Source file src/pkg/crypto/elliptic/elliptic.go

     1	// Copyright 2010 The Go Authors. All rights reserved.
     2	// Use of this source code is governed by a BSD-style
     3	// license that can be found in the LICENSE file.
     4	
     5	// Package elliptic implements several standard elliptic curves over prime
     6	// fields.
     7	package elliptic
     8	
     9	// This package operates, internally, on Jacobian coordinates. For a given
    10	// (x, y) position on the curve, the Jacobian coordinates are (x1, y1, z1)
    11	// where x = x1/z1² and y = y1/z1³. The greatest speedups come when the whole
    12	// calculation can be performed within the transform (as in ScalarMult and
    13	// ScalarBaseMult). But even for Add and Double, it's faster to apply and
    14	// reverse the transform than to operate in affine coordinates.
    15	
    16	import (
    17		"io"
    18		"math/big"
    19		"sync"
    20	)
    21	
    22	// A Curve represents a short-form Weierstrass curve with a=-3.
    23	// See http://www.hyperelliptic.org/EFD/g1p/auto-shortw.html
    24	type Curve interface {
    25		// Params returns the parameters for the curve.
    26		Params() *CurveParams
    27		// IsOnCurve returns true if the given (x,y) lies on the curve.
    28		IsOnCurve(x, y *big.Int) bool
    29		// Add returns the sum of (x1,y1) and (x2,y2)
    30		Add(x1, y1, x2, y2 *big.Int) (x, y *big.Int)
    31		// Double returns 2*(x,y)
    32		Double(x1, y1 *big.Int) (x, y *big.Int)
    33		// ScalarMult returns k*(Bx,By) where k is a number in big-endian form.
    34		ScalarMult(x1, y1 *big.Int, scalar []byte) (x, y *big.Int)
    35		// ScalarBaseMult returns k*G, where G is the base point of the group and k
    36		// is an integer in big-endian form.
    37		ScalarBaseMult(scalar []byte) (x, y *big.Int)
    38	}
    39	
    40	// CurveParams contains the parameters of an elliptic curve and also provides
    41	// a generic, non-constant time implementation of Curve.
    42	type CurveParams struct {
    43		P       *big.Int // the order of the underlying field
    44		N       *big.Int // the order of the base point
    45		B       *big.Int // the constant of the curve equation
    46		Gx, Gy  *big.Int // (x,y) of the base point
    47		BitSize int      // the size of the underlying field
    48	}
    49	
    50	func (curve *CurveParams) Params() *CurveParams {
    51		return curve
    52	}
    53	
    54	func (curve *CurveParams) IsOnCurve(x, y *big.Int) bool {
    55		// y² = x³ - 3x + b
    56		y2 := new(big.Int).Mul(y, y)
    57		y2.Mod(y2, curve.P)
    58	
    59		x3 := new(big.Int).Mul(x, x)
    60		x3.Mul(x3, x)
    61	
    62		threeX := new(big.Int).Lsh(x, 1)
    63		threeX.Add(threeX, x)
    64	
    65		x3.Sub(x3, threeX)
    66		x3.Add(x3, curve.B)
    67		x3.Mod(x3, curve.P)
    68	
    69		return x3.Cmp(y2) == 0
    70	}
    71	
    72	// affineFromJacobian reverses the Jacobian transform. See the comment at the
    73	// top of the file.
    74	func (curve *CurveParams) affineFromJacobian(x, y, z *big.Int) (xOut, yOut *big.Int) {
    75		zinv := new(big.Int).ModInverse(z, curve.P)
    76		zinvsq := new(big.Int).Mul(zinv, zinv)
    77	
    78		xOut = new(big.Int).Mul(x, zinvsq)
    79		xOut.Mod(xOut, curve.P)
    80		zinvsq.Mul(zinvsq, zinv)
    81		yOut = new(big.Int).Mul(y, zinvsq)
    82		yOut.Mod(yOut, curve.P)
    83		return
    84	}
    85	
    86	func (curve *CurveParams) Add(x1, y1, x2, y2 *big.Int) (*big.Int, *big.Int) {
    87		z := new(big.Int).SetInt64(1)
    88		return curve.affineFromJacobian(curve.addJacobian(x1, y1, z, x2, y2, z))
    89	}
    90	
    91	// addJacobian takes two points in Jacobian coordinates, (x1, y1, z1) and
    92	// (x2, y2, z2) and returns their sum, also in Jacobian form.
