main
1package 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
18import (
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
27type 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)
37func (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.
49func (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.
65func (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)
83func (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.
91func (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)
155func (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.
162func (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.
196func (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.
236func (bitCurve *BitCurve) ScalarBaseMult(k []byte) (*big.Int, *big.Int) {
237 return bitCurve.ScalarMult(bitCurve.Gx, bitCurve.Gy, k)
238}
239
240var 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.
245func (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.
267func (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.
282func (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
298var initonce sync.Once
299var secp160k1 *BitCurve
300var secp192k1 *BitCurve
301var secp224k1 *BitCurve
302var secp256k1 *BitCurve
303
304func initAll() {
305 initS160()
306 initS192()
307 initS224()
308 initS256()
309}
310
311func 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
323func 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
335func 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
347func 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)
360func 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)
366func 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)
372func 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)
378func S256() *BitCurve {
379 initonce.Do(initAll)
380 return secp256k1
381}