nbtheory.cpp

// nbtheory.cpp - originally written and placed in the public domain by Wei Dai
 
#include "pch.h"
 
#ifndef CRYPTOPP_IMPORTS
 
#include "nbtheory.h"
#include "integer.h"
#include "modarith.h"
#include "algparam.h"
#include "smartptr.h"
#include "misc.h"
#include "stdcpp.h"
#include "trap.h"
 
#ifdef _OPENMP
# include <omp.h>
#endif
 
NAMESPACE_BEGIN(CryptoPP)
 
const word s_lastSmallPrime = 32719;
 
struct NewPrimeTable
{
    std::vector<word16> * operator()() const
    {
        const unsigned int maxPrimeTableSize = 3511;
 
        member_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(word16(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);
 
    CRYPTOPP_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;
 
    CRYPTOPP_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;
 
    CRYPTOPP_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;
 
    CRYPTOPP_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;
 
    CRYPTOPP_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;
 
    CRYPTOPP_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);
    CRYPTOPP_UNUSED(safe); CRYPTOPP_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
    {
        CRYPTOPP_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)
{
    CRYPTOPP_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;
    }
 
    CRYPTOPP_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)
{
    CRYPTOPP_ASSERT(p < q*q*q);
    CRYPTOPP_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 (exercise ... )
 
    Integer r = (p-1)/q;
    if (((r%q).Squared()-4*(r/q)).IsSquare())
        return false;
 
    unsigned int primeTableSize;
    const word16 * primeTable = GetPrimeTable(primeTableSize);
 
    CRYPTOPP_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 than 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 = std::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)
{
    // Callers must ensure p and q are prime, GH #1249
    CRYPTOPP_ASSERT(IsPrime(p) && IsPrime(q));
 
    // 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)
{
    // Callers must ensure p is prime, GH #1249
    CRYPTOPP_ASSERT(IsPrime(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;
    }
 
    CRYPTOPP_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)
{
    // Callers must ensure p is prime, GH #1249
    CRYPTOPP_ASSERT(IsPrime(p));
 
    Integer D = (b.Squared() - 4*a*c) % p;
    switch (Jacobi(D, p))
    {
    default:
        CRYPTOPP_ASSERT(false); // not reached
        return false;
    case -1:
        return false;
    case 0:
        r1 = r2 = (-b*(a+a).InverseMod(p)) % p;
        CRYPTOPP_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;
        CRYPTOPP_ASSERT(((r1.Squared()*a + r1*b + c) % p).IsZero());
        CRYPTOPP_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)
{
    // Callers must ensure p and q are prime, GH #1249
    CRYPTOPP_ASSERT(IsPrime(p) && IsPrime(q));
 
    // GCC warning bug, https://stackoverflow.com/q/12842306/608639
#ifdef _OPENMP
    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);
        }
#else
    const Integer p2 = ModularExponentiation((a % p), dp, p);
    const Integer q2 = ModularExponentiation((a % q), dq, q);
#endif
 
    return CRT(p2, p, q2, q, u);
}
 
Integer ModularRoot(const Integer &a, const Integer &e,
                    const Integer &p, const Integer &q)
{
    // Callers must ensure p and q are prime, GH #1249
    CRYPTOPP_ASSERT(IsPrime(p) && IsPrime(q));
    
    Integer dp = EuclideanMultiplicativeInverse(e, p-1);
    Integer dq = EuclideanMultiplicativeInverse(e, q-1);
    Integer u = EuclideanMultiplicativeInverse(p, q);
    CRYPTOPP_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;
 
    CRYPTOPP_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:
                CRYPTOPP_ASSERT(false);
        }
    }
}
 
Integer EuclideanMultiplicativeInverse(const Integer &a, const Integer &b)
{
    CRYPTOPP_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)
{
    CRYPTOPP_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;
            }
 
            CRYPTOPP_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)
{
    // Callers must ensure p and q are prime, GH #1249
    CRYPTOPP_ASSERT(IsPrime(p) && IsPrime(q));
 
    // GCC warning bug, https://stackoverflow.com/q/12842306/608639
#ifdef _OPENMP
    Integer d = (m*m-4), 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);
            }
        }
#else
    const Integer d = (m*m-4);
    const Integer t1 = p-Jacobi(d,p);
    const Integer p2 = Lucas(EuclideanMultiplicativeInverse(e,t1), m, p);
 
    const Integer t2 = q-Jacobi(d,q);
    const Integer q2 = Lucas(EuclideanMultiplicativeInverse(e,t2), m, q);
#endif
 
    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 * std::pow((double)n, 1.0/3.0) * std::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 * std::pow((double)n, 1.0/3.0) * std::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
    CRYPTOPP_ASSERT(qbits > 4);
    CRYPTOPP_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))
            {
                CRYPTOPP_ASSERT(IsSmallPrime(p) || SmallDivisorsTest(p));
                q = (p-delta) >> 1;
                CRYPTOPP_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
            CRYPTOPP_ASSERT((p%8==1 || p%8==7) ? g==2 : (p%12==1 || p%12==11) ? g==3 : g==4);
        }
        else
        {
            CRYPTOPP_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);
            CRYPTOPP_ASSERT(a_exp_b_mod_c(g, q, p)==1);
        }
        else
        {
            CRYPTOPP_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);
            CRYPTOPP_ASSERT(Lucas(q, g, p) == 2);
        }
    }
}
 
NAMESPACE_END
 
#endif