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1123 lines
23 KiB
1123 lines
23 KiB
// nbtheory.cpp - written and placed in the public domain by Wei Dai
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#include "pch.h"
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#ifndef CRYPTOPP_IMPORTS
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#include "nbtheory.h"
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#include "modarith.h"
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#include "algparam.h"
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#include <math.h>
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#include <vector>
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#ifdef _OPENMP
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// needed in MSVC 2005 to generate correct manifest
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#include <omp.h>
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#endif
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NAMESPACE_BEGIN(CryptoPP)
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const word s_lastSmallPrime = 32719;
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struct NewPrimeTable
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{
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std::vector<word16> * operator()() const
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{
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const unsigned int maxPrimeTableSize = 3511;
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std::auto_ptr<std::vector<word16> > pPrimeTable(new std::vector<word16>);
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std::vector<word16> &primeTable = *pPrimeTable;
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primeTable.reserve(maxPrimeTableSize);
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primeTable.push_back(2);
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unsigned int testEntriesEnd = 1;
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for (unsigned int p=3; p<=s_lastSmallPrime; p+=2)
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{
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unsigned int j;
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for (j=1; j<testEntriesEnd; j++)
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if (p%primeTable[j] == 0)
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break;
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if (j == testEntriesEnd)
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{
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primeTable.push_back(p);
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testEntriesEnd = UnsignedMin(54U, primeTable.size());
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}
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}
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return pPrimeTable.release();
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}
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};
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const word16 * GetPrimeTable(unsigned int &size)
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{
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const std::vector<word16> &primeTable = Singleton<std::vector<word16>, NewPrimeTable>().Ref();
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size = (unsigned int)primeTable.size();
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return &primeTable[0];
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}
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bool IsSmallPrime(const Integer &p)
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{
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unsigned int primeTableSize;
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const word16 * primeTable = GetPrimeTable(primeTableSize);
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if (p.IsPositive() && p <= primeTable[primeTableSize-1])
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return std::binary_search(primeTable, primeTable+primeTableSize, (word16)p.ConvertToLong());
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else
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return false;
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}
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bool TrialDivision(const Integer &p, unsigned bound)
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{
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unsigned int primeTableSize;
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const word16 * primeTable = GetPrimeTable(primeTableSize);
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assert(primeTable[primeTableSize-1] >= bound);
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unsigned int i;
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for (i = 0; primeTable[i]<bound; i++)
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if ((p % primeTable[i]) == 0)
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return true;
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if (bound == primeTable[i])
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return (p % bound == 0);
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else
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return false;
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}
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bool SmallDivisorsTest(const Integer &p)
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{
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unsigned int primeTableSize;
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const word16 * primeTable = GetPrimeTable(primeTableSize);
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return !TrialDivision(p, primeTable[primeTableSize-1]);
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}
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bool IsFermatProbablePrime(const Integer &n, const Integer &b)
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{
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if (n <= 3)
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return n==2 || n==3;
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assert(n>3 && b>1 && b<n-1);
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return a_exp_b_mod_c(b, n-1, n)==1;
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}
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bool IsStrongProbablePrime(const Integer &n, const Integer &b)
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{
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if (n <= 3)
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return n==2 || n==3;
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assert(n>3 && b>1 && b<n-1);
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if ((n.IsEven() && n!=2) || GCD(b, n) != 1)
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return false;
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Integer nminus1 = (n-1);
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unsigned int a;
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// calculate a = largest power of 2 that divides (n-1)
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for (a=0; ; a++)
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if (nminus1.GetBit(a))
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break;
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Integer m = nminus1>>a;
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Integer z = a_exp_b_mod_c(b, m, n);
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if (z==1 || z==nminus1)
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return true;
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for (unsigned j=1; j<a; j++)
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{
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z = z.Squared()%n;
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if (z==nminus1)
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return true;
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if (z==1)
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return false;
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}
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return false;
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}
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bool RabinMillerTest(RandomNumberGenerator &rng, const Integer &n, unsigned int rounds)
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{
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if (n <= 3)
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return n==2 || n==3;
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assert(n>3);
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Integer b;
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for (unsigned int i=0; i<rounds; i++)
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{
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b.Randomize(rng, 2, n-2);
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if (!IsStrongProbablePrime(n, b))
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return false;
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}
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return true;
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}
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bool IsLucasProbablePrime(const Integer &n)
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{
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if (n <= 1)
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return false;
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if (n.IsEven())
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return n==2;
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assert(n>2);
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Integer b=3;
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unsigned int i=0;
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int j;
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while ((j=Jacobi(b.Squared()-4, n)) == 1)
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{
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if (++i==64 && n.IsSquare()) // avoid infinite loop if n is a square
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return false;
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++b; ++b;
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}
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if (j==0)
