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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 // export a table of small primes
    00012 extern const unsigned int maxPrimeTableSize;
    00013 extern const word lastSmallPrime;
    00014 extern unsigned int primeTableSize;
    00015 extern word primeTable[];
    00016
    00017 // build up the table to maxPrimeTableSize
    00018 CRYPTOPP_DLL void BuildPrimeTable();
    00019
    00020 // ************ primality testing ****************
    00021
    00022 // generate a provable prime
    00023 CRYPTOPP_DLL Integer MaurerProvablePrime(RandomNumberGenerator &rng, unsigned int bits);
    00024 CRYPTOPP_DLL Integer MihailescuProvablePrime(RandomNumberGenerator &rng, unsigned int bits);
    00025
    00026 CRYPTOPP_DLL bool IsSmallPrime(const Integer &p);
    00027
    00028 // returns true if p is divisible by some prime less than bound
    00029 // bound not be greater than the largest entry in the prime table
    00030 CRYPTOPP_DLL bool TrialDivision(const Integer &p, unsigned bound);
    00031
    00032 // returns true if p is NOT divisible by small primes
    00033 CRYPTOPP_DLL bool SmallDivisorsTest(const Integer &p);
    00034
    00035 // These is no reason to use these two, use the ones below instead
    00036 CRYPTOPP_DLL bool IsFermatProbablePrime(const Integer &n, const Integer &b);
    00037 CRYPTOPP_DLL bool IsLucasProbablePrime(const Integer &n);
    00038
    00039 CRYPTOPP_DLL bool IsStrongProbablePrime(const Integer &n, const Integer &b);
    00040 CRYPTOPP_DLL bool IsStrongLucasProbablePrime(const Integer &n);
    00041
    00042 // Rabin-Miller primality test, i.e. repeating the strong probable prime test
    00043 // for several rounds with random bases
    00044 CRYPTOPP_DLL bool RabinMillerTest(RandomNumberGenerator &rng, const Integer &w, unsigned int rounds);
    00045
    00046 // primality test, used to generate primes
    00047 CRYPTOPP_DLL bool IsPrime(const Integer &p);
    00048
    00049 // more reliable than IsPrime(), used to verify primes generated by others
    00050 CRYPTOPP_DLL bool VerifyPrime(RandomNumberGenerator &rng, const Integer &p, unsigned int level = 1);
    00051
    00052 class PrimeSelector
    00053 {
    00054 public:
    00055         const PrimeSelector *GetSelectorPointer() const {return this;}
    00056         virtual bool IsAcceptable(const Integer &candidate) const =0;
    00057 };
    00058
    00059 // use a fast sieve to find the first probable prime in {x | p<=x<=max and x%mod==equiv}
    00060 // returns true iff successful, value of p is undefined if no such prime exists
    00061 CRYPTOPP_DLL bool FirstPrime(Integer &p, const Integer &max, const Integer &equiv, const Integer &mod, const PrimeSelector *pSelector);
    00062
    00063 CRYPTOPP_DLL unsigned int PrimeSearchInterval(const Integer &max);
    00064
    00065 CRYPTOPP_DLL AlgorithmParameters<AlgorithmParameters<AlgorithmParameters<NullNameValuePairs, Integer::RandomNumberType>, Integer>, Integer>
    00066         MakeParametersForTwoPrimesOfEqualSize(unsigned int productBitLength);
    00067
    00068 // ********** other number theoretic functions ************
    00069
    00070 inline Integer GCD(const Integer &a, const Integer &b)
    00071         {return Integer::Gcd(a,b);}
    00072 inline bool RelativelyPrime(const Integer &a, const Integer &b)
    00073         {return Integer::Gcd(a,b) == Integer::One();}
    00074 inline Integer LCM(const Integer &a, const Integer &b)
    00075         {return a/Integer::Gcd(a,b)*b;}
    00076 inline Integer EuclideanMultiplicativeInverse(const Integer &a, const Integer &b)
    00077         {return a.InverseMod(b);}
    00078
    00079 // use Chinese Remainder Theorem to calculate x given x mod p and x mod q
    00080 CRYPTOPP_DLL Integer CRT(const Integer &xp, const Integer &p, const Integer &xq, const Integer &q);
