yaole
619 lines
15 KiB
Go
619 lines
15 KiB
Go
// Package bn256 implements a particular bilinear group.
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//
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// Bilinear groups are the basis of many of the new cryptographic protocols that
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// have been proposed over the past decade. They consist of a triplet of groups
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// (G₁, G₂ and GT) such that there exists a function e(g₁ˣ,g₂ʸ)=gTˣʸ (where gₓ
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// is a generator of the respective group). That function is called a pairing
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// function.
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//
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// This package specifically implements the Optimal Ate pairing over a 256-bit
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// Barreto-Naehrig curve as described in
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// http://cryptojedi.org/papers/dclxvi-20100714.pdf. Its output is compatible
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// with the implementation described in that paper.
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//
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// This package previously claimed to operate at a 128-bit security level.
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// However, recent improvements in attacks mean that is no longer true. See
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// https://moderncrypto.org/mail-archive/curves/2016/000740.html.
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package bn256
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import (
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"io"
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"math/big"
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"xdx.jelly/xgcl/gerrors"
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"xdx.jelly/xgcl/sm/sm9/errors"
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)
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var one = big.NewInt(1)
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// randomK returns a random integer in [1, N-1].
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func randomK(r io.Reader) (k *big.Int, err error) {
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b := make([]byte, numBytes)
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n, err := r.Read(b)
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if err != nil {
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return nil, errors.ErrGenerateRandomFailed
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}
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// it is possible that err != nil but n > 0.
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// In this case, we also consider it succeed.
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if n == 0 {
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return nil, errors.ErrGenerateRandomFailed
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}
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// 0 <= k <= N-2
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k = new(big.Int).SetBytes(b)
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if k.Cmp(nMinusOne) >= 0 {
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k.Sub(k, nMinusOne)
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}
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// 1 <= k <= N-1
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k.Add(k, one)
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return k, nil
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}
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// G1 is an abstract cyclic group. The zero value is suitable for use as the
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// output of an operation, but cannot be used as an input.
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type G1 struct {
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p curvePoint
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}
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// UnmarshalCompressed restore e from x and the LSB of y.
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func (e *G1) UnmarshalCompressed(x []byte, yBit0 byte) (*G1, error) {
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ex := &gfP{}
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ey := &gfP{}
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ex.Unmarshal(x)
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montEncode(ex, ex)
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// y^2 = x^3 + B
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gfpMul(ey, ex, ex)
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gfpMul(ey, ey, ex)
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gfpAdd(ey, ey, curveB)
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if legendre(ey) != 1 {
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return e, gerrors.WithAnnotating(errors.ErrInvalidInput, "sqrt failed, input bytes are not a valid compressed point")
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}
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ey.Sqrt(ey)
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var temp gfP
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montDecode(&temp, ey)
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if yBit0 != byte(temp[0]&1) {
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gfpNeg(ey, ey)
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}
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e.p.x = *ex
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e.p.y = *ey
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e.p.z = r
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e.p.t = r
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return e, nil
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}
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// RandomG1 returns x and g₁ˣ where x is a random, non-zero number read from r.
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func RandomG1(r io.Reader) (*big.Int, *G1, error) {
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k, err := randomK(r)
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if err != nil {
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return nil, nil, gerrors.WithMessage(err, "RandomG1 failed")
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}
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return k, new(G1).ScalarBaseMult(k), nil
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}
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// returns the montgemery domain of x
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func (e *G1) X() *big.Int {
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e.p.MakeAffine()
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return e.p.x.toBigInt()
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}
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// returns the montgemery domain of x
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func (e *G1) Y() *big.Int {
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e.p.MakeAffine()
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return e.p.y.toBigInt()
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}
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// returns the montgemery domain of x
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func (e *G1) AffineX() *big.Int {
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e.p.MakeAffine()
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var x gfP
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montDecode(&x, &e.p.x)
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return x.toBigInt()
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}
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// returns the montgemery domain of x
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func (e *G1) AffineY() *big.Int {
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e.p.MakeAffine()
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var y gfP
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montDecode(&y, &e.p.y)
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return y.toBigInt()
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}
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// String returns a readable string representation of e
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func (e *G1) String() string {
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return "G1" + e.p.String()
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}
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// IsInfinity returns if G1 is the infinity point
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func (e *G1) IsInfinity() bool {
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return e.p.IsInfinity()
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}
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// SetInfinity sets e to the infinity point
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func (e *G1) SetInfinity() {
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e.p.SetInfinity()
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}
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// Equal returns if e equals the other point
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func (e *G1) Equal(other *G1) bool {
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return e.p.Equal(&other.p)
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}
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// IsZero return true if e is infinity
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func (e *G1) IsZero() bool {
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return e.p.IsInfinity()
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}
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// IsValid return if e is a valid point of G1
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func (e *G1) IsValid() bool {
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return e.p.IsOnCurve()
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}
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// ScalarBaseMult sets e to [k]g1 where g1 is the generator of the group and returns e.
