ecp.cpp

// ecp.cpp - originally written and placed in the public domain by Wei Dai
 
#include "pch.h"
 
#ifndef CRYPTOPP_IMPORTS
 
#include "ecp.h"
#include "asn.h"
#include "integer.h"
#include "nbtheory.h"
#include "modarith.h"
#include "filters.h"
#include "algebra.cpp"
 
ANONYMOUS_NAMESPACE_BEGIN
 
using CryptoPP::ECP;
using CryptoPP::Integer;
using CryptoPP::ModularArithmetic;
 
#if defined(HAVE_GCC_INIT_PRIORITY)
    #define INIT_ATTRIBUTE __attribute__ ((init_priority (CRYPTOPP_INIT_PRIORITY + 50)))
    const ECP::Point g_identity INIT_ATTRIBUTE = ECP::Point();
#elif defined(HAVE_MSC_INIT_PRIORITY)
    #pragma warning(disable: 4075)
    #pragma init_seg(".CRT$XCU")
    const ECP::Point g_identity;
    #pragma warning(default: 4075)
#elif defined(HAVE_XLC_INIT_PRIORITY)
    #pragma priority(290)
    const ECP::Point g_identity;
#endif
 
inline ECP::Point ToMontgomery(const ModularArithmetic &mr, const ECP::Point &P)
{
    return P.identity ? P : ECP::Point(mr.ConvertIn(P.x), mr.ConvertIn(P.y));
}
 
inline ECP::Point FromMontgomery(const ModularArithmetic &mr, const ECP::Point &P)
{
    return P.identity ? P : ECP::Point(mr.ConvertOut(P.x), mr.ConvertOut(P.y));
}
 
inline Integer IdentityToInteger(bool val)
{
    return val ? Integer::One() : Integer::Zero();
}
 
struct ProjectivePoint
{
    ProjectivePoint() {}
    ProjectivePoint(const Integer &x, const Integer &y, const Integer &z)
        : x(x), y(y), z(z)  {}
 
    Integer x, y, z;
};
 
ANONYMOUS_NAMESPACE_END
 
NAMESPACE_BEGIN(CryptoPP)
 
ECP::ECP(const ECP &ecp, bool convertToMontgomeryRepresentation)
{
    if (convertToMontgomeryRepresentation && !ecp.GetField().IsMontgomeryRepresentation())
    {
        m_fieldPtr.reset(new MontgomeryRepresentation(ecp.GetField().GetModulus()));
        m_a = GetField().ConvertIn(ecp.m_a);
        m_b = GetField().ConvertIn(ecp.m_b);
    }
    else
        operator=(ecp);
}
 
ECP::ECP(BufferedTransformation &bt)
    : m_fieldPtr(new Field(bt))
{
    BERSequenceDecoder seq(bt);
    GetField().BERDecodeElement(seq, m_a);
    GetField().BERDecodeElement(seq, m_b);
    // skip optional seed
    if (!seq.EndReached())
    {
        SecByteBlock seed;
        unsigned int unused;
        BERDecodeBitString(seq, seed, unused);
    }
    seq.MessageEnd();
}
 
void ECP::DEREncode(BufferedTransformation &bt) const
{
    GetField().DEREncode(bt);
    DERSequenceEncoder seq(bt);
    GetField().DEREncodeElement(seq, m_a);
    GetField().DEREncodeElement(seq, m_b);
    seq.MessageEnd();
}
 
bool ECP::DecodePoint(ECP::Point &P, const byte *encodedPoint, size_t encodedPointLen) const
{
    StringStore store(encodedPoint, encodedPointLen);
    return DecodePoint(P, store, encodedPointLen);
}
 
bool ECP::DecodePoint(ECP::Point &P, BufferedTransformation &bt, size_t encodedPointLen) const
{
    byte type;
    if (encodedPointLen < 1 || !bt.Get(type))
        return false;
 
    switch (type)
    {
    case 0:
        P.identity = true;
        return true;
    case 2:
    case 3:
    {
        if (encodedPointLen != EncodedPointSize(true))
            return false;
 
        Integer p = FieldSize();
 
        P.identity = false;
        P.x.Decode(bt, GetField().MaxElementByteLength());
        P.y = ((P.x*P.x+m_a)*P.x+m_b) % p;
 
        if (Jacobi(P.y, p) !=1)
            return false;
 
