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-rw-r--r--lib/cryptopp/nbtheory.cpp1123
1 files changed, 0 insertions, 1123 deletions
diff --git a/lib/cryptopp/nbtheory.cpp b/lib/cryptopp/nbtheory.cpp
deleted file mode 100644
index 3fdea4e69..000000000
--- a/lib/cryptopp/nbtheory.cpp
+++ /dev/null
@@ -1,1123 +0,0 @@
-// nbtheory.cpp - written and placed in the public domain by Wei Dai
-
-#include "pch.h"
-
-#ifndef CRYPTOPP_IMPORTS
-
-#include "nbtheory.h"
-#include "modarith.h"
-#include "algparam.h"
-
-#include <math.h>
-#include <vector>
-
-#ifdef _OPENMP
-// needed in MSVC 2005 to generate correct manifest
-#include <omp.h>
-#endif
-
-NAMESPACE_BEGIN(CryptoPP)
-
-const word s_lastSmallPrime = 32719;
-
-struct NewPrimeTable
-{
- std::vector<word16> * operator()() const
- {
- const unsigned int maxPrimeTableSize = 3511;
-
- std::auto_ptr<std::vector<word16> > pPrimeTable(new std::vector<word16>);
- std::vector<word16> &primeTable = *pPrimeTable;
- primeTable.reserve(maxPrimeTableSize);
-
- primeTable.push_back(2);
- unsigned int testEntriesEnd = 1;
-
- for (unsigned int p=3; p<=s_lastSmallPrime; p+=2)
- {
- unsigned int j;
- for (j=1; j<testEntriesEnd; j++)
- if (p%primeTable[j] == 0)
- break;
- if (j == testEntriesEnd)
- {
- primeTable.push_back(p);
- testEntriesEnd = UnsignedMin(54U, primeTable.size());
- }
- }
-
- return pPrimeTable.release();
- }
-};
-
-const word16 * GetPrimeTable(unsigned int &size)
-{
- const std::vector<word16> &primeTable = Singleton<std::vector<word16>, NewPrimeTable>().Ref();
- size = (unsigned int)primeTable.size();
- return &primeTable[0];
-}
-
-bool IsSmallPrime(const Integer &p)
-{
- unsigned int primeTableSize;
- const word16 * primeTable = GetPrimeTable(primeTableSize);
-
- if (p.IsPositive() && p <= primeTable[primeTableSize-1])
- return std::binary_search(primeTable, primeTable+primeTableSize, (word16)p.ConvertToLong());
- else
- return false;
-}
-
-bool TrialDivision(const Integer &p, unsigned bound)
-{
- unsigned int primeTableSize;
- const word16 * primeTable = GetPrimeTable(primeTableSize);
-
- assert(primeTable[primeTableSize-1] >= bound);
-
- unsigned int i;
- for (i = 0; primeTable[i]<bound; i++)
- if ((p % primeTable[i]) == 0)
- return true;
-
- if (bound == primeTable[i])
- return (p % bound == 0);
- else
- return false;
-}
-
-bool SmallDivisorsTest(const Integer &p)
-{
- unsigned int primeTableSize;
- const word16 * primeTable = GetPrimeTable(primeTableSize);
- return !TrialDivision(p, primeTable[primeTableSize-1]);
-}
-
-bool IsFermatProbablePrime(const Integer &n, const Integer &b)
-{
- if (n <= 3)
- return n==2 || n==3;
-
- assert(n>3 && b>1 && b<n-1);
- return a_exp_b_mod_c(b, n-1, n)==1;
-}
-
-bool IsStrongProbablePrime(const Integer &n, const Integer &b)
-{
- if (n <= 3)
- return n==2 || n==3;
-
- assert(n>3 && b>1 && b<n-1);
-
- if ((n.IsEven() && n!=2) || GCD(b, n) != 1)
- return false;
-
- Integer nminus1 = (n-1);
- unsigned int a;
-
- // calculate a = largest power of 2 that divides (n-1)
- for (a=0; ; a++)
- if (nminus1.GetBit(a))
- break;
- Integer m = nminus1>>a;
-
- Integer z = a_exp_b_mod_c(b, m, n);
- if (z==1 || z==nminus1)
- return true;
- for (unsigned j=1; j<a; j++)
- {
- z = z.Squared()%n;
- if (z==nminus1)
- return true;
- if (z==1)
- return false;
- }
- return false;
-}
-
