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Source file src/crypto/elliptic/elliptic.go

Documentation: crypto/elliptic

  // Copyright 2010 The Go Authors. All rights reserved.
  // Use of this source code is governed by a BSD-style
  // license that can be found in the LICENSE file.
  
  // Package elliptic implements several standard elliptic curves over prime
  // fields.
  package elliptic
  
  // This package operates, internally, on Jacobian coordinates. For a given
  // (x, y) position on the curve, the Jacobian coordinates are (x1, y1, z1)
  // where x = x1/z1² and y = y1/z1³. The greatest speedups come when the whole
  // calculation can be performed within the transform (as in ScalarMult and
  // ScalarBaseMult). But even for Add and Double, it's faster to apply and
  // reverse the transform than to operate in affine coordinates.
  
  import (
  	"io"
  	"math/big"
  	"sync"
  )
  
  // A Curve represents a short-form Weierstrass curve with a=-3.
  // See http://www.hyperelliptic.org/EFD/g1p/auto-shortw.html
  type Curve interface {
  	// Params returns the parameters for the curve.
  	Params() *CurveParams
  	// IsOnCurve reports whether the given (x,y) lies on the curve.
  	IsOnCurve(x, y *big.Int) bool
  	// Add returns the sum of (x1,y1) and (x2,y2)
  	Add(x1, y1, x2, y2 *big.Int) (x, y *big.Int)
  	// Double returns 2*(x,y)
  	Double(x1, y1 *big.Int) (x, y *big.Int)
  	// ScalarMult returns k*(Bx,By) where k is a number in big-endian form.
  	ScalarMult(x1, y1 *big.Int, k []byte) (x, y *big.Int)
  	// ScalarBaseMult returns k*G, where G is the base point of the group
  	// and k is an integer in big-endian form.
  	ScalarBaseMult(k []byte) (x, y *big.Int)
  }
  
  // CurveParams contains the parameters of an elliptic curve and also provides
  // a generic, non-constant time implementation of Curve.
  type CurveParams struct {
  	P       *big.Int // the order of the underlying field
  	N       *big.Int // the order of the base point
  	B       *big.Int // the constant of the curve equation
  	Gx, Gy  *big.Int // (x,y) of the base point
  	BitSize int      // the size of the underlying field
  	Name    string   // the canonical name of the curve
  }
  
  func (curve *CurveParams) Params() *CurveParams {
  	return curve
  }
  
  func (curve *CurveParams) IsOnCurve(x, y *big.Int) bool {
  	// y² = x³ - 3x + b
  	y2 := new(big.Int).Mul(y, y)
  	y2.Mod(y2, curve.P)
  
  	x3 := new(big.Int).Mul(x, x)
  	x3.Mul(x3, x)
  
  	threeX := new(big.Int).Lsh(x, 1)
  	threeX.Add(threeX, x)
  
  	x3.Sub(x3, threeX)
  	x3.Add(x3, curve.B)
  	x3.Mod(x3, curve.P)
  
  	return x3.Cmp(y2) == 0
  }
  
  // zForAffine returns a Jacobian Z value for the affine point (x, y). If x and
  // y are zero, it assumes that they represent the point at infinity because (0,
  // 0) is not on the any of the curves handled here.
  func zForAffine(x, y *big.Int) *big.Int {
  	z := new(big.Int)
  	if x.Sign() != 0 || y.Sign() != 0 {
  		z.SetInt64(1)
  	}
  	return z
  }
  
  // affineFromJacobian reverses the Jacobian transform. See the comment at the
  // top of the file. If the point is ∞ it returns 0, 0.
  func (curve *CurveParams) affineFromJacobian(x, y, z *big.Int) (xOut, yOut *big.Int) {
  	if z.Sign() == 0 {
  		return new(big.Int), new(big.Int)
  	}
  
  	zinv := new(big.Int).ModInverse(z, curve.P)
  	zinvsq := new(big.Int).Mul(zinv, zinv)
  
  	xOut = new(big.Int).Mul(x, zinvsq)
  	xOut.Mod(xOut, curve.P)
  	zinvsq.Mul(zinvsq, zinv)
  	yOut = new(big.Int).Mul(y, zinvsq)
  	yOut.Mod(yOut, curve.P)
  	return
  }
  
  func (curve *CurveParams) Add(x1, y1, x2, y2 *big.Int) (*big.Int, *big.Int) {
  	z1 := zForAffine(x1, y1)
  	z2 := zForAffine(x2, y2)
  	return curve.affineFromJacobian(curve.addJacobian(x1, y1, z1, x2, y2, z2))
  }
  
