-
Notifications
You must be signed in to change notification settings - Fork 19.9k
/
curve.go
397 lines (345 loc) · 13 KB
/
curve.go
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
// Copyright 2010 The Go Authors. All rights reserved.
// Copyright 2011 ThePiachu. All rights reserved.
//
// Redistribution and use in source and binary forms, with or without
// modification, are permitted provided that the following conditions are
// met:
//
// * Redistributions of source code must retain the above copyright
// notice, this list of conditions and the following disclaimer.
// * Redistributions in binary form must reproduce the above
// copyright notice, this list of conditions and the following disclaimer
// in the documentation and/or other materials provided with the
// distribution.
// * Neither the name of Google Inc. nor the names of its
// contributors may be used to endorse or promote products derived from
// this software without specific prior written permission.
// * The name of ThePiachu may not be used to endorse or promote products
// derived from this software without specific prior written permission.
//
// THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
// "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
// LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
// A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT
// OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
// SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT
// LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
// DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
// THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
// (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
// OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
package crypto
import (
"crypto/elliptic"
"io"
"math/big"
"sync"
)
// This code is from https://github.com/ThePiachu/GoBit and implements
// several Koblitz elliptic curves over prime fields.
//
// The curve methods, 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.
// A BitCurve represents a Koblitz Curve with a=0.
// See http://www.hyperelliptic.org/EFD/g1p/auto-shortw.html
type BitCurve 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 BitCurve equation
Gx, Gy *big.Int // (x,y) of the base point
BitSize int // the size of the underlying field
}
func (BitCurve *BitCurve) Params() *elliptic.CurveParams {
return &elliptic.CurveParams{
P: BitCurve.P,
N: BitCurve.N,
B: BitCurve.B,
Gx: BitCurve.Gx,
Gy: BitCurve.Gy,
BitSize: BitCurve.BitSize,
}
}
// IsOnBitCurve returns true if the given (x,y) lies on the BitCurve.
func (BitCurve *BitCurve) IsOnCurve(x, y *big.Int) bool {
// y² = x³ + b
y2 := new(big.Int).Mul(y, y) //y²
y2.Mod(y2, BitCurve.P) //y²%P
x3 := new(big.Int).Mul(x, x) //x²
x3.Mul(x3, x) //x³
x3.Add(x3, BitCurve.B) //x³+B
x3.Mod(x3, BitCurve.P) //(x³+B)%P
return x3.Cmp(y2) == 0
}
//TODO: double check if the function is okay
// affineFromJacobian reverses the Jacobian transform. See the comment at the
// top of the file.
func (BitCurve *BitCurve) affineFromJacobian(x, y, z *big.Int) (xOut, yOut *big.Int) {
zinv := new(big.Int).ModInverse(z, BitCurve.P)
zinvsq := new(big.Int).Mul(zinv, zinv)
xOut = new(big.Int).Mul(x, zinvsq)
xOut.Mod(xOut, BitCurve.P)
zinvsq.Mul(zinvsq, zinv)
yOut = new(big.Int).Mul(y, zinvsq)
yOut.Mod(yOut, BitCurve.P)
return
}
// Add returns the sum of (x1,y1) and (x2,y2)
func (BitCurve *BitCurve) Add(x1, y1, x2, y2 *big.Int) (*big.Int, *big.Int) {
z := new(big.Int).SetInt64(1)
return BitCurve.affineFromJacobian(BitCurve.addJacobian(x1, y1, z, x2, y2, z))
}
// addJacobian takes two points in Jacobian coordinates, (x1, y1, z1) and
// (x2, y2, z2) and returns their sum, also in Jacobian form.
