---

ec2n.cpp

// ec2n.cpp - written and placed in the public domain by Wei Dai

#include "pch.h"
#include "ec2n.h"
#include "asn.h"
#include "nbtheory.h"   // for primeTable

#include "algebra.cpp"
#include "eprecomp.cpp"

NAMESPACE_BEGIN(CryptoPP)

EC2N::EC2N(BufferedTransformation &bt)
        : m_field(BERDecodeGF2NP(bt))
{
        BERSequenceDecoder seq(bt);
        m_field->BERDecodeElement(seq, m_a);
        m_field->BERDecodeElement(seq, m_b);
        // skip optional seed
        if (!seq.EndReached())
                BERDecodeOctetString(seq, g_bitBucket);
        seq.MessageEnd();
}

void EC2N::DEREncode(BufferedTransformation &bt) const
{
        m_field->DEREncode(bt);
        DERSequenceEncoder seq(bt);
        m_field->DEREncodeElement(seq, m_a);
        m_field->DEREncodeElement(seq, m_b);
        seq.MessageEnd();
}

bool EC2N::DecodePoint(EC2N::Point &P, const byte *encodedPoint, unsigned int encodedPointLen) const
{
        StringStore store(encodedPoint, encodedPointLen);
        return DecodePoint(P, store, encodedPointLen);
}

bool EC2N::DecodePoint(EC2N::Point &P, BufferedTransformation &bt, unsigned int 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;

                P.identity = false;
                P.x.Decode(bt, m_field->MaxElementByteLength()); 

                if (P.x.IsZero())
                {
                        P.y = m_field->SquareRoot(m_b);
                        return true;
                }

                FieldElement z = m_field->Square(P.x);
                assert(P.x == m_field->SquareRoot(z));
                P.y = m_field->Divide(m_field->Add(m_field->Multiply(z, m_field->Add(P.x, m_a)), m_b), z);
                assert(P.x == m_field->Subtract(m_field->Divide(m_field->Subtract(m_field->Multiply(P.y, z), m_b), z), m_a));
                z = m_field->SolveQuadraticEquation(P.y);
                assert(m_field->Add(m_field->Square(z), z) == P.y);
                z.SetCoefficient(0, type & 1);

                P.y = m_field->Multiply(z, P.x);
                return true;
        }
        case 4:
        {
                if (encodedPointLen != EncodedPointSize(false))
                        return false;

                unsigned int len = m_field->MaxElementByteLength();
                P.identity = false;
                P.x.Decode(bt, len);
                P.y.Decode(bt, len);
                return true;
        }
        default:
                return false;
        }
}

void EC2N::EncodePoint(byte *encodedPoint, const Point &P, bool compressed) const
{
        if (P.identity)
                memset(encodedPoint, 0, EncodedPointSize(compressed));
        else if (compressed)
        {
                encodedPoint[0] = 2 + (!P.x ? 0 : m_field->Divide(P.y, P.x).GetBit(0));
                P.x.Encode(encodedPoint+1, m_field->MaxElementByteLength());
        }
        else
        {
                unsigned int len = m_field->MaxElementByteLength();
                encodedPoint[0] = 4;    // uncompressed
                P.x.Encode(encodedPoint+1, len);
                P.y.Encode(encodedPoint+1+len, len);
        }
}

EC2N::Point EC2N::BERDecodePoint(BufferedTransformation &bt) const
{
        SecByteBlock str;
        BERDecodeOctetString(bt, str);
        Point P;
        if (!DecodePoint(P, str, str.size))
                BERDecodeError();
        return P;
}

void EC2N::DEREncodePoint(BufferedTransformation &bt, const Point &P, bool compressed) const
{
        SecByteBlock str(EncodedPointSize(compressed));
        EncodePoint(str, P, compressed);
        DEREncodeOctetString(bt, str);
}

bool EC2N::ValidateParameters(RandomNumberGenerator &rng) const
{
        return m_field->GetModulus().IsIrreducible()
                && m_a.CoefficientCount() <= m_field->MaxElementBitLength()
                && m_b.CoefficientCount() <= m_field->MaxElementBitLength() && !!m_b;
}

bool EC2N::VerifyPoint(const Point &P) const
{
        const FieldElement &x = P.x, &y = P.y;
        return P.identity || 
                (x.CoefficientCount() <= m_field->MaxElementBitLength()
                && y.CoefficientCount() <= m_field->MaxElementBitLength()
                && !(((x+m_a)*x*x+m_b-(x+y)*y)%m_field->GetModulus()));
}

