174 lines
8.4 KiB
C++
174 lines
8.4 KiB
C++
// nbtheory.h - written and placed in the public domain by Wei Dai
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//! \file nbtheory.h
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//! \brief Classes and functions for number theoretic operations
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#ifndef CRYPTOPP_NBTHEORY_H
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#define CRYPTOPP_NBTHEORY_H
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#include "cryptlib.h"
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#include "integer.h"
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#include "algparam.h"
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NAMESPACE_BEGIN(CryptoPP)
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// obtain pointer to small prime table and get its size
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CRYPTOPP_DLL const word16 * CRYPTOPP_API GetPrimeTable(unsigned int &size);
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// ************ primality testing ****************
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//! \brief Generates a provable prime
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//! \param rng a RandomNumberGenerator to produce keying material
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//! \param bits the number of bits in the prime number
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//! \returns Integer() meeting Maurer's tests for primality
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CRYPTOPP_DLL Integer CRYPTOPP_API MaurerProvablePrime(RandomNumberGenerator &rng, unsigned int bits);
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//! \brief Generates a provable prime
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//! \param rng a RandomNumberGenerator to produce keying material
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//! \param bits the number of bits in the prime number
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//! \returns Integer() meeting Mihailescu's tests for primality
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//! \details Mihailescu's methods performs a search using algorithmic progressions.
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CRYPTOPP_DLL Integer CRYPTOPP_API MihailescuProvablePrime(RandomNumberGenerator &rng, unsigned int bits);
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//! \brief Tests whether a number is a small prime
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//! \param p a candidate prime to test
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//! \returns true if p is a small prime, false otherwise
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//! \details Internally, the library maintains a table fo the first 32719 prime numbers
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//! in sorted order. IsSmallPrime() searches the table and returns true if p is
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//! in the table.
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CRYPTOPP_DLL bool CRYPTOPP_API IsSmallPrime(const Integer &p);
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//!
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//! \returns true if p is divisible by some prime less than bound.
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//! \details TrialDivision() true if p is divisible by some prime less than bound. bound not be
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//! greater than the largest entry in the prime table, which is 32719.
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CRYPTOPP_DLL bool CRYPTOPP_API TrialDivision(const Integer &p, unsigned bound);
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// returns true if p is NOT divisible by small primes
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CRYPTOPP_DLL bool CRYPTOPP_API SmallDivisorsTest(const Integer &p);
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// These is no reason to use these two, use the ones below instead
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CRYPTOPP_DLL bool CRYPTOPP_API IsFermatProbablePrime(const Integer &n, const Integer &b);
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CRYPTOPP_DLL bool CRYPTOPP_API IsLucasProbablePrime(const Integer &n);
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CRYPTOPP_DLL bool CRYPTOPP_API IsStrongProbablePrime(const Integer &n, const Integer &b);
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CRYPTOPP_DLL bool CRYPTOPP_API IsStrongLucasProbablePrime(const Integer &n);
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// Rabin-Miller primality test, i.e. repeating the strong probable prime test
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// for several rounds with random bases
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CRYPTOPP_DLL bool CRYPTOPP_API RabinMillerTest(RandomNumberGenerator &rng, const Integer &w, unsigned int rounds);
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//! \brief Verifies a prime number
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//! \param p a candidate prime to test
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//! \returns true if p is a probable prime, false otherwise
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//! \details IsPrime() is suitable for testing candidate primes when creating them. Internally,
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//! IsPrime() utilizes SmallDivisorsTest(), IsStrongProbablePrime() and IsStrongLucasProbablePrime().
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CRYPTOPP_DLL bool CRYPTOPP_API IsPrime(const Integer &p);
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//! \brief Verifies a prime number
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//! \param rng a RandomNumberGenerator for randomized testing
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//! \param p a candidate prime to test
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//! \param level the level of thoroughness of testing
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//! \returns true if p is a strong probable prime, false otherwise
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//! \details VerifyPrime() is suitable for testing candidate primes created by others. Internally,
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//! VerifyPrime() utilizes IsPrime() and one-round RabinMillerTest(). If the candiate passes and
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//! level is greater than 1, then 10 round RabinMillerTest() primality testing is performed.
