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nbtheory.h

    00001 // nbtheory.h - written and placed in the public domain by Wei Dai
    00002
    00003 #ifndef CRYPTOPP_NBTHEORY_H
    00004 #define CRYPTOPP_NBTHEORY_H
    00005
    00006 #include "integer.h"
    00007 #include "algparam.h"
    00008
    00009 NAMESPACE_BEGIN(CryptoPP)
    00010
    00011 // obtain pointer to small prime table and get its size
    00012 CRYPTOPP_DLL const word16 * CRYPTOPP_API GetPrimeTable(unsigned int &size);
    00013
    00014 // ************ primality testing ****************
    00015
    00016 // generate a provable prime
    00017 CRYPTOPP_DLL Integer CRYPTOPP_API MaurerProvablePrime(RandomNumberGenerator &rng, unsigned int bits);
    00018 CRYPTOPP_DLL Integer CRYPTOPP_API MihailescuProvablePrime(RandomNumberGenerator &rng, unsigned int bits);
    00019
    00020 CRYPTOPP_DLL bool CRYPTOPP_API IsSmallPrime(const Integer &p);
    00021
    00022 // returns true if p is divisible by some prime less than bound
    00023 // bound not be greater than the largest entry in the prime table
    00024 CRYPTOPP_DLL bool CRYPTOPP_API TrialDivision(const Integer &p, unsigned bound);
    00025
    00026 // returns true if p is NOT divisible by small primes
    00027 CRYPTOPP_DLL bool CRYPTOPP_API SmallDivisorsTest(const Integer &p);
    00028
    00029 // These is no reason to use these two, use the ones below instead
    00030 CRYPTOPP_DLL bool CRYPTOPP_API IsFermatProbablePrime(const Integer &n, const Integer &b);
    00031 CRYPTOPP_DLL bool CRYPTOPP_API IsLucasProbablePrime(const Integer &n);
    00032
    00033 CRYPTOPP_DLL bool CRYPTOPP_API IsStrongProbablePrime(const Integer &n, const Integer &b);
    00034 CRYPTOPP_DLL bool CRYPTOPP_API IsStrongLucasProbablePrime(const Integer &n);
    00035
    00036 // Rabin-Miller primality test, i.e. repeating the strong probable prime test
    00037 // for several rounds with random bases
    00038 CRYPTOPP_DLL bool CRYPTOPP_API RabinMillerTest(RandomNumberGenerator &rng, const Integer &w, unsigned int rounds);
    00039
    00040 // primality test, used to generate primes
    00041 CRYPTOPP_DLL bool CRYPTOPP_API IsPrime(const Integer &p);
    00042
    00043 // more reliable than IsPrime(), used to verify primes generated by others
    00044 CRYPTOPP_DLL bool CRYPTOPP_API VerifyPrime(RandomNumberGenerator &rng, const Integer &p, unsigned int level = 1);
    00045
    00046 class CRYPTOPP_DLL PrimeSelector
    00047 {
    00048 public:
    00049     const PrimeSelector *GetSelectorPointer() const {return this;}
    00050     virtual bool IsAcceptable(const Integer &candidate) const =0;
    00051 };
    00052
    00053 // use a fast sieve to find the first probable prime in {x | p<=x<=max and x%mod==equiv}
    00054 // returns true iff successful, value of p is undefined if no such prime exists
    00055 CRYPTOPP_DLL bool CRYPTOPP_API FirstPrime(Integer &p, const Integer &max, const Integer &equiv, const Integer &mod, const PrimeSelector *pSelector);
    00056
    00057 CRYPTOPP_DLL unsigned int CRYPTOPP_API PrimeSearchInterval(const Integer &max);
    00058
    00059 CRYPTOPP_DLL AlgorithmParameters CRYPTOPP_API MakeParametersForTwoPrimesOfEqualSize(unsigned int productBitLength);
    00060
    00061 // ********** other number theoretic functions ************
    00062
    00063 inline Integer GCD(const Integer &a, const Integer &b)
    00064     {return Integer::Gcd(a,b);}
    00065 inline bool RelativelyPrime(const Integer &a, const Integer &b)
    00066     {return Integer::Gcd(a,b) == Integer::One();}
    00067 inline Integer LCM(const Integer &a, const Integer &b)
    00068     {return a/Integer::Gcd(a,b)*b;}
    00069 inline Integer EuclideanMultiplicativeInverse(const Integer &a, const Integer &b)