  // addJacobian takes two points in Jacobian coordinates, (x1, y1, z1) and
  // (x2, y2, z2) and returns their sum, also in Jacobian form.
  func (curve *CurveParams) addJacobian(x1, y1, z1, x2, y2, z2 *big.Int) (*big.Int, *big.Int, *big.Int) {
  	// See http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#addition-add-2007-bl
  	x3, y3, z3 := new(big.Int), new(big.Int), new(big.Int)
  	if z1.Sign() == 0 {
  		x3.Set(x2)
  		y3.Set(y2)
  		z3.Set(z2)
  		return x3, y3, z3
  	}
  	if z2.Sign() == 0 {
  		x3.Set(x1)
  		y3.Set(y1)
  		z3.Set(z1)
  		return x3, y3, z3
  	}
  
  	z1z1 := new(big.Int).Mul(z1, z1)
  	z1z1.Mod(z1z1, curve.P)
  	z2z2 := new(big.Int).Mul(z2, z2)
  	z2z2.Mod(z2z2, curve.P)
  
  	u1 := new(big.Int).Mul(x1, z2z2)
  	u1.Mod(u1, curve.P)
  	u2 := new(big.Int).Mul(x2, z1z1)
  	u2.Mod(u2, curve.P)
  	h := new(big.Int).Sub(u2, u1)
  	xEqual := h.Sign() == 0
  	if h.Sign() == -1 {
  		h.Add(h, curve.P)
  	}
  	i := new(big.Int).Lsh(h, 1)
  	i.Mul(i, i)
  	j := new(big.Int).Mul(h, i)
  
  	s1 := new(big.Int).Mul(y1, z2)
  	s1.Mul(s1, z2z2)
  	s1.Mod(s1, curve.P)
  	s2 := new(big.Int).Mul(y2, z1)
  	s2.Mul(s2, z1z1)
  	s2.Mod(s2, curve.P)
  	r := new(big.Int).Sub(s2, s1)
  	if r.Sign() == -1 {
  		r.Add(r, curve.P)
  	}
  	yEqual := r.Sign() == 0
  	if xEqual && yEqual {
  		return curve.doubleJacobian(x1, y1, z1)
  	}
  	r.Lsh(r, 1)
  	v := new(big.Int).Mul(u1, i)
  
  	x3.Set(r)
  	x3.Mul(x3, x3)
  	x3.Sub(x3, j)
  	x3.Sub(x3, v)
  	x3.Sub(x3, v)
  	x3.Mod(x3, curve.P)
  
  	y3.Set(r)
  	v.Sub(v, x3)
  	y3.Mul(y3, v)
  	s1.Mul(s1, j)
  	s1.Lsh(s1, 1)
  	y3.Sub(y3, s1)
  	y3.Mod(y3, curve.P)
  
  	z3.Add(z1, z2)
  	z3.Mul(z3, z3)
  	z3.Sub(z3, z1z1)
  	z3.Sub(z3, z2z2)
  	z3.Mul(z3, h)
  	z3.Mod(z3, curve.P)
  
  	return x3, y3, z3
  }
  
  func (curve *CurveParams) Double(x1, y1 *big.Int) (*big.Int, *big.Int) {
  	z1 := zForAffine(x1, y1)
  	return curve.affineFromJacobian(curve.doubleJacobian(x1, y1, z1))
  }
  
