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
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
|
// Copyright 2012 The Go Authors. All rights reserved.
// Use of this source code is governed by a BSD-style
// license that can be found in the LICENSE file.
package elliptic
// This is a constant-time, 32-bit implementation of P224. See FIPS 186-3,
// section D.2.2.
//
// See http://www.imperialviolet.org/2010/12/04/ecc.html ([1]) for background.
import (
"math/big"
)
var p224 p224Curve
type p224Curve struct {
*CurveParams
gx, gy, b p224FieldElement
}
func initP224() {
// See FIPS 186-3, section D.2.2
p224.CurveParams = &CurveParams{Name: "P-224"}
p224.P, _ = new(big.Int).SetString("26959946667150639794667015087019630673557916260026308143510066298881", 10)
p224.N, _ = new(big.Int).SetString("26959946667150639794667015087019625940457807714424391721682722368061", 10)
p224.B, _ = new(big.Int).SetString("b4050a850c04b3abf54132565044b0b7d7bfd8ba270b39432355ffb4", 16)
p224.Gx, _ = new(big.Int).SetString("b70e0cbd6bb4bf7f321390b94a03c1d356c21122343280d6115c1d21", 16)
p224.Gy, _ = new(big.Int).SetString("bd376388b5f723fb4c22dfe6cd4375a05a07476444d5819985007e34", 16)
p224.BitSize = 224
p224FromBig(&p224.gx, p224.Gx)
p224FromBig(&p224.gy, p224.Gy)
p224FromBig(&p224.b, p224.B)
}
// P224 returns a Curve which implements P-224 (see FIPS 186-3, section D.2.2).
//
// The cryptographic operations are implemented using constant-time algorithms.
func P224() Curve {
initonce.Do(initAll)
return p224
}
func (curve p224Curve) Params() *CurveParams {
return curve.CurveParams
}
func (curve p224Curve) IsOnCurve(bigX, bigY *big.Int) bool {
var x, y p224FieldElement
p224FromBig(&x, bigX)
p224FromBig(&y, bigY)
// y² = x³ - 3x + b
var tmp p224LargeFieldElement
var x3 p224FieldElement
p224Square(&x3, &x, &tmp)
p224Mul(&x3, &x3, &x, &tmp)
for i := 0; i < 8; i++ {
x[i] *= 3
}
p224Sub(&x3, &x3, &x)
p224Reduce(&x3)
p224Add(&x3, &x3, &curve.b)
p224Contract(&x3, &x3)
p224Square(&y, &y, &tmp)
p224Contract(&y, &y)
for i := 0; i < 8; i++ {
if y[i] != x3[i] {
return false
}
}
return true
}
func (p224Curve) Add(bigX1, bigY1, bigX2, bigY2 *big.Int) (x, y *big.Int) {
var x1, y1, z1, x2, y2, z2, x3, y3, z3 p224FieldElement
p224FromBig(&x1, bigX1)
p224FromBig(&y1, bigY1)
if bigX1.Sign() != 0 || bigY1.Sign() != 0 {
z1[0] = 1
}
p224FromBig(&x2, bigX2)
p224FromBig(&y2, bigY2)
if bigX2.Sign() != 0 || bigY2.Sign() != 0 {
z2[0] = 1
}
p224AddJacobian(&x3, &y3, &z3, &x1, &y1, &z1, &x2, &y2, &z2)
return p224ToAffine(&x3, &y3, &z3)
}
func (p224Curve) Double(bigX1, bigY1 *big.Int) (x, y *big.Int) {
var x1, y1, z1, x2, y2, z2 p224FieldElement
p224FromBig(&x1, bigX1)
p224FromBig(&y1, bigY1)
z1[0] = 1
p224DoubleJacobian(&x2, &y2, &z2, &x1, &y1, &z1)
return p224ToAffine(&x2, &y2, &z2)
}
func (p224Curve) ScalarMult(bigX1, bigY1 *big.Int, scalar []byte) (x, y *big.Int) {
var x1, y1, z1, x2, y2, z2 p224FieldElement
p224FromBig(&x1, bigX1)
p224FromBig(&y1, bigY1)
z1[0] = 1
p224ScalarMult(&x2, &y2, &z2, &x1, &y1, &z1, scalar)
return p224ToAffine(&x2, &y2, &z2)
}
func (curve p224Curve) ScalarBaseMult(scalar []byte) (x, y *big.Int) {
var z1, x2, y2, z2 p224FieldElement
z1[0] = 1
p224ScalarMult(&x2, &y2, &z2, &curve.gx, &curve.gy, &z1, scalar)
return p224ToAffine(&x2, &y2, &z2)
}
// Field element functions.