    00070     {return a.InverseMod(b);}
    00071
    00072 // use Chinese Remainder Theorem to calculate x given x mod p and x mod q, and u = inverse of p mod q
    00073 CRYPTOPP_DLL Integer CRYPTOPP_API CRT(const Integer &xp, const Integer &p, const Integer &xq, const Integer &q, const Integer &u);
    00074
    00075 // if b is prime, then Jacobi(a, b) returns 0 if a%b==0, 1 if a is quadratic residue mod b, -1 otherwise
    00076 // check a number theory book for what Jacobi symbol means when b is not prime
    00077 CRYPTOPP_DLL int CRYPTOPP_API Jacobi(const Integer &a, const Integer &b);
    00078
    00079 // calculates the Lucas function V_e(p, 1) mod n
    00080 CRYPTOPP_DLL Integer CRYPTOPP_API Lucas(const Integer &e, const Integer &p, const Integer &n);
    00081 // calculates x such that m==Lucas(e, x, p*q), p q primes, u=inverse of p mod q
    00082 CRYPTOPP_DLL Integer CRYPTOPP_API InverseLucas(const Integer &e, const Integer &m, const Integer &p, const Integer &q, const Integer &u);
    00083
    00084 inline Integer ModularExponentiation(const Integer &a, const Integer &e, const Integer &m)
    00085     {return a_exp_b_mod_c(a, e, m);}
    00086 // returns x such that x*x%p == a, p prime
    00087 CRYPTOPP_DLL Integer CRYPTOPP_API ModularSquareRoot(const Integer &a, const Integer &p);
    00088 // returns x such that a==ModularExponentiation(x, e, p*q), p q primes,
    00089 // and e relatively prime to (p-1)*(q-1)
    00090 // dp=d%(p-1), dq=d%(q-1), (d is inverse of e mod (p-1)*(q-1))
    00091 // and u=inverse of p mod q
    00092 CRYPTOPP_DLL Integer CRYPTOPP_API ModularRoot(const Integer &a, const Integer &dp, const Integer &dq, const Integer &p, const Integer &q, const Integer &u);
    00093
    00094 // find r1 and r2 such that ax^2 + bx + c == 0 (mod p) for x in {r1, r2}, p prime
    00095 // returns true if solutions exist
    00096 CRYPTOPP_DLL bool CRYPTOPP_API SolveModularQuadraticEquation(Integer &r1, Integer &r2, const Integer &a, const Integer &b, const Integer &c, const Integer &p);
    00097
    00098 // returns log base 2 of estimated number of operations to calculate discrete log or factor a number
    00099 CRYPTOPP_DLL unsigned int CRYPTOPP_API DiscreteLogWorkFactor(unsigned int bitlength);
    00100 CRYPTOPP_DLL unsigned int CRYPTOPP_API FactoringWorkFactor(unsigned int bitlength);
    00101
    00102 // ********************************************************
    00103
    00104 //! generator of prime numbers of special forms
    00105 class CRYPTOPP_DLL PrimeAndGenerator
    00106 {
    00107 public:
    00108     PrimeAndGenerator() {}
    00109     // generate a random prime p of the form 2*q+delta, where delta is 1 or -1 and q is also prime
    00110     // Precondition: pbits > 5
    00111     // warning: this is slow, because primes of this form are harder to find
    00112     PrimeAndGenerator(signed int delta, RandomNumberGenerator &rng, unsigned int pbits)
    00113         {Generate(delta, rng, pbits, pbits-1);}
    00114     // generate a random prime p of the form 2*r*q+delta, where q is also prime
    00115     // Precondition: qbits > 4 && pbits > qbits
    00116     PrimeAndGenerator(signed int delta, RandomNumberGenerator &rng, unsigned int pbits, unsigned qbits)
    00117         {Generate(delta, rng, pbits, qbits);}
    00118
    00119     void Generate(signed int delta, RandomNumberGenerator &rng, unsigned int pbits, unsigned qbits);
    00120
    00121     const Integer& Prime() const {return p;}
    00122     const Integer& SubPrime() const {return q;}
    00123     const Integer& Generator() const {return g;}
    00124
    00125 private:
    00126     Integer p, q, g;
    00127 };
    00128
    00129 NAMESPACE_END
    00130
    00131 #endif