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CRYPTOPP_DLL bool CRYPTOPP_API VerifyPrime(RandomNumberGenerator &rng, const Integer &p, unsigned int level = 1);
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//! \class PrimeSelector
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//! \brief Application callback to signal suitability of a cabdidate prime
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class CRYPTOPP_DLL PrimeSelector
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{
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public:
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const PrimeSelector *GetSelectorPointer() const {return this;}
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virtual bool IsAcceptable(const Integer &candidate) const =0;
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};
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//! \brief Finds a random prime of special form
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//! \param p an Integer reference to receive the prime
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//! \param max the maximum value
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//! \param equiv the equivalence class based on the parameter mod
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//! \param mod the modulus used to reduce the equivalence class
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//! \param pSelector pointer to a PrimeSelector function for the application to signal suitability
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//! \returns true if and only if FirstPrime() finds a prime and returns the prime through p. If FirstPrime()
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//! returns false, then no such prime exists and the value of p is undefined
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//! \details FirstPrime() uses a fast sieve to find the first probable prime
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//! in <tt>{x | p<=x<=max and x%mod==equiv}</tt>
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CRYPTOPP_DLL bool CRYPTOPP_API FirstPrime(Integer &p, const Integer &max, const Integer &equiv, const Integer &mod, const PrimeSelector *pSelector);
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CRYPTOPP_DLL unsigned int CRYPTOPP_API PrimeSearchInterval(const Integer &max);
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CRYPTOPP_DLL AlgorithmParameters CRYPTOPP_API MakeParametersForTwoPrimesOfEqualSize(unsigned int productBitLength);
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// ********** other number theoretic functions ************
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inline Integer GCD(const Integer &a, const Integer &b)
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{return Integer::Gcd(a,b);}
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inline bool RelativelyPrime(const Integer &a, const Integer &b)
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{return Integer::Gcd(a,b) == Integer::One();}
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inline Integer LCM(const Integer &a, const Integer &b)
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{return a/Integer::Gcd(a,b)*b;}
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inline Integer EuclideanMultiplicativeInverse(const Integer &a, const Integer &b)
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{return a.InverseMod(b);}
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// use Chinese Remainder Theorem to calculate x given x mod p and x mod q, and u = inverse of p mod q
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CRYPTOPP_DLL Integer CRYPTOPP_API CRT(const Integer &xp, const Integer &p, const Integer &xq, const Integer &q, const Integer &u);
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// if b is prime, then Jacobi(a, b) returns 0 if a%b==0, 1 if a is quadratic residue mod b, -1 otherwise
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// check a number theory book for what Jacobi symbol means when b is not prime
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CRYPTOPP_DLL int CRYPTOPP_API Jacobi(const Integer &a, const Integer &b);
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// calculates the Lucas function V_e(p, 1) mod n
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CRYPTOPP_DLL Integer CRYPTOPP_API Lucas(const Integer &e, const Integer &p, const Integer &n);
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// calculates x such that m==Lucas(e, x, p*q), p q primes, u=inverse of p mod q
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CRYPTOPP_DLL Integer CRYPTOPP_API InverseLucas(const Integer &e, const Integer &m, const Integer &p, const Integer &q, const Integer &u);
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inline Integer ModularExponentiation(const Integer &a, const Integer &e, const Integer &m)
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{return a_exp_b_mod_c(a, e, m);}
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// returns x such that x*x%p == a, p prime
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CRYPTOPP_DLL Integer CRYPTOPP_API ModularSquareRoot(const Integer &a, const Integer &p);
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// returns x such that a==ModularExponentiation(x, e, p*q), p q primes,
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// and e relatively prime to (p-1)*(q-1)
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// dp=d%(p-1), dq=d%(q-1), (d is inverse of e mod (p-1)*(q-1))
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// and u=inverse of p mod q
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CRYPTOPP_DLL Integer CRYPTOPP_API ModularRoot(const Integer &a, const Integer &dp, const Integer &dq, const Integer &p, const Integer &q, const Integer &u);
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// find r1 and r2 such that ax^2 + bx + c == 0 (mod p) for x in {r1, r2}, p prime
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// returns true if solutions exist
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CRYPTOPP_DLL bool CRYPTOPP_API SolveModularQuadraticEquation(Integer &r1, Integer &r2, const Integer &a, const Integer &b, const Integer &c, const Integer &p);
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// returns log base 2 of estimated number of operations to calculate discrete log or factor a number
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CRYPTOPP_DLL unsigned int CRYPTOPP_API DiscreteLogWorkFactor(unsigned int bitlength);
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CRYPTOPP_DLL unsigned int CRYPTOPP_API FactoringWorkFactor(unsigned int bitlength);
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// ********************************************************
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//! generator of prime numbers of special forms
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class CRYPTOPP_DLL PrimeAndGenerator
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{
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public:
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PrimeAndGenerator() {}
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// generate a random prime p of the form 2*q+delta, where delta is 1 or -1 and q is also prime
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// Precondition: pbits > 5
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// warning: this is slow, because primes of this form are harder to find
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PrimeAndGenerator(signed int delta, RandomNumberGenerator &rng, unsigned int pbits)
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{Generate(delta, rng, pbits, pbits-1);}
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// generate a random prime p of the form 2*r*q+delta, where q is also prime
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// Precondition: qbits > 4 && pbits > qbits
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PrimeAndGenerator(signed int delta, RandomNumberGenerator &rng, unsigned int pbits, unsigned qbits)
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{Generate(delta, rng, pbits, qbits);}
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void Generate(signed int delta, RandomNumberGenerator &rng, unsigned int pbits, unsigned qbits);
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const Integer& Prime() const {return p;}
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const Integer& SubPrime() const {return q;}
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const Integer& Generator() const {return g;}
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private:
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Integer p, q, g;
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};
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NAMESPACE_END
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#endif
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