github.com/jonasnick/go-ethereum@v0.7.12-0.20150216215225-22176f05d387/crypto/curve.go (about)

     1  package crypto
     2  
     3  // Copyright 2010 The Go Authors. All rights reserved.
     4  // Copyright 2011 ThePiachu. All rights reserved.
     5  // Use of this source code is governed by a BSD-style
     6  // license that can be found in the LICENSE file.
     7  
     8  // Package bitelliptic implements several Koblitz elliptic curves over prime
     9  // fields.
    10  
    11  // This package operates, internally, on Jacobian coordinates. For a given
    12  // (x, y) position on the curve, the Jacobian coordinates are (x1, y1, z1)
    13  // where x = x1/z1² and y = y1/z1³. The greatest speedups come when the whole
    14  // calculation can be performed within the transform (as in ScalarMult and
    15  // ScalarBaseMult). But even for Add and Double, it's faster to apply and
    16  // reverse the transform than to operate in affine coordinates.
    17  
    18  import (
    19  	"crypto/elliptic"
    20  	"io"
    21  	"math/big"
    22  	"sync"
    23  )
    24  
    25  // A BitCurve represents a Koblitz Curve with a=0.
    26  // See http://www.hyperelliptic.org/EFD/g1p/auto-shortw.html
    27  type BitCurve struct {
    28  	P       *big.Int // the order of the underlying field
    29  	N       *big.Int // the order of the base point
    30  	B       *big.Int // the constant of the BitCurve equation
    31  	Gx, Gy  *big.Int // (x,y) of the base point
    32  	BitSize int      // the size of the underlying field
    33  }
    34  
    35  func (BitCurve *BitCurve) Params() *elliptic.CurveParams {
    36  	return &elliptic.CurveParams{BitCurve.P, BitCurve.N, BitCurve.B, BitCurve.Gx, BitCurve.Gy, BitCurve.BitSize}
    37  }
    38  
    39  // IsOnBitCurve returns true if the given (x,y) lies on the BitCurve.
    40  func (BitCurve *BitCurve) IsOnCurve(x, y *big.Int) bool {
    41  	// y² = x³ + b
    42  	y2 := new(big.Int).Mul(y, y) //y²
    43  	y2.Mod(y2, BitCurve.P)       //y²%P
    44  
    45  	x3 := new(big.Int).Mul(x, x) //x²
    46  	x3.Mul(x3, x)                //x³
    47  
    48  	x3.Add(x3, BitCurve.B) //x³+B
    49  	x3.Mod(x3, BitCurve.P) //(x³+B)%P
    50  
    51  	return x3.Cmp(y2) == 0
    52  }
    53  
    54  //TODO: double check if the function is okay
    55  // affineFromJacobian reverses the Jacobian transform. See the comment at the
    56  // top of the file.
    57  func (BitCurve *BitCurve) affineFromJacobian(x, y, z *big.Int) (xOut, yOut *big.Int) {
    58  	zinv := new(big.Int).ModInverse(z, BitCurve.P)
    59  	zinvsq := new(big.Int).Mul(zinv, zinv)
    60  
    61  	xOut = new(big.Int).Mul(x, zinvsq)
    62  	xOut.Mod(xOut, BitCurve.P)
    63  	zinvsq.Mul(zinvsq, zinv)
    64  	yOut = new(big.Int).Mul(y, zinvsq)
    65  	yOut.Mod(yOut, BitCurve.P)
    66  	return
    67  }
    68  
    69  // Add returns the sum of (x1,y1) and (x2,y2)
    70  func (BitCurve *BitCurve) Add(x1, y1, x2, y2 *big.Int) (*big.Int, *big.Int) {
    71  	z := new(big.Int).SetInt64(1)
    72  	return BitCurve.affineFromJacobian(BitCurve.addJacobian(x1, y1, z, x2, y2, z))
    73  }
    74  
    75  // addJacobian takes two points in Jacobian coordinates, (x1, y1, z1) and
    76  // (x2, y2, z2) and returns their sum, also in Jacobian form.
