-
Notifications
You must be signed in to change notification settings - Fork 21
/
fe.go
1055 lines (929 loc) · 30 KB
/
fe.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
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
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
// Package field implements fast arithmetic modulo 2^448-2^224-1.
// based on https://github.com/golang/crypto/blob/56aed061732aaf690a941aa617c5b0e322727650/curve25519/internal/field/fe.go
//
// original copyright:
// Copyright (c) 2017 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 field
import "math/bits"
// Element represents an element of the field GF(2^448-2^224-1). Note that this
// is not a cryptographically secure group
//
// This type works similarly to math/big.Int, and all arguments and receivers
// are allowed to alias.
//
// The zero value is a valid zero element.
type Element struct {
// An element t represents the integer
// t.l0 + t.l1*2^56 + t.l2*2^112 + t.l3*2^168 + t.l4*2^224
// + t.l5*2^280 + t.l6*2^336 + t.l7*2^392
//
// Between operations, all limbs are expected to be lower than 2^57.
l0 uint64
l1 uint64
l2 uint64
l3 uint64
l4 uint64
l5 uint64
l6 uint64
l7 uint64
}
const maskLow56Bits uint64 = (1 << 56) - 1
var feZero = &Element{0, 0, 0, 0, 0, 0, 0, 0}
// Zero sets v = 0, and returns v.
func (v *Element) Zero() *Element {
*v = *feZero
return v
}
var feOne = &Element{1, 0, 0, 0, 0, 0, 0, 0}
// One sets v = 1, and returns v.
func (v *Element) One() *Element {
*v = *feOne
return v
}
// IsNegative returns 1 if v is negative, and 0 otherwise.
func (v *Element) IsNegative() int {
v0 := *v // shallow copy
v0.reduce()
return int(v0.l0 & 1)
}
func (v *Element) Abs(u *Element) *Element {
var x Element
x.Negate(u)
return v.Select(&x, u, u.IsNegative())
}
// reduce reduces v modulo 2^448 - 2^224 - 1 and returns it.
func (v *Element) reduce() *Element {
v.carryPropagate()
// After the light reduction we now have a field element representation
// v < 2^448 + 2^232 + 2^8, but need v < 2^448 - 2^224 - 1.
// If v >= 2^448 - 2^224 - 1, then v + 2^224 + 1 >= 2^448, which would overflow 2^448 - 1,
// generating a carry. That is, c will be 0 if v < 2^448 - 2^224 - 1, and 1 otherwise.
c := (v.l0 + 1) >> 56
c = (v.l1 + c) >> 56
c = (v.l2 + c) >> 56
c = (v.l3 + c) >> 56
c = (v.l4 + c + 1) >> 56
c = (v.l5 + c) >> 56
c = (v.l6 + c) >> 56
c = (v.l7 + c) >> 56
// If v < 2^448 - 2^224 - 1 and c = 0, this will be a no-op. Otherwise, it's
// effectively applying the reduction identity to the carry.
v.l0 += c
v.l4 += c
v.l1 += v.l0 >> 56
v.l0 &= maskLow56Bits
v.l2 += v.l1 >> 56
v.l1 &= maskLow56Bits
v.l3 += v.l2 >> 56
v.l2 &= maskLow56Bits
v.l4 += v.l3 >> 56
v.l3 &= maskLow56Bits
v.l5 += v.l4 >> 56
v.l4 &= maskLow56Bits
v.l6 += v.l5 >> 56
v.l5 &= maskLow56Bits
v.l7 += v.l6 >> 56
v.l6 &= maskLow56Bits
// no additional carry
v.l7 &= maskLow56Bits
return v
}
// Add sets v = a + b, and returns v.
func (v *Element) Add(a, b *Element) *Element {
v.l0 = a.l0 + b.l0
v.l1 = a.l1 + b.l1
v.l2 = a.l2 + b.l2
v.l3 = a.l3 + b.l3
v.l4 = a.l4 + b.l4
v.l5 = a.l5 + b.l5
v.l6 = a.l6 + b.l6
v.l7 = a.l7 + b.l7
// propagate carry
c0 := v.l0 >> 56
c1 := v.l1 >> 56
c2 := v.l2 >> 56
c3 := v.l3 >> 56
c4 := v.l4 >> 56
c5 := v.l5 >> 56
c6 := v.l6 >> 56
c7 := v.l7 >> 56
// c7 is at most 64 - 56 = 8 bits,
// so the final l0 will be at most 57 bits. Similarly for the rest.
v.l0 = v.l0&maskLow56Bits + c7
v.l1 = v.l1&maskLow56Bits + c0
v.l2 = v.l2&maskLow56Bits + c1
v.l3 = v.l3&maskLow56Bits + c2
v.l4 = v.l4&maskLow56Bits + c3 + c7
v.l5 = v.l5&maskLow56Bits + c4
v.l6 = v.l6&maskLow56Bits + c5
v.l7 = v.l7&maskLow56Bits + c6
return v
}
// Subtract sets v = a - b, and returns v.
func (v *Element) Sub(a, b *Element) *Element {
// We first add 2 * p, to guarantee the subtraction won't underflow, and
// then subtract b (which can be up to 2^448 + 2^232 + 2^8).
