Crypto++
nbtheory.h
1 // nbtheory.h - written and placed in the public domain by Wei Dai
2 
3 #ifndef CRYPTOPP_NBTHEORY_H
4 #define CRYPTOPP_NBTHEORY_H
5 
6 #include "integer.h"
7 #include "algparam.h"
8 
9 NAMESPACE_BEGIN(CryptoPP)
10 
11 // obtain pointer to small prime table and get its size
12 CRYPTOPP_DLL const word16 * CRYPTOPP_API GetPrimeTable(unsigned int &size);
13 
14 // ************ primality testing ****************
15 
16 // generate a provable prime
17 CRYPTOPP_DLL Integer CRYPTOPP_API MaurerProvablePrime(RandomNumberGenerator &rng, unsigned int bits);
18 CRYPTOPP_DLL Integer CRYPTOPP_API MihailescuProvablePrime(RandomNumberGenerator &rng, unsigned int bits);
19 
20 CRYPTOPP_DLL bool CRYPTOPP_API IsSmallPrime(const Integer &p);
21 
22 // returns true if p is divisible by some prime less than bound
23 // bound not be greater than the largest entry in the prime table
24 CRYPTOPP_DLL bool CRYPTOPP_API TrialDivision(const Integer &p, unsigned bound);
25 
26 // returns true if p is NOT divisible by small primes
27 CRYPTOPP_DLL bool CRYPTOPP_API SmallDivisorsTest(const Integer &p);
28 
29 // These is no reason to use these two, use the ones below instead
30 CRYPTOPP_DLL bool CRYPTOPP_API IsFermatProbablePrime(const Integer &n, const Integer &b);
31 CRYPTOPP_DLL bool CRYPTOPP_API IsLucasProbablePrime(const Integer &n);
32 
33 CRYPTOPP_DLL bool CRYPTOPP_API IsStrongProbablePrime(const Integer &n, const Integer &b);
34 CRYPTOPP_DLL bool CRYPTOPP_API IsStrongLucasProbablePrime(const Integer &n);
35 
36 // Rabin-Miller primality test, i.e. repeating the strong probable prime test
37 // for several rounds with random bases
38 CRYPTOPP_DLL bool CRYPTOPP_API RabinMillerTest(RandomNumberGenerator &rng, const Integer &w, unsigned int rounds);
39 
40 // primality test, used to generate primes
41 CRYPTOPP_DLL bool CRYPTOPP_API IsPrime(const Integer &p);
42 
43 // more reliable than IsPrime(), used to verify primes generated by others
44 CRYPTOPP_DLL bool CRYPTOPP_API VerifyPrime(RandomNumberGenerator &rng, const Integer &p, unsigned int level = 1);
45 
46 class CRYPTOPP_DLL PrimeSelector
47 {
48 public:
49  const PrimeSelector *GetSelectorPointer() const {return this;}
50  virtual bool IsAcceptable(const Integer &candidate) const =0;
51 };
52 
53 // use a fast sieve to find the first probable prime in {x | p<=x<=max and x%mod==equiv}
54 // returns true iff successful, value of p is undefined if no such prime exists
55 CRYPTOPP_DLL bool CRYPTOPP_API FirstPrime(Integer &p, const Integer &max, const Integer &equiv, const Integer &mod, const PrimeSelector *pSelector);
56 
57 CRYPTOPP_DLL unsigned int CRYPTOPP_API PrimeSearchInterval(const Integer &max);
58 
59 CRYPTOPP_DLL AlgorithmParameters CRYPTOPP_API MakeParametersForTwoPrimesOfEqualSize(unsigned int productBitLength);
60 
61 // ********** other number theoretic functions ************
62 
63 inline Integer GCD(const Integer &a, const Integer &b)
64  {return Integer::Gcd(a,b);}
65 inline bool RelativelyPrime(const Integer &a, const Integer &b)
66  {return Integer::Gcd(a,b) == Integer::One();}
67 inline Integer LCM(const Integer &a, const Integer &b)
68  {return a/Integer::Gcd(a,b)*b;}
69 inline Integer EuclideanMultiplicativeInverse(const Integer &a, const Integer &b)
70  {return a.InverseMod(b);}
71 
72 // use Chinese Remainder Theorem to calculate x given x mod p and x mod q, and u = inverse of p mod q
73 CRYPTOPP_DLL Integer CRYPTOPP_API CRT(const Integer &xp, const Integer &p, const Integer &xq, const Integer &q, const Integer &u);