    00081 // use this one if u = inverse of p mod q has been precalculated
    00082 CRYPTOPP_DLL Integer CRT(const Integer &xp, const Integer &p, const Integer &xq, const Integer &q, const Integer &u);
    00083
    00084 // if b is prime, then Jacobi(a, b) returns 0 if a%b==0, 1 if a is quadratic residue mod b, -1 otherwise
    00085 // check a number theory book for what Jacobi symbol means when b is not prime
    00086 CRYPTOPP_DLL int Jacobi(const Integer &a, const Integer &b);
    00087
    00088 // calculates the Lucas function V_e(p, 1) mod n
    00089 CRYPTOPP_DLL Integer Lucas(const Integer &e, const Integer &p, const Integer &n);
    00090 // calculates x such that m==Lucas(e, x, p*q), p q primes
    00091 CRYPTOPP_DLL Integer InverseLucas(const Integer &e, const Integer &m, const Integer &p, const Integer &q);
    00092 // use this one if u=inverse of p mod q has been precalculated
    00093 CRYPTOPP_DLL Integer InverseLucas(const Integer &e, const Integer &m, const Integer &p, const Integer &q, const Integer &u);
    00094
    00095 inline Integer ModularExponentiation(const Integer &a, const Integer &e, const Integer &m)
    00096         {return a_exp_b_mod_c(a, e, m);}
    00097 // returns x such that x*x%p == a, p prime
    00098 CRYPTOPP_DLL Integer ModularSquareRoot(const Integer &a, const Integer &p);
    00099 // returns x such that a==ModularExponentiation(x, e, p*q), p q primes,
    00100 // and e relatively prime to (p-1)*(q-1)
    00101 CRYPTOPP_DLL Integer ModularRoot(const Integer &a, const Integer &e, const Integer &p, const Integer &q);
    00102 // use this one if dp=d%(p-1), dq=d%(q-1), (d is inverse of e mod (p-1)*(q-1))
    00103 // and u=inverse of p mod q have been precalculated
    00104 CRYPTOPP_DLL Integer ModularRoot(const Integer &a, const Integer &dp, const Integer &dq, const Integer &p, const Integer &q, const Integer &u);
    00105
    00106 // find r1 and r2 such that ax^2 + bx + c == 0 (mod p) for x in {r1, r2}, p prime
    00107 // returns true if solutions exist
    00108 CRYPTOPP_DLL bool SolveModularQuadraticEquation(Integer &r1, Integer &r2, const Integer &a, const Integer &b, const Integer &c, const Integer &p);
    00109
    00110 // returns log base 2 of estimated number of operations to calculate discrete log or factor a number
    00111 CRYPTOPP_DLL unsigned int DiscreteLogWorkFactor(unsigned int bitlength);
    00112 CRYPTOPP_DLL unsigned int FactoringWorkFactor(unsigned int bitlength);
    00113
    00114 // ********************************************************
    00115
    00116 //! generator of prime numbers of special forms
    00117 class CRYPTOPP_DLL PrimeAndGenerator
    00118 {
    00119 public:
    00120         PrimeAndGenerator() {}
    00121         // generate a random prime p of the form 2*q+delta, where delta is 1 or -1 and q is also prime
    00122         // Precondition: pbits > 5
    00123         // warning: this is slow, because primes of this form are harder to find
    00124         PrimeAndGenerator(signed int delta, RandomNumberGenerator &rng, unsigned int pbits)
    00125                 {Generate(delta, rng, pbits, pbits-1);}
    00126         // generate a random prime p of the form 2*r*q+delta, where q is also prime
    00127         // Precondition: qbits > 4 && pbits > qbits
    00128         PrimeAndGenerator(signed int delta, RandomNumberGenerator &rng, unsigned int pbits, unsigned qbits)
    00129                 {Generate(delta, rng, pbits, qbits);}
    00130
    00131         void Generate(signed int delta, RandomNumberGenerator &rng, unsigned int pbits, unsigned qbits);
    00132
    00133         const Integer& Prime() const {return p;}
    00134         const Integer& SubPrime() const {return q;}
    00135         const Integer& Generator() const {return g;}
    00136
    00137 private:
    00138         Integer p, q, g;
    00139 };
    00140
    00141 NAMESPACE_END
    00142
    00143 #endif