forked from winderica/kryptology
-
Notifications
You must be signed in to change notification settings - Fork 0
/
field.go
388 lines (345 loc) · 11 KB
/
field.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
package native
import (
"encoding/binary"
"fmt"
"math/big"
"github.com/coinbase/kryptology/internal"
)
// FieldLimbs is the number of limbs needed to represent this field
const FieldLimbs = 4
// FieldBytes is the number of bytes needed to represent this field
const FieldBytes = 32
// WideFieldBytes is the number of bytes needed for safe conversion
// to this field to avoid bias when reduced
const WideFieldBytes = 64
// Field represents a field element
type Field struct {
// Value is the field elements value
Value [FieldLimbs]uint64
// Params are the field parameters
Params *FieldParams
// Arithmetic are the field methods
Arithmetic FieldArithmetic
}
// FieldParams are the field parameters
type FieldParams struct {
// R is 2^256 mod Modulus
R [FieldLimbs]uint64
// R2 is 2^512 mod Modulus
R2 [FieldLimbs]uint64
// R3 is 2^768 mod Modulus
R3 [FieldLimbs]uint64
// Modulus of the field
Modulus [FieldLimbs]uint64
// Modulus as big.Int
BiModulus *big.Int
}
// FieldArithmetic are the methods that can be done on a field
type FieldArithmetic interface {
// ToMontgomery converts this field to montgomery form
ToMontgomery(out, arg *[FieldLimbs]uint64)
// FromMontgomery converts this field from montgomery form
FromMontgomery(out, arg *[FieldLimbs]uint64)
// Neg performs modular negation
Neg(out, arg *[FieldLimbs]uint64)
// Square performs modular square
Square(out, arg *[FieldLimbs]uint64)
// Mul performs modular multiplication
Mul(out, arg1, arg2 *[FieldLimbs]uint64)
// Add performs modular addition
Add(out, arg1, arg2 *[FieldLimbs]uint64)
// Sub performs modular subtraction
Sub(out, arg1, arg2 *[FieldLimbs]uint64)
// Sqrt performs modular square root
Sqrt(wasSquare *int, out, arg *[FieldLimbs]uint64)
// Invert performs modular inverse
Invert(wasInverted *int, out, arg *[FieldLimbs]uint64)
// FromBytes converts a little endian byte array into a field element
FromBytes(out *[FieldLimbs]uint64, arg *[FieldBytes]byte)
// ToBytes converts a field element to a little endian byte array
ToBytes(out *[FieldBytes]byte, arg *[FieldLimbs]uint64)
// Selectznz performs conditional select.
// selects arg1 if choice == 0 and arg2 if choice == 1
Selectznz(out, arg1, arg2 *[FieldLimbs]uint64, choice int)
}
// Cmp returns -1 if f < rhs
// 0 if f == rhs
// 1 if f > rhs
func (f *Field) Cmp(rhs *Field) int {
return cmpHelper(&f.Value, &rhs.Value)
}
// cmpHelper returns -1 if lhs < rhs
// -1 if lhs == rhs
// 1 if lhs > rhs
// Public only for convenience for some internal implementations
func cmpHelper(lhs, rhs *[FieldLimbs]uint64) int {
gt := uint64(0)
lt := uint64(0)
for i := 3; i >= 0; i-- {
// convert to two 64-bit numbers where
// the leading bits are zeros and hold no meaning
// so rhs - fp actually means gt
// and fp - rhs actually means lt.