//
// The field that we're dealing with is ℤ/pℤ where p = 2**224 - 2**96 + 1.
//
// Field elements are represented by a FieldElement, which is a typedef to an
// array of 8 uint32's. The value of a FieldElement, a, is:
// a[0] + 2**28·a[1] + 2**56·a[1] + ... + 2**196·a[7]
//
// Using 28-bit limbs means that there's only 4 bits of headroom, which is less
// than we would really like. But it has the useful feature that we hit 2**224
// exactly, making the reflections during a reduce much nicer.
type p224FieldElement [8]uint32
// p224P is the order of the field, represented as a p224FieldElement.
var p224P = [8]uint32{1, 0, 0, 0xffff000, 0xfffffff, 0xfffffff, 0xfffffff, 0xfffffff}
// p224IsZero returns 1 if a == 0 mod p and 0 otherwise.
//
// a[i] < 2**29
func p224IsZero(a *p224FieldElement) uint32 {
// Since a p224FieldElement contains 224 bits there are two possible
// representations of 0: 0 and p.
var minimal p224FieldElement
p224Contract(&minimal, a)
var isZero, isP uint32
for i, v := range minimal {
isZero |= v
isP |= v - p224P[i]
}
// If either isZero or isP is 0, then we should return 1.
isZero |= isZero >> 16
isZero |= isZero >> 8
isZero |= isZero >> 4
isZero |= isZero >> 2
isZero |= isZero >> 1
isP |= isP >> 16
isP |= isP >> 8
isP |= isP >> 4
isP |= isP >> 2
isP |= isP >> 1
// For isZero and isP, the LSB is 0 iff all the bits are zero.
result := isZero & isP
result = (^result) & 1
return result
}
// p224Add computes *out = a+b
//
// a[i] + b[i] < 2**32
func p224Add(out, a, b *p224FieldElement) {
for i := 0; i < 8; i++ {
out[i] = a[i] + b[i]
}
}
const two31p3 = 1<<31 + 1<<3
const two31m3 = 1<<31 - 1<<3
const two31m15m3 = 1<<31 - 1<<15 - 1<<3
// p224ZeroModP31 is 0 mod p where bit 31 is set in all limbs so that we can
// subtract smaller amounts without underflow. See the section "Subtraction" in
// [1] for reasoning.
var p224ZeroModP31 = []uint32{two31p3, two31m3, two31m3, two31m15m3, two31m3, two31m3, two31m3, two31m3}
// p224Sub computes *out = a-b
//
// a[i], b[i] < 2**30
// out[i] < 2**32
func p224Sub(out, a, b *p224FieldElement) {
for i := 0; i < 8; i++ {
out[i] = a[i] + p224ZeroModP31[i] - b[i]
}
}
// LargeFieldElement also represents an element of the field. The limbs are
// still spaced 28-bits apart and in little-endian order. So the limbs are at
// 0, 28, 56, ..., 392 bits, each 64-bits wide.
type p224LargeFieldElement [15]uint64
const two63p35 = 1<<63 + 1<<35
const two63m35 = 1<<63 - 1<<35
const two63m35m19 = 1<<63 - 1<<35 - 1<<19
// p224ZeroModP63 is 0 mod p where bit 63 is set in all limbs. See the section
// "Subtraction" in [1] for why.