bool EC2N::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 (m_field->Equal(P.x,Q.x) && m_field->Equal(P.y,Q.y));
}

const EC2N::Point& EC2N::Inverse(const Point &P) const
{
        if (P.identity)
                return P;
        else
        {
                m_R.identity = false;
                m_R.y = m_field->Add(P.x, P.y);
                m_R.x = P.x;
                return m_R;
        }
}

const EC2N::Point& EC2N::Add(const Point &P, const Point &Q) const
{
        if (P.identity) return Q;
        if (Q.identity) return P;
        if (Equal(P, Q)) return Double(P);
        if (m_field->Equal(P.x, Q.x) && m_field->Equal(P.y, m_field->Add(Q.x, Q.y))) return Zero();

        FieldElement t = m_field->Add(P.y, Q.y);
        t = m_field->Divide(t, m_field->Add(P.x, Q.x));
        FieldElement x = m_field->Square(t);
        m_field->Accumulate(x, t);
        m_field->Accumulate(x, Q.x);
        m_field->Accumulate(x, m_a);
        m_R.y = m_field->Add(P.y, m_field->Multiply(t, x));
        m_field->Accumulate(x, P.x);
        m_field->Accumulate(m_R.y, x);

        m_R.x.swap(x);
        m_R.identity = false;
        return m_R;
}

const EC2N::Point& EC2N::Double(const Point &P) const
{
        if (P.identity) return P;
        if (!m_field->IsUnit(P.x)) return Zero();

        FieldElement t = m_field->Divide(P.y, P.x);
        m_field->Accumulate(t, P.x);
        m_R.y = m_field->Square(P.x);
        m_R.x = m_field->Square(t);
        m_field->Accumulate(m_R.x, t);
        m_field->Accumulate(m_R.x, m_a);
        m_field->Accumulate(m_R.y, m_field->Multiply(t, m_R.x));
        m_field->Accumulate(m_R.y, m_R.x);

        m_R.identity = false;
        return m_R;
}

// ********************************************************

EcPrecomputation<EC2N>& EcPrecomputation<EC2N>::operator=(const EcPrecomputation<EC2N> &rhs)
{
        m_ec = rhs.m_ec;
        m_ep = rhs.m_ep;
        m_ep.m_group = m_ec.get();
        return *this;
}

void EcPrecomputation<EC2N>::SetCurveAndBase(const EC2N &ec, const EC2N::Point &base)
{
        m_ec.reset(new EC2N(ec));
        m_ep.SetGroupAndBase(*m_ec, base);
}

void EcPrecomputation<EC2N>::Precompute(unsigned int maxExpBits, unsigned int storage)
{
        m_ep.Precompute(maxExpBits, storage);
}

void EcPrecomputation<EC2N>::Load(BufferedTransformation &bt)
{
        BERSequenceDecoder seq(bt);
        word32 version;
        BERDecodeUnsigned<word32>(seq, version, INTEGER, 1, 1);
        m_ep.m_exponentBase.BERDecode(seq);
        m_ep.m_windowSize = m_ep.m_exponentBase.BitCount() - 1;
        m_ep.m_bases.clear();
        while (!seq.EndReached())
                m_ep.m_bases.push_back(m_ec->BERDecodePoint(seq));
        seq.MessageEnd();
}

void EcPrecomputation<EC2N>::Save(BufferedTransformation &bt) const
{
        DERSequenceEncoder seq(bt);
        DEREncodeUnsigned<word32>(seq, 1);      // version
        m_ep.m_exponentBase.DEREncode(seq);
        for (unsigned i=0; i<m_ep.m_bases.size(); i++)
                m_ec->DEREncodePoint(seq, m_ep.m_bases[i]);
        seq.MessageEnd();
}

EC2N::Point EcPrecomputation<EC2N>::Multiply(const Integer &exponent) const
{
        return m_ep.Exponentiate(exponent);
}

EC2N::Point EcPrecomputation<EC2N>::CascadeMultiply(const Integer &exponent, const EcPrecomputation<EC2N> &pc2, const Integer &exponent2) const
{
        return m_ep.CascadeExponentiate(exponent, pc2.m_ep, exponent2);
}

NAMESPACE_END

---