rhsH := rhs[i] >> 32
rhsL := rhs[i] & 0xffffffff
lhsH := lhs[i] >> 32
lhsL := lhs[i] & 0xffffffff
// Check the leading bit
// if negative then fp > rhs
// if positive then fp < rhs
gt |= (rhsH - lhsH) >> 32 & 1 &^ lt
lt |= (lhsH - rhsH) >> 32 & 1 &^ gt
gt |= (rhsL - lhsL) >> 32 & 1 &^ lt
lt |= (lhsL - rhsL) >> 32 & 1 &^ gt
}
// Make the result -1 for <, 0 for =, 1 for >
return int(gt) - int(lt)
}
// Equal returns 1 if f == rhs, 0 otherwise
func (f *Field) Equal(rhs *Field) int {
return equalHelper(&f.Value, &rhs.Value)
}
func equalHelper(lhs, rhs *[FieldLimbs]uint64) int {
t := lhs[0] ^ rhs[0]
t |= lhs[1] ^ rhs[1]
t |= lhs[2] ^ rhs[2]
t |= lhs[3] ^ rhs[3]
return int(((int64(t) | int64(-t)) >> 63) + 1)
}
// IsZero returns 1 if f == 0, 0 otherwise
func (f *Field) IsZero() int {
t := f.Value[0]
t |= f.Value[1]
t |= f.Value[2]
t |= f.Value[3]
return int(((int64(t) | int64(-t)) >> 63) + 1)
}
// IsNonZero returns 1 if f != 0, 0 otherwise
func (f *Field) IsNonZero() int {
t := f.Value[0]
t |= f.Value[1]
t |= f.Value[2]
t |= f.Value[3]
return int(-((int64(t) | int64(-t)) >> 63))
}
// IsOne returns 1 if f == 1, 0 otherwise
func (f *Field) IsOne() int {
return equalHelper(&f.Value, &f.Params.R)
}
// Set f = rhs
func (f *Field) Set(rhs *Field) *Field {
f.Value[0] = rhs.Value[0]
f.Value[1] = rhs.Value[1]
f.Value[2] = rhs.Value[2]
f.Value[3] = rhs.Value[3]
f.Params = rhs.Params
f.Arithmetic = rhs.Arithmetic
return f
}
// SetUint64 f = rhs
func (f *Field) SetUint64(rhs uint64) *Field {
t := &[FieldLimbs]uint64{rhs, 0, 0, 0}
f.Arithmetic.ToMontgomery(&f.Value, t)
return f
}
// SetOne f = r
func (f *Field) SetOne() *Field {
f.Value[0] = f.Params.R[0]
f.Value[1] = f.Params.R[1]
f.Value[2] = f.Params.R[2]
f.Value[3] = f.Params.R[3]
return f
}
// SetZero f = 0
func (f *Field) SetZero() *Field {
f.Value[0] = 0
f.Value[1] = 0
f.Value[2] = 0
f.Value[3] = 0
return f
}
// SetBytesWide takes 64 bytes as input and treats them as a 512-bit number.
// Attributed to https://github.com/zcash/pasta_curves/blob/main/src/fields/Fp.rs#L255
// We reduce an arbitrary 512-bit number by decomposing it into two 256-bit digits
// with the higher bits multiplied by 2^256. Thus, we perform two reductions
//
// 1. the lower bits are multiplied by r^2, as normal
// 2. the upper bits are multiplied by r^2 * 2^256 = r^3
//
// and computing their sum in the field. It remains to see that arbitrary 256-bit
// numbers can be placed into Montgomery form safely using the reduction. The
// reduction works so long as the product is less than r=2^256 multiplied by
// the modulus. This holds because for any `c` smaller than the modulus, we have
// that (2^256 - 1)*c is an acceptable product for the reduction. Therefore, the
// reduction always works so long as `c` is in the field; in this case it is either the
// constant `r2` or `r3`.
func (f *Field) SetBytesWide(input *[WideFieldBytes]byte) *Field {
d0 := [FieldLimbs]uint64{
binary.LittleEndian.Uint64(input[:8]),
binary.LittleEndian.Uint64(input[8:16]),
binary.LittleEndian.Uint64(input[16:24]),
binary.LittleEndian.Uint64(input[24:32]),
}
d1 := [FieldLimbs]uint64{
binary.LittleEndian.Uint64(input[32:40]),
binary.LittleEndian.Uint64(input[40:48]),
binary.LittleEndian.Uint64(input[48:56]),
binary.LittleEndian.Uint64(input[56:64]),
}
//f.Arithmetic.ToMontgomery(&d0, &d0)
//f.Arithmetic.Mul(&d1, &d1, &f.Params.R2)
//f.Arithmetic.Add(&f.Value, &d0, &d0)
// Convert to Montgomery form
tv1 := &[FieldLimbs]uint64{}
tv2 := &[FieldLimbs]uint64{}
// d0*r2 + d1*r3
f.Arithmetic.Mul(tv1, &d0, &f.Params.R2)
f.Arithmetic.Mul(tv2, &d1, &f.Params.R3)
f.Arithmetic.Add(&f.Value, tv1, tv2)
return f
}
// SetBytes attempts to convert a little endian byte representation
// of a scalar into a `Fp`, failing if input is not canonical
func (f *Field) SetBytes(input *[FieldBytes]byte) (*Field, error) {
d0 := [FieldLimbs]uint64{0, 0, 0, 0}
f.Arithmetic.FromBytes(&d0, input)
if cmpHelper(&d0, &f.Params.Modulus) != -1 {
return nil, fmt.Errorf("invalid byte sequence")
}
return f.SetLimbs(&d0), nil
}
// SetBigInt initializes an element from big.Int
// The value is reduced by the modulus
