Crypto++ 8.9 Free C++ class library of cryptographic schemes

nbtheory.h
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1 // nbtheory.h - originally written and placed in the public domain by Wei Dai
2
3 /// \file nbtheory.h
4 /// \brief Classes and functions for number theoretic operations
5
6 #ifndef CRYPTOPP_NBTHEORY_H
7 #define CRYPTOPP_NBTHEORY_H
8
9 #include "cryptlib.h"
10 #include "integer.h"
11 #include "algparam.h"
12
13 NAMESPACE_BEGIN(CryptoPP)
14
15 /// \brief The Small Prime table
16 /// \param size number of elements in the table
17 /// \return prime table with /p size elements
18 /// \details GetPrimeTable() obtains pointer to small prime table and provides the size of the table.
19 /// /p size is an out parameter.
20 CRYPTOPP_DLL const word16 * CRYPTOPP_API GetPrimeTable(unsigned int &size);
21
22 // ************ primality testing ****************
23
24 /// \brief Generates a provable prime
25 /// \param rng a RandomNumberGenerator to produce random material
26 /// \param bits the number of bits in the prime number
27 /// \return Integer() meeting Maurer's tests for primality
28 CRYPTOPP_DLL Integer CRYPTOPP_API MaurerProvablePrime(RandomNumberGenerator &rng, unsigned int bits);
29
30 /// \brief Generates a provable prime
31 /// \param rng a RandomNumberGenerator to produce random material
32 /// \param bits the number of bits in the prime number
33 /// \return Integer() meeting Mihailescu's tests for primality
34 /// \details Mihailescu's methods performs a search using algorithmic progressions.
35 CRYPTOPP_DLL Integer CRYPTOPP_API MihailescuProvablePrime(RandomNumberGenerator &rng, unsigned int bits);
36
37 /// \brief Tests whether a number is a small prime
38 /// \param p a candidate prime to test
39 /// \return true if p is a small prime, false otherwise
40 /// \details Internally, the library maintains a table of the first 32719 prime numbers
41 /// in sorted order. IsSmallPrime searches the table and returns true if p is
42 /// in the table.
43 CRYPTOPP_DLL bool CRYPTOPP_API IsSmallPrime(const Integer &p);
44
45 /// \brief Tests whether a number is divisible by a small prime
46 /// \return true if p is divisible by some prime less than bound.
47 /// \details TrialDivision() returns <tt>true</tt> if <tt>p</tt> is divisible by some prime less
48 /// than <tt>bound</tt>. <tt>bound</tt> should not be greater than the largest entry in the
49 /// prime table, which is 32719.
50 CRYPTOPP_DLL bool CRYPTOPP_API TrialDivision(const Integer &p, unsigned bound);
51
52 /// \brief Tests whether a number is divisible by a small prime
53 /// \return true if p is NOT divisible by small primes.
54 /// \details SmallDivisorsTest() returns <tt>true</tt> if <tt>p</tt> is NOT divisible by some
55 /// prime less than 32719.
56 CRYPTOPP_DLL bool CRYPTOPP_API SmallDivisorsTest(const Integer &p);
57
58 /// \brief Determine if a number is probably prime
59 /// \param n the number to test
60 /// \param b the base to exponentiate
61 /// \return true if the number n is probably prime, false otherwise.
62 /// \details IsFermatProbablePrime raises <tt>b</tt> to the <tt>n-1</tt> power and checks if
63 /// the result is congruent to 1 modulo <tt>n</tt>.
64 /// \details These is no reason to use IsFermatProbablePrime, use IsStrongProbablePrime or
65 /// IsStrongLucasProbablePrime instead.
66 /// \sa IsStrongProbablePrime, IsStrongLucasProbablePrime
67 CRYPTOPP_DLL bool CRYPTOPP_API IsFermatProbablePrime(const Integer &n, const Integer &b);
68
69 /// \brief Determine if a number is probably prime
70 /// \param n the number to test
71 /// \return true if the number n is probably prime, false otherwise.
72 /// \details These is no reason to use IsLucasProbablePrime, use IsStrongProbablePrime or
73 /// IsStrongLucasProbablePrime instead.