func (BitCurve *BitCurve) addJacobian(x1, y1, z1, x2, y2, z2 *big.Int) (*big.Int, *big.Int, *big.Int) {
// See http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#addition-add-2007-bl
z1z1 := new(big.Int).Mul(z1, z1)
z1z1.Mod(z1z1, BitCurve.P)
z2z2 := new(big.Int).Mul(z2, z2)
z2z2.Mod(z2z2, BitCurve.P)
u1 := new(big.Int).Mul(x1, z2z2)
u1.Mod(u1, BitCurve.P)
u2 := new(big.Int).Mul(x2, z1z1)
u2.Mod(u2, BitCurve.P)
h := new(big.Int).Sub(u2, u1)
if h.Sign() == -1 {
h.Add(h, BitCurve.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, BitCurve.P)
s2 := new(big.Int).Mul(y2, z1)
s2.Mul(s2, z1z1)
s2.Mod(s2, BitCurve.P)
r := new(big.Int).Sub(s2, s1)
if r.Sign() == -1 {
r.Add(r, BitCurve.P)
}
r.Lsh(r, 1)
v := new(big.Int).Mul(u1, i)
x3 := new(big.Int).Set(r)
x3.Mul(x3, x3)
x3.Sub(x3, j)
x3.Sub(x3, v)
x3.Sub(x3, v)
x3.Mod(x3, BitCurve.P)
y3 := new(big.Int).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, BitCurve.P)
z3 := new(big.Int).Add(z1, z2)
z3.Mul(z3, z3)
z3.Sub(z3, z1z1)
if z3.Sign() == -1 {
z3.Add(z3, BitCurve.P)
}
z3.Sub(z3, z2z2)
if z3.Sign() == -1 {
z3.Add(z3, BitCurve.P)
}
z3.Mul(z3, h)
z3.Mod(z3, BitCurve.P)
return x3, y3, z3
}
// Double returns 2*(x,y)
func (BitCurve *BitCurve) Double(x1, y1 *big.Int) (*big.Int, *big.Int) {
z1 := new(big.Int).SetInt64(1)
return BitCurve.affineFromJacobian(BitCurve.doubleJacobian(x1, y1, z1))
}
// doubleJacobian takes a point in Jacobian coordinates, (x, y, z), and
// returns its double, also in Jacobian form.
func (BitCurve *BitCurve) doubleJacobian(x, y, z *big.Int) (*big.Int, *big.Int, *big.Int) {
// See http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#doubling-dbl-2009-l
a := new(big.Int).Mul(x, x) //X1²
b := new(big.Int).Mul(y, y) //Y1²
c := new(big.Int).Mul(b, b) //B²
d := new(big.Int).Add(x, b) //X1+B
d.Mul(d, d) //(X1+B)²
d.Sub(d, a) //(X1+B)²-A
d.Sub(d, c) //(X1+B)²-A-C
d.Mul(d, big.NewInt(2)) //2*((X1+B)²-A-C)
e := new(big.Int).Mul(big.NewInt(3), a) //3*A
f := new(big.Int).Mul(e, e) //E²
x3 := new(big.Int).Mul(big.NewInt(2), d) //2*D
x3.Sub(f, x3) //F-2*D
x3.Mod(x3, BitCurve.P)
y3 := new(big.Int).Sub(d, x3) //D-X3
y3.Mul(e, y3) //E*(D-X3)
y3.Sub(y3, new(big.Int).Mul(big.NewInt(8), c)) //E*(D-X3)-8*C
y3.Mod(y3, BitCurve.P)
z3 := new(big.Int).Mul(y, z) //Y1*Z1
z3.Mul(big.NewInt(2), z3) //3*Y1*Z1
z3.Mod(z3, BitCurve.P)
return x3, y3, z3
}
//TODO: double check if it is okay
// ScalarMult returns k*(Bx,By) where k is a number in big-endian form.
func (BitCurve *BitCurve) ScalarMult(Bx, By *big.Int, k []byte) (*big.Int, *big.Int) {
// We have a slight problem in that the identity of the group (the
// point at infinity) cannot be represented in (x, y) form on a finite
// machine. Thus the standard add/double algorithm has to be tweaked
// slightly: our initial state is not the identity, but x, and we
// ignore the first true bit in |k|. If we don't find any true bits in
// |k|, then we return nil, nil, because we cannot return the identity
// element.
Bz := new(big.Int).SetInt64(1)
x := Bx
y := By
z := Bz
seenFirstTrue := false
for _, byte := range k {
for bitNum := 0; bitNum < 8; bitNum++ {
if seenFirstTrue {
x, y, z = BitCurve.doubleJacobian(x, y, z)
}
if byte&0x80 == 0x80 {
if !seenFirstTrue {
seenFirstTrue = true
} else {
x, y, z = BitCurve.addJacobian(Bx, By, Bz, x, y, z)
}
}
byte <<= 1
}
}
if !seenFirstTrue {
return nil, nil
}
return BitCurve.affineFromJacobian(x, y, z)
}
// ScalarBaseMult returns k*G, where G is the base point of the group and k is
// an integer in big-endian form.
func (BitCurve *BitCurve) ScalarBaseMult(k []byte) (*big.Int, *big.Int) {
return BitCurve.ScalarMult(BitCurve.Gx, BitCurve.Gy, k)
}
var mask = []byte{0xff, 0x1, 0x3, 0x7, 0xf, 0x1f, 0x3f, 0x7f}
//TODO: double check if it is okay
// GenerateKey returns a public/private key pair. The private key is generated
// using the given reader, which must return random data.
func (BitCurve *BitCurve) GenerateKey(rand io.Reader) (priv []byte, x, y *big.Int, err error) {
byteLen := (BitCurve.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[BitCurve.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
x, y = BitCurve.ScalarBaseMult(priv)
}
return
}
// Marshal converts a point into the form specified in section 4.3.6 of ANSI
// X9.62.