    93	func (curve *CurveParams) addJacobian(x1, y1, z1, x2, y2, z2 *big.Int) (*big.Int, *big.Int, *big.Int) {
    94		// See http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#addition-add-2007-bl
    95		z1z1 := new(big.Int).Mul(z1, z1)
    96		z1z1.Mod(z1z1, curve.P)
    97		z2z2 := new(big.Int).Mul(z2, z2)
    98		z2z2.Mod(z2z2, curve.P)
    99	
   100		u1 := new(big.Int).Mul(x1, z2z2)
   101		u1.Mod(u1, curve.P)
   102		u2 := new(big.Int).Mul(x2, z1z1)
   103		u2.Mod(u2, curve.P)
   104		h := new(big.Int).Sub(u2, u1)
   105		if h.Sign() == -1 {
   106			h.Add(h, curve.P)
   107		}
   108		i := new(big.Int).Lsh(h, 1)
   109		i.Mul(i, i)
   110		j := new(big.Int).Mul(h, i)
   111	
   112		s1 := new(big.Int).Mul(y1, z2)
   113		s1.Mul(s1, z2z2)
   114		s1.Mod(s1, curve.P)
   115		s2 := new(big.Int).Mul(y2, z1)
   116		s2.Mul(s2, z1z1)
   117		s2.Mod(s2, curve.P)
   118		r := new(big.Int).Sub(s2, s1)
   119		if r.Sign() == -1 {
   120			r.Add(r, curve.P)
   121		}
   122		r.Lsh(r, 1)
   123		v := new(big.Int).Mul(u1, i)
   124	
   125		x3 := new(big.Int).Set(r)
   126		x3.Mul(x3, x3)
   127		x3.Sub(x3, j)
   128		x3.Sub(x3, v)
   129		x3.Sub(x3, v)
   130		x3.Mod(x3, curve.P)
   131	
   132		y3 := new(big.Int).Set(r)
   133		v.Sub(v, x3)
   134		y3.Mul(y3, v)
   135		s1.Mul(s1, j)
   136		s1.Lsh(s1, 1)
   137		y3.Sub(y3, s1)
   138		y3.Mod(y3, curve.P)
   139	
   140		z3 := new(big.Int).Add(z1, z2)
   141		z3.Mul(z3, z3)
   142		z3.Sub(z3, z1z1)
   143		if z3.Sign() == -1 {
   144			z3.Add(z3, curve.P)
   145		}
   146		z3.Sub(z3, z2z2)
   147		if z3.Sign() == -1 {
   148			z3.Add(z3, curve.P)
   149		}
   150		z3.Mul(z3, h)
   151		z3.Mod(z3, curve.P)
   152	
   153		return x3, y3, z3
   154	}
   155	
   156	func (curve *CurveParams) Double(x1, y1 *big.Int) (*big.Int, *big.Int) {
   157		z1 := new(big.Int).SetInt64(1)
   158		return curve.affineFromJacobian(curve.doubleJacobian(x1, y1, z1))
   159	}
   160	
   161	// doubleJacobian takes a point in Jacobian coordinates, (x, y, z), and
   162	// returns its double, also in Jacobian form.
   163	func (curve *CurveParams) doubleJacobian(x, y, z *big.Int) (*big.Int, *big.Int, *big.Int) {
   164		// See http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#doubling-dbl-2001-b
   165		delta := new(big.Int).Mul(z, z)
   166		delta.Mod(delta, curve.P)
   167		gamma := new(big.Int).Mul(y, y)
   168		gamma.Mod(gamma, curve.P)
   169		alpha := new(big.Int).Sub(x, delta)
   170		if alpha.Sign() == -1 {
   171			alpha.Add(alpha, curve.P)
   172		}
   173		alpha2 := new(big.Int).Add(x, delta)
   174		alpha.Mul(alpha, alpha2)
   175		alpha2.Set(alpha)
   176		alpha.Lsh(alpha, 1)
   177		alpha.Add(alpha, alpha2)
   178	
   179		beta := alpha2.Mul(x, gamma)
   180	
   181		x3 := new(big.Int).Mul(alpha, alpha)
   182		beta8 := new(big.Int).Lsh(beta, 3)
   183		x3.Sub(x3, beta8)
   184		for x3.Sign() == -1 {
   185			x3.Add(x3, curve.P)
   186		}
   187		x3.Mod(x3, curve.P)
   188	
   189		z3 := new(big.Int).Add(y, z)
   190		z3.Mul(z3, z3)
   191		z3.Sub(z3, gamma)
   192		if z3.Sign() == -1 {
   193			z3.Add(z3, curve.P)
   194		}
   195		z3.Sub(z3, delta)
   196		if z3.Sign() == -1 {
   197			z3.Add(z3, curve.P)
   198		}
   199		z3.Mod(z3, curve.P)
   200	
   201		beta.Lsh(beta, 2)
   202		beta.Sub(beta, x3)
   203		if beta.Sign() == -1 {
   204			beta.Add(beta, curve.P)
   205		}
   206		y3 := alpha.Mul(alpha, beta)
   207	
   208		gamma.Mul(gamma, gamma)
   209		gamma.Lsh(gamma, 3)
   210		gamma.Mod(gamma, curve.P)
   211	
   212		y3.Sub(y3, gamma)
   213		if y3.Sign() == -1 {
   214			y3.Add(y3, curve.P)
   215		}
   216		y3.Mod(y3, curve.P)
   217	
   218		return x3, y3, z3
   219	}
   220	
   221	func (curve *CurveParams) ScalarMult(Bx, By *big.Int, k []byte) (*big.Int, *big.Int) {
   222		// We have a slight problem in that the identity of the group (the
   223		// point at infinity) cannot be represented in (x, y) form on a finite
   224		// machine. Thus the standard add/double algorithm has to be tweaked
   225		// slightly: our initial state is not the identity, but x, and we
   226		// ignore the first true bit in |k|.  If we don't find any true bits in
   227		// |k|, then we return nil, nil, because we cannot return the identity
   228		// element.
