Crypto++
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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