func (BitCurve *BitCurve) Marshal(x, y *big.Int) []byte {
byteLen := (BitCurve.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, serialised by Marshal, into an x, y pair. On
// error, x = nil.
func (BitCurve *BitCurve) Unmarshal(data []byte) (x, y *big.Int) {
byteLen := (BitCurve.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:])
return
}
//curve parameters taken from:
//http://www.secg.org/collateral/sec2_final.pdf
var initonce sync.Once
var ecp160k1 *BitCurve
var ecp192k1 *BitCurve
var ecp224k1 *BitCurve
var ecp256k1 *BitCurve
func initAll() {
initS160()
initS192()
initS224()
initS256()
}
func initS160() {
// See SEC 2 section 2.4.1
ecp160k1 = new(BitCurve)
ecp160k1.P, _ = new(big.Int).SetString("FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFAC73", 16)
ecp160k1.N, _ = new(big.Int).SetString("0100000000000000000001B8FA16DFAB9ACA16B6B3", 16)
ecp160k1.B, _ = new(big.Int).SetString("0000000000000000000000000000000000000007", 16)
ecp160k1.Gx, _ = new(big.Int).SetString("3B4C382CE37AA192A4019E763036F4F5DD4D7EBB", 16)
ecp160k1.Gy, _ = new(big.Int).SetString("938CF935318FDCED6BC28286531733C3F03C4FEE", 16)
ecp160k1.BitSize = 160
}
func initS192() {
// See SEC 2 section 2.5.1
ecp192k1 = new(BitCurve)
ecp192k1.P, _ = new(big.Int).SetString("FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFEE37", 16)
ecp192k1.N, _ = new(big.Int).SetString("FFFFFFFFFFFFFFFFFFFFFFFE26F2FC170F69466A74DEFD8D", 16)
ecp192k1.B, _ = new(big.Int).SetString("000000000000000000000000000000000000000000000003", 16)
ecp192k1.Gx, _ = new(big.Int).SetString("DB4FF10EC057E9AE26B07D0280B7F4341DA5D1B1EAE06C7D", 16)
ecp192k1.Gy, _ = new(big.Int).SetString("9B2F2F6D9C5628A7844163D015BE86344082AA88D95E2F9D", 16)
ecp192k1.BitSize = 192
}
func initS224() {
// See SEC 2 section 2.6.1
ecp224k1 = new(BitCurve)
ecp224k1.P, _ = new(big.Int).SetString("FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFE56D", 16)
ecp224k1.N, _ = new(big.Int).SetString("010000000000000000000000000001DCE8D2EC6184CAF0A971769FB1F7", 16)
ecp224k1.B, _ = new(big.Int).SetString("00000000000000000000000000000000000000000000000000000005", 16)
ecp224k1.Gx, _ = new(big.Int).SetString("A1455B334DF099DF30FC28A169A467E9E47075A90F7E650EB6B7A45C", 16)
ecp224k1.Gy, _ = new(big.Int).SetString("7E089FED7FBA344282CAFBD6F7E319F7C0B0BD59E2CA4BDB556D61A5", 16)
ecp224k1.BitSize = 224
}
func initS256() {
// See SEC 2 section 2.7.1
ecp256k1 = new(BitCurve)
ecp256k1.P, _ = new(big.Int).SetString("FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC2F", 16)
ecp256k1.N, _ = new(big.Int).SetString("FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE6AF48A03BBFD25E8CD0364141", 16)
ecp256k1.B, _ = new(big.Int).SetString("0000000000000000000000000000000000000000000000000000000000000007", 16)
ecp256k1.Gx, _ = new(big.Int).SetString("79BE667EF9DCBBAC55A06295CE870B07029BFCDB2DCE28D959F2815B16F81798", 16)
ecp256k1.Gy, _ = new(big.Int).SetString("483ADA7726A3C4655DA4FBFC0E1108A8FD17B448A68554199C47D08FFB10D4B8", 16)
ecp256k1.BitSize = 256
}
// S160 returns a BitCurve which implements secp160k1 (see SEC 2 section 2.4.1)
func S160() *BitCurve {
initonce.Do(initAll)
return ecp160k1
}
// S192 returns a BitCurve which implements secp192k1 (see SEC 2 section 2.5.1)
func S192() *BitCurve {
initonce.Do(initAll)
return ecp192k1
}
// S224 returns a BitCurve which implements secp224k1 (see SEC 2 section 2.6.1)
func S224() *BitCurve {
initonce.Do(initAll)
return ecp224k1
}
// S256 returns a BitCurve which implements secp256k1 (see SEC 2 section 2.7.1)
func S256() *BitCurve {
initonce.Do(initAll)
return ecp256k1
}