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Adam Langley71484c72012-07-27 12:54:55 -04001// Copyright 2012 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
5package bn256
6
7// For details of the algorithms used, see "Multiplication and Squaring on
8// Pairing-Friendly Fields, Devegili et al.
9// http://eprint.iacr.org/2006/471.pdf.
10
11import (
12 "math/big"
13)
14
15// gfP2 implements a field of size p² as a quadratic extension of the base
16// field where i²=-1.
17type gfP2 struct {
18 x, y *big.Int // value is xi+y.
19}
20
21func newGFp2(pool *bnPool) *gfP2 {
22 return &gfP2{pool.Get(), pool.Get()}
23}
24
25func (e *gfP2) String() string {
26 x := new(big.Int).Mod(e.x, p)
27 y := new(big.Int).Mod(e.y, p)
28 return "(" + x.String() + "," + y.String() + ")"
29}
30
31func (e *gfP2) Put(pool *bnPool) {
32 pool.Put(e.x)
33 pool.Put(e.y)
34}
35
36func (e *gfP2) Set(a *gfP2) *gfP2 {
37 e.x.Set(a.x)
38 e.y.Set(a.y)
39 return e
40}
41
42func (e *gfP2) SetZero() *gfP2 {
43 e.x.SetInt64(0)
44 e.y.SetInt64(0)
45 return e
46}
47
48func (e *gfP2) SetOne() *gfP2 {
49 e.x.SetInt64(0)
50 e.y.SetInt64(1)
51 return e
52}
53
54func (e *gfP2) Minimal() {
55 if e.x.Sign() < 0 || e.x.Cmp(p) >= 0 {
56 e.x.Mod(e.x, p)
57 }
58 if e.y.Sign() < 0 || e.y.Cmp(p) >= 0 {
59 e.y.Mod(e.y, p)
60 }
61}
62
63func (e *gfP2) IsZero() bool {
64 return e.x.Sign() == 0 && e.y.Sign() == 0
65}
66
67func (e *gfP2) IsOne() bool {
68 if e.x.Sign() != 0 {
69 return false
70 }
71 words := e.y.Bits()
72 return len(words) == 1 && words[0] == 1
73}
74
75func (e *gfP2) Conjugate(a *gfP2) *gfP2 {
76 e.y.Set(a.y)
77 e.x.Neg(a.x)
78 return e
79}
80
81func (e *gfP2) Negative(a *gfP2) *gfP2 {
82 e.x.Neg(a.x)
83 e.y.Neg(a.y)
84 return e
85}
86
87func (e *gfP2) Add(a, b *gfP2) *gfP2 {
88 e.x.Add(a.x, b.x)
89 e.y.Add(a.y, b.y)
90 return e
91}
92
93func (e *gfP2) Sub(a, b *gfP2) *gfP2 {
94 e.x.Sub(a.x, b.x)
95 e.y.Sub(a.y, b.y)
96 return e
97}
98
99func (e *gfP2) Double(a *gfP2) *gfP2 {
100 e.x.Lsh(a.x, 1)
101 e.y.Lsh(a.y, 1)
102 return e
103}
104
105func (c *gfP2) Exp(a *gfP2, power *big.Int, pool *bnPool) *gfP2 {
106 sum := newGFp2(pool)
107 sum.SetOne()
108 t := newGFp2(pool)
109
110 for i := power.BitLen() - 1; i >= 0; i-- {
111 t.Square(sum, pool)
112 if power.Bit(i) != 0 {
113 sum.Mul(t, a, pool)
114 } else {
115 sum.Set(t)
116 }
117 }
118
119 c.Set(sum)
120
121 sum.Put(pool)
122 t.Put(pool)
123
124 return c
125}
126
127// See "Multiplication and Squaring in Pairing-Friendly Fields",
128// http://eprint.iacr.org/2006/471.pdf
129func (e *gfP2) Mul(a, b *gfP2, pool *bnPool) *gfP2 {
130 tx := pool.Get().Mul(a.x, b.y)
131 t := pool.Get().Mul(b.x, a.y)
132 tx.Add(tx, t)
133 tx.Mod(tx, p)
134
135 ty := pool.Get().Mul(a.y, b.y)
136 t.Mul(a.x, b.x)
137 ty.Sub(ty, t)
138 e.y.Mod(ty, p)
139 e.x.Set(tx)
140
141 pool.Put(tx)
142 pool.Put(ty)
143 pool.Put(t)
144
145 return e
146}
147
148func (e *gfP2) MulScalar(a *gfP2, b *big.Int) *gfP2 {
149 e.x.Mul(a.x, b)
150 e.y.Mul(a.y, b)
151 return e
152}
153
154// MulXi sets e=ξa where ξ=i+3 and then returns e.
155func (e *gfP2) MulXi(a *gfP2, pool *bnPool) *gfP2 {
156 // (xi+y)(i+3) = (3x+y)i+(3y-x)
157 tx := pool.Get().Lsh(a.x, 1)
158 tx.Add(tx, a.x)
159 tx.Add(tx, a.y)
160
161 ty := pool.Get().Lsh(a.y, 1)
162 ty.Add(ty, a.y)
163 ty.Sub(ty, a.x)
164
165 e.x.Set(tx)
166 e.y.Set(ty)
167
168 pool.Put(tx)
169 pool.Put(ty)
170
171 return e
172}
173
174func (e *gfP2) Square(a *gfP2, pool *bnPool) *gfP2 {
175 // Complex squaring algorithm:
176 // (xi+b)² = (x+y)(y-x) + 2*i*x*y
177 t1 := pool.Get().Sub(a.y, a.x)
178 t2 := pool.Get().Add(a.x, a.y)
179 ty := pool.Get().Mul(t1, t2)
180 ty.Mod(ty, p)
181
182 t1.Mul(a.x, a.y)
183 t1.Lsh(t1, 1)
184
185 e.x.Mod(t1, p)
186 e.y.Set(ty)
187
188 pool.Put(t1)
189 pool.Put(t2)
190 pool.Put(ty)
191
192 return e
193}
194
195func (e *gfP2) Invert(a *gfP2, pool *bnPool) *gfP2 {
196 // See "Implementing cryptographic pairings", M. Scott, section 3.2.
197 // ftp://136.206.11.249/pub/crypto/pairings.pdf
198 t := pool.Get()
199 t.Mul(a.y, a.y)
200 t2 := pool.Get()
201 t2.Mul(a.x, a.x)
202 t.Add(t, t2)
203
204 inv := pool.Get()
205 inv.ModInverse(t, p)
206
207 e.x.Neg(a.x)
208 e.x.Mul(e.x, inv)
209 e.x.Mod(e.x, p)
210
211 e.y.Mul(a.y, inv)
212 e.y.Mod(e.y, p)
213
214 pool.Put(t)
215 pool.Put(t2)
216 pool.Put(inv)
217
218 return e
219}