1// nbtheory.h - written and placed in the public domain by Wei Dai 2 3#ifndef CRYPTOPP_NBTHEORY_H 4#define CRYPTOPP_NBTHEORY_H 5 6#include "integer.h" 7#include "algparam.h" 8 9NAMESPACE_BEGIN(CryptoPP) 10 11// obtain pointer to small prime table and get its size 12CRYPTOPP_DLL const word16 * CRYPTOPP_API GetPrimeTable(unsigned int &size); 13 14// ************ primality testing **************** 15 16// generate a provable prime 17CRYPTOPP_DLL Integer CRYPTOPP_API MaurerProvablePrime(RandomNumberGenerator &rng, unsigned int bits); 18CRYPTOPP_DLL Integer CRYPTOPP_API MihailescuProvablePrime(RandomNumberGenerator &rng, unsigned int bits); 19 20CRYPTOPP_DLL bool CRYPTOPP_API IsSmallPrime(const Integer &p); 21 22// returns true if p is divisible by some prime less than bound 23// bound not be greater than the largest entry in the prime table 24CRYPTOPP_DLL bool CRYPTOPP_API TrialDivision(const Integer &p, unsigned bound); 25 26// returns true if p is NOT divisible by small primes 27CRYPTOPP_DLL bool CRYPTOPP_API SmallDivisorsTest(const Integer &p); 28 29// These is no reason to use these two, use the ones below instead 30CRYPTOPP_DLL bool CRYPTOPP_API IsFermatProbablePrime(const Integer &n, const Integer &b); 31CRYPTOPP_DLL bool CRYPTOPP_API IsLucasProbablePrime(const Integer &n); 32 33CRYPTOPP_DLL bool CRYPTOPP_API IsStrongProbablePrime(const Integer &n, const Integer &b); 34CRYPTOPP_DLL bool CRYPTOPP_API IsStrongLucasProbablePrime(const Integer &n); 35 36// Rabin-Miller primality test, i.e. repeating the strong probable prime test 37// for several rounds with random bases 38CRYPTOPP_DLL bool CRYPTOPP_API RabinMillerTest(RandomNumberGenerator &rng, const Integer &w, unsigned int rounds); 39 40// primality test, used to generate primes 41CRYPTOPP_DLL bool CRYPTOPP_API IsPrime(const Integer &p); 42 43// more reliable than IsPrime(), used to verify primes generated by others 44CRYPTOPP_DLL bool CRYPTOPP_API VerifyPrime(RandomNumberGenerator &rng, const Integer &p, unsigned int level = 1); 45 46class CRYPTOPP_DLL PrimeSelector 47{ 48public: 49 const PrimeSelector *GetSelectorPointer() const {return this;} 50 virtual bool IsAcceptable(const Integer &candidate) const =0; 51}; 52 53// use a fast sieve to find the first probable prime in {x | p<=x<=max and x%mod==equiv} 54// returns true iff successful, value of p is undefined if no such prime exists 55CRYPTOPP_DLL bool CRYPTOPP_API FirstPrime(Integer &p, const Integer &max, const Integer &equiv, const Integer &mod, const PrimeSelector *pSelector); 56 57CRYPTOPP_DLL unsigned int CRYPTOPP_API PrimeSearchInterval(const Integer &max); 58 59CRYPTOPP_DLL AlgorithmParameters CRYPTOPP_API MakeParametersForTwoPrimesOfEqualSize(unsigned int productBitLength); 60 61// ********** other number theoretic functions ************ 62 63inline Integer GCD(const Integer &a, const Integer &b) 64 {return Integer::Gcd(a,b);} 65inline bool RelativelyPrime(const Integer &a, const Integer &b) 66 {return Integer::Gcd(a,b) == Integer::One();} 67inline Integer LCM(const Integer &a, const Integer &b) 68 {return a/Integer::Gcd(a,b)*b;} 69inline Integer EuclideanMultiplicativeInverse(const Integer &a, const Integer &b) 70 {return a.InverseMod(b);} 71 72// use