nbtheory.cpp

// 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<AlgorithmParameters<AlgorithmParameters<NullNameValuePairs, Integer::RandomNumberType>, Integer>, Integer>
        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 CRT(const Integer &xp, const Integer &p, const Integer &xq, const Integer &q)
{
        return CRT(xp, p, xq, q, EuclideanMultiplicativeInverse(p, q));
}

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);
}

Integer InverseLucas(const Integer &e, const Integer &m, const Integer &p, const Integer &q)
{
        return InverseLucas(e, m, p, q, EuclideanMultiplicativeInverse(p, q));
}

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

---