74 /// \sa IsStrongProbablePrime, IsStrongLucasProbablePrime
75 CRYPTOPP_DLL bool CRYPTOPP_API IsLucasProbablePrime(const Integer &n);
76
77 /// \brief Determine if a number is probably prime
78 /// \param n the number to test
79 /// \param b the base to exponentiate
80 /// \return true if the number n is probably prime, false otherwise.
81 CRYPTOPP_DLL bool CRYPTOPP_API IsStrongProbablePrime(const Integer &n, const Integer &b);
82
83 /// \brief Determine if a number is probably prime
84 /// \param n the number to test
85 /// \return true if the number n is probably prime, false otherwise.
86 CRYPTOPP_DLL bool CRYPTOPP_API IsStrongLucasProbablePrime(const Integer &n);
87
88 /// \brief Determine if a number is probably prime
89 /// \param rng a RandomNumberGenerator to produce random material
90 /// \param n the number to test
91 /// \param rounds the number of tests to perform
92 /// \details This is the Rabin-Miller primality test, i.e. repeating the strong probable prime
93 /// test for several rounds with random bases
94 /// \sa <A HREF="https://crypto.stackexchange.com/q/17707/10496">Trial divisions before
95 /// Miller-Rabin checks?</A> on Crypto Stack Exchange
96 CRYPTOPP_DLL bool CRYPTOPP_API RabinMillerTest(RandomNumberGenerator &rng, const Integer &n, unsigned int rounds);
97
98 /// \brief Verifies a number is probably prime
99 /// \param p a candidate prime to test
100 /// \return true if p is a probable prime, false otherwise
101 /// \details IsPrime() is suitable for testing candidate primes when creating them. Internally,
102 /// IsPrime() utilizes SmallDivisorsTest(), IsStrongProbablePrime() and IsStrongLucasProbablePrime().
103 CRYPTOPP_DLL bool CRYPTOPP_API IsPrime(const Integer &p);
104
105 /// \brief Verifies a number is probably prime
106 /// \param rng a RandomNumberGenerator for randomized testing
107 /// \param p a candidate prime to test
108 /// \param level the level of thoroughness of testing
109 /// \return true if p is a strong probable prime, false otherwise
110 /// \details VerifyPrime() is suitable for testing candidate primes created by others. Internally,
111 /// VerifyPrime() utilizes IsPrime() and one-round RabinMillerTest(). If the candidate passes and
112 /// level is greater than 1, then 10 round RabinMillerTest() primality testing is performed.
113 CRYPTOPP_DLL bool CRYPTOPP_API VerifyPrime(RandomNumberGenerator &rng, const Integer &p, unsigned int level = 1);
114
115 /// \brief Application callback to signal suitability of a candidate prime
116 class CRYPTOPP_DLL PrimeSelector
117 {
118 public:
119 virtual ~PrimeSelector() {}
120 const PrimeSelector *GetSelectorPointer() const {return this;}
121 virtual bool IsAcceptable(const Integer &candidate) const =0;
122 };
123
124 /// \brief Finds a random prime of special form
125 /// \param p an Integer reference to receive the prime
126 /// \param max the maximum value
127 /// \param equiv the equivalence class based on the parameter mod
128 /// \param mod the modulus used to reduce the equivalence class
129 /// \param pSelector pointer to a PrimeSelector function for the application to signal suitability
130 /// \return true if and only if FirstPrime() finds a prime and returns the prime through p. If FirstPrime()
131 /// returns false, then no such prime exists and the value of p is undefined
132 /// \details FirstPrime() uses a fast sieve to find the first probable prime
133 /// in <tt>{x | p<=x<=max and x%mod==equiv}</tt>
134 CRYPTOPP_DLL bool CRYPTOPP_API FirstPrime(Integer &p, const Integer &max, const Integer &equiv, const Integer &mod, const PrimeSelector *pSelector);
135
136 CRYPTOPP_DLL unsigned int CRYPTOPP_API PrimeSearchInterval(const Integer &max);
137
138 CRYPTOPP_DLL AlgorithmParameters CRYPTOPP_API MakeParametersForTwoPrimesOfEqualSize(unsigned int productBitLength);
139
140 // ********** other number theoretic functions ************
141
142 /// \brief Calculate the greatest common divisor
143 /// \param a the first term
144 /// \param b the second term
145 /// \return the greatest common divisor if one exists, 0 otherwise.