  // doubleJacobian takes a point in Jacobian coordinates, (x, y, z), and
  // returns its double, also in Jacobian form.
  func (curve *CurveParams) doubleJacobian(x, y, z *big.Int) (*big.Int, *big.Int, *big.Int) {
  	// See http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#doubling-dbl-2001-b
  	delta := new(big.Int).Mul(z, z)
  	delta.Mod(delta, curve.P)
  	gamma := new(big.Int).Mul(y, y)
  	gamma.Mod(gamma, curve.P)
  	alpha := new(big.Int).Sub(x, delta)
  	if alpha.Sign() == -1 {
  		alpha.Add(alpha, curve.P)
  	}
  	alpha2 := new(big.Int).Add(x, delta)
  	alpha.Mul(alpha, alpha2)
  	alpha2.Set(alpha)
  	alpha.Lsh(alpha, 1)
  	alpha.Add(alpha, alpha2)
  
  	beta := alpha2.Mul(x, gamma)
  
  	x3 := new(big.Int).Mul(alpha, alpha)
  	beta8 := new(big.Int).Lsh(beta, 3)
  	x3.Sub(x3, beta8)
  	for x3.Sign() == -1 {
  		x3.Add(x3, curve.P)
  	}
  	x3.Mod(x3, curve.P)
  
  	z3 := new(big.Int).Add(y, z)
  	z3.Mul(z3, z3)
  	z3.Sub(z3, gamma)
  	if z3.Sign() == -1 {
  		z3.Add(z3, curve.P)
  	}
  	z3.Sub(z3, delta)
  	if z3.Sign() == -1 {
  		z3.Add(z3, curve.P)
  	}
  	z3.Mod(z3, curve.P)
  
  	beta.Lsh(beta, 2)
  	beta.Sub(beta, x3)
  	if beta.Sign() == -1 {
  		beta.Add(beta, curve.P)
  	}
  	y3 := alpha.Mul(alpha, beta)
  
  	gamma.Mul(gamma, gamma)
  	gamma.Lsh(gamma, 3)
  	gamma.Mod(gamma, curve.P)
  
  	y3.Sub(y3, gamma)
  	if y3.Sign() == -1 {
  		y3.Add(y3, curve.P)
  	}
  	y3.Mod(y3, curve.P)
  
  	return x3, y3, z3
  }
  
  func (curve *CurveParams) ScalarMult(Bx, By *big.Int, k []byte) (*big.Int, *big.Int) {
  	Bz := new(big.Int).SetInt64(1)
  	x, y, z := new(big.Int), new(big.Int), new(big.Int)
  
  	for _, byte := range k {
  		for bitNum := 0; bitNum < 8; bitNum++ {
  			x, y, z = curve.doubleJacobian(x, y, z)
  			if byte&0x80 == 0x80 {
  				x, y, z = curve.addJacobian(Bx, By, Bz, x, y, z)
  			}
  			byte <<= 1
  		}
  	}
  
  	return curve.affineFromJacobian(x, y, z)
  }
  
  func (curve *CurveParams) ScalarBaseMult(k []byte) (*big.Int, *big.Int) {
  	return curve.ScalarMult(curve.Gx, curve.Gy, k)
  }
  
  var mask = []byte{0xff, 0x1, 0x3, 0x7, 0xf, 0x1f, 0x3f, 0x7f}
  
  // GenerateKey returns a public/private key pair. The private key is
  // generated using the given reader, which must return random data.
  func GenerateKey(curve Curve, rand io.Reader) (priv []byte, x, y *big.Int, err error) {
  	N := curve.Params().N
  	bitSize := N.BitLen()
  	byteLen := (bitSize + 7) >> 3
  	priv = make([]byte, byteLen)
  
  	for x == nil {
  		_, err = io.ReadFull(rand, priv)
  		if err != nil {
  			return
  		}
  		// We have to mask off any excess bits in the case that the size of the
  		// underlying field is not a whole number of bytes.
  		priv[0] &= mask[bitSize%8]
  		// This is because, in tests, rand will return all zeros and we don't
  		// want to get the point at infinity and loop forever.
  		priv[1] ^= 0x42
  
  		// If the scalar is out of range, sample another random number.
  		if new(big.Int).SetBytes(priv).Cmp(N) >= 0 {
  			continue
  		}
  
  		x, y = curve.ScalarBaseMult(priv)
  	}
  	return
  }
  
  // Marshal converts a point into the form specified in section 4.3.6 of ANSI X9.62.
  func Marshal(curve Curve, x, y *big.Int) []byte {
  	byteLen := (curve.Params().BitSize + 7) >> 3
  
  	ret := make([]byte, 1+2*byteLen)
  	ret[0] = 4 // uncompressed point
  
  	xBytes := x.Bytes()
  	copy(ret[1+byteLen-len(xBytes):], xBytes)
  	yBytes := y.Bytes()
  	copy(ret[1+2*byteLen-len(yBytes):], yBytes)
  	return ret
  }
  
