bitcurve.go

Documentation: github.com/ProtonMail/go-crypto/bitcurves

     1  package bitcurves
     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  	Name    string
    29  	P       *big.Int // the order of the underlying field
    30  	N       *big.Int // the order of the base point
    31  	B       *big.Int // the constant of the BitCurve equation
    32  	Gx, Gy  *big.Int // (x,y) of the base point
    33  	BitSize int      // the size of the underlying field
    34  }
    35  
    36  // Params returns the parameters of the given BitCurve (see BitCurve struct)
    37  func (bitCurve *BitCurve) Params() (cp *elliptic.CurveParams) {
    38  	cp = new(elliptic.CurveParams)
    39  	cp.Name = bitCurve.Name
    40  	cp.P = bitCurve.P
    41  	cp.N = bitCurve.N
    42  	cp.Gx = bitCurve.Gx
    43  	cp.Gy = bitCurve.Gy
    44  	cp.BitSize = bitCurve.BitSize
    45  	return cp
    46  }
    47  
    48  // IsOnCurve returns true if the given (x,y) lies on the BitCurve.
    49  func (bitCurve *BitCurve) IsOnCurve(x, y *big.Int) bool {
    50  	// y² = x³ + b
    51  	y2 := new(big.Int).Mul(y, y) //y²
    52  	y2.Mod(y2, bitCurve.P)       //y²%P
    53  
    54  	x3 := new(big.Int).Mul(x, x) //x²
    55  	x3.Mul(x3, x)                //x³
    56  
    57  	x3.Add(x3, bitCurve.B) //x³+B
    58  	x3.Mod(x3, bitCurve.P) //(x³+B)%P
    59  
    60  	return x3.Cmp(y2) == 0
    61  }
    62  
    63  // affineFromJacobian reverses the Jacobian transform. See the comment at the
    64  // top of the file.
    65  func (bitCurve *BitCurve) affineFromJacobian(x, y, z *big.Int) (xOut, yOut *big.Int) {
    66  	if z.Cmp(big.NewInt(0)) == 0 {
    67  		panic("bitcurve: Can't convert to affine with Jacobian Z = 0")
    68  	}
    69  	// x = YZ^2 mod P
    70  	zinv := new(big.Int).ModInverse(z, bitCurve.P)
    71  	zinvsq := new(big.Int).Mul(zinv, zinv)
    72  
    73  	xOut = new(big.Int).Mul(x, zinvsq)
    74  	xOut.Mod(xOut, bitCurve.P)
    75  	// y = YZ^3 mod P
    76  	zinvsq.Mul(zinvsq, zinv)
    77  	yOut = new(big.Int).Mul(y, zinvsq)
    78  	yOut.Mod(yOut, bitCurve.P)
    79  	return xOut, yOut
    80  }
    81  
    82  // Add returns the sum of (x1,y1) and (x2,y2)
    83  func (bitCurve *BitCurve) Add(x1, y1, x2, y2 *big.Int) (*big.Int, *big.Int) {
    84  	z := new(big.Int).SetInt64(1)
    85  	x, y, z := bitCurve.addJacobian(x1, y1, z, x2, y2, z)
    86  	return bitCurve.affineFromJacobian(x, y, z)
    87  }
    88  
    89  // addJacobian takes two points in Jacobian coordinates, (x1, y1, z1) and
    90  // (x2, y2, z2) and returns their sum, also in Jacobian form.