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return false;
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else
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return Lucas(n+1, b, n)==2;
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}
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bool IsStrongLucasProbablePrime(const Integer &n)
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{
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if (n <= 1)
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return false;
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if (n.IsEven())
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return n==2;
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assert(n>2);
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Integer b=3;
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unsigned int i=0;
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int j;
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while ((j=Jacobi(b.Squared()-4, n)) == 1)
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{
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if (++i==64 && n.IsSquare()) // avoid infinite loop if n is a square
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return false;
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++b; ++b;
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}
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if (j==0)
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return false;
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Integer n1 = n+1;
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unsigned int a;
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// calculate a = largest power of 2 that divides n1
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for (a=0; ; a++)
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if (n1.GetBit(a))
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break;
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Integer m = n1>>a;
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Integer z = Lucas(m, b, n);
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if (z==2 || z==n-2)
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return true;
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for (i=1; i<a; i++)
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{
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z = (z.Squared()-2)%n;
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if (z==n-2)
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return true;
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if (z==2)
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return false;
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}
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return false;
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}
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struct NewLastSmallPrimeSquared
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{
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Integer * operator()() const
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{
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return new Integer(Integer(s_lastSmallPrime).Squared());
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}
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};
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bool IsPrime(const Integer &p)
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{
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if (p <= s_lastSmallPrime)
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return IsSmallPrime(p);
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else if (p <= Singleton<Integer, NewLastSmallPrimeSquared>().Ref())
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return SmallDivisorsTest(p);
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else
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return SmallDivisorsTest(p) && IsStrongProbablePrime(p, 3) && IsStrongLucasProbablePrime(p);
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}
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bool VerifyPrime(RandomNumberGenerator &rng, const Integer &p, unsigned int level)
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{
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bool pass = IsPrime(p) && RabinMillerTest(rng, p, 1);
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if (level >= 1)
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pass = pass && RabinMillerTest(rng, p, 10);
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return pass;
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}
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unsigned int PrimeSearchInterval(const Integer &max)
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{
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return max.BitCount();
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}
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static inline bool FastProbablePrimeTest(const Integer &n)
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{
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return IsStrongProbablePrime(n,2);
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}
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AlgorithmParameters MakeParametersForTwoPrimesOfEqualSize(unsigned int productBitLength)
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{
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if (productBitLength < 16)
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throw InvalidArgument("invalid bit length");
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Integer minP, maxP;
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if (productBitLength%2==0)
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{
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minP = Integer(182) << (productBitLength/2-8);
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maxP = Integer::Power2(productBitLength/2)-1;
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}
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else
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{
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minP = Integer::Power2((productBitLength-1)/2);
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maxP = Integer(181) << ((productBitLength+1)/2-8);
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}
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return MakeParameters("RandomNumberType", Integer::PRIME)("Min", minP)("Max", maxP);
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}
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class PrimeSieve
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{
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public:
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// delta == 1 or -1 means double sieve with p = 2*q + delta
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PrimeSieve(const Integer &first, const Integer &last, const Integer &step, signed int delta=0);
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bool NextCandidate(Integer &c);
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void DoSieve();
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static void SieveSingle(std::vector<bool> &sieve, word16 p, const Integer &first, const Integer &step, word16 stepInv);
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Integer m_first, m_last, m_step;
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signed int m_delta;
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word m_next;
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std::vector<bool> m_sieve;
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};
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PrimeSieve::PrimeSieve(const Integer &first, const Integer &last, const Integer &step, signed int delta)
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: m_first(first), m_last(last), m_step(step), m_delta(delta), m_next(0)
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{
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DoSieve();
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}
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bool PrimeSieve::NextCandidate(Integer &c)
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{
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bool safe = SafeConvert(std::find(m_sieve.begin()+m_next, m_sieve.end(), false) - m_sieve.begin(), m_next);
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assert(safe);
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if (m_next == m_sieve.size())
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{
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m_first += long(m_sieve.size())*m_step;
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if (m_first > m_last)
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return false;
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else
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{
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m_next = 0;
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DoSieve();
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return NextCandidate(c);
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}
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}
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else
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{
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c = m_first + long(m_next)*m_step;
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++m_next;
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return true;
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}
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}
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void PrimeSieve::SieveSingle(std::vector<bool> &sieve, word16 p, const Integer &first, const Integer &step, word16 stepInv)
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{
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if (stepInv)
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{
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size_t sieveSize = sieve.size();
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size_t j = (word32(p-(first%p))*stepInv) % p;
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// if the first multiple of p is p, skip it
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if (first.WordCount() <= 1 && first + step*long(j) == p)