146 inline Integer GCD(const Integer &a, const Integer &b)
147 {return Integer::Gcd(a,b);}
148
149 /// \brief Determine relative primality
150 /// \param a the first term
151 /// \param b the second term
152 /// \return true if <tt>a</tt> and <tt>b</tt> are relatively prime, false otherwise.
153 inline bool RelativelyPrime(const Integer &a, const Integer &b)
154 {return Integer::Gcd(a,b) == Integer::One();}
155
156 /// \brief Calculate the least common multiple
157 /// \param a the first term
158 /// \param b the second term
159 /// \return the least common multiple of <tt>a</tt> and <tt>b</tt>.
160 inline Integer LCM(const Integer &a, const Integer &b)
161 {return a/Integer::Gcd(a,b)*b;}
162
163 /// \brief Calculate multiplicative inverse
164 /// \param a the number to test
165 /// \param b the modulus
166 /// \return an Integer <tt>(a ^ -1) % n</tt> or 0 if none exists.
167 /// \details EuclideanMultiplicativeInverse returns the multiplicative inverse of the Integer
168 /// <tt>*a</tt> modulo the Integer <tt>b</tt>. If no Integer exists then Integer 0 is returned.
169 inline Integer EuclideanMultiplicativeInverse(const Integer &a, const Integer &b)
170 {return a.InverseMod(b);}
171
172
173 /// \brief Chinese Remainder Theorem
174 /// \param xp the first number, mod p
175 /// \param p the first prime modulus
176 /// \param xq the second number, mod q
177 /// \param q the second prime modulus
178 /// \param u inverse of p mod q
179 /// \return the CRT value of the parameters
180 /// \details CRT uses the Chinese Remainder Theorem to calculate <tt>x</tt> given
181 /// <tt>x mod p</tt> and <tt>x mod q</tt>, and <tt>u</tt> the inverse of <tt>p mod q</tt>.
182 CRYPTOPP_DLL Integer CRYPTOPP_API CRT(const Integer &xp, const Integer &p, const Integer &xq, const Integer &q, const Integer &u);
183
184 /// \brief Calculate the Jacobi symbol
185 /// \param a the first term
186 /// \param b the second term
187 /// \return the Jacobi symbol.
188 /// \details Jacobi symbols are calculated using the following rules:
189 /// -# if <tt>b</tt> is prime, then <tt>Jacobi(a, b)</tt>, then return 0
190 /// -# if <tt>a%b</tt>==0 AND <tt>a</tt> is quadratic residue <tt>mod b</tt>, then return 1
191 /// -# return -1 otherwise
192 /// \details Refer to a number theory book for what Jacobi symbol means when <tt>b</tt> is not prime.
193 CRYPTOPP_DLL int CRYPTOPP_API Jacobi(const Integer &a, const Integer &b);
194
195 /// \brief Calculate the Lucas value
196 /// \return the Lucas value
197 /// \details Lucas() calculates the Lucas function <tt>V_e(p, 1) mod n</tt>.
198 CRYPTOPP_DLL Integer CRYPTOPP_API Lucas(const Integer &e, const Integer &p, const Integer &n);
199
200 /// \brief Calculate the inverse Lucas value
201 /// \return the inverse Lucas value
202 /// \details InverseLucas() calculates <tt>x</tt> such that <tt>m==Lucas(e, x, p*q)</tt>,
203 /// <tt>p q</tt> primes, <tt>u</tt> is inverse of <tt>p mod q</tt>.
204 CRYPTOPP_DLL Integer CRYPTOPP_API InverseLucas(const Integer &e, const Integer &m, const Integer &p, const Integer &q, const Integer &u);
205
206 /// \brief Modular multiplication
207 /// \param x the first term
208 /// \param y the second term
209 /// \param m the modulus
210 /// \return an Integer <tt>(x * y) % m</tt>.
211 inline Integer ModularMultiplication(const Integer &x, const Integer &y, const Integer &m)
212 {return a_times_b_mod_c(x, y, m);}
213
214 /// \brief Modular exponentiation
215 /// \param x the base
216 /// \param e the exponent
217 /// \param m the modulus
218 /// \return an Integer <tt>(a ^ b) % m</tt>.