    91  func (bitCurve *BitCurve) addJacobian(x1, y1, z1, x2, y2, z2 *big.Int) (*big.Int, *big.Int, *big.Int) {
    92  	// See http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#addition-add-2007-bl
    93  	z1z1 := new(big.Int).Mul(z1, z1)
    94  	z1z1.Mod(z1z1, bitCurve.P)
    95  	z2z2 := new(big.Int).Mul(z2, z2)
    96  	z2z2.Mod(z2z2, bitCurve.P)
    97  
    98  	u1 := new(big.Int).Mul(x1, z2z2)
    99  	u1.Mod(u1, bitCurve.P)
   100  	u2 := new(big.Int).Mul(x2, z1z1)
   101  	u2.Mod(u2, bitCurve.P)
   102  	h := new(big.Int).Sub(u2, u1)
   103  	if h.Sign() == -1 {
   104  		h.Add(h, bitCurve.P)
   105  	}
   106  	i := new(big.Int).Lsh(h, 1)
   107  	i.Mul(i, i)
   108  	j := new(big.Int).Mul(h, i)
   109  
   110  	s1 := new(big.Int).Mul(y1, z2)
   111  	s1.Mul(s1, z2z2)
   112  	s1.Mod(s1, bitCurve.P)
   113  	s2 := new(big.Int).Mul(y2, z1)
   114  	s2.Mul(s2, z1z1)
   115  	s2.Mod(s2, bitCurve.P)
   116  	r := new(big.Int).Sub(s2, s1)
   117  	if r.Sign() == -1 {
   118  		r.Add(r, bitCurve.P)
   119  	}
   120  	r.Lsh(r, 1)
   121  	v := new(big.Int).Mul(u1, i)
   122  
   123  	x3 := new(big.Int).Set(r)
   124  	x3.Mul(x3, x3)
   125  	x3.Sub(x3, j)
   126  	x3.Sub(x3, v)
   127  	x3.Sub(x3, v)
   128  	x3.Mod(x3, bitCurve.P)
   129  
   130  	y3 := new(big.Int).Set(r)
   131  	v.Sub(v, x3)
   132  	y3.Mul(y3, v)
   133  	s1.Mul(s1, j)
   134  	s1.Lsh(s1, 1)
   135  	y3.Sub(y3, s1)
   136  	y3.Mod(y3, bitCurve.P)
   137  
   138  	z3 := new(big.Int).Add(z1, z2)
   139  	z3.Mul(z3, z3)
   140  	z3.Sub(z3, z1z1)
   141  	if z3.Sign() == -1 {
   142  		z3.Add(z3, bitCurve.P)
   143  	}
   144  	z3.Sub(z3, z2z2)
   145  	if z3.Sign() == -1 {
   146  		z3.Add(z3, bitCurve.P)
   147  	}
   148  	z3.Mul(z3, h)
   149  	z3.Mod(z3, bitCurve.P)
   150  
   151  	return x3, y3, z3
   152  }
   153  
   154  // Double returns 2*(x,y)
   155  func (bitCurve *BitCurve) Double(x1, y1 *big.Int) (*big.Int, *big.Int) {
   156  	z1 := new(big.Int).SetInt64(1)
   157  	return bitCurve.affineFromJacobian(bitCurve.doubleJacobian(x1, y1, z1))
   158  }
   159  
   160  // doubleJacobian takes a point in Jacobian coordinates, (x, y, z), and
   161  // returns its double, also in Jacobian form.
   162  func (bitCurve *BitCurve) doubleJacobian(x, y, z *big.Int) (*big.Int, *big.Int, *big.Int) {
   163  	// See http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#doubling-dbl-2009-l
   164  
   165  	a := new(big.Int).Mul(x, x) //X1²
   166  	b := new(big.Int).Mul(y, y) //Y1²
   167  	c := new(big.Int).Mul(b, b) //B²
   168  
   169  	d := new(big.Int).Add(x, b) //X1+B
   170  	d.Mul(d, d)                 //(X1+B)²
   171  	d.Sub(d, a)                 //(X1+B)²-A
   172  	d.Sub(d, c)                 //(X1+B)²-A-C
   173  	d.Mul(d, big.NewInt(2))     //2*((X1+B)²-A-C)
   174  
   175  	e := new(big.Int).Mul(big.NewInt(3), a) //3*A
   176  	f := new(big.Int).Mul(e, e)             //E²
   177  
   178  	x3 := new(big.Int).Mul(big.NewInt(2), d) //2*D
   179  	x3.Sub(f, x3)                            //F-2*D
   180  	x3.Mod(x3, bitCurve.P)
   181  
   182  	y3 := new(big.Int).Sub(d, x3)                  //D-X3
   183  	y3.Mul(e, y3)                                  //E*(D-X3)
   184  	y3.Sub(y3, new(big.Int).Mul(big.NewInt(8), c)) //E*(D-X3)-8*C
   185  	y3.Mod(y3, bitCurve.P)
   186  
   187  	z3 := new(big.Int).Mul(y, z) //Y1*Z1
   188  	z3.Mul(big.NewInt(2), z3)    //3*Y1*Z1
   189  	z3.Mod(z3, bitCurve.P)
   190  
   191  	return x3, y3, z3
   192  }
   193  
   194  // TODO: double check if it is okay
   195  // ScalarMult returns k*(Bx,By) where k is a number in big-endian form.
   196  func (bitCurve *BitCurve) ScalarMult(Bx, By *big.Int, k []byte) (*big.Int, *big.Int) {
   197  	// We have a slight problem in that the identity of the group (the
   198  	// point at infinity) cannot be represented in (x, y) form on a finite
   199  	// machine. Thus the standard add/double algorithm has to be tweaked
   200  	// slightly: our initial state is not the identity, but x, and we
   201  	// ignore the first true bit in |k|.  If we don't find any true bits in
   202  	// |k|, then we return nil, nil, because we cannot return the identity
   203  	// element.
