forked from cloudflare/circl
-
Notifications
You must be signed in to change notification settings - Fork 1
/
fp.go
263 lines (240 loc) · 7.28 KB
/
fp.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
package ff
import (
"io"
"github.com/ReallyMeLabs/circl/internal/conv"
)
// FpSize is the length in bytes of an Fp element.
const FpSize = 48
// fpMont represents an element in the Montgomery domain (little-endian).
type fpMont = [FpSize / 8]uint64
// fpRaw represents an element in the integers domain (little-endian).
type fpRaw = [FpSize / 8]uint64
// Fp represents prime field elements as positive integers less than FpOrder.
type Fp struct{ i fpMont }
func (z Fp) String() string { x := z.fromMont(); return conv.Uint64Le2Hex(x[:]) }
func (z *Fp) SetUint64(n uint64) { z.toMont(&fpRaw{n}) }
func (z *Fp) SetOne() { z.SetUint64(1) }
func (z *Fp) Random(r io.Reader) error { return randomInt(z.i[:], r, fpOrder[:]) }
// IsNegative returns 0 if the least absolute residue for z is in [0,(p-1)/2],
// and 1 otherwise. Equivalently, this function returns 1 if z is
// lexicographically larger than -z.
func (z Fp) IsNegative() int {
b, _ := z.MarshalBinary()
return 1 - isLessThan(b, fpOrderPlus1Div2[:])
}
// IsZero returns 1 if z == 0 and 0 otherwise.
func (z Fp) IsZero() int { return ctUint64Eq(z.i[:], (&fpMont{})[:]) }
// IsEqual returns 1 if z == x and 0 otherwise.
func (z Fp) IsEqual(x *Fp) int { return ctUint64Eq(z.i[:], x.i[:]) }
func (z *Fp) Neg() { fiatFpMontSub(&z.i, &fpMont{}, &z.i) }
func (z *Fp) Add(x, y *Fp) { fiatFpMontAdd(&z.i, &x.i, &y.i) }
func (z *Fp) Sub(x, y *Fp) { fiatFpMontSub(&z.i, &x.i, &y.i) }
func (z *Fp) Mul(x, y *Fp) { fiatFpMontMul(&z.i, &x.i, &y.i) }
func (z *Fp) Sqr(x *Fp) { fiatFpMontSquare(&z.i, &x.i) }
func (z *Fp) toMont(in *fpRaw) { fiatFpMontMul(&z.i, in, &fpRSquare) }
func (z Fp) fromMont() (out fpRaw) { fiatFpMontMul(&out, &z.i, &fpMont{1}); return }
func (z Fp) Sgn0() int { return int(z.fromMont()[0]) & 1 }
// Sqrt returns 1 and sets z=sqrt(x) only if x is a quadratic-residue; otherwise, returns 0 and z is unmodified.
func (z *Fp) Sqrt(x *Fp) int {
var y, y2 Fp
y.ExpVarTime(x, fpOrderPlus1Div4[:])
y2.Sqr(&y)
isQR := y2.IsEqual(x)
z.CMov(z, &y, isQR)
return isQR
}
// CMov sets z=x if b == 0 and z=y if b == 1. Its behavior is undefined if b takes any other value.
func (z *Fp) CMov(x, y *Fp, b int) {
mask := -uint64(b & 0x1)
for i := 0; i < FpSize/8; i++ {
z.i[i] = (x.i[i] &^ mask) | (y.i[i] & mask)
}
}
// FpOrder is the order of the base field for towering returned as a big-endian slice.
//
// FpOrder = 0x1a0111ea397fe69a4b1ba7b6434bacd764774b84f38512bf6730d2a0f6b0f6241eabfffeb153ffffb9feffffffffaaab.
func FpOrder() []byte { o := fpOrder; return o[:] }
// ExpVarTime calculates z=x^n, where n is the exponent in big-endian order.
func (z *Fp) ExpVarTime(x *Fp, n []byte) {
zz := new(Fp)
zz.SetOne()
N := 8 * len(n)
for i := 0; i < N; i++ {
zz.Sqr(zz)
bit := 0x1 & (n[i/8] >> uint(7-i%8))
if bit != 0 {
zz.Mul(zz, x)
}
}
*z = *zz
}
// SetBytes assigns to z the number modulo FpOrder stored in the slice
// (in big-endian order).
func (z *Fp) SetBytes(data []byte) {
in64 := setBytesUnbounded(data, fpOrder[:])
s := &fpRaw{}
copy(s[:], in64[:FpSize/8])
z.toMont(s)
}
// MarshalBinary returns a slice of FpSize bytes that contains the minimal
// residue of z such that 0 <= z < FpOrder (in big-endian order).
func (z *Fp) MarshalBinary() ([]byte, error) {
x := z.fromMont()
return conv.Uint64Le2BytesBe(x[:]), nil
}
// UnmarshalBinary reconstructs a Fp from a slice that must have at least
// FpSize bytes and contain a number (in big-endian order) from 0
// to FpOrder-1.
func (z *Fp) UnmarshalBinary(b []byte) error {
if len(b) < FpSize {
return errInputLength
}
in64, err := setBytesBounded(b[:FpSize], fpOrder[:])
if err == nil {
s := &fpRaw{}
copy(s[:], in64[:FpSize/8])
z.toMont(s)
}
return err
}
// SetString reconstructs a Fp from a numeric string from 0 to FpOrder-1.