        P.y = ModularSquareRoot(P.y, p);
 
        if ((type & 1) != P.y.GetBit(0))
            P.y = p-P.y;
 
        return true;
    }
    case 4:
    {
        if (encodedPointLen != EncodedPointSize(false))
            return false;
 
        unsigned int len = GetField().MaxElementByteLength();
        P.identity = false;
        P.x.Decode(bt, len);
        P.y.Decode(bt, len);
        return true;
    }
    default:
        return false;
    }
}
 
void ECP::EncodePoint(BufferedTransformation &bt, const Point &P, bool compressed) const
{
    if (P.identity)
        NullStore().TransferTo(bt, EncodedPointSize(compressed));
    else if (compressed)
    {
        bt.Put((byte)(2U + P.y.GetBit(0)));
        P.x.Encode(bt, GetField().MaxElementByteLength());
    }
    else
    {
        unsigned int len = GetField().MaxElementByteLength();
        bt.Put(4U); // uncompressed
        P.x.Encode(bt, len);
        P.y.Encode(bt, len);
    }
}
 
void ECP::EncodePoint(byte *encodedPoint, const Point &P, bool compressed) const
{
    ArraySink sink(encodedPoint, EncodedPointSize(compressed));
    EncodePoint(sink, P, compressed);
    CRYPTOPP_ASSERT(sink.TotalPutLength() == EncodedPointSize(compressed));
}
 
ECP::Point ECP::BERDecodePoint(BufferedTransformation &bt) const
{
    SecByteBlock str;
    BERDecodeOctetString(bt, str);
    Point P;
    if (!DecodePoint(P, str, str.size()))
        BERDecodeError();
    return P;
}
 
void ECP::DEREncodePoint(BufferedTransformation &bt, const Point &P, bool compressed) const
{
    SecByteBlock str(EncodedPointSize(compressed));
    EncodePoint(str, P, compressed);
    DEREncodeOctetString(bt, str);
}
 
bool ECP::ValidateParameters(RandomNumberGenerator &rng, unsigned int level) const
{
    Integer p = FieldSize();
 
    bool pass = p.IsOdd();
    pass = pass && !m_a.IsNegative() && m_a<p && !m_b.IsNegative() && m_b<p;
 
    if (level >= 1)
        pass = pass && ((4*m_a*m_a*m_a+27*m_b*m_b)%p).IsPositive();
 
    if (level >= 2)
        pass = pass && VerifyPrime(rng, p);
 
    return pass;
}
 
bool ECP::VerifyPoint(const Point &P) const
{
    const FieldElement &x = P.x, &y = P.y;
    Integer p = FieldSize();
    return P.identity ||
        (!x.IsNegative() && x<p && !y.IsNegative() && y<p
        && !(((x*x+m_a)*x+m_b-y*y)%p));
}
 
bool ECP::Equal(const Point &P, const Point &Q) const
{
    if (P.identity && Q.identity)
        return true;
 
    if (P.identity && !Q.identity)
        return false;
 
    if (!P.identity && Q.identity)
        return false;
 
    return (GetField().Equal(P.x,Q.x) && GetField().Equal(P.y,Q.y));
}
 
const ECP::Point& ECP::Identity() const
{
#if defined(HAVE_GCC_INIT_PRIORITY) || defined(HAVE_MSC_INIT_PRIORITY) || defined(HAVE_XLC_INIT_PRIORITY)
    return g_identity;
#elif defined(CRYPTOPP_CXX11_STATIC_INIT)
    static const ECP::Point g_identity;
    return g_identity;
#else
    return Singleton<Point>().Ref();
#endif
}
 
const ECP::Point& ECP::Inverse(const Point &P) const
{
    if (P.identity)
        return P;
    else
    {
        m_R.identity = false;
        m_R.x = P.x;
        m_R.y = GetField().Inverse(P.y);
        return m_R;
    }
}
 
const ECP::Point& ECP::Add(const Point &P, const Point &Q) const
{
    if (P.identity) return Q;
    if (Q.identity) return P;
    if (GetField().Equal(P.x, Q.x))
        return GetField().Equal(P.y, Q.y) ? Double(P) : Identity();
 