var p224ZeroModP63 = [8]uint64{two63p35, two63m35, two63m35, two63m35, two63m35m19, two63m35, two63m35, two63m35}
const bottom12Bits = 0xfff
const bottom28Bits = 0xfffffff
// p224Mul computes *out = a*b
//
// a[i] < 2**29, b[i] < 2**30 (or vice versa)
// out[i] < 2**29
func p224Mul(out, a, b *p224FieldElement, tmp *p224LargeFieldElement) {
for i := 0; i < 15; i++ {
tmp[i] = 0
}
for i := 0; i < 8; i++ {
for j := 0; j < 8; j++ {
tmp[i+j] += uint64(a[i]) * uint64(b[j])
}
}
p224ReduceLarge(out, tmp)
}
// Square computes *out = a*a
//
// a[i] < 2**29
// out[i] < 2**29
func p224Square(out, a *p224FieldElement, tmp *p224LargeFieldElement) {
for i := 0; i < 15; i++ {
tmp[i] = 0
}
for i := 0; i < 8; i++ {
for j := 0; j <= i; j++ {
r := uint64(a[i]) * uint64(a[j])
if i == j {
tmp[i+j] += r
} else {
tmp[i+j] += r << 1
}
}
}
p224ReduceLarge(out, tmp)
}
// ReduceLarge converts a p224LargeFieldElement to a p224FieldElement.
//
// in[i] < 2**62
func p224ReduceLarge(out *p224FieldElement, in *p224LargeFieldElement) {
for i := 0; i < 8; i++ {
in[i] += p224ZeroModP63[i]
}
// Eliminate the coefficients at 2**224 and greater.
for i := 14; i >= 8; i-- {
in[i-8] -= in[i]
in[i-5] += (in[i] & 0xffff) << 12
in[i-4] += in[i] >> 16
}
in[8] = 0
// in[0..8] < 2**64
// As the values become small enough, we start to store them in |out|
// and use 32-bit operations.
for i := 1; i < 8; i++ {
in[i+1] += in[i] >> 28
out[i] = uint32(in[i] & bottom28Bits)
}
in[0] -= in[8]
out[3] += uint32(in[8]&0xffff) << 12
out[4] += uint32(in[8] >> 16)
// in[0] < 2**64
// out[3] < 2**29
// out[4] < 2**29
// out[1,2,5..7] < 2**28
out[0] = uint32(in[0] & bottom28Bits)
out[1] += uint32((in[0] >> 28) & bottom28Bits)
out[2] += uint32(in[0] >> 56)
// out[0] < 2**28
// out[1..4] < 2**29
// out[5..7] < 2**28
}
// Reduce reduces the coefficients of a to smaller bounds.
//
// On entry: a[i] < 2**31 + 2**30
// On exit: a[i] < 2**29
func p224Reduce(a *p224FieldElement) {
for i := 0; i < 7; i++ {
a[i+1] += a[i] >> 28
a[i] &= bottom28Bits
}
top := a[7] >> 28
a[7] &= bottom28Bits
// top < 2**4
mask := top
mask |= mask >> 2
mask |= mask >> 1
mask <<= 31
mask = uint32(int32(mask) >> 31)
// Mask is all ones if top != 0, all zero otherwise
a[0] -= top
a[3] += top << 12
// We may have just made a[0] negative but, if we did, then we must
// have added something to a[3], this it's > 2**12. Therefore we can
// carry down to a[0].
a[3] -= 1 & mask
a[2] += mask & (1<<28 - 1)
a[1] += mask & (1<<28 - 1)
a[0] += mask & (1 << 28)
}
// p224Invert calculates *out = in**-1 by computing in**(2**224 - 2**96 - 1),
// i.e. Fermat's little theorem.
