algebra.cpp

// algebra.cpp - originally written and placed in the public domain by Wei Dai
 
#include "pch.h"
 
#ifndef CRYPTOPP_ALGEBRA_CPP    // SunCC workaround: compiler could cause this file to be included twice
#define CRYPTOPP_ALGEBRA_CPP
 
#include "algebra.h"
#include "integer.h"
 
#include <vector>
 
NAMESPACE_BEGIN(CryptoPP)
 
template <class T> const T& AbstractGroup<T>::Double(const Element &a) const
{
    return this->Add(a, a);
}
 
template <class T> const T& AbstractGroup<T>::Subtract(const Element &a, const Element &b) const
{
    // make copy of a in case Inverse() overwrites it
    Element a1(a);
    return this->Add(a1, Inverse(b));
}
 
template <class T> T& AbstractGroup<T>::Accumulate(Element &a, const Element &b) const
{
    return a = this->Add(a, b);
}
 
template <class T> T& AbstractGroup<T>::Reduce(Element &a, const Element &b) const
{
    return a = this->Subtract(a, b);
}
 
template <class T> const T& AbstractRing<T>::Square(const Element &a) const
{
    return this->Multiply(a, a);
}
 
template <class T> const T& AbstractRing<T>::Divide(const Element &a, const Element &b) const
{
    // make copy of a in case MultiplicativeInverse() overwrites it
    Element a1(a);
    return this->Multiply(a1, this->MultiplicativeInverse(b));
}
 
template <class T> const T& AbstractEuclideanDomain<T>::Mod(const Element &a, const Element &b) const
{
    Element q;
    this->DivisionAlgorithm(result, q, a, b);
    return result;
}
 
template <class T> const T& AbstractEuclideanDomain<T>::Gcd(const Element &a, const Element &b) const
{
    Element g[3]={b, a};
    unsigned int i0=0, i1=1, i2=2;
 
    while (!this->Equal(g[i1], this->Identity()))
    {
        g[i2] = this->Mod(g[i0], g[i1]);
        unsigned int t = i0; i0 = i1; i1 = i2; i2 = t;
    }
 
    return result = g[i0];
}
 
template <class T> const typename QuotientRing<T>::Element& QuotientRing<T>::MultiplicativeInverse(const Element &a) const
{
    Element g[3]={m_modulus, a};
    Element v[3]={m_domain.Identity(), m_domain.MultiplicativeIdentity()};
    Element y;
    unsigned int i0=0, i1=1, i2=2;
 
    while (!this->Equal(g[i1], this->Identity()))
    {
        // y = g[i0] / g[i1];
        // g[i2] = g[i0] % g[i1];
        m_domain.DivisionAlgorithm(g[i2], y, g[i0], g[i1]);
        // v[i2] = v[i0] - (v[i1] * y);
        v[i2] = m_domain.Subtract(v[i0], m_domain.Multiply(v[i1], y));
        unsigned int t = i0; i0 = i1; i1 = i2; i2 = t;
    }
 
    return m_domain.IsUnit(g[i0]) ? m_domain.Divide(v[i0], g[i0]) : m_domain.Identity();
}
 
template <class T> T AbstractGroup<T>::ScalarMultiply(const Element &base, const Integer &exponent) const
{
    Element result;
    this->SimultaneousMultiply(&result, base, &exponent, 1);
    return result;
}
 
template <class T> T AbstractGroup<T>::CascadeScalarMultiply(const Element &x, const Integer &e1, const Element &y, const Integer &e2) const
{
    const unsigned expLen = STDMAX(e1.BitCount(), e2.BitCount());
    if (expLen==0)
        return this->Identity();
 
    const unsigned w = (expLen <= 46 ? 1 : (expLen <= 260 ? 2 : 3));
    const unsigned tableSize = 1<<w;
    std::vector<Element> powerTable(tableSize << w);
 