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j += p;
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for (; j < sieveSize; j += p)
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sieve[j] = true;
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}
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}
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void PrimeSieve::DoSieve()
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{
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unsigned int primeTableSize;
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const word16 * primeTable = GetPrimeTable(primeTableSize);
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const unsigned int maxSieveSize = 32768;
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unsigned int sieveSize = STDMIN(Integer(maxSieveSize), (m_last-m_first)/m_step+1).ConvertToLong();
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m_sieve.clear();
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m_sieve.resize(sieveSize, false);
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if (m_delta == 0)
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{
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for (unsigned int i = 0; i < primeTableSize; ++i)
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SieveSingle(m_sieve, primeTable[i], m_first, m_step, (word16)m_step.InverseMod(primeTable[i]));
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}
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else
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{
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assert(m_step%2==0);
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Integer qFirst = (m_first-m_delta) >> 1;
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Integer halfStep = m_step >> 1;
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for (unsigned int i = 0; i < primeTableSize; ++i)
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{
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word16 p = primeTable[i];
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word16 stepInv = (word16)m_step.InverseMod(p);
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SieveSingle(m_sieve, p, m_first, m_step, stepInv);
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word16 halfStepInv = 2*stepInv < p ? 2*stepInv : 2*stepInv-p;
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SieveSingle(m_sieve, p, qFirst, halfStep, halfStepInv);
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}
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}
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}
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bool FirstPrime(Integer &p, const Integer &max, const Integer &equiv, const Integer &mod, const PrimeSelector *pSelector)
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{
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assert(!equiv.IsNegative() && equiv < mod);
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Integer gcd = GCD(equiv, mod);
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if (gcd != Integer::One())
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{
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// the only possible prime p such that p%mod==equiv where GCD(mod,equiv)!=1 is GCD(mod,equiv)
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if (p <= gcd && gcd <= max && IsPrime(gcd) && (!pSelector || pSelector->IsAcceptable(gcd)))
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{
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p = gcd;
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return true;
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}
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else
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return false;
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}
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unsigned int primeTableSize;
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const word16 * primeTable = GetPrimeTable(primeTableSize);
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if (p <= primeTable[primeTableSize-1])
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{
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const word16 *pItr;
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--p;
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if (p.IsPositive())
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pItr = std::upper_bound(primeTable, primeTable+primeTableSize, (word)p.ConvertToLong());
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else
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pItr = primeTable;
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while (pItr < primeTable+primeTableSize && !(*pItr%mod == equiv && (!pSelector || pSelector->IsAcceptable(*pItr))))
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++pItr;
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if (pItr < primeTable+primeTableSize)
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{
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p = *pItr;
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return p <= max;
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}
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p = primeTable[primeTableSize-1]+1;
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}
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assert(p > primeTable[primeTableSize-1]);
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if (mod.IsOdd())
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return FirstPrime(p, max, CRT(equiv, mod, 1, 2, 1), mod<<1, pSelector);
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p += (equiv-p)%mod;
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if (p>max)
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return false;
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PrimeSieve sieve(p, max, mod);
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while (sieve.NextCandidate(p))
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{
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if ((!pSelector || pSelector->IsAcceptable(p)) && FastProbablePrimeTest(p) && IsPrime(p))
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return true;
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}
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return false;
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}
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// the following two functions are based on code and comments provided by Preda Mihailescu
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static bool ProvePrime(const Integer &p, const Integer &q)
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{
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assert(p < q*q*q);
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assert(p % q == 1);
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// this is the Quisquater test. Numbers p having passed the Lucas - Lehmer test
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// for q and verifying p < q^3 can only be built up of two factors, both = 1 mod q,
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// or be prime. The next two lines build the discriminant of a quadratic equation
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// which holds iff p is built up of two factors (excercise ... )
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Integer r = (p-1)/q;
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if (((r%q).Squared()-4*(r/q)).IsSquare())
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return false;
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unsigned int primeTableSize;
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const word16 * primeTable = GetPrimeTable(primeTableSize);
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assert(primeTableSize >= 50);
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for (int i=0; i<50; i++)
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{
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Integer b = a_exp_b_mod_c(primeTable[i], r, p);
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if (b != 1)
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return a_exp_b_mod_c(b, q, p) == 1;
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}
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return false;
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}
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Integer MihailescuProvablePrime(RandomNumberGenerator &rng, unsigned int pbits)
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{
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Integer p;
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Integer minP = Integer::Power2(pbits-1);
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Integer maxP = Integer::Power2(pbits) - 1;
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if (maxP <= Integer(s_lastSmallPrime).Squared())
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{
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// Randomize() will generate a prime provable by trial division
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p.Randomize(rng, minP, maxP, Integer::PRIME);
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return p;
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}
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unsigned int qbits = (pbits+2)/3 + 1 + rng.GenerateWord32(0, pbits/36);
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Integer q = MihailescuProvablePrime(rng, qbits);
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Integer q2 = q<<1;
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while (true)
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{
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// this initializes the sieve to search in the arithmetic
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// progression p = p_0 + \lambda * q2 = p_0 + 2 * \lambda * q,
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// with q the recursively generated prime above. We will be able
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// to use Lucas tets for proving primality. A trick of Quisquater
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// allows taking q > cubic_root(p) rather then square_root: this
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// decreases the recursion.