func p224Invert(out, in *p224FieldElement) {
var f1, f2, f3, f4 p224FieldElement
var c p224LargeFieldElement
p224Square(&f1, in, &c) // 2
p224Mul(&f1, &f1, in, &c) // 2**2 - 1
p224Square(&f1, &f1, &c) // 2**3 - 2
p224Mul(&f1, &f1, in, &c) // 2**3 - 1
p224Square(&f2, &f1, &c) // 2**4 - 2
p224Square(&f2, &f2, &c) // 2**5 - 4
p224Square(&f2, &f2, &c) // 2**6 - 8
p224Mul(&f1, &f1, &f2, &c) // 2**6 - 1
p224Square(&f2, &f1, &c) // 2**7 - 2
for i := 0; i < 5; i++ { // 2**12 - 2**6
p224Square(&f2, &f2, &c)
}
p224Mul(&f2, &f2, &f1, &c) // 2**12 - 1
p224Square(&f3, &f2, &c) // 2**13 - 2
for i := 0; i < 11; i++ { // 2**24 - 2**12
p224Square(&f3, &f3, &c)
}
p224Mul(&f2, &f3, &f2, &c) // 2**24 - 1
p224Square(&f3, &f2, &c) // 2**25 - 2
for i := 0; i < 23; i++ { // 2**48 - 2**24
p224Square(&f3, &f3, &c)
}
p224Mul(&f3, &f3, &f2, &c) // 2**48 - 1
p224Square(&f4, &f3, &c) // 2**49 - 2
for i := 0; i < 47; i++ { // 2**96 - 2**48
p224Square(&f4, &f4, &c)
}
p224Mul(&f3, &f3, &f4, &c) // 2**96 - 1
p224Square(&f4, &f3, &c) // 2**97 - 2
for i := 0; i < 23; i++ { // 2**120 - 2**24
p224Square(&f4, &f4, &c)
}
p224Mul(&f2, &f4, &f2, &c) // 2**120 - 1
for i := 0; i < 6; i++ { // 2**126 - 2**6
p224Square(&f2, &f2, &c)
}
p224Mul(&f1, &f1, &f2, &c) // 2**126 - 1
p224Square(&f1, &f1, &c) // 2**127 - 2
p224Mul(&f1, &f1, in, &c) // 2**127 - 1
for i := 0; i < 97; i++ { // 2**224 - 2**97
p224Square(&f1, &f1, &c)
}
p224Mul(out, &f1, &f3, &c) // 2**224 - 2**96 - 1
}
// p224Contract converts a FieldElement to its unique, minimal form.
//
// On entry, in[i] < 2**29
// On exit, in[i] < 2**28
func p224Contract(out, in *p224FieldElement) {
copy(out[:], in[:])
for i := 0; i < 7; i++ {
out[i+1] += out[i] >> 28
out[i] &= bottom28Bits
}
top := out[7] >> 28
out[7] &= bottom28Bits
out[0] -= top
out[3] += top << 12
// We may just have made out[i] negative. So we carry down. If we made
// out[0] negative then we know that out[3] is sufficiently positive
// because we just added to it.
for i := 0; i < 3; i++ {
mask := uint32(int32(out[i]) >> 31)
out[i] += (1 << 28) & mask
out[i+1] -= 1 & mask
}
// We might have pushed out[3] over 2**28 so we perform another, partial,
// carry chain.
for i := 3; i < 7; i++ {
out[i+1] += out[i] >> 28
out[i] &= bottom28Bits
}
top = out[7] >> 28
out[7] &= bottom28Bits
// Eliminate top while maintaining the same value mod p.
out[0] -= top
out[3] += top << 12
// There are two cases to consider for out[3]:
// 1) The first time that we eliminated top, we didn't push out[3] over
// 2**28. In this case, the partial carry chain didn't change any values
// and top is zero.
// 2) We did push out[3] over 2**28 the first time that we eliminated top.
// The first value of top was in [0..16), therefore, prior to eliminating
// the first top, 0xfff1000 <= out[3] <= 0xfffffff. Therefore, after
// overflowing and being reduced by the second carry chain, out[3] <=
// 0xf000. Thus it cannot have overflowed when we eliminated top for the
// second time.
// Again, we may just have made out[0] negative, so do the same carry down.
// As before, if we made out[0] negative then we know that out[3] is
// sufficiently positive.
for i := 0; i < 3; i++ {
mask := uint32(int32(out[i]) >> 31)
out[i] += (1 << 28) & mask
out[i+1] -= 1 & mask
}
// Now we see if the value is >= p and, if so, subtract p.