    powerTable[1] = x;
    powerTable[tableSize] = y;
    if (w==1)
        powerTable[3] = this->Add(x,y);
    else
    {
        powerTable[2] = this->Double(x);
        powerTable[2*tableSize] = this->Double(y);
 
        unsigned i, j;
 
        for (i=3; i<tableSize; i+=2)
            powerTable[i] = Add(powerTable[i-2], powerTable[2]);
        for (i=1; i<tableSize; i+=2)
            for (j=i+tableSize; j<(tableSize<<w); j+=tableSize)
                powerTable[j] = Add(powerTable[j-tableSize], y);
 
        for (i=3*tableSize; i<(tableSize<<w); i+=2*tableSize)
            powerTable[i] = Add(powerTable[i-2*tableSize], powerTable[2*tableSize]);
        for (i=tableSize; i<(tableSize<<w); i+=2*tableSize)
            for (j=i+2; j<i+tableSize; j+=2)
                powerTable[j] = Add(powerTable[j-1], x);
    }
 
    Element result;
    unsigned power1 = 0, power2 = 0, prevPosition = expLen-1;
    bool firstTime = true;
 
    for (int i = expLen-1; i>=0; i--)
    {
        power1 = 2*power1 + e1.GetBit(i);
        power2 = 2*power2 + e2.GetBit(i);
 
        if (i==0 || 2*power1 >= tableSize || 2*power2 >= tableSize)
        {
            unsigned squaresBefore = prevPosition-i;
            unsigned squaresAfter = 0;
            prevPosition = i;
            while ((power1 || power2) && power1%2 == 0 && power2%2==0)
            {
                power1 /= 2;
                power2 /= 2;
                squaresBefore--;
                squaresAfter++;
            }
            if (firstTime)
            {
                result = powerTable[(power2<<w) + power1];
                firstTime = false;
            }
            else
            {
                while (squaresBefore--)
                    result = this->Double(result);
                if (power1 || power2)
                    Accumulate(result, powerTable[(power2<<w) + power1]);
            }
            while (squaresAfter--)
                result = this->Double(result);
            power1 = power2 = 0;
        }
    }
    return result;
}
 
template <class Element, class Iterator> Element GeneralCascadeMultiplication(const AbstractGroup<Element> &group, Iterator begin, Iterator end)
{
    if (end-begin == 1)
        return group.ScalarMultiply(begin->base, begin->exponent);
    else if (end-begin == 2)
        return group.CascadeScalarMultiply(begin->base, begin->exponent, (begin+1)->base, (begin+1)->exponent);
    else
    {
        Integer q, t;
        Iterator last = end;
        --last;
 
        std::make_heap(begin, end);
        std::pop_heap(begin, end);
 
        while (!!begin->exponent)
        {
            // last->exponent is largest exponent, begin->exponent is next largest
            t = last->exponent;
            Integer::Divide(last->exponent, q, t, begin->exponent);
 
            if (q == Integer::One())
                group.Accumulate(begin->base, last->base);  // avoid overhead of ScalarMultiply()
            else
                group.Accumulate(begin->base, group.ScalarMultiply(last->base, q));
 
            std::push_heap(begin, end);
            std::pop_heap(begin, end);
        }
 
        return group.ScalarMultiply(last->base, last->exponent);
    }
}
 
struct WindowSlider
{
    WindowSlider(const Integer &expIn, bool fastNegate, unsigned int windowSizeIn=0)
        : exp(expIn), windowModulus(Integer::One()), windowSize(windowSizeIn), windowBegin(0), expWindow(0)
        , fastNegate(fastNegate), negateNext(false), firstTime(true), finished(false)
    {
        if (windowSize == 0)
        {
            unsigned int expLen = exp.BitCount();
            windowSize = expLen <= 17 ? 1 : (expLen <= 24 ? 2 : (expLen <= 70 ? 3 : (expLen <= 197 ? 4 : (expLen <= 539 ? 5 : (expLen <= 1434 ? 6 : 7)))));
        }
        windowModulus <<= windowSize;
    }
 