   229	
   230		Bz := new(big.Int).SetInt64(1)
   231		x := Bx
   232		y := By
   233		z := Bz
   234	
   235		seenFirstTrue := false
   236		for _, byte := range k {
   237			for bitNum := 0; bitNum < 8; bitNum++ {
   238				if seenFirstTrue {
   239					x, y, z = curve.doubleJacobian(x, y, z)
   240				}
   241				if byte&0x80 == 0x80 {
   242					if !seenFirstTrue {
   243						seenFirstTrue = true
   244					} else {
   245						x, y, z = curve.addJacobian(Bx, By, Bz, x, y, z)
   246					}
   247				}
   248				byte <<= 1
   249			}
   250		}
   251	
   252		if !seenFirstTrue {
   253			return nil, nil
   254		}
   255	
   256		return curve.affineFromJacobian(x, y, z)
   257	}
   258	
   259	func (curve *CurveParams) ScalarBaseMult(k []byte) (*big.Int, *big.Int) {
   260		return curve.ScalarMult(curve.Gx, curve.Gy, k)
   261	}
   262	
   263	var mask = []byte{0xff, 0x1, 0x3, 0x7, 0xf, 0x1f, 0x3f, 0x7f}
   264	
   265	// GenerateKey returns a public/private key pair. The private key is
   266	// generated using the given reader, which must return random data.
   267	func GenerateKey(curve Curve, rand io.Reader) (priv []byte, x, y *big.Int, err error) {
   268		bitSize := curve.Params().BitSize
   269		byteLen := (bitSize + 7) >> 3
   270		priv = make([]byte, byteLen)
   271	
   272		for x == nil {
   273			_, err = io.ReadFull(rand, priv)
   274			if err != nil {
   275				return
   276			}
   277			// We have to mask off any excess bits in the case that the size of the
   278			// underlying field is not a whole number of bytes.
   279			priv[0] &= mask[bitSize%8]
   280			// This is because, in tests, rand will return all zeros and we don't
   281			// want to get the point at infinity and loop forever.
   282			priv[1] ^= 0x42
   283			x, y = curve.ScalarBaseMult(priv)
   284		}
   285		return
   286	}
   287	
   288	// Marshal converts a point into the form specified in section 4.3.6 of ANSI X9.62.
   289	func Marshal(curve Curve, x, y *big.Int) []byte {
   290		byteLen := (curve.Params().BitSize + 7) >> 3
   291	
   292		ret := make([]byte, 1+2*byteLen)
   293		ret[0] = 4 // uncompressed point
   294	
   295		xBytes := x.Bytes()
   296		copy(ret[1+byteLen-len(xBytes):], xBytes)
   297		yBytes := y.Bytes()
   298		copy(ret[1+2*byteLen-len(yBytes):], yBytes)
   299		return ret
   300	}
   301	
   302	// Unmarshal converts a point, serialized by Marshal, into an x, y pair. On error, x = nil.