-bool RabinMillerTest(RandomNumberGenerator &rng, const Integer &n, unsigned int rounds)
-{
- if (n <= 3)
- return n==2 || n==3;
-
- assert(n>3);
-
- Integer b;
- for (unsigned int i=0; i<rounds; i++)
- {
- b.Randomize(rng, 2, n-2);
- if (!IsStrongProbablePrime(n, b))
- return false;
- }
- return true;
-}
-
-bool IsLucasProbablePrime(const Integer &n)
-{
- if (n <= 1)
- return false;
-
- if (n.IsEven())
- return n==2;
-
- assert(n>2);
-
- Integer b=3;
- unsigned int i=0;
- int j;
-
- while ((j=Jacobi(b.Squared()-4, n)) == 1)
- {
- if (++i==64 && n.IsSquare()) // avoid infinite loop if n is a square
- return false;
- ++b; ++b;
- }
-
- if (j==0)
- return false;
- else
- return Lucas(n+1, b, n)==2;
-}
-
-bool IsStrongLucasProbablePrime(const Integer &n)
-{
- if (n <= 1)
- return false;
-
- if (n.IsEven())
- return n==2;
-
- assert(n>2);
-
- Integer b=3;
- unsigned int i=0;
- int j;
-
- while ((j=Jacobi(b.Squared()-4, n)) == 1)
- {
- if (++i==64 && n.IsSquare()) // avoid infinite loop if n is a square
- return false;
- ++b; ++b;
- }
-
- if (j==0)
- return false;
-
- Integer n1 = n+1;
- unsigned int a;
-
- // calculate a = largest power of 2 that divides n1
- for (a=0; ; a++)
- if (n1.GetBit(a))
- break;
- Integer m = n1>>a;
-
- Integer z = Lucas(m, b, n);
- if (z==2 || z==n-2)
- return true;
- for (i=1; i<a; i++)
- {
- z = (z.Squared()-2)%n;
- if (z==n-2)
- return true;
- if (z==2)
- return false;
- }
- return false;
-}
-
-struct NewLastSmallPrimeSquared
-{
- Integer * operator()() const
- {
- return new Integer(Integer(s_lastSmallPrime).Squared());
- }
-};
-
-bool IsPrime(const Integer &p)
-{
- if (p <= s_lastSmallPrime)
- return IsSmallPrime(p);
- else if (p <= Singleton<Integer, NewLastSmallPrimeSquared>().Ref())
- return SmallDivisorsTest(p);
- else
- return SmallDivisorsTest(p) && IsStrongProbablePrime(p, 3) && IsStrongLucasProbablePrime(p);
-}
-
-bool VerifyPrime(RandomNumberGenerator &rng, const Integer &p, unsigned int level)
-{
- bool pass = IsPrime(p) && RabinMillerTest(rng, p, 1);
- if (level >= 1)
- pass = pass && RabinMillerTest(rng, p, 10);
- return pass;
-}
-
-unsigned int PrimeSearchInterval(const Integer &max)
-{
- return max.BitCount();
-}
-
-static inline bool FastProbablePrimeTest(const Integer &n)
-{
- return IsStrongProbablePrime(n,2);
-}
-
-AlgorithmParameters MakeParametersForTwoPrimesOfEqualSize(unsigned int productBitLength)
-{
- if (productBitLength < 16)
- throw InvalidArgument("invalid bit length");
-
- Integer minP, maxP;
-
- if (productBitLength%2==0)
- {
- minP = Integer(182) << (productBitLength/2-8);
- maxP = Integer::Power2(productBitLength/2)-1;
- }
- else
- {
- minP = Integer::Power2((productBitLength-1)/2);
- maxP = Integer(181) << ((productBitLength+1)/2-8);
- }
-
- return MakeParameters("RandomNumberType", Integer::PRIME)("Min", minP)("Max", maxP);
-}
-
-class PrimeSieve
-{
-public:
- // delta == 1 or -1 means double sieve with p = 2*q + delta
- PrimeSieve(const Integer &first, const Integer &last, const Integer &step, signed int delta=0);
- bool NextCandidate(Integer &c);
-
- void DoSieve();
- static void SieveSingle(std::vector<bool> &sieve, word16 p, const Integer &first, const Integer &step, word16 stepInv);
-
- Integer m_first, m_last, m_step;
- signed int m_delta;
- word m_next;
- std::vector<bool> m_sieve;
-};
-