// First we build a mask from the top four limbs, which must all be
// equal to bottom28Bits if the whole value is >= p. If top4AllOnes
// ends up with any zero bits in the bottom 28 bits, then this wasn't
// true.
top4AllOnes := uint32(0xffffffff)
for i := 4; i < 8; i++ {
top4AllOnes &= out[i]
}
top4AllOnes |= 0xf0000000
// Now we replicate any zero bits to all the bits in top4AllOnes.
top4AllOnes &= top4AllOnes >> 16
top4AllOnes &= top4AllOnes >> 8
top4AllOnes &= top4AllOnes >> 4
top4AllOnes &= top4AllOnes >> 2
top4AllOnes &= top4AllOnes >> 1
top4AllOnes = uint32(int32(top4AllOnes<<31) >> 31)
// Now we test whether the bottom three limbs are non-zero.
bottom3NonZero := out[0] | out[1] | out[2]
bottom3NonZero |= bottom3NonZero >> 16
bottom3NonZero |= bottom3NonZero >> 8
bottom3NonZero |= bottom3NonZero >> 4
bottom3NonZero |= bottom3NonZero >> 2
bottom3NonZero |= bottom3NonZero >> 1
bottom3NonZero = uint32(int32(bottom3NonZero<<31) >> 31)
// Everything depends on the value of out[3].
// If it's > 0xffff000 and top4AllOnes != 0 then the whole value is >= p
// If it's = 0xffff000 and top4AllOnes != 0 and bottom3NonZero != 0,
// then the whole value is >= p
// If it's < 0xffff000, then the whole value is < p
n := out[3] - 0xffff000
out3Equal := n
out3Equal |= out3Equal >> 16
out3Equal |= out3Equal >> 8
out3Equal |= out3Equal >> 4
out3Equal |= out3Equal >> 2
out3Equal |= out3Equal >> 1
out3Equal = ^uint32(int32(out3Equal<<31) >> 31)
// If out[3] > 0xffff000 then n's MSB will be zero.
out3GT := ^uint32(int32(n) >> 31)
mask := top4AllOnes & ((out3Equal & bottom3NonZero) | out3GT)
out[0] -= 1 & mask
out[3] -= 0xffff000 & mask
out[4] -= 0xfffffff & mask
out[5] -= 0xfffffff & mask
out[6] -= 0xfffffff & mask
out[7] -= 0xfffffff & mask
}
// Group element functions.
//
// These functions deal with group elements. The group is an elliptic curve
// group with a = -3 defined in FIPS 186-3, section D.2.2.
// p224AddJacobian computes *out = a+b where a != b.
func p224AddJacobian(x3, y3, z3, x1, y1, z1, x2, y2, z2 *p224FieldElement) {
// See http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#addition-p224Add-2007-bl
var z1z1, z2z2, u1, u2, s1, s2, h, i, j, r, v p224FieldElement
var c p224LargeFieldElement
z1IsZero := p224IsZero(z1)
z2IsZero := p224IsZero(z2)
// Z1Z1 = Z1²
p224Square(&z1z1, z1, &c)
// Z2Z2 = Z2²
p224Square(&z2z2, z2, &c)
// U1 = X1*Z2Z2
p224Mul(&u1, x1, &z2z2, &c)
// U2 = X2*Z1Z1
p224Mul(&u2, x2, &z1z1, &c)
// S1 = Y1*Z2*Z2Z2
p224Mul(&s1, z2, &z2z2, &c)
p224Mul(&s1, y1, &s1, &c)
// S2 = Y2*Z1*Z1Z1
p224Mul(&s2, z1, &z1z1, &c)
p224Mul(&s2, y2, &s2, &c)
// H = U2-U1
p224Sub(&h, &u2, &u1)
p224Reduce(&h)
xEqual := p224IsZero(&h)
// I = (2*H)²
for j := 0; j < 8; j++ {
i[j] = h[j] << 1
}
p224Reduce(&i)
p224Square(&i, &i, &c)
// J = H*I
p224Mul(&j, &h, &i, &c)
// r = 2*(S2-S1)
p224Sub(&r, &s2, &s1)
p224Reduce(&r)
yEqual := p224IsZero(&r)
if xEqual == 1 && yEqual == 1 && z1IsZero == 0 && z2IsZero == 0 {