   303	func Unmarshal(curve Curve, data []byte) (x, y *big.Int) {
   304		byteLen := (curve.Params().BitSize + 7) >> 3
   305		if len(data) != 1+2*byteLen {
   306			return
   307		}
   308		if data[0] != 4 { // uncompressed form
   309			return
   310		}
   311		x = new(big.Int).SetBytes(data[1 : 1+byteLen])
   312		y = new(big.Int).SetBytes(data[1+byteLen:])
   313		return
   314	}
   315	
   316	var initonce sync.Once
   317	var p256 *CurveParams
   318	var p384 *CurveParams
   319	var p521 *CurveParams
   320	
   321	func initAll() {
   322		initP224()
   323		initP256()
   324		initP384()
   325		initP521()
   326	}
   327	
   328	func initP256() {
   329		// See FIPS 186-3, section D.2.3
   330		p256 = new(CurveParams)
   331		p256.P, _ = new(big.Int).SetString("115792089210356248762697446949407573530086143415290314195533631308867097853951", 10)
   332		p256.N, _ = new(big.Int).SetString("115792089210356248762697446949407573529996955224135760342422259061068512044369", 10)
   333		p256.B, _ = new(big.Int).SetString("5ac635d8aa3a93e7b3ebbd55769886bc651d06b0cc53b0f63bce3c3e27d2604b", 16)
   334		p256.Gx, _ = new(big.Int).SetString("6b17d1f2e12c4247f8bce6e563a440f277037d812deb33a0f4a13945d898c296", 16)
   335		p256.Gy, _ = new(big.Int).SetString("4fe342e2fe1a7f9b8ee7eb4a7c0f9e162bce33576b315ececbb6406837bf51f5", 16)
   336		p256.BitSize = 256
   337	}
   338	
   339	func initP384() {
   340		// See FIPS 186-3, section D.2.4
   341		p384 = new(CurveParams)
   342		p384.P, _ = new(big.Int).SetString("39402006196394479212279040100143613805079739270465446667948293404245721771496870329047266088258938001861606973112319", 10)
   343		p384.N, _ = new(big.Int).SetString("39402006196394479212279040100143613805079739270465446667946905279627659399113263569398956308152294913554433653942643", 10)
   344		p384.B, _ = new(big.Int).SetString("b3312fa7e23ee7e4988e056be3f82d19181d9c6efe8141120314088f5013875ac656398d8a2ed19d2a85c8edd3ec2aef", 16)
   345		p384.Gx, _ = new(big.Int).SetString("aa87ca22be8b05378eb1c71ef320ad746e1d3b628ba79b9859f741e082542a385502f25dbf55296c3a545e3872760ab7", 16)
   346		p384.Gy, _ = new(big.Int).SetString("3617de4a96262c6f5d9e98bf9292dc29f8f41dbd289a147ce9da3113b5f0b8c00a60b1ce1d7e819d7a431d7c90ea0e5f", 16)
   347		p384.BitSize = 384
   348	}
   349	
   350	func initP521() {
   351		// See FIPS 186-3, section D.2.5
   352		p521 = new(CurveParams)
   353		p521.P, _ = new(big.Int).SetString("6864797660130609714981900799081393217269435300143305409394463459185543183397656052122559640661454554977296311391480858037121987999716643812574028291115057151", 10)
   354		p521.N, _ = new(big.Int).SetString("6864797660130609714981900799081393217269435300143305409394463459185543183397655394245057746333217197532963996371363321113864768612440380340372808892707005449", 10)
   355		p521.B, _ = new(big.Int).SetString("051953eb9618e1c9a1f929a21a0b68540eea2da725b99b315f3b8b489918ef109e156193951ec7e937b1652c0bd3bb1bf073573df883d2c34f1ef451fd46b503f00", 16)
   356		p521.Gx, _ = new(big.Int).SetString("c6858e06b70404e9cd9e3ecb662395b4429c648139053fb521f828af606b4d3dbaa14b5e77efe75928fe1dc127a2ffa8de3348b3c1856a429bf97e7e31c2e5bd66", 16)
   357		p521.Gy, _ = new(big.Int).SetString("11839296a789a3bc0045c8a5fb42c7d1bd998f54449579b446817afbd17273e662c97ee72995ef42640c550b9013fad0761353c7086a272c24088be94769fd16650", 16)
   358		p521.BitSize = 521
   359	}
   360	
   361	// P256 returns a Curve which implements P-256 (see FIPS 186-3, section D.2.3)
   362	func P256() Curve {
   363		initonce.Do(initAll)
   364		return p256
   365	}
   366	
   367	// P384 returns a Curve which implements P-384 (see FIPS 186-3, section D.2.4)
   368	func P384() Curve {
   369		initonce.Do(initAll)
   370		return p384
   371	}
   372	
   373	// P256 returns a Curve which implements P-521 (see FIPS 186-3, section D.2.5)
   374	func P521() Curve {
   375		initonce.Do(initAll)
   376		return p521
   377	}