    FieldElement t = GetField().Subtract(Q.y, P.y);
    t = GetField().Divide(t, GetField().Subtract(Q.x, P.x));
    FieldElement x = GetField().Subtract(GetField().Subtract(GetField().Square(t), P.x), Q.x);
    m_R.y = GetField().Subtract(GetField().Multiply(t, GetField().Subtract(P.x, x)), P.y);
 
    m_R.x.swap(x);
    m_R.identity = false;
    return m_R;
}
 
const ECP::Point& ECP::Double(const Point &P) const
{
    if (P.identity || P.y==GetField().Identity()) return Identity();
 
    FieldElement t = GetField().Square(P.x);
    t = GetField().Add(GetField().Add(GetField().Double(t), t), m_a);
    t = GetField().Divide(t, GetField().Double(P.y));
    FieldElement x = GetField().Subtract(GetField().Subtract(GetField().Square(t), P.x), P.x);
    m_R.y = GetField().Subtract(GetField().Multiply(t, GetField().Subtract(P.x, x)), P.y);
 
    m_R.x.swap(x);
    m_R.identity = false;
    return m_R;
}
 
template <class T, class Iterator> void ParallelInvert(const AbstractRing<T> &ring, Iterator begin, Iterator end)
{
    size_t n = end-begin;
    if (n == 1)
        *begin = ring.MultiplicativeInverse(*begin);
    else if (n > 1)
    {
        std::vector<T> vec((n+1)/2);
        unsigned int i;
        Iterator it;
 
        for (i=0, it=begin; i<n/2; i++, it+=2)
            vec[i] = ring.Multiply(*it, *(it+1));
        if (n%2 == 1)
            vec[n/2] = *it;
 
        ParallelInvert(ring, vec.begin(), vec.end());
 
        for (i=0, it=begin; i<n/2; i++, it+=2)
        {
            if (!vec[i])
            {
                *it = ring.MultiplicativeInverse(*it);
                *(it+1) = ring.MultiplicativeInverse(*(it+1));
            }
            else
            {
                std::swap(*it, *(it+1));
                *it = ring.Multiply(*it, vec[i]);
                *(it+1) = ring.Multiply(*(it+1), vec[i]);
            }
        }
        if (n%2 == 1)
            *it = vec[n/2];
    }
}
 
class ProjectiveDoubling
{
public:
    ProjectiveDoubling(const ModularArithmetic &m_mr, const Integer &m_a, const Integer &m_b, const ECPPoint &Q)
        : mr(m_mr)
    {
        CRYPTOPP_UNUSED(m_b);
        if (Q.identity)
        {
            sixteenY4 = P.x = P.y = mr.MultiplicativeIdentity();
            aZ4 = P.z = mr.Identity();
        }
        else
        {
            P.x = Q.x;
            P.y = Q.y;
            sixteenY4 = P.z = mr.MultiplicativeIdentity();
            aZ4 = m_a;
        }
    }
 
    void Double()
    {
        twoY = mr.Double(P.y);
        P.z = mr.Multiply(P.z, twoY);
        fourY2 = mr.Square(twoY);
        S = mr.Multiply(fourY2, P.x);
        aZ4 = mr.Multiply(aZ4, sixteenY4);
        M = mr.Square(P.x);
        M = mr.Add(mr.Add(mr.Double(M), M), aZ4);
        P.x = mr.Square(M);
        mr.Reduce(P.x, S);
        mr.Reduce(P.x, S);
        mr.Reduce(S, P.x);
        P.y = mr.Multiply(M, S);
        sixteenY4 = mr.Square(fourY2);
        mr.Reduce(P.y, mr.Half(sixteenY4));
    }
 
    const ModularArithmetic &mr;
    ProjectivePoint P;
    Integer sixteenY4, aZ4, twoY, fourY2, S, M;
};
 
struct ZIterator
{
    ZIterator() {}
    ZIterator(std::vector<ProjectivePoint>::iterator it) : it(it) {}
    Integer& operator*() {return it->z;}
    int operator-(ZIterator it2) {return int(it-it2.it);}
    ZIterator operator+(int i) {return ZIterator(it+i);}
    ZIterator& operator+=(int i) {it+=i; return *this;}
    std::vector<ProjectivePoint>::iterator it;
};
 