219 inline Integer ModularExponentiation(const Integer &x, const Integer &e, const Integer &m)
220 {return a_exp_b_mod_c(x, e, m);}
221
222 /// \brief Extract a modular square root
223 /// \param a the number to extract square root
224 /// \param p the prime modulus
225 /// \return the modular square root if it exists
226 /// \details ModularSquareRoot returns <tt>x</tt> such that <tt>x*x%p == a</tt>, <tt>p</tt> prime
227 CRYPTOPP_DLL Integer CRYPTOPP_API ModularSquareRoot(const Integer &a, const Integer &p);
228
229 /// \brief Extract a modular root
230 /// \return a modular root if it exists
231 /// \details ModularRoot returns <tt>x</tt> such that <tt>a==ModularExponentiation(x, e, p*q)</tt>,
232 /// <tt>p</tt> <tt>q</tt> primes, and <tt>e</tt> relatively prime to <tt>(p-1)*(q-1)</tt>,
233 /// <tt>dp=d%(p-1)</tt>, <tt>dq=d%(q-1)</tt>, (d is inverse of <tt>e mod (p-1)*(q-1)</tt>)
234 /// and <tt>u=inverse of p mod q</tt>.
235 CRYPTOPP_DLL Integer CRYPTOPP_API ModularRoot(const Integer &a, const Integer &dp, const Integer &dq, const Integer &p, const Integer &q, const Integer &u);
236
237 /// \brief Solve a Modular Quadratic Equation
238 /// \param r1 the first residue
239 /// \param r2 the second residue
240 /// \param a the first coefficient
241 /// \param b the second coefficient
242 /// \param c the third constant
243 /// \param p the prime modulus
244 /// \return true if solutions exist
245 /// \details SolveModularQuadraticEquation() finds <tt>r1</tt> and <tt>r2</tt> such that <tt>ax^2 +
246 /// bx + c == 0 (mod p)</tt> for x in <tt>{r1, r2}</tt>, <tt>p</tt> prime.
247 CRYPTOPP_DLL bool CRYPTOPP_API SolveModularQuadraticEquation(Integer &r1, Integer &r2, const Integer &a, const Integer &b, const Integer &c, const Integer &p);
248
249 /// \brief Estimate work factor
250 /// \param bitlength the size of the number, in bits
251 /// \return the estimated work factor, in operations
252 /// \details DiscreteLogWorkFactor returns log base 2 of estimated number of operations to
253 /// calculate discrete log or factor a number.
254 CRYPTOPP_DLL unsigned int CRYPTOPP_API DiscreteLogWorkFactor(unsigned int bitlength);
255
256 /// \brief Estimate work factor
257 /// \param bitlength the size of the number, in bits
258 /// \return the estimated work factor, in operations
259 /// \details FactoringWorkFactor returns log base 2 of estimated number of operations to
260 /// calculate discrete log or factor a number.
261 CRYPTOPP_DLL unsigned int CRYPTOPP_API FactoringWorkFactor(unsigned int bitlength);
262
263 // ********************************************************
264
265 /// \brief Generator of prime numbers of special forms
266 class CRYPTOPP_DLL PrimeAndGenerator
267 {
268 public:
269 /// \brief Construct a PrimeAndGenerator
270 PrimeAndGenerator() {}
271
272 /// \brief Construct a PrimeAndGenerator
273 /// \param delta +1 or -1
274 /// \param rng a RandomNumberGenerator derived class
275 /// \param pbits the number of bits in the prime p
276 /// \details PrimeAndGenerator() generates a random prime p of the form <tt>2*q+delta</tt>, where delta is 1 or -1 and q is
277 /// also prime. Internally the constructor calls <tt>Generate(delta, rng, pbits, pbits-1)</tt>.
278 /// \pre <tt>pbits > 5</tt>
279 /// \warning This PrimeAndGenerator() is slow because primes of this form are harder to find.
280 PrimeAndGenerator(signed int delta, RandomNumberGenerator &rng, unsigned int pbits)
281 {Generate(delta, rng, pbits, pbits-1);}
282
283 /// \brief Construct a PrimeAndGenerator
284 /// \param delta +1 or -1
285 /// \param rng a RandomNumberGenerator derived class
286 /// \param pbits the number of bits in the prime p
287 /// \param qbits the number of bits in the prime q
288 /// \details PrimeAndGenerator() generates a random prime p of the form <tt>2*r*q+delta</tt>, where q is also prime.