-PrimeSieve::PrimeSieve(const Integer &first, const Integer &last, const Integer &step, signed int delta)
- : m_first(first), m_last(last), m_step(step), m_delta(delta), m_next(0)
-{
- DoSieve();
-}
-
-bool PrimeSieve::NextCandidate(Integer &c)
-{
- bool safe = SafeConvert(std::find(m_sieve.begin()+m_next, m_sieve.end(), false) - m_sieve.begin(), m_next);
- assert(safe);
- if (m_next == m_sieve.size())
- {
- m_first += long(m_sieve.size())*m_step;
- if (m_first > m_last)
- return false;
- else
- {
- m_next = 0;
- DoSieve();
- return NextCandidate(c);
- }
- }
- else
- {
- c = m_first + long(m_next)*m_step;
- ++m_next;
- return true;
- }
-}
-
-void PrimeSieve::SieveSingle(std::vector<bool> &sieve, word16 p, const Integer &first, const Integer &step, word16 stepInv)
-{
- if (stepInv)
- {
- size_t sieveSize = sieve.size();
- size_t j = (word32(p-(first%p))*stepInv) % p;
- // if the first multiple of p is p, skip it
- if (first.WordCount() <= 1 && first + step*long(j) == p)
- j += p;
- for (; j < sieveSize; j += p)
- sieve[j] = true;
- }
-}
-
-void PrimeSieve::DoSieve()
-{
- unsigned int primeTableSize;
- const word16 * primeTable = GetPrimeTable(primeTableSize);
-
- const unsigned int maxSieveSize = 32768;
- unsigned int sieveSize = STDMIN(Integer(maxSieveSize), (m_last-m_first)/m_step+1).ConvertToLong();
-
- m_sieve.clear();
- m_sieve.resize(sieveSize, false);
-
- if (m_delta == 0)
- {
- for (unsigned int i = 0; i < primeTableSize; ++i)
- SieveSingle(m_sieve, primeTable[i], m_first, m_step, (word16)m_step.InverseMod(primeTable[i]));
- }
- else
- {
- assert(m_step%2==0);
- Integer qFirst = (m_first-m_delta) >> 1;
- Integer halfStep = m_step >> 1;
- for (unsigned int i = 0; i < primeTableSize; ++i)
- {
- word16 p = primeTable[i];
- word16 stepInv = (word16)m_step.InverseMod(p);
- SieveSingle(m_sieve, p, m_first, m_step, stepInv);
-
- word16 halfStepInv = 2*stepInv < p ? 2*stepInv : 2*stepInv-p;
- SieveSingle(m_sieve, p, qFirst, halfStep, halfStepInv);
- }
- }
-}
-
-bool FirstPrime(Integer &p, const Integer &max, const Integer &equiv, const Integer &mod, const PrimeSelector *pSelector)
-{
- assert(!equiv.IsNegative() && equiv < mod);
-
- Integer gcd = GCD(equiv, mod);
- if (gcd != Integer::One())
- {
- // the only possible prime p such that p%mod==equiv where GCD(mod,equiv)!=1 is GCD(mod,equiv)
- if (p <= gcd && gcd <= max && IsPrime(gcd) && (!pSelector || pSelector->IsAcceptable(gcd)))
- {
- p = gcd;
- return true;
- }
- else
- return false;
- }
-
- unsigned int primeTableSize;
- const word16 * primeTable = GetPrimeTable(primeTableSize);
-
- if (p <= primeTable[primeTableSize-1])
- {
- const word16 *pItr;
-
- --p;
- if (p.IsPositive())
- pItr = std::upper_bound(primeTable, primeTable+primeTableSize, (word)p.ConvertToLong());
- else
- pItr = primeTable;
-
- while (pItr < primeTable+primeTableSize && !(*pItr%mod == equiv && (!pSelector || pSelector->IsAcceptable(*pItr))))
- ++pItr;
-
- if (pItr < primeTable+primeTableSize)
- {
- p = *pItr;
- return p <= max;
- }
-
- p = primeTable[primeTableSize-1]+1;
- }
-
- assert(p > primeTable[primeTableSize-1]);
-
- if (mod.IsOdd())
- return FirstPrime(p, max, CRT(equiv, mod, 1, 2, 1), mod<<1, pSelector);
-
- p += (equiv-p)%mod;
-
- if (p>max)
- return false;
-
- PrimeSieve sieve(p, max, mod);
-
- while (sieve.NextCandidate(p))
- {