v.l0 = (a.l0 + 2*0xFF_FFFF_FFFF_FFFF) - b.l0
v.l1 = (a.l1 + 2*0xFF_FFFF_FFFF_FFFF) - b.l1
v.l2 = (a.l2 + 2*0xFF_FFFF_FFFF_FFFF) - b.l2
v.l3 = (a.l3 + 2*0xFF_FFFF_FFFF_FFFF) - b.l3
v.l4 = (a.l4 + 2*0xFF_FFFF_FFFF_FFFE) - b.l4
v.l5 = (a.l5 + 2*0xFF_FFFF_FFFF_FFFF) - b.l5
v.l6 = (a.l6 + 2*0xFF_FFFF_FFFF_FFFF) - b.l6
v.l7 = (a.l7 + 2*0xFF_FFFF_FFFF_FFFF) - b.l7
// propagate carry
c0 := v.l0 >> 56
c1 := v.l1 >> 56
c2 := v.l2 >> 56
c3 := v.l3 >> 56
c4 := v.l4 >> 56
c5 := v.l5 >> 56
c6 := v.l6 >> 56
c7 := v.l7 >> 56
// c7 is at most 64 - 56 = 8 bits,
// so the final l0 will be at most 57 bits. Similarly for the rest.
v.l0 = v.l0&maskLow56Bits + c7
v.l1 = v.l1&maskLow56Bits + c0
v.l2 = v.l2&maskLow56Bits + c1
v.l3 = v.l3&maskLow56Bits + c2
v.l4 = v.l4&maskLow56Bits + c3 + c7
v.l5 = v.l5&maskLow56Bits + c4
v.l6 = v.l6&maskLow56Bits + c5
v.l7 = v.l7&maskLow56Bits + c6
return v
}
// Negate sets v = -a, and returns v.
func (v *Element) Negate(a *Element) *Element {
// We first add 2 * p, to guarantee the subtraction won't underflow, and
// then subtract b (which can be up to 2^448 + 2^232 + 2^8).
v.l0 = 2*0xFF_FFFF_FFFF_FFFF - a.l0
v.l1 = 2*0xFF_FFFF_FFFF_FFFF - a.l1
v.l2 = 2*0xFF_FFFF_FFFF_FFFF - a.l2
v.l3 = 2*0xFF_FFFF_FFFF_FFFF - a.l3
v.l4 = 2*0xFF_FFFF_FFFF_FFFE - a.l4
v.l5 = 2*0xFF_FFFF_FFFF_FFFF - a.l5
v.l6 = 2*0xFF_FFFF_FFFF_FFFF - a.l6
v.l7 = 2*0xFF_FFFF_FFFF_FFFF - a.l7
// propagate carry
c0 := v.l0 >> 56
c1 := v.l1 >> 56
c2 := v.l2 >> 56
c3 := v.l3 >> 56
c4 := v.l4 >> 56
c5 := v.l5 >> 56
c6 := v.l6 >> 56
c7 := v.l7 >> 56
// c7 is at most 64 - 56 = 8 bits,
// so the final l0 will be at most 57 bits. Similarly for the rest.
v.l0 = v.l0&maskLow56Bits + c7
v.l1 = v.l1&maskLow56Bits + c0
v.l2 = v.l2&maskLow56Bits + c1
v.l3 = v.l3&maskLow56Bits + c2
v.l4 = v.l4&maskLow56Bits + c3 + c7
v.l5 = v.l5&maskLow56Bits + c4
v.l6 = v.l6&maskLow56Bits + c5
v.l7 = v.l7&maskLow56Bits + c6
return v
}
// Set sets v = a, and returns v.
func (v *Element) Set(a *Element) *Element {
*v = *a
return v
}
// SetBytes sets v to x, which must be a 56-byte little-endian encoding.
func (v *Element) SetBytes(x []byte) *Element {
if len(x) != 56 {
panic("curve448: invalid field element input size")
}
_ = x[55] // bounds check hint to compiler; see golang.org/issue/14808
v.l0 = uint64(x[0]) +
uint64(x[1])<<8 +
uint64(x[2])<<16 +
uint64(x[3])<<24 +
uint64(x[4])<<32 +
uint64(x[5])<<40 +
uint64(x[6])<<48
v.l1 = uint64(x[7]) +
uint64(x[8])<<8 +
uint64(x[9])<<16 +
uint64(x[10])<<24 +
uint64(x[11])<<32 +
uint64(x[12])<<40 +
uint64(x[13])<<48
v.l2 = uint64(x[14]) +
uint64(x[15])<<8 +
uint64(x[16])<<16 +
uint64(x[17])<<24 +
uint64(x[18])<<32 +
uint64(x[19])<<40 +
uint64(x[20])<<48
v.l3 = uint64(x[21]) +
uint64(x[22])<<8 +
uint64(x[23])<<16 +
uint64(x[24])<<24 +
uint64(x[25])<<32 +
uint64(x[26])<<40 +
uint64(x[27])<<48
v.l4 = uint64(x[28]) +
uint64(x[29])<<8 +
uint64(x[30])<<16 +
uint64(x[31])<<24 +
uint64(x[32])<<32 +
uint64(x[33])<<40 +
uint64(x[34])<<48
v.l5 = uint64(x[35]) +
uint64(x[36])<<8 +