289 /// Internally the constructor calls <tt>Generate(delta, rng, pbits, qbits)</tt>.
290 /// \pre <tt>qbits > 4 && pbits > qbits</tt>
291 PrimeAndGenerator(signed int delta, RandomNumberGenerator &rng, unsigned int pbits, unsigned qbits)
292 {Generate(delta, rng, pbits, qbits);}
293
294 /// \brief Generate a Prime and Generator
295 /// \param delta +1 or -1
296 /// \param rng a RandomNumberGenerator derived class
297 /// \param pbits the number of bits in the prime p
298 /// \param qbits the number of bits in the prime q
299 /// \details Generate() generates a random prime p of the form <tt>2*r*q+delta</tt>, where q is also prime.
300 void Generate(signed int delta, RandomNumberGenerator &rng, unsigned int pbits, unsigned qbits);
301
302 /// \brief Retrieve first prime
303 /// \return Prime() returns the prime p.
304 const Integer& Prime() const {return p;}
305
306 /// \brief Retrieve second prime
307 /// \return SubPrime() returns the prime q.
308 const Integer& SubPrime() const {return q;}
309
310 /// \brief Retrieve the generator
311 /// \return Generator() returns the generator g.
312 const Integer& Generator() const {return g;}
313
314 private:
315 Integer p, q, g;
316 };
317
318 NAMESPACE_END
319
320 #endif
algparam.h
Classes for working with NameValuePairs.
AlgorithmParameters
An object that implements NameValuePairs.
Definition: algparam.h:426
Integer
Multiple precision integer with arithmetic operations.
Definition: integer.h:50
Integer::One
static const Integer & One()
Integer representing 1.
Integer::Gcd
static Integer Gcd(const Integer &a, const Integer &n)
Calculate greatest common divisor.
PrimeAndGenerator
Generator of prime numbers of special forms.
Definition: nbtheory.h:267
PrimeAndGenerator::Generator
const Integer & Generator() const
Retrieve the generator.
Definition: nbtheory.h:312
PrimeAndGenerator::PrimeAndGenerator
PrimeAndGenerator(signed int delta, RandomNumberGenerator &rng, unsigned int pbits)
Construct a PrimeAndGenerator.
Definition: nbtheory.h:280
PrimeAndGenerator::PrimeAndGenerator
PrimeAndGenerator()
Construct a PrimeAndGenerator.
Definition: nbtheory.h:270
PrimeAndGenerator::Generate
void Generate(signed int delta, RandomNumberGenerator &rng, unsigned int pbits, unsigned qbits)
Generate a Prime and Generator.
PrimeAndGenerator::Prime
const Integer & Prime() const
Retrieve first prime.
Definition: nbtheory.h:304
PrimeAndGenerator::SubPrime
const Integer & SubPrime() const
Retrieve second prime.
Definition: nbtheory.h:308
PrimeAndGenerator::PrimeAndGenerator
PrimeAndGenerator(signed int delta, RandomNumberGenerator &rng, unsigned int pbits, unsigned qbits)
Construct a PrimeAndGenerator.
Definition: nbtheory.h:291
PrimeSelector
Application callback to signal suitability of a candidate prime.
Definition: nbtheory.h:117
RandomNumberGenerator
Interface for random number generators.
Definition: cryptlib.h:1440
CRYPTOPP_API
#define CRYPTOPP_API
Win32 calling convention.
Definition: config_dll.h:119
word16
unsigned short word16
16-bit unsigned datatype
Definition: config_int.h:69
cryptlib.h
Abstract base classes that provide a uniform interface to this library.
integer.h
Multiple precision integer with arithmetic operations.
CryptoPP
Crypto++ library namespace.
Jacobi
CRYPTOPP_DLL int Jacobi(const Integer &a, const Integer &b)
Calculate the Jacobi symbol.
IsPrime
CRYPTOPP_DLL bool IsPrime(const Integer &p)
Verifies a number is probably prime.
RelativelyPrime
bool RelativelyPrime(const Integer &a, const Integer &b)
Determine relative primality.
Definition: nbtheory.h:153
ModularMultiplication
Integer ModularMultiplication(const Integer &x, const Integer &y, const Integer &m)
Modular multiplication.