   204  
   205  	Bz := new(big.Int).SetInt64(1)
   206  	x := Bx
   207  	y := By
   208  	z := Bz
   209  
   210  	seenFirstTrue := false
   211  	for _, byte := range k {
   212  		for bitNum := 0; bitNum < 8; bitNum++ {
   213  			if seenFirstTrue {
   214  				x, y, z = bitCurve.doubleJacobian(x, y, z)
   215  			}
   216  			if byte&0x80 == 0x80 {
   217  				if !seenFirstTrue {
   218  					seenFirstTrue = true
   219  				} else {
   220  					x, y, z = bitCurve.addJacobian(Bx, By, Bz, x, y, z)
   221  				}
   222  			}
   223  			byte <<= 1
   224  		}
   225  	}
   226  
   227  	if !seenFirstTrue {
   228  		return nil, nil
   229  	}
   230  
   231  	return bitCurve.affineFromJacobian(x, y, z)
   232  }
   233  
   234  // ScalarBaseMult returns k*G, where G is the base point of the group and k is
   235  // an integer in big-endian form.
   236  func (bitCurve *BitCurve) ScalarBaseMult(k []byte) (*big.Int, *big.Int) {
   237  	return bitCurve.ScalarMult(bitCurve.Gx, bitCurve.Gy, k)
   238  }
   239  
   240  var mask = []byte{0xff, 0x1, 0x3, 0x7, 0xf, 0x1f, 0x3f, 0x7f}
   241  
   242  // TODO: double check if it is okay
   243  // GenerateKey returns a public/private key pair. The private key is generated
   244  // using the given reader, which must return random data.
   245  func (bitCurve *BitCurve) GenerateKey(rand io.Reader) (priv []byte, x, y *big.Int, err error) {
   246  	byteLen := (bitCurve.BitSize + 7) >> 3
   247  	priv = make([]byte, byteLen)
   248  
   249  	for x == nil {
   250  		_, err = io.ReadFull(rand, priv)
   251  		if err != nil {
   252  			return
   253  		}
   254  		// We have to mask off any excess bits in the case that the size of the
   255  		// underlying field is not a whole number of bytes.
   256  		priv[0] &= mask[bitCurve.BitSize%8]
   257  		// This is because, in tests, rand will return all zeros and we don't
   258  		// want to get the point at infinity and loop forever.
   259  		priv[1] ^= 0x42
   260  		x, y = bitCurve.ScalarBaseMult(priv)
   261  	}
   262  	return
   263  }
   264  
   265  // Marshal converts a point into the form specified in section 4.3.6 of ANSI
   266  // X9.62.
   267  func (bitCurve *BitCurve) Marshal(x, y *big.Int) []byte {
   268  	byteLen := (bitCurve.BitSize + 7) >> 3
   269  
   270  	ret := make([]byte, 1+2*byteLen)
   271  	ret[0] = 4 // uncompressed point
   272  
   273  	xBytes := x.Bytes()
   274  	copy(ret[1+byteLen-len(xBytes):], xBytes)
   275  	yBytes := y.Bytes()
   276  	copy(ret[1+2*byteLen-len(yBytes):], yBytes)
   277  	return ret
   278  }
   279  
   280  // Unmarshal converts a point, serialised by Marshal, into an x, y pair. On
   281  // error, x = nil.