uint64(x[37])<<16 +
uint64(x[38])<<24 +
uint64(x[39])<<32 +
uint64(x[40])<<40 +
uint64(x[41])<<48
v.l6 = uint64(x[42]) +
uint64(x[43])<<8 +
uint64(x[44])<<16 +
uint64(x[45])<<24 +
uint64(x[46])<<32 +
uint64(x[47])<<40 +
uint64(x[48])<<48
v.l7 = uint64(x[49]) +
uint64(x[50])<<8 +
uint64(x[51])<<16 +
uint64(x[52])<<24 +
uint64(x[53])<<32 +
uint64(x[54])<<40 +
uint64(x[55])<<48
return v
}
func (v *Element) Bytes() []byte {
var out [56]byte
return v.bytes(&out)
}
func (v *Element) bytes(out *[56]byte) []byte {
t := *v
t.reduce()
out[0] = byte(t.l0)
out[1] = byte(t.l0 >> 8)
out[2] = byte(t.l0 >> 16)
out[3] = byte(t.l0 >> 24)
out[4] = byte(t.l0 >> 32)
out[5] = byte(t.l0 >> 40)
out[6] = byte(t.l0 >> 48)
out[7] = byte(t.l1)
out[8] = byte(t.l1 >> 8)
out[9] = byte(t.l1 >> 16)
out[10] = byte(t.l1 >> 24)
out[11] = byte(t.l1 >> 32)
out[12] = byte(t.l1 >> 40)
out[13] = byte(t.l1 >> 48)
out[14] = byte(t.l2)
out[15] = byte(t.l2 >> 8)
out[16] = byte(t.l2 >> 16)
out[17] = byte(t.l2 >> 24)
out[18] = byte(t.l2 >> 32)
out[19] = byte(t.l2 >> 40)
out[20] = byte(t.l2 >> 48)
out[21] = byte(t.l3)
out[22] = byte(t.l3 >> 8)
out[23] = byte(t.l3 >> 16)
out[24] = byte(t.l3 >> 24)
out[25] = byte(t.l3 >> 32)
out[26] = byte(t.l3 >> 40)
out[27] = byte(t.l3 >> 48)
out[28] = byte(t.l4)
out[29] = byte(t.l4 >> 8)
out[30] = byte(t.l4 >> 16)
out[31] = byte(t.l4 >> 24)
out[32] = byte(t.l4 >> 32)
out[33] = byte(t.l4 >> 40)
out[34] = byte(t.l4 >> 48)
out[35] = byte(t.l5)
out[36] = byte(t.l5 >> 8)
out[37] = byte(t.l5 >> 16)
out[38] = byte(t.l5 >> 24)
out[39] = byte(t.l5 >> 32)
out[40] = byte(t.l5 >> 40)
out[41] = byte(t.l5 >> 48)
out[42] = byte(t.l6)
out[43] = byte(t.l6 >> 8)
out[44] = byte(t.l6 >> 16)
out[45] = byte(t.l6 >> 24)
out[46] = byte(t.l6 >> 32)
out[47] = byte(t.l6 >> 40)
out[48] = byte(t.l6 >> 48)
out[49] = byte(t.l7)
out[50] = byte(t.l7 >> 8)
out[51] = byte(t.l7 >> 16)
out[52] = byte(t.l7 >> 24)
out[53] = byte(t.l7 >> 32)
out[54] = byte(t.l7 >> 40)
out[55] = byte(t.l7 >> 48)
return out[:]
}
// Equal returns 1 if v and u are equal, and 0 otherwise.
func (v *Element) Equal(u *Element) int {
u0 := *u // shallow copy
v0 := *v // shallow copy
u0.reduce()
v0.reduce()
c := v0.l0 ^ u0.l0
c |= v0.l1 ^ u0.l1
c |= v0.l2 ^ u0.l2
c |= v0.l3 ^ u0.l3
c |= v0.l4 ^ u0.l4
c |= v0.l5 ^ u0.l5
c |= v0.l6 ^ u0.l6
c |= v0.l7 ^ u0.l7
c = (c & 0xFFFFFFFF) | (c >> 32)
c--
return int(c >> 63)
}
// mask64Bits returns 0xffffffff if cond is 1, and 0 otherwise.
func mask64Bits(cond int) uint64 { return uint64(0) - uint64(cond) }
// Select sets v to a if cond == 1, and to b if cond == 0.
func (v *Element) Select(a, b *Element, cond int) *Element {
m := mask64Bits(cond)
v.l0 = (m & a.l0) | (^m & b.l0)
v.l1 = (m & a.l1) | (^m & b.l1)
v.l2 = (m & a.l2) | (^m & b.l2)
v.l3 = (m & a.l3) | (^m & b.l3)
v.l4 = (m & a.l4) | (^m & b.l4)
v.l5 = (m & a.l5) | (^m & b.l5)
v.l6 = (m & a.l6) | (^m & b.l6)
v.l7 = (m & a.l7) | (^m & b.l7)
return v
}
// Swap swaps v and u if cond == 1 or leaves them unchanged if cond == 0.