    77  func (BitCurve *BitCurve) addJacobian(x1, y1, z1, x2, y2, z2 *big.Int) (*big.Int, *big.Int, *big.Int) {
    78  	// See http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#addition-add-2007-bl
    79  	z1z1 := new(big.Int).Mul(z1, z1)
    80  	z1z1.Mod(z1z1, BitCurve.P)
    81  	z2z2 := new(big.Int).Mul(z2, z2)
    82  	z2z2.Mod(z2z2, BitCurve.P)
    83  
    84  	u1 := new(big.Int).Mul(x1, z2z2)
    85  	u1.Mod(u1, BitCurve.P)
    86  	u2 := new(big.Int).Mul(x2, z1z1)
    87  	u2.Mod(u2, BitCurve.P)
    88  	h := new(big.Int).Sub(u2, u1)
    89  	if h.Sign() == -1 {
    90  		h.Add(h, BitCurve.P)
    91  	}
    92  	i := new(big.Int).Lsh(h, 1)
    93  	i.Mul(i, i)
    94  	j := new(big.Int).Mul(h, i)
    95  
    96  	s1 := new(big.Int).Mul(y1, z2)
    97  	s1.Mul(s1, z2z2)
    98  	s1.Mod(s1, BitCurve.P)
    99  	s2 := new(big.Int).Mul(y2, z1)
   100  	s2.Mul(s2, z1z1)
   101  	s2.Mod(s2, BitCurve.P)
   102  	r := new(big.Int).Sub(s2, s1)
   103  	if r.Sign() == -1 {
   104  		r.Add(r, BitCurve.P)
   105  	}
   106  	r.Lsh(r, 1)
   107  	v := new(big.Int).Mul(u1, i)
   108  
   109  	x3 := new(big.Int).Set(r)
   110  	x3.Mul(x3, x3)
   111  	x3.Sub(x3, j)
   112  	x3.Sub(x3, v)
   113  	x3.Sub(x3, v)
   114  	x3.Mod(x3, BitCurve.P)
   115  
   116  	y3 := new(big.Int).Set(r)
   117  	v.Sub(v, x3)
   118  	y3.Mul(y3, v)
   119  	s1.Mul(s1, j)
   120  	s1.Lsh(s1, 1)
   121  	y3.Sub(y3, s1)
   122  	y3.Mod(y3, BitCurve.P)
   123  
   124  	z3 := new(big.Int).Add(z1, z2)
   125  	z3.Mul(z3, z3)
   126  	z3.Sub(z3, z1z1)
   127  	if z3.Sign() == -1 {
   128  		z3.Add(z3, BitCurve.P)
   129  	}
   130  	z3.Sub(z3, z2z2)
   131  	if z3.Sign() == -1 {
   132  		z3.Add(z3, BitCurve.P)
   133  	}
   134  	z3.Mul(z3, h)
   135  	z3.Mod(z3, BitCurve.P)
   136  
   137  	return x3, y3, z3
   138  }
   139  
   140  // Double returns 2*(x,y)
   141  func (BitCurve *BitCurve) Double(x1, y1 *big.Int) (*big.Int, *big.Int) {
   142  	z1 := new(big.Int).SetInt64(1)
   143  	return BitCurve.affineFromJacobian(BitCurve.doubleJacobian(x1, y1, z1))
   144  }
   145  
   146  // doubleJacobian takes a point in Jacobian coordinates, (x, y, z), and
   147  // returns its double, also in Jacobian form.
   148  func (BitCurve *BitCurve) doubleJacobian(x, y, z *big.Int) (*big.Int, *big.Int, *big.Int) {
   149  	// See http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#doubling-dbl-2009-l
   150  
   151  	a := new(big.Int).Mul(x, x) //X1²
   152  	b := new(big.Int).Mul(y, y) //Y1²
   153  	c := new(big.Int).Mul(b, b) //B²
   154  
   155  	d := new(big.Int).Add(x, b) //X1+B
   156  	d.Mul(d, d)                 //(X1+B)²
   157  	d.Sub(d, a)                 //(X1+B)²-A
   158  	d.Sub(d, c)                 //(X1+B)²-A-C
   159  	d.Mul(d, big.NewInt(2))     //2*((X1+B)²-A-C)
   160  
   161  	e := new(big.Int).Mul(big.NewInt(3), a) //3*A
   162  	f := new(big.Int).Mul(e, e)             //E²
   163  
   164  	x3 := new(big.Int).Mul(big.NewInt(2), d) //2*D
   165  	x3.Sub(f, x3)                            //F-2*D
   166  	x3.Mod(x3, BitCurve.P)
   167  
   168  	y3 := new(big.Int).Sub(d, x3)                  //D-X3
   169  	y3.Mul(e, y3)                                  //E*(D-X3)
   170  	y3.Sub(y3, new(big.Int).Mul(big.NewInt(8), c)) //E*(D-X3)-8*C
   171  	y3.Mod(y3, BitCurve.P)
   172  
   173  	z3 := new(big.Int).Mul(y, z) //Y1*Z1
   174  	z3.Mul(big.NewInt(2), z3)    //3*Y1*Z1
   175  	z3.Mod(z3, BitCurve.P)
   176  
   177  	return x3, y3, z3
   178  }
   179  
   180  //TODO: double check if it is okay
   181  // ScalarMult returns k*(Bx,By) where k is a number in big-endian form.