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func (e *G1) ScalarBaseMult(k *big.Int) *G1 {
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e.p.MulBase(k, curverBasePrecompted8)
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return e
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}
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// ScalarMult sets e to [k]a and then returns e.
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func (e *G1) ScalarMult(a *G1, k *big.Int) *G1 {
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e.p.Mul(&a.p, k)
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return e
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}
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// Add sets e to a+b and then returns e.
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func (e *G1) Add(a, b *G1) *G1 {
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e.p.Add(&a.p, &b.p)
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return e
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}
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// Neg sets e to -a and then returns e.
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func (e *G1) Neg(a *G1) *G1 {
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e.p.Neg(&a.p)
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return e
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}
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// Set sets e to a and then returns e.
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func (e *G1) Set(a *G1) *G1 {
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e.p.Set(&a.p)
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return e
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}
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// FillBytes fills a point in G1 to a byte slice.
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//
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// The Input slice must be of exactly 64 bytes, i.e., len(b) == 64.
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// Also assume the point is not infinity. But if so, then set b to 0s and return.
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func (e *G1) FillBytes(b []byte) {
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if e.IsInfinity() {
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for i := 0; i < len(b); i++ {
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b[i] = 0
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}
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return
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}
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e.p.MakeAffine()
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temp := &gfP{}
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montDecode(temp, &e.p.x)
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temp.Marshal(b)
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montDecode(temp, &e.p.y)
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temp.Marshal(b[numBytes:])
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}
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// Marshal converts a point in G1 to a byte slice of length 64.
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// The slice is filled with x || y.
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func (e *G1) Marshal() []byte {
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ret := make([]byte, numBytes*2)
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if e.p.IsInfinity() {
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return ret
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}
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e.FillBytes(ret)
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return ret
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}
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// Unmarshal sets e to the result of converting the output of Marshal back into
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// a group element and then returns e.
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// The input byte slice m must at least 64 bytes long.
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func (e *G1) Unmarshal(m []byte) ([]byte, error) {
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if len(m) < 2*numBytes {
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return m, gerrors.WithAnnotating(errors.ErrInvalidInput, "not enough data to unmarshal to G1")
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}
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e.p.x, e.p.y = gfP{0}, gfP{0}
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e.p.x.Unmarshal(m)
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e.p.y.Unmarshal(m[numBytes:])
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montEncode(&e.p.x, &e.p.x)
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montEncode(&e.p.y, &e.p.y)
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zero := gfP{0}
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if e.p.x == zero && e.p.y == zero {
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// The point at infinity.
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e.p.y = gfPOne
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e.p.z = gfP{0}
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e.p.t = gfP{0}
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} else {
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e.p.z = gfPOne
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e.p.t = gfPOne
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if !e.p.IsOnCurve() {
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return nil, gerrors.WithAnnotating(errors.ErrInvalidPoint, "point is not a valid point on curve")
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}
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}
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return m[2*numBytes:], nil
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}
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// G2 is an abstract cyclic group. The zero value is suitable for use as the
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// output of an operation, but cannot be used as an input.
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type G2 struct {
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p twistPoint
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}
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// RandomG2 returns x and g₂ˣ where x is a random, non-zero number read from r.
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func RandomG2(r io.Reader) (*big.Int, *G2, error) {
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k, err := randomK(r)
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if err != nil {
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return nil, nil, gerrors.WithMessage(err, "RandomG2 failed")
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}
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return k, new(G2).ScalarBaseMult(k), nil
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}
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func (e *G2) X() (*big.Int, *big.Int) {
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e.p.MakeAffine()
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return e.p.x.x.toBigInt(), e.p.x.y.toBigInt()
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}
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func (e *G2) Y() (*big.Int, *big.Int) {
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e.p.MakeAffine()
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return e.p.y.x.toBigInt(), e.p.y.y.toBigInt()
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}
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func (e *G2) IsInfinity() bool {
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return e.p.IsInfinity()
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}
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func (e *G2) SetInfinity() {
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e.p.SetInfinity()
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}
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func (e *G2) Equal(other *G2) bool {
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return e.p.Equal(&other.p)
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}
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// FromX restore e from x and the LSB of y. yBit0 can only be 0 or 1.