func (v *Element) Swap(u *Element, cond int) {
m := mask64Bits(cond)
t := m & (v.l0 ^ u.l0)
v.l0 ^= t
u.l0 ^= t
t = m & (v.l1 ^ u.l1)
v.l1 ^= t
u.l1 ^= t
t = m & (v.l2 ^ u.l2)
v.l2 ^= t
u.l2 ^= t
t = m & (v.l3 ^ u.l3)
v.l3 ^= t
u.l3 ^= t
t = m & (v.l4 ^ u.l4)
v.l4 ^= t
u.l4 ^= t
t = m & (v.l5 ^ u.l5)
v.l5 ^= t
u.l5 ^= t
t = m & (v.l6 ^ u.l6)
v.l6 ^= t
u.l6 ^= t
t = m & (v.l7 ^ u.l7)
v.l7 ^= t
u.l7 ^= t
}
// carryPropagate brings the limbs below 56 bits by applying the reduction
// identity (a * 2^448 + b = a * 2^224 + a + b) to the l7 carry. TODO inline
func (v *Element) carryPropagate() *Element {
c0 := v.l0 >> 56
c1 := v.l1 >> 56
c2 := v.l2 >> 56
c3 := v.l3 >> 56
c4 := v.l4 >> 56
c5 := v.l5 >> 56
c6 := v.l6 >> 56
c7 := v.l7 >> 56
// c7 is at most 64 - 56 = 8 bits,
// so the final l0 will be at most 57 bits. Similarly for the rest.
v.l0 = v.l0&maskLow56Bits + c7
v.l1 = v.l1&maskLow56Bits + c0
v.l2 = v.l2&maskLow56Bits + c1
v.l3 = v.l3&maskLow56Bits + c2
v.l4 = v.l4&maskLow56Bits + c3 + c7
v.l5 = v.l5&maskLow56Bits + c4
v.l6 = v.l6&maskLow56Bits + c5
v.l7 = v.l7&maskLow56Bits + c6
return v
}
func (v *Element) Mul32(x *Element, y uint32) *Element {
x0lo, x0hi := mul56(x.l0, y)
x1lo, x1hi := mul56(x.l1, y)
x2lo, x2hi := mul56(x.l2, y)
x3lo, x3hi := mul56(x.l3, y)
x4lo, x4hi := mul56(x.l4, y)
x5lo, x5hi := mul56(x.l5, y)
x6lo, x6hi := mul56(x.l6, y)
x7lo, x7hi := mul56(x.l7, y)
v.l0 = x0lo + x7hi
v.l1 = x1lo + x0hi
v.l2 = x2lo + x1hi
v.l3 = x3lo + x2hi
v.l4 = x4lo + x3hi + x7hi
v.l5 = x5lo + x4hi
v.l6 = x6lo + x5hi
v.l7 = x7lo + x6hi
// The hi portions are going to be only 32 bits, plus any previous excess,
// so we can skip the carry propagation.
return v
}
// mul56 returns lo + hi * 2^56 = a * b
func mul56(a uint64, b uint32) (lo uint64, hi uint64) {
h, l := bits.Mul64(a, uint64(b))
lo = l & maskLow56Bits
hi = (h << 8) | (l >> 56)
return
}
type uint128 struct {
lo, hi uint64
}
// mul64 returns a * b.
func mul64(a, b uint64) uint128 {
hi, lo := bits.Mul64(a, b)
return uint128{lo, hi}
}
// addMul64 returns v + a * b.
func addMul64(v uint128, a, b uint64) uint128 {
hi, lo := bits.Mul64(a, b)
lo, c := bits.Add64(lo, v.lo, 0)
hi, _ = bits.Add64(hi, v.hi, c)
return uint128{lo, hi}
}
func add128(a, b uint128) uint128 {
lo, c := bits.Add64(a.lo, b.lo, 0)
hi, _ := bits.Add64(a.hi, b.hi, c)
return uint128{lo, hi}
}
func lsh128(a uint128) uint128 {
return uint128{a.lo << 1, (a.hi << 1) | (a.lo >> 63)}
}
// shiftRightBy56 returns a >> 56. a is assumed to be at most 117 bits.
func shiftRightBy56(a uint128) uint64 {
return (a.hi << (64 - 56)) | (a.lo >> 56)
}
// b7 b6 b5 b4 b3 b2 b1 b0 =
// a7 a6 a5 a4 a3 a2 a1 a0 x
// ----------------------------------------
// a7b0 a6b0 a5b0 a4b0 a3b0 a2b0 a1b0 a0b0 +
// a7b1 a6b1 a5b1 a4b1 a3b1 a2b1 a1b1 a0b1 +
// a7b2 a6b2 a5b2 a4b2 a3b2 a2b2 a1b2 a0b2 +
// a7b3 a6b3 a5b3 a4b3 a3b3 a2b3 a1b3 a0b3 +
// a7b4 a6b4 a5b4 a4b4 a3b4 a2b4 a1b4 a0b4 +
// a7b5 a6b5 a5b5 a4b5 a3b5 a2b5 a1b5 a0b5 +
// a7b6 a6b6 a5b6 a4b6 a3b6 a2b6 a1b6 a0b6 +
// a7b7 a6b7 a5b7 a4b7 a3b7 a2b7 a1b7 a0b7 =
// -----------------------------------------------------------------------------
// r14 r13 r12 r11 r10 r9 r8 r7 r6 r5 r4 r3 r2 r1 r0
//
// We can then use the reduction identity (a * 2^448 + b = a * 2^224 + a + b) to
// reduce the limbs that would overflow 448 bits. r8 * 2^448 becomes r8 * 2^224 + r8,
// r9 * 2^504 becomes r9 * 2^56 + r9, etc.