    void FindNextWindow()
    {
        unsigned int expLen = exp.WordCount() * WORD_BITS;
        unsigned int skipCount = firstTime ? 0 : windowSize;
        firstTime = false;
        while (!exp.GetBit(skipCount))
        {
            if (skipCount >= expLen)
            {
                finished = true;
                return;
            }
            skipCount++;
        }
 
        exp >>= skipCount;
        windowBegin += skipCount;
        expWindow = word32(exp % (word(1) << windowSize));
 
        if (fastNegate && exp.GetBit(windowSize))
        {
            negateNext = true;
            expWindow = (word32(1) << windowSize) - expWindow;
            exp += windowModulus;
        }
        else
            negateNext = false;
    }
 
    Integer exp, windowModulus;
    unsigned int windowSize, windowBegin;
    word32 expWindow;
    bool fastNegate, negateNext, firstTime, finished;
};
 
template <class T>
void AbstractGroup<T>::SimultaneousMultiply(T *results, const T &base, const Integer *expBegin, unsigned int expCount) const
{
    std::vector<std::vector<Element> > buckets(expCount);
    std::vector<WindowSlider> exponents;
    exponents.reserve(expCount);
    unsigned int i;
 
    for (i=0; expBegin && i<expCount; i++)
    {
        CRYPTOPP_ASSERT(expBegin->NotNegative());
        exponents.push_back(WindowSlider(*expBegin++, InversionIsFast(), 0));
        exponents[i].FindNextWindow();
        buckets[i].resize(((size_t) 1) << (exponents[i].windowSize-1), Identity());
    }
 
    unsigned int expBitPosition = 0;
    Element g = base;
    bool notDone = true;
 
    while (notDone)
    {
        notDone = false;
        for (i=0; i<expCount; i++)
        {
            if (!exponents[i].finished && expBitPosition == exponents[i].windowBegin)
            {
                Element &bucket = buckets[i][exponents[i].expWindow/2];
                if (exponents[i].negateNext)
                    Accumulate(bucket, Inverse(g));
                else
                    Accumulate(bucket, g);
                exponents[i].FindNextWindow();
            }
            notDone = notDone || !exponents[i].finished;
        }
 
        if (notDone)
        {
            g = Double(g);
            expBitPosition++;
        }
    }
 
    for (i=0; i<expCount; i++)
    {
        Element &r = *results++;
        r = buckets[i][buckets[i].size()-1];
        if (buckets[i].size() > 1)
        {
            for (int j = (int)buckets[i].size()-2; j >= 1; j--)
            {
                Accumulate(buckets[i][j], buckets[i][j+1]);
                Accumulate(r, buckets[i][j]);
            }
            Accumulate(buckets[i][0], buckets[i][1]);
            r = Add(Double(r), buckets[i][0]);
        }
    }
}
 
template <class T> T AbstractRing<T>::Exponentiate(const Element &base, const Integer &exponent) const
{
    Element result;
    SimultaneousExponentiate(&result, base, &exponent, 1);
    return result;
}
 
template <class T> T AbstractRing<T>::CascadeExponentiate(const Element &x, const Integer &e1, const Element &y, const Integer &e2) const
{
    return MultiplicativeGroup().AbstractGroup<T>::CascadeScalarMultiply(x, e1, y, e2);
}
 
template <class Element, class Iterator> Element GeneralCascadeExponentiation(const AbstractRing<Element> &ring, Iterator begin, Iterator end)
{
    return GeneralCascadeMultiplication<Element>(ring.MultiplicativeGroup(), begin, end);
}
 
template <class T>
void AbstractRing<T>::SimultaneousExponentiate(T *results, const T &base, const Integer *exponents, unsigned int expCount) const
{
    MultiplicativeGroup().AbstractGroup<T>::SimultaneousMultiply(results, base, exponents, expCount);
}
 
NAMESPACE_END
 
#endif