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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 void BuildPrimeTable();
00019 
00020 // ************ primality testing ****************
00021 
00022 // generate a provable prime
00023 Integer MaurerProvablePrime(RandomNumberGenerator &rng, unsigned int bits);
00024 Integer MihailescuProvablePrime(RandomNumberGenerator &rng, unsigned int bits);
00025 
00026 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 bool TrialDivision(const Integer &p, unsigned bound);
00031 
00032 // returns true if p is NOT divisible by small primes
00033 bool SmallDivisorsTest(const Integer &p);
00034 
00035 // These is no reason to use these two, use the ones below instead
00036 bool IsFermatProbablePrime(const Integer &n, const Integer &b);
00037 bool IsLucasProbablePrime(const Integer &n);
00038 
00039 bool IsStrongProbablePrime(const Integer &n, const Integer &b);
00040 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 bool RabinMillerTest(RandomNumberGenerator &rng, const Integer &w, unsigned int rounds);
00045 
00046 // primality test, used to generate primes
00047 bool IsPrime(const Integer &p);
00048 
00049 // more reliable than IsPrime(), used to verify primes generated by others
00050 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 bool FirstPrime(Integer &p, const Integer &max, const Integer &equiv, const Integer &mod, const PrimeSelector *pSelector);
00062 
00063 unsigned int PrimeSearchInterval(const Integer &max);
00064 
00065 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 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 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 int Jacobi(const Integer &a, const Integer &b);
00087 
00088 // calculates the Lucas function V_e(p, 1) mod n
00089 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 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 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 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 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 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 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 unsigned int DiscreteLogWorkFactor(unsigned int bitlength);
00112 unsigned int FactoringWorkFactor(unsigned int bitlength);
00113 
00114 // ********************************************************
00115 
00116 //! generator of prime numbers of special forms
00117 class 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

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