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func (e *G2) UnmarshalCompressed(x0, x1 []byte, yBit0 byte) (*G2, error) {
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if len(x0) < numBytes || len(x1) < numBytes {
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return nil, gerrors.WithAnnotating(errors.ErrInvalidInput, "point is not a valid point on curve")
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}
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if yBit0&byte(0xfe) != 0 {
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return nil, gerrors.WithAnnotatingf(errors.ErrInvalidInput, "yBit0 can only be 0 or 1, but it's %d", yBit0)
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}
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ex := &gfP2{}
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ey := &gfP2{}
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ex0 := &ex.x
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ex1 := &ex.y
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ex0.Unmarshal(x0)
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montEncode(ex0, ex0)
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ex1.Unmarshal(x1)
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montEncode(ex1, ex1)
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ey.Mul(ex, ex)
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ey.Mul(ey, ex)
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ey.Add(ey, twistB) // ey = x^3 + B
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if !ey.Sqrt(ey) {
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return e, gerrors.WithAnnotatingf(errors.ErrInvalidPoint, "sqrt failed, input bytes are not a valid compressed point")
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}
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tmp := &gfP{}
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montDecode(tmp, &ey.y)
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if yBit0 != byte(tmp[0]&1) {
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ey.Neg(ey)
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}
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e.p.x = *ex
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e.p.y = *ey
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e.p.t = gfP2{y: r}
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e.p.z = gfP2{y: r}
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return e, nil
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}
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func (e *G2) String() string {
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return "G2" + e.p.String()
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}
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// ScalarBaseMult sets e to [k]g where g is the generator of the group and then
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// returns out.
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func (e *G2) ScalarBaseMult(k *big.Int) *G2 {
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e.p.MulBase(k, twistBasePrecomputed8)
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return e
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}
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// ScalarMult sets e to [k]a and then returns e.
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func (e *G2) ScalarMult(a *G2, k *big.Int) *G2 {
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e.p.Mul(&a.p, k)
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return e
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}
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// Add sets e to a+b and then returns e.
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func (e *G2) Add(a, b *G2) *G2 {
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e.p.Add(&a.p, &b.p)
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return e
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}
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// Neg sets e to -a and then returns e.
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func (e *G2) Neg(a *G2) *G2 {
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e.p.Neg(&a.p)
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return e
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}
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// Set sets e to a and then returns e.
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func (e *G2) Set(a *G2) *G2 {
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e.p.Set(&a.p)
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return e
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}
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// FillBytes fills a point in G2 to a byte slice.
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//
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// The Input slice must be of exactly 128 bytes, i.e., len(b) == 128.
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// Also assume the point is not infinity. But if so, then set b to 0s and return.
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func (e *G2) FillBytes(b []byte) {
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if e.IsInfinity() {
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for i := 0; i < len(b); i++ {
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b[i] = 0
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}
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return
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}
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e.p.MakeAffine()
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temp := &gfP{}
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montDecode(temp, &e.p.x.x)
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temp.Marshal(b)
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montDecode(temp, &e.p.x.y)
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temp.Marshal(b[numBytes:])
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montDecode(temp, &e.p.y.x)
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temp.Marshal(b[2*numBytes:])
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montDecode(temp, &e.p.y.y)
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temp.Marshal(b[3*numBytes:])
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}
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// Marshal converts e into a byte slice.
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func (e *G2) Marshal() []byte {
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ret := make([]byte, numBytes*4)
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if e.p.IsInfinity() {
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return ret
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}
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e.FillBytes(ret)
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return ret
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}
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// Unmarshal sets e to the result of converting the output of Marshal back into
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// a group element and then returns e.
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func (e *G2) Unmarshal(m []byte) ([]byte, error) {
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if len(m) < 4*numBytes {
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return m, gerrors.WithAnnotatingf(errors.ErrInvalidInput, "not enough data to unmarshal to G1")
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}
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e.p.x.x.Unmarshal(m)
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e.p.x.y.Unmarshal(m[numBytes:])
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e.p.y.x.Unmarshal(m[2*numBytes:])
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e.p.y.y.Unmarshal(m[3*numBytes:])
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montEncode(&e.p.x.x, &e.p.x.x)
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montEncode(&e.p.x.y, &e.p.x.y)
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montEncode(&e.p.y.x, &e.p.y.x)
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montEncode(&e.p.y.y, &e.p.y.y)
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if e.p.x.IsZero() && e.p.y.IsZero() {
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// This is the point at infinity.