   282  func (bitCurve *BitCurve) Unmarshal(data []byte) (x, y *big.Int) {
   283  	byteLen := (bitCurve.BitSize + 7) >> 3
   284  	if len(data) != 1+2*byteLen {
   285  		return
   286  	}
   287  	if data[0] != 4 { // uncompressed form
   288  		return
   289  	}
   290  	x = new(big.Int).SetBytes(data[1 : 1+byteLen])
   291  	y = new(big.Int).SetBytes(data[1+byteLen:])
   292  	return
   293  }
   294  
   295  //curve parameters taken from:
   296  //http://www.secg.org/collateral/sec2_final.pdf
   297  
   298  var initonce sync.Once
   299  var secp160k1 *BitCurve
   300  var secp192k1 *BitCurve
   301  var secp224k1 *BitCurve
   302  var secp256k1 *BitCurve
   303  
   304  func initAll() {
   305  	initS160()
   306  	initS192()
   307  	initS224()
   308  	initS256()
   309  }
   310  
   311  func initS160() {
   312  	// See SEC 2 section 2.4.1
   313  	secp160k1 = new(BitCurve)
   314  	secp160k1.Name = "secp160k1"
   315  	secp160k1.P, _ = new(big.Int).SetString("FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFAC73", 16)
   316  	secp160k1.N, _ = new(big.Int).SetString("0100000000000000000001B8FA16DFAB9ACA16B6B3", 16)
   317  	secp160k1.B, _ = new(big.Int).SetString("0000000000000000000000000000000000000007", 16)
   318  	secp160k1.Gx, _ = new(big.Int).SetString("3B4C382CE37AA192A4019E763036F4F5DD4D7EBB", 16)
   319  	secp160k1.Gy, _ = new(big.Int).SetString("938CF935318FDCED6BC28286531733C3F03C4FEE", 16)
   320  	secp160k1.BitSize = 160
   321  }
   322  
   323  func initS192() {
   324  	// See SEC 2 section 2.5.1
   325  	secp192k1 = new(BitCurve)
   326  	secp192k1.Name = "secp192k1"
   327  	secp192k1.P, _ = new(big.Int).SetString("FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFEE37", 16)
   328  	secp192k1.N, _ = new(big.Int).SetString("FFFFFFFFFFFFFFFFFFFFFFFE26F2FC170F69466A74DEFD8D", 16)
   329  	secp192k1.B, _ = new(big.Int).SetString("000000000000000000000000000000000000000000000003", 16)
   330  	secp192k1.Gx, _ = new(big.Int).SetString("DB4FF10EC057E9AE26B07D0280B7F4341DA5D1B1EAE06C7D", 16)
   331  	secp192k1.Gy, _ = new(big.Int).SetString("9B2F2F6D9C5628A7844163D015BE86344082AA88D95E2F9D", 16)
   332  	secp192k1.BitSize = 192
   333  }
   334  
   335  func initS224() {
   336  	// See SEC 2 section 2.6.1
   337  	secp224k1 = new(BitCurve)
   338  	secp224k1.Name = "secp224k1"
   339  	secp224k1.P, _ = new(big.Int).SetString("FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFE56D", 16)
   340  	secp224k1.N, _ = new(big.Int).SetString("010000000000000000000000000001DCE8D2EC6184CAF0A971769FB1F7", 16)
   341  	secp224k1.B, _ = new(big.Int).SetString("00000000000000000000000000000000000000000000000000000005", 16)
   342  	secp224k1.Gx, _ = new(big.Int).SetString("A1455B334DF099DF30FC28A169A467E9E47075A90F7E650EB6B7A45C", 16)
   343  	secp224k1.Gy, _ = new(big.Int).SetString("7E089FED7FBA344282CAFBD6F7E319F7C0B0BD59E2CA4BDB556D61A5", 16)
   344  	secp224k1.BitSize = 224
   345  }
   346  
   347  func initS256() {
   348  	// See SEC 2 section 2.7.1
   349  	secp256k1 = new(BitCurve)
   350  	secp256k1.Name = "secp256k1"
   351  	secp256k1.P, _ = new(big.Int).SetString("FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC2F", 16)
   352  	secp256k1.N, _ = new(big.Int).SetString("FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE6AF48A03BBFD25E8CD0364141", 16)
   353  	secp256k1.B, _ = new(big.Int).SetString("0000000000000000000000000000000000000000000000000000000000000007", 16)
   354  	secp256k1.Gx, _ = new(big.Int).SetString("79BE667EF9DCBBAC55A06295CE870B07029BFCDB2DCE28D959F2815B16F81798", 16)
   355  	secp256k1.Gy, _ = new(big.Int).SetString("483ADA7726A3C4655DA4FBFC0E1108A8FD17B448A68554199C47D08FFB10D4B8", 16)
   356  	secp256k1.BitSize = 256
   357  }
   358  
   359  // S160 returns a BitCurve which implements secp160k1 (see SEC 2 section 2.4.1)
   360  func S160() *BitCurve {
   361  	initonce.Do(initAll)
   362  	return secp160k1
   363  }
   364  
   365  // S192 returns a BitCurve which implements secp192k1 (see SEC 2 section 2.5.1)
   366  func S192() *BitCurve {
   367  	initonce.Do(initAll)
   368  	return secp192k1
   369  }
   370  
   371  // S224 returns a BitCurve which implements secp224k1 (see SEC 2 section 2.6.1)
   372  func S224() *BitCurve {
   373  	initonce.Do(initAll)
   374  	return secp224k1
   375  }
   376  
   377  // S256 returns a BitCurve which implements bitcurves (see SEC 2 section 2.7.1)
   378  func S256() *BitCurve {
   379  	initonce.Do(initAll)
   380  	return secp256k1
   381  }
   382
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