//
// r12 * 2^672 = r12 * 2^448 + r12 * 2^224
// = r12 * 2^224 + r12 + r14 * 2^224
// = 2 * r12 * 2^224 + r14
//
// b7 b6 b5 b4 b3 b2 b1 b0 =
// a7 a6 a5 a4 a3 a2 a1 a0 x
// ------------------- ---------- ---------- ---------- ----- -----
// a7b0 a6b0 a5b0 a4b0 a3b0 a2b0 a1b0 a0b0 +
// a6b1 a5b1 a4b1 a3b1+a7b1 a2b1 a1b1 a0b1 a7b1 +
// a5b2 a4b2 a3b2+a7b2 a2b2+a6b2 a1b2 a0b2 a7b2 a6b2 +
// a4b3 a3b3+a7b3 a2b3+a6b3 a1b3+a5b3 a0b3 a7b3 a6b3 a5b3 +
// a3b4+a7b4 a2b4+a6b4 a1b4+a5b4 a0b4+a4b4 a7b4 a6b4 a5b4 a4b4 +
// a2b5+a6b5 a1b5+a5b5 a0b5+a4b5 a3b5+a7b5*2 a6b5 a5b5 a4b5 a3b5+a7b5 +
// a1b6+a5b6 a0b6+a4b6 a3b6+a7b6*2 a2b6+a6b6*2 a5b6 a4b6 a3b6+a7b6 a2b6+a6b6 +
// a0b7+a4b7 a3b7+a7b7*2 a2b7+a6b7*2 a1b7+a5b7*2 a4b7 a3b7+a7b7 a2b7+a6b7 a1b7+a5b7 =
// --------------------------------------------------------------------------------
// r7 r6 r5 r4 r3 r2 r1 r0
func (v *Element) Mul(a, b *Element) *Element {
a0 := a.l0
a1 := a.l1
a2 := a.l2
a3 := a.l3
a4 := a.l4
a5 := a.l5
a6 := a.l6
a7 := a.l7
b0 := b.l0
b1 := b.l1
b2 := b.l2
b3 := b.l3
b4 := b.l4
b5 := b.l5
b6 := b.l6
b7 := b.l7
a7b1 := mul64(a7, b1)
a6b2 := mul64(a6, b2)
a5b3 := mul64(a5, b3)
a4b4 := mul64(a4, b4)
a3b5 := mul64(a3, b5)
a2b6 := mul64(a2, b6)
a1b7 := mul64(a1, b7)
a7b2 := mul64(a7, b2)
a6b3 := mul64(a6, b3)
a5b4 := mul64(a5, b4)
a4b5 := mul64(a4, b5)
a3b6 := mul64(a3, b6)
a2b7 := mul64(a2, b7)
a7b3 := mul64(a7, b3)
a6b4 := mul64(a6, b4)
a5b5 := mul64(a5, b5)
a4b6 := mul64(a4, b6)
a3b7 := mul64(a3, b7)
a7b4 := mul64(a7, b4)
a6b5 := mul64(a6, b5)
a5b6 := mul64(a5, b6)
a4b7 := mul64(a4, b7)
a7b5 := mul64(a7, b5)
a6b6 := mul64(a6, b6)
a5b7 := mul64(a5, b7)
a7b6 := mul64(a7, b6)
a6b7 := mul64(a6, b7)
a7b7 := mul64(a7, b7)
// r0 = a0b0 + a7b1 + a6b2 + a5b3 + a4b4 + a3b5+a7b5 + a2b6+a6b6 + a1b7+a5b7
r0 := mul64(a0, b0)
r0 = add128(r0, a7b1)
r0 = add128(r0, a6b2)
r0 = add128(r0, a5b3)
r0 = add128(r0, a4b4)
r0 = add128(r0, a3b5)
r0 = add128(r0, a7b5)
r0 = add128(r0, a2b6)
r0 = add128(r0, a6b6)
r0 = add128(r0, a1b7)
r0 = add128(r0, a5b7)
// r1 = a1b0 + a0b1 + a7b2 + a6b3 + a5b4 + a4b5 + a3b6+a7b6 + a2b7+a6b7
r1 := mul64(a1, b0)
r1 = addMul64(r1, a0, b1)
r1 = addMul64(r1, a7, b2)
r1 = addMul64(r1, a6, b3)
r1 = addMul64(r1, a5, b4)
r1 = addMul64(r1, a4, b5)
r1 = addMul64(r1, a3, b6)
r1 = add128(r1, a7b6)
r1 = addMul64(r1, a2, b7)
r1 = add128(r1, a6b7)
// r2 = a2b0 + a1b1 + a0b2 + a7b3 + a6b4 + a5b5 + a4b6 + a3b7+a7b7
r2 := mul64(a2, b0)
r2 = addMul64(r2, a1, b1)
r2 = addMul64(r2, a0, b2)
r2 = add128(r2, a7b3)
r2 = add128(r2, a6b4)
r2 = add128(r2, a5b5)
r2 = add128(r2, a4b6)
r2 = add128(r2, a3b7)
r2 = add128(r2, a7b7)
// r3 = a3b0 + a2b1 + a1b2 + a0b3 + a7b4 + a6b5 + a5b6 + a4b7
