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321 lines
15 KiB
C++
321 lines
15 KiB
C++
// nbtheory.h - originally written and placed in the public domain by Wei Dai
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/// \file nbtheory.h
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/// \brief Classes and functions for number theoretic operations
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#ifndef CRYPTOPP_NBTHEORY_H
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#define CRYPTOPP_NBTHEORY_H
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#include "cryptlib.h"
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#include "integer.h"
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#include "algparam.h"
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NAMESPACE_BEGIN(CryptoPP)
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/// \brief The Small Prime table
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/// \param size number of elements in the table
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/// \return prime table with /p size elements
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/// \details GetPrimeTable() obtains pointer to small prime table and provides the size of the table.
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/// /p size is an out parameter.
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CRYPTOPP_DLL const word16 * CRYPTOPP_API GetPrimeTable(unsigned int &size);
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// ************ primality testing ****************
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/// \brief Generates a provable prime
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/// \param rng a RandomNumberGenerator to produce random material
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/// \param bits the number of bits in the prime number
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/// \return Integer() meeting Maurer's tests for primality
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CRYPTOPP_DLL Integer CRYPTOPP_API MaurerProvablePrime(RandomNumberGenerator &rng, unsigned int bits);
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/// \brief Generates a provable prime
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/// \param rng a RandomNumberGenerator to produce random material
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/// \param bits the number of bits in the prime number
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/// \return Integer() meeting Mihailescu's tests for primality
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/// \details Mihailescu's methods performs a search using algorithmic progressions.
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CRYPTOPP_DLL Integer CRYPTOPP_API MihailescuProvablePrime(RandomNumberGenerator &rng, unsigned int bits);
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/// \brief Tests whether a number is a small prime
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/// \param p a candidate prime to test
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/// \return true if p is a small prime, false otherwise
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/// \details Internally, the library maintains a table of the first 32719 prime numbers
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/// in sorted order. IsSmallPrime searches the table and returns true if p is
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/// in the table.
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CRYPTOPP_DLL bool CRYPTOPP_API IsSmallPrime(const Integer &p);
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/// \brief Tests whether a number is divisible by a small prime
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/// \return true if p is divisible by some prime less than bound.
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/// \details TrialDivision() returns <tt>true</tt> if <tt>p</tt> is divisible by some prime less
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/// than <tt>bound</tt>. <tt>bound</tt> should not be greater than the largest entry in the
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/// prime table, which is 32719.
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CRYPTOPP_DLL bool CRYPTOPP_API TrialDivision(const Integer &p, unsigned bound);
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/// \brief Tests whether a number is divisible by a small prime
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/// \return true if p is NOT divisible by small primes.
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/// \details SmallDivisorsTest() returns <tt>true</tt> if <tt>p</tt> is NOT divisible by some
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/// prime less than 32719.
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CRYPTOPP_DLL bool CRYPTOPP_API SmallDivisorsTest(const Integer &p);
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/// \brief Determine if a number is probably prime
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/// \param n the number to test
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/// \param b the base to exponentiate
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/// \return true if the number n is probably prime, false otherwise.
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/// \details IsFermatProbablePrime raises <tt>b</tt> to the <tt>n-1</tt> power and checks if
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/// the result is congruent to 1 modulo <tt>n</tt>.
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/// \details These is no reason to use IsFermatProbablePrime, use IsStrongProbablePrime or
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/// IsStrongLucasProbablePrime instead.
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/// \sa IsStrongProbablePrime, IsStrongLucasProbablePrime
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CRYPTOPP_DLL bool CRYPTOPP_API IsFermatProbablePrime(const Integer &n, const Integer &b);
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/// \brief Determine if a number is probably prime
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/// \param n the number to test
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/// \return true if the number n is probably prime, false otherwise.
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/// \details These is no reason to use IsLucasProbablePrime, use IsStrongProbablePrime or
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/// IsStrongLucasProbablePrime instead.
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/// \sa IsStrongProbablePrime, IsStrongLucasProbablePrime
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CRYPTOPP_DLL bool CRYPTOPP_API IsLucasProbablePrime(const Integer &n);
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/// \brief Determine if a number is probably prime
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/// \param n the number to test
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/// \param b the base to exponentiate
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/// \return true if the number n is probably prime, false otherwise.