p224DoubleJacobian(x3, y3, z3, x1, y1, z1)
return
}
for i := 0; i < 8; i++ {
r[i] <<= 1
}
p224Reduce(&r)
// V = U1*I
p224Mul(&v, &u1, &i, &c)
// Z3 = ((Z1+Z2)²-Z1Z1-Z2Z2)*H
p224Add(&z1z1, &z1z1, &z2z2)
p224Add(&z2z2, z1, z2)
p224Reduce(&z2z2)
p224Square(&z2z2, &z2z2, &c)
p224Sub(z3, &z2z2, &z1z1)
p224Reduce(z3)
p224Mul(z3, z3, &h, &c)
// X3 = r²-J-2*V
for i := 0; i < 8; i++ {
z1z1[i] = v[i] << 1
}
p224Add(&z1z1, &j, &z1z1)
p224Reduce(&z1z1)
p224Square(x3, &r, &c)
p224Sub(x3, x3, &z1z1)
p224Reduce(x3)
// Y3 = r*(V-X3)-2*S1*J
for i := 0; i < 8; i++ {
s1[i] <<= 1
}
p224Mul(&s1, &s1, &j, &c)
p224Sub(&z1z1, &v, x3)
p224Reduce(&z1z1)
p224Mul(&z1z1, &z1z1, &r, &c)
p224Sub(y3, &z1z1, &s1)
p224Reduce(y3)
p224CopyConditional(x3, x2, z1IsZero)
p224CopyConditional(x3, x1, z2IsZero)
p224CopyConditional(y3, y2, z1IsZero)
p224CopyConditional(y3, y1, z2IsZero)
p224CopyConditional(z3, z2, z1IsZero)
p224CopyConditional(z3, z1, z2IsZero)
}
// p224DoubleJacobian computes *out = a+a.
func p224DoubleJacobian(x3, y3, z3, x1, y1, z1 *p224FieldElement) {
var delta, gamma, beta, alpha, t p224FieldElement
var c p224LargeFieldElement
p224Square(&delta, z1, &c)
p224Square(&gamma, y1, &c)
p224Mul(&beta, x1, &gamma, &c)
// alpha = 3*(X1-delta)*(X1+delta)
p224Add(&t, x1, &delta)
for i := 0; i < 8; i++ {
t[i] += t[i] << 1
}
p224Reduce(&t)
p224Sub(&alpha, x1, &delta)
p224Reduce(&alpha)
p224Mul(&alpha, &alpha, &t, &c)
// Z3 = (Y1+Z1)²-gamma-delta
p224Add(z3, y1, z1)
p224Reduce(z3)
p224Square(z3, z3, &c)
p224Sub(z3, z3, &gamma)
p224Reduce(z3)
p224Sub(z3, z3, &delta)
p224Reduce(z3)
// X3 = alpha²-8*beta
for i := 0; i < 8; i++ {
delta[i] = beta[i] << 3
}
p224Reduce(&delta)
p224Square(x3, &alpha, &c)
p224Sub(x3, x3, &delta)
p224Reduce(x3)
// Y3 = alpha*(4*beta-X3)-8*gamma²
for i := 0; i < 8; i++ {
beta[i] <<= 2
}
p224Sub(&beta, &beta, x3)
p224Reduce(&beta)
p224Square(&gamma, &gamma, &c)
for i := 0; i < 8; i++ {
gamma[i] <<= 3
}
p224Reduce(&gamma)
p224Mul(y3, &alpha, &beta, &c)
p224Sub(y3, y3, &gamma)
p224Reduce(y3)
}
// p224CopyConditional sets *out = *in iff the least-significant-bit of control
// is true, and it runs in constant time.
func p224CopyConditional(out, in *p224FieldElement, control uint32) {
control <<= 31
control = uint32(int32(control) >> 31)
for i := 0; i < 8; i++ {
out[i] ^= (out[i] ^ in[i]) & control
}
}
func p224ScalarMult(outX, outY, outZ, inX, inY, inZ *p224FieldElement, scalar []byte) {
var xx, yy, zz p224FieldElement
for i := 0; i < 8; i++ {
outX[i] = 0
outY[i] = 0
outZ[i] = 0
}
for _, byte := range scalar {
for bitNum := uint(0); bitNum < 8; bitNum++ {
p224DoubleJacobian(outX, outY, outZ, outX, outY, outZ)
bit := uint32((byte >> (7 - bitNum)) & 1)
p224AddJacobian(&xx, &yy, &zz, inX, inY, inZ, outX, outY, outZ)
p224CopyConditional(outX, &xx, bit)
p224CopyConditional(outY, &yy, bit)
p224CopyConditional(outZ, &zz, bit)
}
}
}
// p224ToAffine converts from Jacobian to affine form.