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p.Randomize(rng, minP, maxP, Integer::ANY, 1, q2);
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PrimeSieve sieve(p, STDMIN(p+PrimeSearchInterval(maxP)*q2, maxP), q2);
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while (sieve.NextCandidate(p))
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{
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if (FastProbablePrimeTest(p) && ProvePrime(p, q))
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return p;
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}
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}
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// not reached
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return p;
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}
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Integer MaurerProvablePrime(RandomNumberGenerator &rng, unsigned int bits)
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{
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const unsigned smallPrimeBound = 29, c_opt=10;
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Integer p;
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unsigned int primeTableSize;
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const word16 * primeTable = GetPrimeTable(primeTableSize);
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if (bits < smallPrimeBound)
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{
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do
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p.Randomize(rng, Integer::Power2(bits-1), Integer::Power2(bits)-1, Integer::ANY, 1, 2);
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while (TrialDivision(p, 1 << ((bits+1)/2)));
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}
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else
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{
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const unsigned margin = bits > 50 ? 20 : (bits-10)/2;
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double relativeSize;
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do
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relativeSize = pow(2.0, double(rng.GenerateWord32())/0xffffffff - 1);
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while (bits * relativeSize >= bits - margin);
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Integer a,b;
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Integer q = MaurerProvablePrime(rng, unsigned(bits*relativeSize));
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Integer I = Integer::Power2(bits-2)/q;
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Integer I2 = I << 1;
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unsigned int trialDivisorBound = (unsigned int)STDMIN((unsigned long)primeTable[primeTableSize-1], (unsigned long)bits*bits/c_opt);
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bool success = false;
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while (!success)
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{