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e.p.y.SetOne()
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e.p.z.SetZero()
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e.p.t.SetZero()
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} else {
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e.p.z.SetOne()
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e.p.t.SetOne()
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if !e.p.IsOnCurve() {
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return m, gerrors.WithAnnotatingf(errors.ErrInvalidPoint, "unmarshaled point is not a valid point on curve")
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}
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}
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return m[4*numBytes:], nil
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}
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// GT is an abstract cyclic group. The zero value is suitable for use as the
|
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// output of an operation, but cannot be used as an input.
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type GT struct {
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p gfP12
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}
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func (e *GT) Equal(other *GT) bool {
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return e.p.Equal(&other.p)
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}
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// Order12412 change e to 1-2-4-12 field extension represent
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func (e *GT) Order12412() {
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e.p.x.y.x, e.p.x.y.y, e.p.y.y.x, e.p.y.y.y, e.p.x.z.x, e.p.x.z.y, e.p.y.x.x, e.p.y.x.y =
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e.p.y.y.x, e.p.y.y.y, e.p.x.y.x, e.p.x.y.y, e.p.y.x.x, e.p.y.x.y, e.p.x.z.x, e.p.x.z.y
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}
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|
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// RandomGT returns x and e(g₁, g₂)ˣ where x is a random, non-zero number read
|
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// from r.
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func RandomGT(r io.Reader) (*big.Int, *GT, error) {
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k, err := randomK(r)
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if err != nil {
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return nil, nil, gerrors.WithMessage(err, "RandomGT failed")
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}
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return k, new(GT).ScalarBaseMult(k), nil
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}
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// Pair calculates an Optimal Ate pairing.
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func Pair(g1 *G1, g2 *G2) *GT {
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var e GT
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optimalAte(&e.p, &g2.p, &g1.p)
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return &e
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}
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// Pair calculates an Optimal Ate pairing.
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func PairLol(e *GT, g1 *G1, g2 *G2) {
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optimalAte(&e.p, &g2.p, &g1.p)
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}
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// Miller applies Miller's algorithm, which is a bilinear function from the
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// source groups to F_p^12. Miller(g1, g2).Finalize() is equivalent to Pair(g1,
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// g2).
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func Miller(g1 *G1, g2 *G2) *GT {
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var e GT
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miller(&e.p, &g2.p, &g1.p)
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return &e
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}
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func (e *GT) String() string {
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p := *e
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p.Order12412()
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return "GT" + gfP12Decode(&p.p).String()
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}
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|
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// ScalarBaseMult sets e to g*k where g is the generator of the group and then
|
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// returns out.
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func (e *GT) ScalarBaseMult(k *big.Int) *GT {
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if useLattice {
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e.p.latticeExp(gfP12Gen, k)
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return e
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} else {
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return e.ScalarMultSimple(>{*gfP12Gen}, k)
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}
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}
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|
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// ScalarMult sets e to a*k and then returns e. (If e is not guaranteed to be an element of the group because it is the
|
|
// output of Miller(), use ScalarMultSimple.)
|
|
func (e *GT) ScalarMult(a *GT, k *big.Int) *GT {
|
|
if useLattice {
|
|
e.p.latticeExp(&a.p, k)
|
|
return e
|
|
} else {
|
|
return e.ScalarMultSimple(a, k)
|
|
}
|
|
}
|
|
|
|
// ScalarMultSimple sets e to a*k and then returns e.
|
|
func (e *GT) ScalarMultSimple(a *GT, k *big.Int) *GT {
|
|
e.p.Exp(&a.p, k)
|
|
return e
|
|
}
|
|
|
|
func (e *GT) Mul(a, b *GT) *GT {
|
|
e.p.Mul(&a.p, &b.p)
|
|
return e
|
|
}
|
|
|
|
// Add sets e to a+b and then returns e.
|
|
func (e *GT) Add(a, b *GT) *GT {
|
|
e.p.Mul(&a.p, &b.p)
|
|
return e
|
|
}
|
|
|
|
// Neg sets e to -a and then returns e.
|
|
func (e *GT) Neg(a *GT) *GT {
|
|
e.p.Conjugate(&a.p)
|
|
return e
|
|
}
|
|
|
|
// Set sets e to a and then returns e.
|
|
func (e *GT) Set(a *GT) *GT {
|
|
e.p.Set(&a.p)
|
|
return e
|
|
}
|
|
|
|
// Set sets e to a and then returns e.
|
|
func (e *GT) SetOne() *GT {
|
|
e.p.SetOne()
|
|
return e
|
|
}
|
|
|
|
// Set sets e to a and then returns e.