Chinese Remainder Theorem to calculate x given x mod p and x mod q, and u = inverse of p mod q 73CRYPTOPP_DLL Integer CRYPTOPP_API CRT(const Integer &xp, const Integer &p, const Integer &xq, const Integer &q, const Integer &u); 74 75// if b is prime, then Jacobi(a, b) returns 0 if a%b==0, 1 if a is quadratic residue mod b, -1 otherwise 76// check a number theory book for what Jacobi symbol means when b is not prime 77CRYPTOPP_DLL int CRYPTOPP_API Jacobi(const Integer &a, const Integer &b); 78 79// calculates the Lucas function V_e(p, 1) mod n 80CRYPTOPP_DLL Integer CRYPTOPP_API Lucas(const Integer &e, const Integer &p, const Integer &n); 81// calculates x such that m==Lucas(e, x, p*q), p q primes, u=inverse of p mod q 82CRYPTOPP_DLL Integer CRYPTOPP_API InverseLucas(const Integer &e, const Integer &m, const Integer &p, const Integer &q, const Integer &u); 83 84inline Integer ModularExponentiation(const Integer &a, const Integer &e, const Integer &m) 85 {return a_exp_b_mod_c(a, e, m);} 86// returns x such that x*x%p == a, p prime 87CRYPTOPP_DLL Integer CRYPTOPP_API ModularSquareRoot(const Integer &a, const Integer &p); 88// returns x such that a==ModularExponentiation(x, e, p*q), p q primes, 89// and e relatively prime to (p-1)*(q-1) 90// dp=d%(p-1), dq=d%(q-1), (d is inverse of e mod (p-1)*(q-1)) 91// and u=inverse of p mod q 92CRYPTOPP_DLL Integer CRYPTOPP_API ModularRoot(const Integer &a, const Integer &dp, const Integer &dq, const Integer &p, const Integer &q, const Integer &u); 93 94// find r1 and r2 such that ax^2 + bx + c == 0 (mod p) for x in {r1, r2}, p prime 95// returns true if solutions exist 96CRYPTOPP_DLL bool CRYPTOPP_API SolveModularQuadraticEquation(Integer &r1, Integer &r2, const Integer &a, const Integer &b, const Integer &c, const Integer &p); 97 98// returns log base 2 of estimated number of operations to calculate discrete log or factor a number 99CRYPTOPP_DLL unsigned int CRYPTOPP_API DiscreteLogWorkFactor(unsigned int bitlength); 100CRYPTOPP_DLL unsigned int CRYPTOPP_API FactoringWorkFactor(unsigned int bitlength); 101 102// ******************************************************** 103 104//! generator of prime numbers of special forms 105class CRYPTOPP_DLL PrimeAndGenerator 106{ 107public: 108 PrimeAndGenerator() {} 109 // generate a random prime p of the form 2*q+delta, where delta is 1 or -1 and q is also prime 110 // Precondition: pbits > 5 111 // warning: this is slow, because primes of this form are harder to find 112 PrimeAndGenerator(signed int delta, RandomNumberGenerator &rng, unsigned int pbits) 113 {Generate(delta, rng, pbits, pbits-1);} 114 // generate a random prime p of the form 2*r*q+delta, where q is also prime 115 // Precondition: qbits > 4 && pbits > qbits 116 PrimeAndGenerator(signed int delta, RandomNumberGenerator &rng, unsigned int pbits, unsigned qbits) 117 {Generate(delta, rng, pbits, qbits);} 118 119 void Generate(signed int delta, RandomNumberGenerator &rng, unsigned int pbits, unsigned qbits); 120 121 const Integer& Prime() const {return p;} 122 const Integer& SubPrime() const {return q;} 123 const Integer& Generator() const {return g;} 124 125private: 126 Integer p, q, g; 127}; 128 129NAMESPACE_END 130 131#endif 132