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p.Randomize(rng, I, I2, Integer::ANY);
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p *= q; p <<= 1; ++p;
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if (!TrialDivision(p, trialDivisorBound))
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{
|
|
a.Randomize(rng, 2, p-1, Integer::ANY);
|
|
b = a_exp_b_mod_c(a, (p-1)/q, p);
|
|
success = (GCD(b-1, p) == 1) && (a_exp_b_mod_c(b, q, p) == 1);
|
|
}
|
|
}
|
|
}
|
|
return p;
|
|
}
|
|
|
|
Integer CRT(const Integer &xp, const Integer &p, const Integer &xq, const Integer &q, const Integer &u)
|
|
{
|
|
// isn't operator overloading great?
|
|
return p * (u * (xq-xp) % q) + xp;
|
|
/*
|
|
Integer t1 = xq-xp;
|
|
cout << hex << t1 << endl;
|
|
Integer t2 = u * t1;
|
|
cout << hex << t2 << endl;
|
|
Integer t3 = t2 % q;
|
|
cout << hex << t3 << endl;
|
|
Integer t4 = p * t3;
|
|
cout << hex << t4 << endl;
|
|
Integer t5 = t4 + xp;
|
|
cout << hex << t5 << endl;
|
|
return t5;
|
|
*/
|
|
}
|
|
|
|
Integer ModularSquareRoot(const Integer &a, const Integer &p)
|
|
{
|
|
if (p%4 == 3)
|
|
return a_exp_b_mod_c(a, (p+1)/4, p);
|
|
|
|
Integer q=p-1;
|
|
unsigned int r=0;
|
|
while (q.IsEven())
|
|
{
|
|
r++;
|
|
q >>= 1;
|
|
}
|
|
|
|
Integer n=2;
|
|
while (Jacobi(n, p) != -1)
|
|
++n;
|
|
|
|
Integer y = a_exp_b_mod_c(n, q, p);
|
|
Integer x = a_exp_b_mod_c(a, (q-1)/2, p);
|
|
Integer b = (x.Squared()%p)*a%p;
|
|
x = a*x%p;
|
|
Integer tempb, t;
|
|
|
|
while (b != 1)
|
|
{
|
|
unsigned m=0;
|
|
tempb = b;
|
|
do
|
|
{
|
|
m++;
|
|
b = b.Squared()%p;
|
|
if (m==r)
|
|
return Integer::Zero();
|
|
}
|
|
while (b != 1);
|
|
|
|
t = y;
|
|
for (unsigned i=0; i<r-m-1; i++)
|
|
t = t.Squared()%p;
|
|
y = t.Squared()%p;
|
|
r = m;
|
|
x = x*t%p;
|
|
b = tempb*y%p;
|
|
}
|
|
|
|
assert(x.Squared()%p == a);
|
|
return x;
|
|
}
|
|
|
|
bool SolveModularQuadraticEquation(Integer &r1, Integer &r2, const Integer &a, const Integer &b, const Integer &c, const Integer &p)
|
|
{
|
|
Integer D = (b.Squared() - 4*a*c) % p;
|
|
switch (Jacobi(D, p))
|
|
{
|
|
default:
|
|
assert(false); // not reached
|
|
return false;
|
|
case -1:
|
|
return false;
|
|
case 0:
|
|
r1 = r2 = (-b*(a+a).InverseMod(p)) % p;
|
|
assert(((r1.Squared()*a + r1*b + c) % p).IsZero());
|
|
return true;
|
|
case 1:
|
|
Integer s = ModularSquareRoot(D, p);
|
|
Integer t = (a+a).InverseMod(p);
|
|
r1 = (s-b)*t % p;
|
|
r2 = (-s-b)*t % p;
|
|
assert(((r1.Squared()*a + r1*b + c) % p).IsZero());
|
|
assert(((r2.Squared()*a + r2*b + c) % p).IsZero());
|
|
return true;
|
|
}
|
|
}
|
|
|
|
Integer ModularRoot(const Integer &a, const Integer &dp, const Integer &dq,
|
|
const Integer &p, const Integer &q, const Integer &u)
|
|
{
|
|
Integer p2, q2;
|
|
#pragma omp parallel
|
|
#pragma omp sections
|
|
{
|
|
#pragma omp section
|
|
p2 = ModularExponentiation((a % p), dp, p);
|
|
#pragma omp section
|
|
q2 = ModularExponentiation((a % q), dq, q);
|
|
}
|
|
return CRT(p2, p, q2, q, u);
|
|
}
|
|
|
|
Integer ModularRoot(const Integer &a, const Integer &e,
|
|
const Integer &p, const Integer &q)
|
|
{
|
|
Integer dp = EuclideanMultiplicativeInverse(e, p-1);
|
|
Integer dq = EuclideanMultiplicativeInverse(e, q-1);
|
|
Integer u = EuclideanMultiplicativeInverse(p, q);