   182  func (BitCurve *BitCurve) ScalarMult(Bx, By *big.Int, k []byte) (*big.Int, *big.Int) {
   183  	// We have a slight problem in that the identity of the group (the
   184  	// point at infinity) cannot be represented in (x, y) form on a finite
   185  	// machine. Thus the standard add/double algorithm has to be tweaked
   186  	// slightly: our initial state is not the identity, but x, and we
   187  	// ignore the first true bit in |k|.  If we don't find any true bits in
   188  	// |k|, then we return nil, nil, because we cannot return the identity
   189  	// element.
   190  
   191  	Bz := new(big.Int).SetInt64(1)
   192  	x := Bx
   193  	y := By
   194  	z := Bz
   195  
   196  	seenFirstTrue := false
   197  	for _, byte := range k {
   198  		for bitNum := 0; bitNum < 8; bitNum++ {
   199  			if seenFirstTrue {
   200  				x, y, z = BitCurve.doubleJacobian(x, y, z)
   201  			}
   202  			if byte&0x80 == 0x80 {
   203  				if !seenFirstTrue {
   204  					seenFirstTrue = true
   205  				} else {
   206  					x, y, z = BitCurve.addJacobian(Bx, By, Bz, x, y, z)
   207  				}
   208  			}
   209  			byte <<= 1
   210  		}
   211  	}
   212  
   213  	if !seenFirstTrue {
   214  		return nil, nil
   215  	}
   216  
   217  	return BitCurve.affineFromJacobian(x, y, z)
   218  }
   219  
   220  // ScalarBaseMult returns k*G, where G is the base point of the group and k is
   221  // an integer in big-endian form.
   222  func (BitCurve *BitCurve) ScalarBaseMult(k []byte) (*big.Int, *big.Int) {
   223  	return BitCurve.ScalarMult(BitCurve.Gx, BitCurve.Gy, k)
   224  }
   225  
   226  var mask = []byte{0xff, 0x1, 0x3, 0x7, 0xf, 0x1f, 0x3f, 0x7f}
   227  
   228  //TODO: double check if it is okay
   229  // GenerateKey returns a public/private key pair. The private key is generated
   230  // using the given reader, which must return random data.
   231  func (BitCurve *BitCurve) GenerateKey(rand io.Reader) (priv []byte, x, y *big.Int, err error) {
   232  	byteLen := (BitCurve.BitSize + 7) >> 3
   233  	priv = make([]byte, byteLen)
   234  
   235  	for x == nil {
   236  		_, err = io.ReadFull(rand, priv)
   237  		if err != nil {
   238  			return
   239  		}
   240  		// We have to mask off any excess bits in the case that the size of the
   241  		// underlying field is not a whole number of bytes.
   242  		priv[0] &= mask[BitCurve.BitSize%8]
   243  		// This is because, in tests, rand will return all zeros and we don't
   244  		// want to get the point at infinity and loop forever.
   245  		priv[1] ^= 0x42
   246  		x, y = BitCurve.ScalarBaseMult(priv)
   247  	}
   248  	return
   249  }
   250  
   251  // Marshal converts a point into the form specified in section 4.3.6 of ANSI
   252  // X9.62.
   253  func (BitCurve *BitCurve) Marshal(x, y *big.Int) []byte {
   254  	byteLen := (BitCurve.BitSize + 7) >> 3
   255  
   256  	ret := make([]byte, 1+2*byteLen)
   257  	ret[0] = 4 // uncompressed point
   258  
   259  	xBytes := x.Bytes()
   260  	copy(ret[1+byteLen-len(xBytes):], xBytes)
   261  	yBytes := y.Bytes()
   262  	copy(ret[1+2*byteLen-len(yBytes):], yBytes)
   263  	return ret
   264  }
   265  
   266  // Unmarshal converts a point, serialised by Marshal, into an x, y pair. On
   267  // error, x = nil.