r3 := mul64(a3, b0)
r3 = addMul64(r3, a2, b1)
r3 = addMul64(r3, a1, b2)
r3 = addMul64(r3, a0, b3)
r3 = add128(r3, a7b4)
r3 = add128(r3, a6b5)
r3 = add128(r3, a5b6)
r3 = add128(r3, a4b7)
// r4 = a4b0 + a3b1+a7b1 + a2b2+a6b2 + a1b3+a5b3 + a0b4+a4b4 + a3b5+a7b5*2 + a2b6+a6b6*2 + a1b7+a5b7*2
r4 := add128(a7b5, a6b6)
r4 = add128(r4, a5b7)
r4 = lsh128(r4)
r4 = addMul64(r4, a4, b0)
r4 = addMul64(r4, a3, b1)
r4 = add128(r4, a7b1)
r4 = addMul64(r4, a2, b2)
r4 = add128(r4, a6b2)
r4 = addMul64(r4, a1, b3)
r4 = add128(r4, a5b3)
r4 = addMul64(r4, a0, b4)
r4 = addMul64(r4, a4, b4)
r4 = addMul64(r4, a3, b5)
r4 = addMul64(r4, a2, b6)
r4 = addMul64(r4, a1, b7)
// r5 = a5b0 + a4b1 + a3b2+a7b2 + a2b3+a6b3 + a1b4+a5b4 + a0b5+a4b5 + a3b6+a7b6*2 + a2b7+a6b7*2
r5 := lsh128(add128(a7b6, a6b7))
r5 = addMul64(r5, a5, b0)
r5 = addMul64(r5, a4, b1)
r5 = addMul64(r5, a3, b2)
r5 = add128(r5, a7b2)
r5 = addMul64(r5, a2, b3)
r5 = add128(r5, a6b3)
r5 = addMul64(r5, a1, b4)
r5 = add128(r5, a5b4)
r5 = addMul64(r5, a0, b5)
r5 = add128(r5, a4b5)
r5 = add128(r5, a3b6)
r5 = add128(r5, a2b7)
// r6 = a6b0 + a5b1 + a4b2 + a3b3+a7b3 + a2b4+a6b4 + a1b5+a5b5 + a0b6+a4b6 + a3b7+a7b7*2
r6 := mul64(a6, b0)
r6 = addMul64(r6, a5, b1)
r6 = addMul64(r6, a4, b2)
r6 = addMul64(r6, a3, b3)
r6 = add128(r6, a7b3)
r6 = addMul64(r6, a2, b4)
r6 = add128(r6, a6b4)
r6 = addMul64(r6, a1, b5)
r6 = add128(r6, a5b5)
r6 = addMul64(r6, a0, b6)
r6 = add128(r6, a4b6)
r6 = addMul64(r6, a3, b7)
r6 = add128(r6, lsh128(a7b7))
// r7 = a7b0 + a6b1 + a5b2 + a4b3 + a3b4+a7b4 + a2b5+a6b5 + a1b6+a5b6 + a0b7+a4b7
r7 := mul64(a7, b0)
r7 = addMul64(r7, a6, b1)
r7 = addMul64(r7, a5, b2)
r7 = addMul64(r7, a4, b3)
r7 = addMul64(r7, a3, b4)
r7 = add128(r7, a7b4)
r7 = addMul64(r7, a2, b5)
r7 = add128(r7, a6b5)
r7 = addMul64(r7, a1, b6)
r7 = add128(r7, a5b6)
r7 = addMul64(r7, a0, b7)
r7 = add128(r7, a4b7)
c0 := shiftRightBy56(r0)
c1 := shiftRightBy56(r1)
c2 := shiftRightBy56(r2)
c3 := shiftRightBy56(r3)
c4 := shiftRightBy56(r4)
c5 := shiftRightBy56(r5)
c6 := shiftRightBy56(r6)
c7 := shiftRightBy56(r7)
rr0 := r0.lo&maskLow56Bits + c7
rr1 := r1.lo&maskLow56Bits + c0
rr2 := r2.lo&maskLow56Bits + c1
rr3 := r3.lo&maskLow56Bits + c2
rr4 := r4.lo&maskLow56Bits + c3 + c7
rr5 := r5.lo&maskLow56Bits + c4
rr6 := r6.lo&maskLow56Bits + c5
rr7 := r7.lo&maskLow56Bits + c6
*v = Element{rr0, rr1, rr2, rr3, rr4, rr5, rr6, rr7}
// propagate carry
c0 = v.l0 >> 56
c1 = v.l1 >> 56
c2 = v.l2 >> 56
c3 = v.l3 >> 56
c4 = v.l4 >> 56
c5 = v.l5 >> 56
c6 = v.l6 >> 56
c7 = v.l7 >> 56
// c7 is at most 64 - 56 = 8 bits,
// so the final l0 will be at most 57 bits. Similarly for the rest.