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CRYPTOPP_DLL bool CRYPTOPP_API IsStrongProbablePrime(const Integer &n, const Integer &b);
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/// \brief Determine if a number is probably prime
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/// \param n the number to test
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/// \return true if the number n is probably prime, false otherwise.
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CRYPTOPP_DLL bool CRYPTOPP_API IsStrongLucasProbablePrime(const Integer &n);
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/// \brief Determine if a number is probably prime
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/// \param rng a RandomNumberGenerator to produce random material
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/// \param n the number to test
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/// \param rounds the number of tests to perform
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/// \details This is the Rabin-Miller primality test, i.e. repeating the strong probable prime
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/// test for several rounds with random bases
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/// \sa <A HREF="https://crypto.stackexchange.com/q/17707/10496">Trial divisions before
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/// Miller-Rabin checks?</A> on Crypto Stack Exchange
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CRYPTOPP_DLL bool CRYPTOPP_API RabinMillerTest(RandomNumberGenerator &rng, const Integer &n, unsigned int rounds);
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/// \brief Verifies a number is probably prime
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/// \param p a candidate prime to test
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/// \return true if p is a probable prime, false otherwise
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/// \details IsPrime() is suitable for testing candidate primes when creating them. Internally,
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/// IsPrime() utilizes SmallDivisorsTest(), IsStrongProbablePrime() and IsStrongLucasProbablePrime().
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CRYPTOPP_DLL bool CRYPTOPP_API IsPrime(const Integer &p);
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/// \brief Verifies a number is probably prime
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/// \param rng a RandomNumberGenerator for randomized testing
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/// \param p a candidate prime to test
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/// \param level the level of thoroughness of testing
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/// \return true if p is a strong probable prime, false otherwise
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/// \details VerifyPrime() is suitable for testing candidate primes created by others. Internally,
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/// VerifyPrime() utilizes IsPrime() and one-round RabinMillerTest(). If the candidate passes and
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/// level is greater than 1, then 10 round RabinMillerTest() primality testing is performed.
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CRYPTOPP_DLL bool CRYPTOPP_API VerifyPrime(RandomNumberGenerator &rng, const Integer &p, unsigned int level = 1);
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/// \brief Application callback to signal suitability of a cabdidate prime
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class CRYPTOPP_DLL PrimeSelector
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{
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public:
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virtual ~PrimeSelector() {}
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const PrimeSelector *GetSelectorPointer() const {return this;}
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virtual bool IsAcceptable(const Integer &candidate) const =0;
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};
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/// \brief Finds a random prime of special form
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/// \param p an Integer reference to receive the prime
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/// \param max the maximum value
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/// \param equiv the equivalence class based on the parameter mod
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/// \param mod the modulus used to reduce the equivalence class
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/// \param pSelector pointer to a PrimeSelector function for the application to signal suitability
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/// \return true if and only if FirstPrime() finds a prime and returns the prime through p. If FirstPrime()
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/// returns false, then no such prime exists and the value of p is undefined
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/// \details FirstPrime() uses a fast sieve to find the first probable prime
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/// in <tt>{x | p<=x<=max and x%mod==equiv}</tt>
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CRYPTOPP_DLL bool CRYPTOPP_API FirstPrime(Integer &p, const Integer &max, const Integer &equiv, const Integer &mod, const PrimeSelector *pSelector);
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CRYPTOPP_DLL unsigned int CRYPTOPP_API PrimeSearchInterval(const Integer &max);
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CRYPTOPP_DLL AlgorithmParameters CRYPTOPP_API MakeParametersForTwoPrimesOfEqualSize(unsigned int productBitLength);
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// ********** other number theoretic functions ************
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/// \brief Calculate the greatest common divisor
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/// \param a the first term
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/// \param b the second term
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/// \return the greatest common divisor if one exists, 0 otherwise.
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inline Integer GCD(const Integer &a, const Integer &b)
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{return Integer::Gcd(a,b);}
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/// \brief Determine relative primality
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/// \param a the first term
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/// \param b the second term
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/// \return true if <tt>a</tt> and <tt>b</tt> are relatively prime, false otherwise.
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inline bool RelativelyPrime(const Integer &a, const Integer &b)
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{return Integer::Gcd(a,b) == Integer::One();}
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/// \brief Calculate the least common multiple
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/// \param a the first term
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/// \param b the second term
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/// \return the least common multiple of <tt>a</tt> and <tt>b</tt>.