|
|
assert(!!dp && !!dq && !!u);
|
|
return ModularRoot(a, dp, dq, p, q, u);
|
|
}
|
|
|
|
/*
|
|
Integer GCDI(const Integer &x, const Integer &y)
|
|
{
|
|
Integer a=x, b=y;
|
|
unsigned k=0;
|
|
|
|
assert(!!a && !!b);
|
|
|
|
while (a[0]==0 && b[0]==0)
|
|
{
|
|
a >>= 1;
|
|
b >>= 1;
|
|
k++;
|
|
}
|
|
|
|
while (a[0]==0)
|
|
a >>= 1;
|
|
|
|
while (b[0]==0)
|
|
b >>= 1;
|
|
|
|
while (1)
|
|
{
|
|
switch (a.Compare(b))
|
|
{
|
|
case -1:
|
|
b -= a;
|
|
while (b[0]==0)
|
|
b >>= 1;
|
|
break;
|
|
|
|
case 0:
|
|
return (a <<= k);
|
|
|
|
case 1:
|
|
a -= b;
|
|
while (a[0]==0)
|
|
a >>= 1;
|
|
break;
|
|
|
|
default:
|
|
assert(false);
|
|
}
|
|
}
|
|
}
|
|
|
|
Integer EuclideanMultiplicativeInverse(const Integer &a, const Integer &b)
|
|
{
|
|
assert(b.Positive());
|
|
|
|
if (a.Negative())
|
|
return EuclideanMultiplicativeInverse(a%b, b);
|
|
|
|
if (b[0]==0)
|
|
{
|
|
if (!b || a[0]==0)
|
|
return Integer::Zero(); // no inverse
|
|
if (a==1)
|
|
return 1;
|
|
Integer u = EuclideanMultiplicativeInverse(b, a);
|
|
if (!u)
|
|
return Integer::Zero(); // no inverse
|
|
else
|
|
return (b*(a-u)+1)/a;
|
|
}
|
|
|
|
Integer u=1, d=a, v1=b, v3=b, t1, t3, b2=(b+1)>>1;
|
|
|
|
if (a[0])
|
|
{
|
|
t1 = Integer::Zero();
|
|
t3 = -b;
|
|
}
|
|
else
|
|
{
|
|
t1 = b2;
|
|
t3 = a>>1;
|
|
}
|
|
|
|
while (!!t3)
|
|
{
|
|
while (t3[0]==0)
|
|
{
|
|
t3 >>= 1;
|
|
if (t1[0]==0)
|
|
t1 >>= 1;
|
|
else
|
|
{
|
|
t1 >>= 1;
|
|
t1 += b2;
|
|
}
|
|
}
|
|
if (t3.Positive())
|
|
{
|
|
u = t1;
|
|
d = t3;
|
|
}
|
|
else
|
|
{
|
|
v1 = b-t1;
|
|
v3 = -t3;
|
|
}
|
|
t1 = u-v1;
|
|
t3 = d-v3;
|
|
if (t1.Negative())
|
|
t1 += b;
|
|
}
|
|
if (d==1)
|
|
return u;
|
|
else
|
|
return Integer::Zero(); // no inverse
|
|
}
|
|
*/
|
|
|
|
int Jacobi(const Integer &aIn, const Integer &bIn)
|
|
{
|
|
assert(bIn.IsOdd());
|
|
|
|
Integer b = bIn, a = aIn%bIn;
|
|
int result = 1;
|
|
|
|
while (!!a)
|
|
{
|
|
unsigned i=0;
|
|
while (a.GetBit(i)==0)
|
|
i++;
|
|
a>>=i;
|
|
|
|
if (i%2==1 && (b%8==3 || b%8==5))
|
|
result = -result;
|
|
|
|
if (a%4==3 && b%4==3)
|
|
result = -result;
|
|
|
|
std::swap(a, b);
|
|
a %= b;
|
|
}
|
|
|
|
return (b==1) ? result : 0;
|
|
}
|
|
|
|
Integer Lucas(const Integer &e, const Integer &pIn, const Integer &n)
|
|
{
|
|
unsigned i = e.BitCount();
|
|
if (i==0)
|
|
return Integer::Two();
|
|
|
|
MontgomeryRepresentation m(n);
|
|
Integer p=m.ConvertIn(pIn%n), two=m.ConvertIn(Integer::Two());
|
|
Integer v=p, v1=m.Subtract(m.Square(p), two);
|
|
|
|
i--;
|
|
while (i--)
|
|
{
|
|
if (e.GetBit(i))
|
|
{
|
|
// v = (v*v1 - p) % m;
|
|
v = m.Subtract(m.Multiply(v,v1), p);
|
|
// v1 = (v1*v1 - 2) % m;
|
|
v1 = m.Subtract(m.Square(v1), two);
|
|
}
|
|
else
|
|
{
|
|
// v1 = (v*v1 - p) % m;
|
|
v1 = m.Subtract(m.Multiply(v,v1), p);
|
|
// v = (v*v - 2) % m;
|
|
v = m.Subtract(m.Square(v), two);
|
|
}
|
|
}
|
|
return m.ConvertOut(v);
|
|
}
|
|
|
|
// This is Peter Montgomery's unpublished Lucas sequence evalutation algorithm.
|
|
// The total number of multiplies and squares used is less than the binary
|
|
// algorithm (see above). Unfortunately I can't get it to run as fast as
|
|
// the binary algorithm because of the extra overhead.