Definition: nbtheory.h:211
MihailescuProvablePrime
CRYPTOPP_DLL Integer MihailescuProvablePrime(RandomNumberGenerator &rng, unsigned int bits)
Generates a provable prime.
IsStrongLucasProbablePrime
CRYPTOPP_DLL bool IsStrongLucasProbablePrime(const Integer &n)
Determine if a number is probably prime.
DiscreteLogWorkFactor
CRYPTOPP_DLL unsigned int DiscreteLogWorkFactor(unsigned int bitlength)
Estimate work factor.
ModularExponentiation
Integer ModularExponentiation(const Integer &x, const Integer &e, const Integer &m)
Modular exponentiation.
Definition: nbtheory.h:219
ModularSquareRoot
CRYPTOPP_DLL Integer ModularSquareRoot(const Integer &a, const Integer &p)
Extract a modular square root.
GetPrimeTable
CRYPTOPP_DLL const word16 * GetPrimeTable(unsigned int &size)
The Small Prime table.
IsSmallPrime
CRYPTOPP_DLL bool IsSmallPrime(const Integer &p)
Tests whether a number is a small prime.
SolveModularQuadraticEquation
CRYPTOPP_DLL bool SolveModularQuadraticEquation(Integer &r1, Integer &r2, const Integer &a, const Integer &b, const Integer &c, const Integer &p)
Solve a Modular Quadratic Equation.
RabinMillerTest
CRYPTOPP_DLL bool RabinMillerTest(RandomNumberGenerator &rng, const Integer &n, unsigned int rounds)
Determine if a number is probably prime.
MaurerProvablePrime
CRYPTOPP_DLL Integer MaurerProvablePrime(RandomNumberGenerator &rng, unsigned int bits)
Generates a provable prime.
Lucas
CRYPTOPP_DLL Integer Lucas(const Integer &e, const Integer &p, const Integer &n)
Calculate the Lucas value.
InverseLucas
CRYPTOPP_DLL Integer InverseLucas(const Integer &e, const Integer &m, const Integer &p, const Integer &q, const Integer &u)
Calculate the inverse Lucas value.
VerifyPrime
CRYPTOPP_DLL bool VerifyPrime(RandomNumberGenerator &rng, const Integer &p, unsigned int level=1)
Verifies a number is probably prime.
ModularRoot
CRYPTOPP_DLL Integer ModularRoot(const Integer &a, const Integer &dp, const Integer &dq, const Integer &p, const Integer &q, const Integer &u)
Extract a modular root.
EuclideanMultiplicativeInverse
Integer EuclideanMultiplicativeInverse(const Integer &a, const Integer &b)
Calculate multiplicative inverse.
Definition: nbtheory.h:169
SmallDivisorsTest
CRYPTOPP_DLL bool SmallDivisorsTest(const Integer &p)
Tests whether a number is divisible by a small prime.
IsLucasProbablePrime
CRYPTOPP_DLL bool IsLucasProbablePrime(const Integer &n)
Determine if a number is probably prime.
GCD
Integer GCD(const Integer &a, const Integer &b)
Calculate the greatest common divisor.
Definition: nbtheory.h:146
TrialDivision
CRYPTOPP_DLL bool TrialDivision(const Integer &p, unsigned bound)
Tests whether a number is divisible by a small prime.
FactoringWorkFactor
CRYPTOPP_DLL unsigned int FactoringWorkFactor(unsigned int bitlength)
Estimate work factor.
IsFermatProbablePrime
CRYPTOPP_DLL bool IsFermatProbablePrime(const Integer &n, const Integer &b)
Determine if a number is probably prime.
LCM
Integer LCM(const Integer &a, const Integer &b)
Calculate the least common multiple.
Definition: nbtheory.h:160
CRT
CRYPTOPP_DLL Integer CRT(const Integer &xp, const Integer &p, const Integer &xq, const Integer &q, const Integer &u)
Chinese Remainder Theorem.
IsStrongProbablePrime
CRYPTOPP_DLL bool IsStrongProbablePrime(const Integer &n, const Integer &b)
Determine if a number is probably prime.
FirstPrime
CRYPTOPP_DLL bool FirstPrime(Integer &p, const Integer &max, const Integer &equiv, const Integer &mod, const PrimeSelector *pSelector)
Finds a random prime of special form.