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inline Integer LCM(const Integer &a, const Integer &b)
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{return a/Integer::Gcd(a,b)*b;}
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/// \brief Calculate multiplicative inverse
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/// \param a the number to test
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/// \param b the modulus
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/// \return an Integer <tt>(a ^ -1) % n</tt> or 0 if none exists.
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/// \details EuclideanMultiplicativeInverse returns the multiplicative inverse of the Integer
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/// <tt>*a</tt> modulo the Integer <tt>b</tt>. If no Integer exists then Integer 0 is returned.
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inline Integer EuclideanMultiplicativeInverse(const Integer &a, const Integer &b)
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{return a.InverseMod(b);}
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/// \brief Chinese Remainder Theorem
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/// \param xp the first number, mod p
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/// \param p the first prime modulus
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/// \param xq the second number, mod q
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/// \param q the second prime modulus
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/// \param u inverse of p mod q
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/// \return the CRT value of the parameters
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/// \details CRT uses the Chinese Remainder Theorem to calculate <tt>x</tt> given
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/// <tt>x mod p</tt> and <tt>x mod q</tt>, and <tt>u</tt> the inverse of <tt>p mod q</tt>.
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CRYPTOPP_DLL Integer CRYPTOPP_API CRT(const Integer &xp, const Integer &p, const Integer &xq, const Integer &q, const Integer &u);
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/// \brief Calculate the Jacobi symbol
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/// \param a the first term
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/// \param b the second term
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/// \return the Jacobi symbol.
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/// \details Jacobi symbols are calculated using the following rules:
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/// -# if <tt>b</tt> is prime, then <tt>Jacobi(a, b)</tt>, then return 0
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/// -# if <tt>a%b</tt>==0 AND <tt>a</tt> is quadratic residue <tt>mod b</tt>, then return 1
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/// -# return -1 otherwise
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/// \details Refer to a number theory book for what Jacobi symbol means when <tt>b</tt> is not prime.
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CRYPTOPP_DLL int CRYPTOPP_API Jacobi(const Integer &a, const Integer &b);
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/// \brief Calculate the Lucas value
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/// \return the Lucas value
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/// \details Lucas() calculates the Lucas function <tt>V_e(p, 1) mod n</tt>.
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CRYPTOPP_DLL Integer CRYPTOPP_API Lucas(const Integer &e, const Integer &p, const Integer &n);
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/// \brief Calculate the inverse Lucas value
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/// \return the inverse Lucas value
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/// \details InverseLucas() calculates <tt>x</tt> such that <tt>m==Lucas(e, x, p*q)</tt>,
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/// <tt>p q</tt> primes, <tt>u</tt> is inverse of <tt>p mod q</tt>.
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CRYPTOPP_DLL Integer CRYPTOPP_API InverseLucas(const Integer &e, const Integer &m, const Integer &p, const Integer &q, const Integer &u);
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/// \brief Modular multiplication
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/// \param x the first term
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/// \param y the second term
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/// \param m the modulus
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/// \return an Integer <tt>(x * y) % m</tt>.
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inline Integer ModularMultiplication(const Integer &x, const Integer &y, const Integer &m)
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{return a_times_b_mod_c(x, y, m);}
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/// \brief Modular exponentiation
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/// \param x the base
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/// \param e the exponent
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/// \param m the modulus
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/// \return an Integer <tt>(a ^ b) % m</tt>.
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inline Integer ModularExponentiation(const Integer &x, const Integer &e, const Integer &m)
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{return a_exp_b_mod_c(x, e, m);}
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/// \brief Extract a modular square root
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/// \param a the number to extract square root
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/// \param p the prime modulus
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/// \return the modular square root if it exists
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/// \details ModularSquareRoot returns <tt>x</tt> such that <tt>x*x%p == a</tt>, <tt>p</tt> prime
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CRYPTOPP_DLL Integer CRYPTOPP_API ModularSquareRoot(const Integer &a, const Integer &p);
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/// \brief Extract a modular root
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/// \return a modular root if it exists
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/// \details ModularRoot returns <tt>x</tt> such that <tt>a==ModularExponentiation(x, e, p*q)</tt>,
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/// <tt>p</tt> <tt>q</tt> primes, and <tt>e</tt> relatively prime to <tt>(p-1)*(q-1)</tt>,
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/// <tt>dp=d%(p-1)</tt>, <tt>dq=d%(q-1)</tt>, (d is inverse of <tt>e mod (p-1)*(q-1)</tt>)
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/// and <tt>u=inverse of p mod q</tt>.