func (f *Field) SetBigInt(bi *big.Int) *Field {
var buffer [FieldBytes]byte
t := new(big.Int).Set(bi)
t.Mod(t, f.Params.BiModulus)
t.FillBytes(buffer[:])
copy(buffer[:], internal.ReverseScalarBytes(buffer[:]))
_, _ = f.SetBytes(&buffer)
return f
}
// SetRaw converts a raw array into a field element
// Assumes input is already in montgomery form
func (f *Field) SetRaw(input *[FieldLimbs]uint64) *Field {
f.Value[0] = input[0]
f.Value[1] = input[1]
f.Value[2] = input[2]
f.Value[3] = input[3]
return f
}
// SetLimbs converts an array into a field element
// by converting to montgomery form
func (f *Field) SetLimbs(input *[FieldLimbs]uint64) *Field {
f.Arithmetic.ToMontgomery(&f.Value, input)
return f
}
// Bytes converts this element into a byte representation
// in little endian byte order
func (f *Field) Bytes() [FieldBytes]byte {
var output [FieldBytes]byte
tv := &[FieldLimbs]uint64{}
f.Arithmetic.FromMontgomery(tv, &f.Value)
f.Arithmetic.ToBytes(&output, tv)
return output
}
// BigInt converts this element into the big.Int struct
func (f *Field) BigInt() *big.Int {
buffer := f.Bytes()
return new(big.Int).SetBytes(internal.ReverseScalarBytes(buffer[:]))
}
// Raw converts this element into the a [FieldLimbs]uint64
func (f *Field) Raw() [FieldLimbs]uint64 {
res := &[FieldLimbs]uint64{}
f.Arithmetic.FromMontgomery(res, &f.Value)
return *res
}
// Double this element
func (f *Field) Double(a *Field) *Field {
f.Arithmetic.Add(&f.Value, &a.Value, &a.Value)
return f
}
// Square this element
func (f *Field) Square(a *Field) *Field {
f.Arithmetic.Square(&f.Value, &a.Value)
return f
}
// Sqrt this element, if it exists. If true, then value
// is a square root. If false, value is a QNR
func (f *Field) Sqrt(a *Field) (*Field, bool) {
wasSquare := 0
f.Arithmetic.Sqrt(&wasSquare, &f.Value, &a.Value)
return f, wasSquare == 1
}
// Invert this element i.e. compute the multiplicative inverse
// return false, zero if this element is zero.
func (f *Field) Invert(a *Field) (*Field, bool) {
wasInverted := 0
f.Arithmetic.Invert(&wasInverted, &f.Value, &a.Value)
return f, wasInverted == 1
}
// Mul returns the result from multiplying this element by rhs
func (f *Field) Mul(lhs, rhs *Field) *Field {
f.Arithmetic.Mul(&f.Value, &lhs.Value, &rhs.Value)
return f
}
// Sub returns the result from subtracting rhs from this element
func (f *Field) Sub(lhs, rhs *Field) *Field {
f.Arithmetic.Sub(&f.Value, &lhs.Value, &rhs.Value)
return f
}
// Add returns the result from adding rhs to this element
func (f *Field) Add(lhs, rhs *Field) *Field {
f.Arithmetic.Add(&f.Value, &lhs.Value, &rhs.Value)
return f
}
// Neg returns negation of this element
func (f *Field) Neg(input *Field) *Field {
f.Arithmetic.Neg(&f.Value, &input.Value)
return f
}
// Exp raises base^exp
func (f *Field) Exp(base, exp *Field) *Field {
e := [FieldLimbs]uint64{}
f.Arithmetic.FromMontgomery(&e, &exp.Value)
Pow(&f.Value, &base.Value, &e, f.Params, f.Arithmetic)
return f
}
// CMove sets f = lhs if choice == 0 and f = rhs if choice == 1
func (f *Field) CMove(lhs, rhs *Field, choice int) *Field {
f.Arithmetic.Selectznz(&f.Value, &lhs.Value, &rhs.Value, choice)
return f
}
// Pow raises base^exp. The result is written to out.
// Public only for convenience for some internal implementations
func Pow(out, base, exp *[FieldLimbs]uint64, params *FieldParams, arithmetic FieldArithmetic) {
res := [FieldLimbs]uint64{params.R[0], params.R[1], params.R[2], params.R[3]}
tmp := [FieldLimbs]uint64{}
for i := len(exp) - 1; i >= 0; i-- {
for j := 63; j >= 0; j-- {
arithmetic.Square(&res, &res)
arithmetic.Mul(&tmp, &res, base)
arithmetic.Selectznz(&res, &res, &tmp, int(exp[i]>>j)&1)
}
}
out[0] = res[0]
out[1] = res[1]
out[2] = res[2]
out[3] = res[3]
}
// Pow2k raises arg to the power `2^k`. This result is written to out.
// Public only for convenience for some internal implementations
func Pow2k(out, arg *[FieldLimbs]uint64, k int, arithmetic FieldArithmetic) {
var t [FieldLimbs]uint64
t[0] = arg[0]
t[1] = arg[1]
t[2] = arg[2]
t[3] = arg[3]
for i := 0; i < k; i++ {
arithmetic.Square(&t, &t)
}
out[0] = t[0]
out[1] = t[1]
out[2] = t[2]
out[3] = t[3]
}