v.l0 = v.l0&maskLow56Bits + c7
v.l1 = v.l1&maskLow56Bits + c0
v.l2 = v.l2&maskLow56Bits + c1
v.l3 = v.l3&maskLow56Bits + c2
v.l4 = v.l4&maskLow56Bits + c3 + c7
v.l5 = v.l5&maskLow56Bits + c4
v.l6 = v.l6&maskLow56Bits + c5
v.l7 = v.l7&maskLow56Bits + c6
return v
}
// a7 a6 a5 a4 a3 a2 a1 a0 =
// a7 a6 a5 a4 a3 a2 a1 a0 x
// ----------------------------------------
// a7a0 a6a0 a5a0 a4a0 a3a0 a2a0 a1a0 a0a0 +
// a7a1 a6a1 a5a1 a4a1 a3a1 a2a1 a1a1 a0a1 +
// a7a2 a6a2 a5a2 a4a2 a3a2 a2a2 a1a2 a0a2 +
// a7a3 a6a3 a5a3 a4a3 a3a3 a2a3 a1a3 a0a3 +
// a7a4 a6a4 a5a4 a4a4 a3a4 a2a4 a1a4 a0a4 +
// a7a5 a6a5 a5a5 a4a5 a3a5 a2a5 a1a5 a0a5 +
// a7a6 a6a6 a5a6 a4a6 a3a6 a2a6 a1a6 a0a6 +
// a7a7 a6a7 a5a7 a4a7 a3a7 a2a7 a1a7 a0a7 =
// -----------------------------------------------------------------------------
// r14 r13 r12 r11 r10 r9 r8 r7 r6 r5 r4 r3 r2 r1 r0
//
// We can then use the reduction identity (a * 2^448 + b = a * 2^224 + a + b) to
// reduce the limbs that would overflow 448 bits. r8 * 2^448 becomes r8 * 2^224 + r8,
// r9 * 2^504 becomes r9 * 2^56 + r9, etc.
//
// r12 * 2^672 = r12 * 2^448 + r12 * 2^224
// = r12 * 2^224 + r12 + r14 * 2^224
// = 2 * r12 * 2^224 + r14
//
// a7 a6 a5 a4 a3 a2 a1 a0 =
// a7 a6 a5 a4 a3 a2 a1 a0 x
// ------------------- ---------- ---------- ---------- ----- -----
// a0a7 a0a6 a0a5 a0a4 a0a3 a0a2 a0a1 a0a0 +
// a1a6 a1a5 a1a4 a1a3+a1a7 a1a2 a1a1 a0a1 a1a7 +
// a2a5 a2a4 a2a3+a2a7 a2a2+a2a6 a1a2 a0a2 a2a7 a2a6 +
// a3a4 a3a3+a3a7 a2a3+a3a6 a1a3+a3a5 a0a3 a3a7 a3a6 a3a5 +
// a3a4+a4a7 a2a4+a4a6 a1a4+a4a5 a0a4+a4a4 a4a7 a4a6 a4a5 a4a4 +
// a2a5+a5a6 a1a5+a5a5 a0a5+a4a5 a3a5+a5a7*2 a5a6 a5a5 a4a5 a3a5+a5a7 +
// a1a6+a5a6 a0a6+a4a6 a3a6+a6a7*2 a2a6+a6a6*2 a5a6 a4a6 a3a6+a6a7 a2a6+a6a6 +
// a0a7+a4a7 a3a7+a7a7*2 a2a7+a6a7*2 a1a7+a5a7*2 a4a7 a3a7+a7a7 a2a7+a6a7 a1a7+a5a7 =
// --------------------------------------------------------------------------------
// r7 r6 r5 r4 r3 r2 r1 r0
func (v *Element) Square(a *Element) *Element {
a0 := a.l0
a1 := a.l1
a2 := a.l2
a3 := a.l3
a4 := a.l4
a5 := a.l5
a6 := a.l6
a7 := a.l7
a1a7 := mul64(a1, a7)
a2a6 := mul64(a2, a6)
a2a7 := mul64(a2, a7)
a3a5 := mul64(a3, a5)
a3a6 := mul64(a3, a6)
a3a7 := mul64(a3, a7)
a4a4 := mul64(a4, a4)
a4a5 := mul64(a4, a5)
a4a6 := mul64(a4, a6)
a4a7 := mul64(a4, a7)
a5a5 := mul64(a5, a5)
a5a6 := mul64(a5, a6)
a5a7 := mul64(a5, a7)
a6a6 := mul64(a6, a6)
a6a7 := mul64(a6, a7)
a7a7 := mul64(a7, a7)
// r0 = a0a0 + a1a7 + a2a6 + a3a5 + a4a4 + a3a5+a5a7 + a2a6+a6a6 + a1a7+a5a7
r0 := a1a7
r0 = add128(r0, a2a6)
r0 = add128(r0, a3a5)
r0 = add128(r0, a5a7)
r0 = lsh128(r0)
r0 = addMul64(r0, a0, a0)
r0 = add128(r0, a4a4)
r0 = add128(r0, a6a6)
// r1 = a0a1 + a0a1 + a2a7 + a3a6 + a4a5 + a4a5 + a3a6+a6a7 + a2a7+a6a7
r1 := mul64(a0, a1)
r1 = add128(r1, a2a7)
r1 = add128(r1, a3a6)
r1 = add128(r1, a4a5)
r1 = add128(r1, a6a7)
r1 = lsh128(r1)
// r2 = a0a2 + a1a1 + a0a2 + a3a7 + a4a6 + a5a5 + a4a6 + a3a7+a7a7
r2 := mul64(a0, a2)
r2 = add128(r2, a3a7)
r2 = add128(r2, a4a6)
r2 = lsh128(r2)
r2 = addMul64(r2, a1, a1)
r2 = add128(r2, a5a5)
r2 = addMul64(r2, a7, a7)
// r3 = a0a3 + a1a2 + a1a2 + a0a3 + a4a7 + a5a6 + a5a6 + a4a7