func (z *Fp) SetString(s string) error {
in64, err := setString(s, fpOrder[:])
if err == nil {
s := &fpRaw{}
copy(s[:], in64[:FpSize/8])
z.toMont(s)
}
return err
}
func fiatFpMontCmovznzU64(z *uint64, b, x, y uint64) { cselectU64(z, b, x, y) }
func (z *Fp) Inv(x *Fp) {
// Addition chain found using mmcloughlin/addchain: v0.3.0
// McLoughlin, Michael Ben. (2021). https://doi.org/10.5281/zenodo.4758226
var i2, i4, i8, i9, i11, i13, i17, i20, i25, i26, i52, i54, i55, i77, i79,
i86, i93, i103, i105, i119, i123, i137, i149, i151, i169, i177, i191,
i195, i208, i215, i225, i229, i235, i245, i255 Fp
i2.Sqr(x)
i4.Sqr(&i2)
i8.Sqr(&i4)
i9.Mul(&i8, x)
i11.Mul(&i9, &i2)
i13.Mul(&i11, &i2)
i17.Mul(&i13, &i4)
i20.Mul(&i11, &i9)
i25.Mul(&i17, &i8)
i26.Mul(&i25, x)
i52.Sqr(&i26)
i54.Mul(&i52, &i2)
i55.Mul(&i54, x)
i77.Mul(&i52, &i25)
i79.Mul(&i77, &i2)
i86.Mul(&i77, &i8)
i93.Mul(&i86, &i8)
i103.Mul(&i77, &i26)
i105.Mul(&i103, &i2)
i119.Mul(&i93, &i26)
i123.Mul(&i119, &i4)
i137.Mul(&i86, &i52)
i149.Mul(&i123, &i26)
i151.Mul(&i149, &i2)
i169.Mul(&i149, &i20)
i177.Mul(&i169, &i8)
i191.Mul(&i137, &i54)
i195.Mul(&i191, &i4)
i208.Mul(&i195, &i13)
i215.Mul(&i195, &i20)
i225.Mul(&i208, &i17)
i229.Mul(&i225, &i4)
i235.Mul(&i215, &i20)
i245.Mul(&i225, &i20)
i255.Mul(&i235, &i20)
z.Mul(&i225, &i191)
for _, s := range []struct {
l int
x *Fp
}{
{8, &i17},
{11, &i245},
{11, &i229},
{8, &i255},
{7, &i77},
{9, &i105},
{10, &i177},
{7, &i93},
{9, &i123},
{6, &i25},
{11, &i105},
{9, &i235},
{10, &i215},
{6, &i25},
{10, &i119},
{9, &i151},
{11, &i79},
{10, &i225},
{9, &i137},
{9, &i191},
{8, &i103},
{10, &i195},
{9, &i149},
{12, &i123},
{5, &i11},
{11, &i123},
{7, &i9},
{13, &i245},
{9, &i191},
{8, &i255},
{8, &i235},
{11, &i169},
{8, &i255},
{8, &i255},
{6, &i55},
{10, &i255},
{9, &i255},
{8, &i255},
{8, &i255},
{8, &i255},
{7, &i86},
{9, &i169},
} {
for i := 0; i < s.l; i++ {
z.Sqr(z)
}
z.Mul(z, s.x)
}
}
var (
// fpOrder is the order of the Fp field (big-endian).
fpOrder = [FpSize]byte{
0x1a, 0x01, 0x11, 0xea, 0x39, 0x7f, 0xe6, 0x9a,
0x4b, 0x1b, 0xa7, 0xb6, 0x43, 0x4b, 0xac, 0xd7,
0x64, 0x77, 0x4b, 0x84, 0xf3, 0x85, 0x12, 0xbf,
0x67, 0x30, 0xd2, 0xa0, 0xf6, 0xb0, 0xf6, 0x24,
0x1e, 0xab, 0xff, 0xfe, 0xb1, 0x53, 0xff, 0xff,
0xb9, 0xfe, 0xff, 0xff, 0xff, 0xff, 0xaa, 0xab,
}
// fpOrderPlus1Div2 is the half of (fpOrder plus one) used for lexicographically order (big-endian).
fpOrderPlus1Div2 = [FpSize]byte{
0x0d, 0x00, 0x88, 0xf5, 0x1c, 0xbf, 0xf3, 0x4d,
0x25, 0x8d, 0xd3, 0xdb, 0x21, 0xa5, 0xd6, 0x6b,
0xb2, 0x3b, 0xa5, 0xc2, 0x79, 0xc2, 0x89, 0x5f,
0xb3, 0x98, 0x69, 0x50, 0x7b, 0x58, 0x7b, 0x12,
0x0f, 0x55, 0xff, 0xff, 0x58, 0xa9, 0xff, 0xff,
0xdc, 0xff, 0x7f, 0xff, 0xff, 0xff, 0xd5, 0x56,
}
// fpOrderPlus1Div4 is (fpOrder plus one) divided by four used for square-roots (big-endian).
fpOrderPlus1Div4 = [FpSize]byte{
0x06, 0x80, 0x44, 0x7a, 0x8e, 0x5f, 0xf9, 0xa6,
0x92, 0xc6, 0xe9, 0xed, 0x90, 0xd2, 0xeb, 0x35,
0xd9, 0x1d, 0xd2, 0xe1, 0x3c, 0xe1, 0x44, 0xaf,
0xd9, 0xcc, 0x34, 0xa8, 0x3d, 0xac, 0x3d, 0x89,
0x07, 0xaa, 0xff, 0xff, 0xac, 0x54, 0xff, 0xff,
0xee, 0x7f, 0xbf, 0xff, 0xff, 0xff, 0xea, 0xab,
}
// fpRSquare is R^2 mod fpOrder, where R=2^384 (little-endian).
fpRSquare = fpMont{
0xf4df1f341c341746, 0x0a76e6a609d104f1,
0x8de5476c4c95b6d5, 0x67eb88a9939d83c0,
0x9a793e85b519952d, 0x11988fe592cae3aa,
}
)