func p224ToAffine(x, y, z *p224FieldElement) (*big.Int, *big.Int) {
var zinv, zinvsq, outx, outy p224FieldElement
var tmp p224LargeFieldElement
if isPointAtInfinity := p224IsZero(z); isPointAtInfinity == 1 {
return new(big.Int), new(big.Int)
}
p224Invert(&zinv, z)
p224Square(&zinvsq, &zinv, &tmp)
p224Mul(x, x, &zinvsq, &tmp)
p224Mul(&zinvsq, &zinvsq, &zinv, &tmp)
p224Mul(y, y, &zinvsq, &tmp)
p224Contract(&outx, x)
p224Contract(&outy, y)
return p224ToBig(&outx), p224ToBig(&outy)
}
// get28BitsFromEnd returns the least-significant 28 bits from buf>>shift,
// where buf is interpreted as a big-endian number.
func get28BitsFromEnd(buf []byte, shift uint) (uint32, []byte) {
var ret uint32
for i := uint(0); i < 4; i++ {
var b byte
if l := len(buf); l > 0 {
b = buf[l-1]
// We don't remove the byte if we're about to return and we're not
// reading all of it.
if i != 3 || shift == 4 {
buf = buf[:l-1]
}
}
ret |= uint32(b) << (8 * i) >> shift
}
ret &= bottom28Bits
return ret, buf
}
// p224FromBig sets *out = *in.
func p224FromBig(out *p224FieldElement, in *big.Int) {
bytes := in.Bytes()
out[0], bytes = get28BitsFromEnd(bytes, 0)
out[1], bytes = get28BitsFromEnd(bytes, 4)
out[2], bytes = get28BitsFromEnd(bytes, 0)
out[3], bytes = get28BitsFromEnd(bytes, 4)
out[4], bytes = get28BitsFromEnd(bytes, 0)
out[5], bytes = get28BitsFromEnd(bytes, 4)
out[6], bytes = get28BitsFromEnd(bytes, 0)
out[7], bytes = get28BitsFromEnd(bytes, 4)
}
// p224ToBig returns in as a big.Int.
func p224ToBig(in *p224FieldElement) *big.Int {
var buf [28]byte
buf[27] = byte(in[0])
buf[26] = byte(in[0] >> 8)
buf[25] = byte(in[0] >> 16)
buf[24] = byte(((in[0] >> 24) & 0x0f) | (in[1]<<4)&0xf0)
buf[23] = byte(in[1] >> 4)
buf[22] = byte(in[1] >> 12)
buf[21] = byte(in[1] >> 20)
buf[20] = byte(in[2])
buf[19] = byte(in[2] >> 8)
buf[18] = byte(in[2] >> 16)
buf[17] = byte(((in[2] >> 24) & 0x0f) | (in[3]<<4)&0xf0)
buf[16] = byte(in[3] >> 4)
buf[15] = byte(in[3] >> 12)
buf[14] = byte(in[3] >> 20)
buf[13] = byte(in[4])
buf[12] = byte(in[4] >> 8)
buf[11] = byte(in[4] >> 16)
buf[10] = byte(((in[4] >> 24) & 0x0f) | (in[5]<<4)&0xf0)
buf[9] = byte(in[5] >> 4)
buf[8] = byte(in[5] >> 12)
buf[7] = byte(in[5] >> 20)
buf[6] = byte(in[6])
buf[5] = byte(in[6] >> 8)
buf[4] = byte(in[6] >> 16)
buf[3] = byte(((in[6] >> 24) & 0x0f) | (in[7]<<4)&0xf0)
buf[2] = byte(in[7] >> 4)
buf[1] = byte(in[7] >> 12)
buf[0] = byte(in[7] >> 20)
return new(big.Int).SetBytes(buf[:])
}
|