r3 := mul64(a0, a3)
r3 = addMul64(r3, a1, a2)
r3 = add128(r3, a4a7)
r3 = add128(r3, a5a6)
r3 = lsh128(r3)
// r4 = a0a4 + a1a3+a1a7 + a2a2+a2a6 + a1a3+a3a5 + a0a4+a4a4 + a3a5+a5a7*2 + a2a6+a6a6*2 + a1a7+a5a7*2
r4 := lsh128(a5a7)
r4 = add128(r4, a6a6)
r4 = addMul64(r4, a0, a4)
r4 = addMul64(r4, a1, a3)
r4 = add128(r4, a1a7)
r4 = add128(r4, a2a6)
r4 = add128(r4, a3a5)
r4 = lsh128(r4)
r4 = addMul64(r4, a2, a2)
r4 = add128(r4, a4a4)
// r5 = a0a5 + a1a4 + a2a3+a2a7 + a2a3+a3a6 + a1a4+a4a5 + a0a5+a4a5 + a3a6+a6a7*2 + a2a7+a6a7*2
r5 := lsh128(a6a7)
r5 = addMul64(r5, a0, a5)
r5 = addMul64(r5, a1, a4)
r5 = addMul64(r5, a2, a3)
r5 = add128(r5, a2a7)
r5 = add128(r5, a3a6)
r5 = add128(r5, a4a5)
r5 = lsh128(r5)
// r6 = a0a6 + a1a5 + a2a4 + a3a3+a3a7 + a2a4+a4a6 + a1a5+a5a5 + a0a6+a4a6 + a3a7+a7a7*2
r6 := a7a7
r6 = addMul64(r6, a0, a6)
r6 = addMul64(r6, a1, a5)
r6 = addMul64(r6, a2, a4)
r6 = add128(r6, a3a7)
r6 = add128(r6, a4a6)
r6 = lsh128(r6)
r6 = addMul64(r6, a3, a3)
r6 = add128(r6, a5a5)
// r7 = a0a7 + a1a6 + a2a5 + a3a4 + a3a4+a4a7 + a2a5+a5a6 + a1a6+a5a6 + a0a7+a4a7
r7 := mul64(a0, a7)
r7 = addMul64(r7, a1, a6)
r7 = addMul64(r7, a2, a5)
r7 = addMul64(r7, a3, a4)
r7 = add128(r7, a4a7)
r7 = add128(r7, a5a6)
r7 = lsh128(r7)
c0 := shiftRightBy56(r0)
c1 := shiftRightBy56(r1)
c2 := shiftRightBy56(r2)
c3 := shiftRightBy56(r3)
c4 := shiftRightBy56(r4)
c5 := shiftRightBy56(r5)
c6 := shiftRightBy56(r6)
c7 := shiftRightBy56(r7)
rr0 := r0.lo&maskLow56Bits + c7
rr1 := r1.lo&maskLow56Bits + c0
rr2 := r2.lo&maskLow56Bits + c1
rr3 := r3.lo&maskLow56Bits + c2
rr4 := r4.lo&maskLow56Bits + c3 + c7
rr5 := r5.lo&maskLow56Bits + c4
rr6 := r6.lo&maskLow56Bits + c5
rr7 := r7.lo&maskLow56Bits + c6
*v = Element{rr0, rr1, rr2, rr3, rr4, rr5, rr6, rr7}
// propagate carry
c0 = v.l0 >> 56
c1 = v.l1 >> 56
c2 = v.l2 >> 56
c3 = v.l3 >> 56
c4 = v.l4 >> 56
c5 = v.l5 >> 56
c6 = v.l6 >> 56
c7 = v.l7 >> 56
// c7 is at most 64 - 56 = 8 bits,
// so the final l0 will be at most 57 bits. Similarly for the rest.
v.l0 = v.l0&maskLow56Bits + c7
v.l1 = v.l1&maskLow56Bits + c0
v.l2 = v.l2&maskLow56Bits + c1
v.l3 = v.l3&maskLow56Bits + c2
v.l4 = v.l4&maskLow56Bits + c3 + c7
v.l5 = v.l5&maskLow56Bits + c4
v.l6 = v.l6&maskLow56Bits + c5
v.l7 = v.l7&maskLow56Bits + c6
return v
}
// Inv sets v = 1/z mod p, and returns v.
func (v *Element) Inv(z *Element) *Element {
// Inversion is implemented as exponentiation with exponent p − 2.
var x Element
x.Power446(z)
x.Square(&x)
x.Square(&x) // 2^448 - 2^224 - 4
return v.Mul(&x, z) // 2^448 - 2^224 - 3
}
// Power446 sets v = v ^ ((p-3)/4) mod p, and returns v.
// (p-3)/4 is 2^446 - 2^222 - 1.
func (v *Element) Power446(z *Element) *Element {
var z1, z2, z3 Element
z1.Square(z) // 2^1
z2.Square(&z1) // 2^2
z3.Mul(z, &z1)
z3.Mul(&z3, &z2) // 2^3 - 1
var z6 Element
z6.Square(&z3)
z6.Square(&z6)
z6.Square(&z6)
z6.Mul(&z6, &z3) // 2^6 - 1
var z9 Element
z9.Square(&z6)
z9.Square(&z9)
z9.Square(&z9)
z9.Mul(&z9, &z3) // 2^9 - 1
var z18 Element
z18.Square(&z9)
for i := 1; i < 9; i++ {
z18.Square(&z18)
}
z18.Mul(&z18, &z9) // 2^18 - 1
var z37 Element
z37.Square(&z18)
for i := 1; i < 18; i++ {
z37.Square(&z37)