  // Unmarshal converts a point, serialized by Marshal, into an x, y pair.
  // It is an error if the point is not on the curve. On error, x = nil.
  func Unmarshal(curve Curve, data []byte) (x, y *big.Int) {
  	byteLen := (curve.Params().BitSize + 7) >> 3
  	if len(data) != 1+2*byteLen {
  		return
  	}
  	if data[0] != 4 { // uncompressed form
  		return
  	}
  	x = new(big.Int).SetBytes(data[1 : 1+byteLen])
  	y = new(big.Int).SetBytes(data[1+byteLen:])
  	if !curve.IsOnCurve(x, y) {
  		x, y = nil, nil
  	}
  	return
  }
  
  var initonce sync.Once
  var p384 *CurveParams
  var p521 *CurveParams
  
  func initAll() {
  	initP224()
  	initP256()
  	initP384()
  	initP521()
  }
  
  func initP384() {
  	// See FIPS 186-3, section D.2.4
  	p384 = &CurveParams{Name: "P-384"}
  	p384.P, _ = new(big.Int).SetString("39402006196394479212279040100143613805079739270465446667948293404245721771496870329047266088258938001861606973112319", 10)
  	p384.N, _ = new(big.Int).SetString("39402006196394479212279040100143613805079739270465446667946905279627659399113263569398956308152294913554433653942643", 10)
  	p384.B, _ = new(big.Int).SetString("b3312fa7e23ee7e4988e056be3f82d19181d9c6efe8141120314088f5013875ac656398d8a2ed19d2a85c8edd3ec2aef", 16)
  	p384.Gx, _ = new(big.Int).SetString("aa87ca22be8b05378eb1c71ef320ad746e1d3b628ba79b9859f741e082542a385502f25dbf55296c3a545e3872760ab7", 16)
  	p384.Gy, _ = new(big.Int).SetString("3617de4a96262c6f5d9e98bf9292dc29f8f41dbd289a147ce9da3113b5f0b8c00a60b1ce1d7e819d7a431d7c90ea0e5f", 16)
  	p384.BitSize = 384
  }
  
  func initP521() {
  	// See FIPS 186-3, section D.2.5
  	p521 = &CurveParams{Name: "P-521"}
  	p521.P, _ = new(big.Int).SetString("6864797660130609714981900799081393217269435300143305409394463459185543183397656052122559640661454554977296311391480858037121987999716643812574028291115057151", 10)
  	p521.N, _ = new(big.Int).SetString("6864797660130609714981900799081393217269435300143305409394463459185543183397655394245057746333217197532963996371363321113864768612440380340372808892707005449", 10)
  	p521.B, _ = new(big.Int).SetString("051953eb9618e1c9a1f929a21a0b68540eea2da725b99b315f3b8b489918ef109e156193951ec7e937b1652c0bd3bb1bf073573df883d2c34f1ef451fd46b503f00", 16)
  	p521.Gx, _ = new(big.Int).SetString("c6858e06b70404e9cd9e3ecb662395b4429c648139053fb521f828af606b4d3dbaa14b5e77efe75928fe1dc127a2ffa8de3348b3c1856a429bf97e7e31c2e5bd66", 16)
  	p521.Gy, _ = new(big.Int).SetString("11839296a789a3bc0045c8a5fb42c7d1bd998f54449579b446817afbd17273e662c97ee72995ef42640c550b9013fad0761353c7086a272c24088be94769fd16650", 16)
  	p521.BitSize = 521
  }
  
  // P256 returns a Curve which implements P-256 (see FIPS 186-3, section D.2.3)
  //
  // The cryptographic operations are implemented using constant-time algorithms.
  func P256() Curve {
  	initonce.Do(initAll)
  	return p256
  }
  
  // P384 returns a Curve which implements P-384 (see FIPS 186-3, section D.2.4)
  //
  // The cryptographic operations do not use constant-time algorithms.
  func P384() Curve {
  	initonce.Do(initAll)
  	return p384
  }
  
  // P521 returns a Curve which implements P-521 (see FIPS 186-3, section D.2.5)
  //
  // The cryptographic operations do not use constant-time algorithms.
  func P521() Curve {
  	initonce.Do(initAll)
  	return p521
  }
  

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