- if ((!pSelector || pSelector->IsAcceptable(p)) && FastProbablePrimeTest(p) && IsPrime(p))
- return true;
- }
-
- return false;
-}
-
-// the following two functions are based on code and comments provided by Preda Mihailescu
-static bool ProvePrime(const Integer &p, const Integer &q)
-{
- assert(p < q*q*q);
- assert(p % q == 1);
-
-// this is the Quisquater test. Numbers p having passed the Lucas - Lehmer test
-// for q and verifying p < q^3 can only be built up of two factors, both = 1 mod q,
-// or be prime. The next two lines build the discriminant of a quadratic equation
-// which holds iff p is built up of two factors (excercise ... )
-
- Integer r = (p-1)/q;
- if (((r%q).Squared()-4*(r/q)).IsSquare())
- return false;
-
- unsigned int primeTableSize;
- const word16 * primeTable = GetPrimeTable(primeTableSize);
-
- assert(primeTableSize >= 50);
- for (int i=0; i<50; i++)
- {
- Integer b = a_exp_b_mod_c(primeTable[i], r, p);
- if (b != 1)
- return a_exp_b_mod_c(b, q, p) == 1;
- }
- return false;
-}
-
-Integer MihailescuProvablePrime(RandomNumberGenerator &rng, unsigned int pbits)
-{
- Integer p;
- Integer minP = Integer::Power2(pbits-1);
- Integer maxP = Integer::Power2(pbits) - 1;
-
- if (maxP <= Integer(s_lastSmallPrime).Squared())
- {
- // Randomize() will generate a prime provable by trial division
- p.Randomize(rng, minP, maxP, Integer::PRIME);
- return p;
- }
-
- unsigned int qbits = (pbits+2)/3 + 1 + rng.GenerateWord32(0, pbits/36);
- Integer q = MihailescuProvablePrime(rng, qbits);
- Integer q2 = q<<1;
-
- while (true)
- {
- // this initializes the sieve to search in the arithmetic
- // progression p = p_0 + \lambda * q2 = p_0 + 2 * \lambda * q,
- // with q the recursively generated prime above. We will be able
- // to use Lucas tets for proving primality. A trick of Quisquater
- // allows taking q > cubic_root(p) rather then square_root: this
- // decreases the recursion.
-
- p.Randomize(rng, minP, maxP, Integer::ANY, 1, q2);
- PrimeSieve sieve(p, STDMIN(p+PrimeSearchInterval(maxP)*q2, maxP), q2);
-
- while (sieve.NextCandidate(p))
- {
- if (FastProbablePrimeTest(p) && ProvePrime(p, q))
- return p;
- }
- }
-
- // not reached
- return p;
-}
-
-Integer MaurerProvablePrime(RandomNumberGenerator &rng, unsigned int bits)
-{
- const unsigned smallPrimeBound = 29, c_opt=10;
- Integer p;
-
- unsigned int primeTableSize;
- const word16 * primeTable = GetPrimeTable(primeTableSize);
-
- if (bits < smallPrimeBound)
- {
- do
- p.Randomize(rng, Integer::Power2(bits-1), Integer::Power2(bits)-1, Integer::ANY, 1, 2);
- while (TrialDivision(p, 1 << ((bits+1)/2)));
- }
- else
- {
- const unsigned margin = bits > 50 ? 20 : (bits-10)/2;
- double relativeSize;
- do
- relativeSize = pow(2.0, double(rng.GenerateWord32())/0xffffffff - 1);
- while (bits * relativeSize >= bits - margin);
-
- Integer a,b;
- Integer q = MaurerProvablePrime(rng, unsigned(bits*relativeSize));
- Integer I = Integer::Power2(bits-2)/q;
- Integer I2 = I << 1;
- unsigned int trialDivisorBound = (unsigned int)STDMIN((unsigned long)primeTable[primeTableSize-1], (unsigned long)bits*bits/c_opt);
- bool success = false;
- while (!success)
- {
- p.Randomize(rng, I, I2, Integer::ANY);
- p *= q; p <<= 1; ++p;
- if (!TrialDivision(p, trialDivisorBound))
- {
- 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