   268  func (BitCurve *BitCurve) Unmarshal(data []byte) (x, y *big.Int) {
   269  	byteLen := (BitCurve.BitSize + 7) >> 3
   270  	if len(data) != 1+2*byteLen {
   271  		return
   272  	}
   273  	if data[0] != 4 { // uncompressed form
   274  		return
   275  	}
   276  	x = new(big.Int).SetBytes(data[1 : 1+byteLen])
   277  	y = new(big.Int).SetBytes(data[1+byteLen:])
   278  	return
   279  }
   280  
   281  //curve parameters taken from:
   282  //http://www.secg.org/collateral/sec2_final.pdf
   283  
   284  var initonce sync.Once
   285  var ecp160k1 *BitCurve
   286  var ecp192k1 *BitCurve
   287  var ecp224k1 *BitCurve
   288  var ecp256k1 *BitCurve
   289  
   290  func initAll() {
   291  	initS160()
   292  	initS192()
   293  	initS224()
   294  	initS256()
   295  }
   296  
   297  func initS160() {
   298  	// See SEC 2 section 2.4.1
   299  	ecp160k1 = new(BitCurve)
   300  	ecp160k1.P, _ = new(big.Int).SetString("FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFAC73", 16)
   301  	ecp160k1.N, _ = new(big.Int).SetString("0100000000000000000001B8FA16DFAB9ACA16B6B3", 16)
   302  	ecp160k1.B, _ = new(big.Int).SetString("0000000000000000000000000000000000000007", 16)
   303  	ecp160k1.Gx, _ = new(big.Int).SetString("3B4C382CE37AA192A4019E763036F4F5DD4D7EBB", 16)
   304  	ecp160k1.Gy, _ = new(big.Int).SetString("938CF935318FDCED6BC28286531733C3F03C4FEE", 16)
   305  	ecp160k1.BitSize = 160
   306  }
   307  
   308  func initS192() {
   309  	// See SEC 2 section 2.5.1
   310  	ecp192k1 = new(BitCurve)
   311  	ecp192k1.P, _ = new(big.Int).SetString("FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFEE37", 16)
   312  	ecp192k1.N, _ = new(big.Int).SetString("FFFFFFFFFFFFFFFFFFFFFFFE26F2FC170F69466A74DEFD8D", 16)
   313  	ecp192k1.B, _ = new(big.Int).SetString("000000000000000000000000000000000000000000000003", 16)
   314  	ecp192k1.Gx, _ = new(big.Int).SetString("DB4FF10EC057E9AE26B07D0280B7F4341DA5D1B1EAE06C7D", 16)
   315  	ecp192k1.Gy, _ = new(big.Int).SetString("9B2F2F6D9C5628A7844163D015BE86344082AA88D95E2F9D", 16)
   316  	ecp192k1.BitSize = 192
   317  }
   318  
   319  func initS224() {
   320  	// See SEC 2 section 2.6.1
   321  	ecp224k1 = new(BitCurve)
   322  	ecp224k1.P, _ = new(big.Int).SetString("FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFE56D", 16)
   323  	ecp224k1.N, _ = new(big.Int).SetString("010000000000000000000000000001DCE8D2EC6184CAF0A971769FB1F7", 16)
   324  	ecp224k1.B, _ = new(big.Int).SetString("00000000000000000000000000000000000000000000000000000005", 16)
   325  	ecp224k1.Gx, _ = new(big.Int).SetString("A1455B334DF099DF30FC28A169A467E9E47075A90F7E650EB6B7A45C", 16)
   326  	ecp224k1.Gy, _ = new(big.Int).SetString("7E089FED7FBA344282CAFBD6F7E319F7C0B0BD59E2CA4BDB556D61A5", 16)
   327  	ecp224k1.BitSize = 224
   328  }
   329  
   330  func initS256() {
   331  	// See SEC 2 section 2.7.1
   332  	ecp256k1 = new(BitCurve)
   333  	ecp256k1.P, _ = new(big.Int).SetString("FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC2F", 16)
   334  	ecp256k1.N, _ = new(big.Int).SetString("FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE6AF48A03BBFD25E8CD0364141", 16)
   335  	ecp256k1.B, _ = new(big.Int).SetString("0000000000000000000000000000000000000000000000000000000000000007", 16)
   336  	ecp256k1.Gx, _ = new(big.Int).SetString("79BE667EF9DCBBAC55A06295CE870B07029BFCDB2DCE28D959F2815B16F81798", 16)
   337  	ecp256k1.Gy, _ = new(big.Int).SetString("483ADA7726A3C4655DA4FBFC0E1108A8FD17B448A68554199C47D08FFB10D4B8", 16)
   338  	ecp256k1.BitSize = 256
   339  }
   340  
   341  // S160 returns a BitCurve which implements secp160k1 (see SEC 2 section 2.4.1)
   342  func S160() *BitCurve {
   343  	initonce.Do(initAll)
   344  	return ecp160k1
   345  }
   346  
   347  // S192 returns a BitCurve which implements secp192k1 (see SEC 2 section 2.5.1)
   348  func S192() *BitCurve {
   349  	initonce.Do(initAll)
   350  	return ecp192k1
   351  }
   352  
   353  // S224 returns a BitCurve which implements secp224k1 (see SEC 2 section 2.6.1)
   354  func S224() *BitCurve {
   355  	initonce.Do(initAll)
   356  	return ecp224k1
   357  }
   358  
   359  // S256 returns a BitCurve which implements secp256k1 (see SEC 2 section 2.7.1)
   360  func S256() *BitCurve {
   361  	initonce.Do(initAll)
   362  	return ecp256k1
   363  }