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

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