74 
75 // if b is prime, then Jacobi(a, b) returns 0 if a%b==0, 1 if a is quadratic residue mod b, -1 otherwise
76 // check a number theory book for what Jacobi symbol means when b is not prime
77 CRYPTOPP_DLL int CRYPTOPP_API Jacobi(const Integer &a, const Integer &b);
78 
79 // calculates the Lucas function V_e(p, 1) mod n
80 CRYPTOPP_DLL Integer CRYPTOPP_API Lucas(const Integer &e, const Integer &p, const Integer &n);
81 // calculates x such that m==Lucas(e, x, p*q), p q primes, u=inverse of p mod q
82 CRYPTOPP_DLL Integer CRYPTOPP_API InverseLucas(const Integer &e, const Integer &m, const Integer &p, const Integer &q, const Integer &u);
83 
84 inline Integer ModularExponentiation(const Integer &a, const Integer &e, const Integer &m)
85  {return a_exp_b_mod_c(a, e, m);}
86 // returns x such that x*x%p == a, p prime
87 CRYPTOPP_DLL Integer CRYPTOPP_API ModularSquareRoot(const Integer &a, const Integer &p);
88 // returns x such that a==ModularExponentiation(x, e, p*q), p q primes,
89 // and e relatively prime to (p-1)*(q-1)
90 // dp=d%(p-1), dq=d%(q-1), (d is inverse of e mod (p-1)*(q-1))
91 // and u=inverse of p mod q
92 CRYPTOPP_DLL Integer CRYPTOPP_API ModularRoot(const Integer &a, const Integer &dp, const Integer &dq, const Integer &p, const Integer &q, const Integer &u);
93 
94 // find r1 and r2 such that ax^2 + bx + c == 0 (mod p) for x in {r1, r2}, p prime
95 // returns true if solutions exist
96 CRYPTOPP_DLL bool CRYPTOPP_API SolveModularQuadraticEquation(Integer &r1, Integer &r2, const Integer &a, const Integer &b, const Integer &c, const Integer &p);
97 
98 // returns log base 2 of estimated number of operations to calculate discrete log or factor a number
99 CRYPTOPP_DLL unsigned int CRYPTOPP_API DiscreteLogWorkFactor(unsigned int bitlength);
100 CRYPTOPP_DLL unsigned int CRYPTOPP_API FactoringWorkFactor(unsigned int bitlength);
101 
102 // ********************************************************
103 
104 //! generator of prime numbers of special forms
105 class CRYPTOPP_DLL PrimeAndGenerator
106 {
107 public:
108  PrimeAndGenerator() {}
109  // generate a random prime p of the form 2*q+delta, where delta is 1 or -1 and q is also prime
110  // Precondition: pbits > 5
111  // warning: this is slow, because primes of this form are harder to find
112  PrimeAndGenerator(signed int delta, RandomNumberGenerator &rng, unsigned int pbits)
113  {Generate(delta, rng, pbits, pbits-1);}
114  // generate a random prime p of the form 2*r*q+delta, where q is also prime
115  // Precondition: qbits > 4 && pbits > qbits
116  PrimeAndGenerator(signed int delta, RandomNumberGenerator &rng, unsigned int pbits, unsigned qbits)
117  {Generate(delta, rng, pbits, qbits);}
118 
119  void Generate(signed int delta, RandomNumberGenerator &rng, unsigned int pbits, unsigned qbits);
120 
121  const Integer& Prime() const {return p;}
122  const Integer& SubPrime() const {return q;}
123  const Integer& Generator() const {return g;}
124 
125 private:
126  Integer p, q, g;
127 };
128 
129 NAMESPACE_END
130 
131 #endif
static const Integer & One()
avoid calling constructors for these frequently used integers
static Integer Gcd(const Integer &a, const Integer &n)
greatest common divisor
interface for random number generators
Definition: cryptlib.h:668
generator of prime numbers of special forms
Definition: nbtheory.h:105
multiple precision integer and basic arithmetics
Definition: integer.h:26
Integer InverseMod(const Integer &n) const
calculate multiplicative inverse of *this mod n