ECP::Point ECP::ScalarMultiply(const Point &P, const Integer &k) const
{
    Element result;
    if (k.BitCount() <= 5)
        AbstractGroup<ECPPoint>::SimultaneousMultiply(&result, P, &k, 1);
    else
        ECP::SimultaneousMultiply(&result, P, &k, 1);
    return result;
}
 
void ECP::SimultaneousMultiply(ECP::Point *results, const ECP::Point &P, const Integer *expBegin, unsigned int expCount) const
{
    if (!GetField().IsMontgomeryRepresentation())
    {
        ECP ecpmr(*this, true);
        const ModularArithmetic &mr = ecpmr.GetField();
        ecpmr.SimultaneousMultiply(results, ToMontgomery(mr, P), expBegin, expCount);
        for (unsigned int i=0; i<expCount; i++)
            results[i] = FromMontgomery(mr, results[i]);
        return;
    }
 
    ProjectiveDoubling rd(GetField(), m_a, m_b, P);
    std::vector<ProjectivePoint> bases;
    std::vector<WindowSlider> exponents;
    exponents.reserve(expCount);
    std::vector<std::vector<word32> > baseIndices(expCount);
    std::vector<std::vector<bool> > negateBase(expCount);
    std::vector<std::vector<word32> > exponentWindows(expCount);
    unsigned int i;
 
    for (i=0; i<expCount; i++)
    {
        CRYPTOPP_ASSERT(expBegin->NotNegative());
        exponents.push_back(WindowSlider(*expBegin++, InversionIsFast(), 5));
        exponents[i].FindNextWindow();
    }
 
    unsigned int expBitPosition = 0;
    bool notDone = true;
 
    while (notDone)
    {
        notDone = false;
        bool baseAdded = false;
        for (i=0; i<expCount; i++)
        {
            if (!exponents[i].finished && expBitPosition == exponents[i].windowBegin)
            {
                if (!baseAdded)
                {
                    bases.push_back(rd.P);
                    baseAdded =true;
                }
 
                exponentWindows[i].push_back(exponents[i].expWindow);
                baseIndices[i].push_back((word32)bases.size()-1);
                negateBase[i].push_back(exponents[i].negateNext);
 
                exponents[i].FindNextWindow();
            }
            notDone = notDone || !exponents[i].finished;
        }
 
        if (notDone)
        {
            rd.Double();
            expBitPosition++;
        }
    }
 
    // convert from projective to affine coordinates
    ParallelInvert(GetField(), ZIterator(bases.begin()), ZIterator(bases.end()));
    for (i=0; i<bases.size(); i++)
    {
        if (bases[i].z.NotZero())
        {
            bases[i].y = GetField().Multiply(bases[i].y, bases[i].z);
            bases[i].z = GetField().Square(bases[i].z);
            bases[i].x = GetField().Multiply(bases[i].x, bases[i].z);
            bases[i].y = GetField().Multiply(bases[i].y, bases[i].z);
        }
    }
 
    std::vector<BaseAndExponent<Point, Integer> > finalCascade;
    for (i=0; i<expCount; i++)
    {
        finalCascade.resize(baseIndices[i].size());
        for (unsigned int j=0; j<baseIndices[i].size(); j++)
        {
            ProjectivePoint &base = bases[baseIndices[i][j]];
            if (base.z.IsZero())
                finalCascade[j].base.identity = true;
            else
            {
                finalCascade[j].base.identity = false;
                finalCascade[j].base.x = base.x;
                if (negateBase[i][j])
                    finalCascade[j].base.y = GetField().Inverse(base.y);
                else
                    finalCascade[j].base.y = base.y;
            }
            finalCascade[j].exponent = Integer(Integer::POSITIVE, 0, exponentWindows[i][j]);
        }
        results[i] = GeneralCascadeMultiplication(*this, finalCascade.begin(), finalCascade.end());
    }
}
 
ECP::Point ECP::CascadeScalarMultiply(const Point &P, const Integer &k1, const Point &Q, const Integer &k2) const
{
    if (!GetField().IsMontgomeryRepresentation())
    {
        ECP ecpmr(*this, true);
        const ModularArithmetic &mr = ecpmr.GetField();
        return FromMontgomery(mr, ecpmr.CascadeScalarMultiply(ToMontgomery(mr, P), k1, ToMontgomery(mr, Q), k2));
    }
    else
        return AbstractGroup<Point>::CascadeScalarMultiply(P, k1, Q, k2);
}
 
NAMESPACE_END
 
#endif