|
|
/*
|
|
Integer Lucas(const Integer &n, const Integer &P, const Integer &modulus)
|
|
{
|
|
if (!n)
|
|
return 2;
|
|
|
|
#define f(A, B, C) m.Subtract(m.Multiply(A, B), C)
|
|
#define X2(A) m.Subtract(m.Square(A), two)
|
|
#define X3(A) m.Multiply(A, m.Subtract(m.Square(A), three))
|
|
|
|
MontgomeryRepresentation m(modulus);
|
|
Integer two=m.ConvertIn(2), three=m.ConvertIn(3);
|
|
Integer A=m.ConvertIn(P), B, C, p, d=n, e, r, t, T, U;
|
|
|
|
while (d!=1)
|
|
{
|
|
p = d;
|
|
unsigned int b = WORD_BITS * p.WordCount();
|
|
Integer alpha = (Integer(5)<<(2*b-2)).SquareRoot() - Integer::Power2(b-1);
|
|
r = (p*alpha)>>b;
|
|
e = d-r;
|
|
B = A;
|
|
C = two;
|
|
d = r;
|
|
|
|
while (d!=e)
|
|
{
|
|
if (d<e)
|
|
{
|
|
swap(d, e);
|
|
swap(A, B);
|
|
}
|
|
|
|
unsigned int dm2 = d[0], em2 = e[0];
|
|
unsigned int dm3 = d%3, em3 = e%3;
|
|
|
|
// if ((dm6+em6)%3 == 0 && d <= e + (e>>2))
|
|
if ((dm3+em3==0 || dm3+em3==3) && (t = e, t >>= 2, t += e, d <= t))
|
|
{
|
|
// #1
|
|
// t = (d+d-e)/3;
|
|
// t = d; t += d; t -= e; t /= 3;
|
|
// e = (e+e-d)/3;
|
|
// e += e; e -= d; e /= 3;
|
|
// d = t;
|
|
|
|
// t = (d+e)/3
|
|
t = d; t += e; t /= 3;
|
|
e -= t;
|
|
d -= t;
|
|
|
|
T = f(A, B, C);
|
|
U = f(T, A, B);
|
|
B = f(T, B, A);
|
|
A = U;
|
|
continue;
|
|
}
|
|
|
|
// if (dm6 == em6 && d <= e + (e>>2))
|
|
if (dm3 == em3 && dm2 == em2 && (t = e, t >>= 2, t += e, d <= t))
|
|
{
|
|
// #2
|
|
// d = (d-e)>>1;
|
|
d -= e; d >>= 1;
|
|
B = f(A, B, C);
|
|
A = X2(A);
|
|
continue;
|
|
}
|
|
|
|
// if (d <= (e<<2))
|
|
if (d <= (t = e, t <<= 2))
|
|
{
|
|
// #3
|
|
d -= e;
|
|
C = f(A, B, C);
|
|
swap(B, C);
|
|
continue;
|
|
}
|
|
|
|
if (dm2 == em2)
|
|
{
|
|
// #4
|
|
// d = (d-e)>>1;
|
|
d -= e; d >>= 1;
|
|
B = f(A, B, C);
|
|
A = X2(A);
|
|
continue;
|
|
}
|
|
|
|
if (dm2 == 0)
|
|
{
|
|
// #5
|
|
d >>= 1;
|
|
C = f(A, C, B);
|
|
A = X2(A);
|
|
continue;
|
|
}
|
|
|
|
if (dm3 == 0)
|
|
{
|
|
// #6
|
|
// d = d/3 - e;
|
|
d /= 3; d -= e;
|
|
T = X2(A);
|
|
C = f(T, f(A, B, C), C);
|
|
swap(B, C);
|
|
A = f(T, A, A);
|
|
continue;
|
|
}
|
|
|
|
if (dm3+em3==0 || dm3+em3==3)
|
|
{
|
|
// #7
|
|
// d = (d-e-e)/3;
|
|
d -= e; d -= e; d /= 3;
|
|
T = f(A, B, C);
|
|
B = f(T, A, B);
|
|
A = X3(A);
|
|
continue;
|
|
}
|
|
|
|
if (dm3 == em3)
|
|
{
|
|
// #8
|
|
// d = (d-e)/3;
|
|
d -= e; d /= 3;
|
|
T = f(A, B, C);
|
|
C = f(A, C, B);
|
|
B = T;
|
|
A = X3(A);
|
|
continue;
|
|
}
|
|
|
|
assert(em2 == 0);
|
|
// #9
|
|
e >>= 1;
|
|
C = f(C, B, A);
|
|
B = X2(B);
|
|
}
|
|
|
|
A = f(A, B, C);
|
|
}
|
|
|
|
#undef f
|
|
#undef X2
|
|
#undef X3
|
|
|
|
return m.ConvertOut(A);
|
|
}
|
|
*/
|
|
|
|
Integer InverseLucas(const Integer &e, const Integer &m, const Integer &p, const Integer &q, const Integer &u)
|
|
{
|
|
Integer d = (m*m-4);
|
|
Integer p2, q2;
|
|
#pragma omp parallel
|
|
#pragma omp sections
|
|
{
|
|
#pragma omp section
|
|
{
|
|
p2 = p-Jacobi(d,p);
|
|
p2 = Lucas(EuclideanMultiplicativeInverse(e,p2), m, p);
|
|
}
|
|
#pragma omp section
|
|
{
|
|
q2 = q-Jacobi(d,q);
|
|
q2 = Lucas(EuclideanMultiplicativeInverse(e,q2), m, q);
|
