/**
* The size of the PRIMES[] array
*/
const size_t PRIME_TABLE_SIZE = 6541;

/**
* A const array of all primes less than 65535
*/
extern const uint16_t BOTAN_PUBLIC_API(2,0) PRIMES[];

}

#endif