|
|
func (e *GT) Invert(a *GT) *GT {
|
|
e.p.Invert(&a.p)
|
|
return e
|
|
}
|
|
|
|
// Finalize is a linear function from F_p^12 to GT.
|
|
// func (e *GT) Finalize() *GT {
|
|
// finalExponentiation(&e.p, &e.p)
|
|
// return e
|
|
// }
|
|
|
|
// Marshal converts e into a byte slice.
|
|
func (e *GT) Marshal() []byte {
|
|
p := *e
|
|
p.Order12412()
|
|
|
|
ret := make([]byte, numBytes*12)
|
|
temp := &gfP{}
|
|
|
|
montDecode(temp, &p.p.x.x.x)
|
|
temp.Marshal(ret)
|
|
montDecode(temp, &p.p.x.x.y)
|
|
temp.Marshal(ret[numBytes:])
|
|
montDecode(temp, &p.p.x.y.x)
|
|
temp.Marshal(ret[2*numBytes:])
|
|
montDecode(temp, &p.p.x.y.y)
|
|
temp.Marshal(ret[3*numBytes:])
|
|
montDecode(temp, &p.p.x.z.x)
|
|
temp.Marshal(ret[4*numBytes:])
|
|
montDecode(temp, &p.p.x.z.y)
|
|
temp.Marshal(ret[5*numBytes:])
|
|
montDecode(temp, &p.p.y.x.x)
|
|
temp.Marshal(ret[6*numBytes:])
|
|
montDecode(temp, &p.p.y.x.y)
|
|
temp.Marshal(ret[7*numBytes:])
|
|
montDecode(temp, &p.p.y.y.x)
|
|
temp.Marshal(ret[8*numBytes:])
|
|
montDecode(temp, &p.p.y.y.y)
|
|
temp.Marshal(ret[9*numBytes:])
|
|
montDecode(temp, &p.p.y.z.x)
|
|
temp.Marshal(ret[10*numBytes:])
|
|
montDecode(temp, &p.p.y.z.y)
|
|
temp.Marshal(ret[11*numBytes:])
|
|
|
|
return ret
|
|
}
|
|
|
|
// Unmarshal sets e to the result of converting the output of Marshal back into
|
|
// a group element and then returns e.
|
|
func (e *GT) Unmarshal(m []byte) ([]byte, error) {
|
|
if len(m) < 12*numBytes {
|
|
return m, gerrors.WithAnnotating(errors.ErrInvalidInput, "not enough data to unmarshal to GT")
|
|
}
|
|
|
|
e.p.x.x.x.Unmarshal(m)
|
|
e.p.x.x.y.Unmarshal(m[numBytes:])
|
|
e.p.x.y.x.Unmarshal(m[2*numBytes:])
|
|
e.p.x.y.y.Unmarshal(m[3*numBytes:])
|
|
e.p.x.z.x.Unmarshal(m[4*numBytes:])
|
|
e.p.x.z.y.Unmarshal(m[5*numBytes:])
|
|
e.p.y.x.x.Unmarshal(m[6*numBytes:])
|
|
e.p.y.x.y.Unmarshal(m[7*numBytes:])
|
|
e.p.y.y.x.Unmarshal(m[8*numBytes:])
|
|
e.p.y.y.y.Unmarshal(m[9*numBytes:])
|
|
e.p.y.z.x.Unmarshal(m[10*numBytes:])
|
|
e.p.y.z.y.Unmarshal(m[11*numBytes:])
|
|
montEncode(&e.p.x.x.x, &e.p.x.x.x)
|
|
montEncode(&e.p.x.x.y, &e.p.x.x.y)
|
|
montEncode(&e.p.x.y.x, &e.p.x.y.x)
|
|
montEncode(&e.p.x.y.y, &e.p.x.y.y)
|
|
montEncode(&e.p.x.z.x, &e.p.x.z.x)
|
|
montEncode(&e.p.x.z.y, &e.p.x.z.y)
|
|
montEncode(&e.p.y.x.x, &e.p.y.x.x)
|
|
montEncode(&e.p.y.x.y, &e.p.y.x.y)
|
|
montEncode(&e.p.y.y.x, &e.p.y.y.x)
|
|
montEncode(&e.p.y.y.y, &e.p.y.y.y)
|
|
montEncode(&e.p.y.z.x, &e.p.y.z.x)
|
|
montEncode(&e.p.y.z.y, &e.p.y.z.y)
|
|
|
|
e.Order12412()
|
|
return m[12*numBytes:], nil
|
|
}
|