|
}
|
|
}
|
|
return CRT(p2, p, q2, q, u);
|
|
}
|
|
|
|
unsigned int FactoringWorkFactor(unsigned int n)
|
|
{
|
|
// extrapolated from the table in Odlyzko's "The Future of Integer Factorization"
|
|
// updated to reflect the factoring of RSA-130
|
|
if (n<5) return 0;
|
|
else return (unsigned int)(2.4 * pow((double)n, 1.0/3.0) * pow(log(double(n)), 2.0/3.0) - 5);
|
|
}
|
|
|
|
unsigned int DiscreteLogWorkFactor(unsigned int n)
|
|
{
|
|
// assuming discrete log takes about the same time as factoring
|
|
if (n<5) return 0;
|
|
else return (unsigned int)(2.4 * pow((double)n, 1.0/3.0) * pow(log(double(n)), 2.0/3.0) - 5);
|
|
}
|
|
|
|
// ********************************************************
|
|
|
|
void PrimeAndGenerator::Generate(signed int delta, RandomNumberGenerator &rng, unsigned int pbits, unsigned int qbits)
|
|
{
|
|
// no prime exists for delta = -1, qbits = 4, and pbits = 5
|
|
assert(qbits > 4);
|
|
assert(pbits > qbits);
|
|
|
|
if (qbits+1 == pbits)
|
|
{
|
|
Integer minP = Integer::Power2(pbits-1);
|
|
Integer maxP = Integer::Power2(pbits) - 1;
|
|
bool success = false;
|
|
|
|
while (!success)
|
|
{
|
|
p.Randomize(rng, minP, maxP, Integer::ANY, 6+5*delta, 12);
|
|
PrimeSieve sieve(p, STDMIN(p+PrimeSearchInterval(maxP)*12, maxP), 12, delta);
|
|
|
|
while (sieve.NextCandidate(p))
|
|
{
|
|
assert(IsSmallPrime(p) || SmallDivisorsTest(p));
|
|
q = (p-delta) >> 1;
|
|
assert(IsSmallPrime(q) || SmallDivisorsTest(q));
|
|
if (FastProbablePrimeTest(q) && FastProbablePrimeTest(p) && IsPrime(q) && IsPrime(p))
|
|
{
|
|
success = true;
|
|
break;
|
|
}
|
|
}
|
|
}
|
|
|
|
if (delta == 1)
|
|
{
|
|
// find g such that g is a quadratic residue mod p, then g has order q
|
|
// g=4 always works, but this way we get the smallest quadratic residue (other than 1)
|
|
for (g=2; Jacobi(g, p) != 1; ++g) {}
|
|
// contributed by Walt Tuvell: g should be the following according to the Law of Quadratic Reciprocity
|
|
assert((p%8==1 || p%8==7) ? g==2 : (p%12==1 || p%12==11) ? g==3 : g==4);
|
|
}
|
|
else
|
|
{
|
|
assert(delta == -1);
|
|
// find g such that g*g-4 is a quadratic non-residue,
|
|
// and such that g has order q
|
|
for (g=3; ; ++g)
|
|
if (Jacobi(g*g-4, p)==-1 && Lucas(q, g, p)==2)
|
|
break;
|
|
}
|
|
}
|
|
else
|
|
{
|
|
Integer minQ = Integer::Power2(qbits-1);
|
|
Integer maxQ = Integer::Power2(qbits) - 1;
|
|
Integer minP = Integer::Power2(pbits-1);
|
|
Integer maxP = Integer::Power2(pbits) - 1;
|
|
|
|
do
|
|
{
|
|
q.Randomize(rng, minQ, maxQ, Integer::PRIME);
|
|
} while (!p.Randomize(rng, minP, maxP, Integer::PRIME, delta%q, q));
|
|
|
|
// find a random g of order q
|
|
if (delta==1)
|
|
{
|
|
do
|
|
{
|
|
Integer h(rng, 2, p-2, Integer::ANY);
|
|
g = a_exp_b_mod_c(h, (p-1)/q, p);
|
|
} while (g <= 1);
|
|
assert(a_exp_b_mod_c(g, q, p)==1);
|
|
}
|
|
else
|
|
{
|
|
assert(delta==-1);
|
|
do
|
|
{
|
|
Integer h(rng, 3, p-1, Integer::ANY);
|
|
if (Jacobi(h*h-4, p)==1)
|
|
continue;
|
|
g = Lucas((p+1)/q, h, p);
|
|
} while (g <= 2);
|
|
assert(Lucas(q, g, p) == 2);
|
|
}
|
|
}
|
|
}
|
|
|
|
NAMESPACE_END
|
|
|
|
#endif
|