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CRYPTOPP_DLL Integer CRYPTOPP_API ModularRoot(const Integer &a, const Integer &dp, const Integer &dq, const Integer &p, const Integer &q, const Integer &u);
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/// \brief Solve a Modular Quadratic Equation
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/// \param r1 the first residue
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/// \param r2 the second residue
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/// \param a the first coefficient
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/// \param b the second coefficient
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/// \param c the third constant
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/// \param p the prime modulus
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/// \return true if solutions exist
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/// \details SolveModularQuadraticEquation() finds <tt>r1</tt> and <tt>r2</tt> such that <tt>ax^2 +
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/// bx + c == 0 (mod p)</tt> for x in <tt>{r1, r2}</tt>, <tt>p</tt> prime.
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CRYPTOPP_DLL bool CRYPTOPP_API SolveModularQuadraticEquation(Integer &r1, Integer &r2, const Integer &a, const Integer &b, const Integer &c, const Integer &p);
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/// \brief Estimate work factor
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/// \param bitlength the size of the number, in bits
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/// \return the estimated work factor, in operations
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/// \details DiscreteLogWorkFactor returns log base 2 of estimated number of operations to
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/// calculate discrete log or factor a number.
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CRYPTOPP_DLL unsigned int CRYPTOPP_API DiscreteLogWorkFactor(unsigned int bitlength);
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/// \brief Estimate work factor
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/// \param bitlength the size of the number, in bits
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/// \return the estimated work factor, in operations
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/// \details FactoringWorkFactor returns log base 2 of estimated number of operations to
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/// calculate discrete log or factor a number.
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CRYPTOPP_DLL unsigned int CRYPTOPP_API FactoringWorkFactor(unsigned int bitlength);
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// ********************************************************
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/// \brief Generator of prime numbers of special forms
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class CRYPTOPP_DLL PrimeAndGenerator
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{
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public:
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/// \brief Construct a PrimeAndGenerator
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PrimeAndGenerator() {}
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/// \brief Construct a PrimeAndGenerator
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/// \param delta +1 or -1
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/// \param rng a RandomNumberGenerator derived class
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/// \param pbits the number of bits in the prime p
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/// \details PrimeAndGenerator() generates a random prime p of the form <tt>2*q+delta</tt>, where delta is 1 or -1 and q is
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/// also prime. Internally the constructor calls <tt>Generate(delta, rng, pbits, pbits-1)</tt>.
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/// \pre <tt>pbits > 5</tt>
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/// \warning This PrimeAndGenerator() is slow because primes of this form are harder to find.
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PrimeAndGenerator(signed int delta, RandomNumberGenerator &rng, unsigned int pbits)
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{Generate(delta, rng, pbits, pbits-1);}
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/// \brief Construct a PrimeAndGenerator
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/// \param delta +1 or -1
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/// \param rng a RandomNumberGenerator derived class
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/// \param pbits the number of bits in the prime p
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/// \param qbits the number of bits in the prime q
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/// \details PrimeAndGenerator() generates a random prime p of the form <tt>2*r*q+delta</tt>, where q is also prime.
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/// Internally the constructor calls <tt>Generate(delta, rng, pbits, qbits)</tt>.
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/// \pre <tt>qbits > 4 && pbits > qbits</tt>
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PrimeAndGenerator(signed int delta, RandomNumberGenerator &rng, unsigned int pbits, unsigned qbits)
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{Generate(delta, rng, pbits, qbits);}
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/// \brief Generate a Prime and Generator
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/// \param delta +1 or -1
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/// \param rng a RandomNumberGenerator derived class
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/// \param pbits the number of bits in the prime p
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/// \param qbits the number of bits in the prime q
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/// \details Generate() generates a random prime p of the form <tt>2*r*q+delta</tt>, where q is also prime.
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void Generate(signed int delta, RandomNumberGenerator &rng, unsigned int pbits, unsigned qbits);
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/// \brief Retrieve first prime
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/// \return Prime() returns the prime p.
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const Integer& Prime() const {return p;}
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/// \brief Retrieve second prime
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/// \return SubPrime() returns the prime q.
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const Integer& SubPrime() const {return q;}
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/// \brief Retrieve the generator
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/// \return Generator() returns the generator g.
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const Integer& Generator() const {return g;}
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private:
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Integer p, q, g;
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};
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NAMESPACE_END
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#endif
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