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
1001
1002
1003
1004
1005
1006
1007
1008
1009
1010
1011
1012
1013
1014
1015
1016
1017
1018
1019
1020
1021
1022
1023
1024
1025
1026
1027
1028
1029
1030
1031
1032
1033
1034
1035
1036
1037
1038
1039
1040
1041
1042
1043
1044
1045
1046
1047
1048
1049
1050
1051
1052
1053
1054
1055
1056
1057
1058
1059
1060
1061
1062
1063
1064
1065
1066
1067
1068
1069
1070
1071
1072
1073
1074
1075
1076
1077
1078
1079
1080
1081
1082
1083
1084
1085
1086
1087
1088
1089
1090
1091
1092
1093
1094
1095
1096
1097
1098
1099
1100
1101
1102
1103
1104
1105
1106
1107
1108
1109
1110
1111
1112
1113
1114
1115
1116
1117
1118
1119
1120
1121
1122
1123
1124
1125
1126
1127
1128
1129
1130
1131
1132
1133
1134
1135
1136
1137
1138
1139
1140
1141
1142
1143
1144
1145
1146
1147
1148
1149
1150
1151
1152
1153
1154
1155
1156
1157
1158
1159
1160
1161
1162
1163
1164
1165
1166
1167
|
#include <assert.h>
#include "ssh.h"
#include "mpint.h"
#include "ecc.h"
/* ----------------------------------------------------------------------
* Weierstrass curves.
*/
struct WeierstrassPoint {
/*
* Internally, we represent a point using 'Jacobian coordinates',
* which are three values X,Y,Z whose relation to the affine
* coordinates x,y is that x = X/Z^2 and y = Y/Z^3.
*
* This allows us to do most of our calculations without having to
* take an inverse mod p: every time the obvious affine formulae
* would need you to divide by something, you instead multiply it
* into the 'denominator' coordinate Z. You only have to actually
* take the inverse of Z when you need to get the affine
* coordinates back out, which means you do it once after your
* entire computation instead of at every intermediate step.
*
* The point at infinity is represented by setting all three
* coordinates to zero.
*
* These values are also stored in the Montgomery-multiplication
* transformed representation.
*/
mp_int *X, *Y, *Z;
WeierstrassCurve *wc;
};
struct WeierstrassCurve {
/* Prime modulus of the finite field. */
mp_int *p;
/* Persistent Montgomery context for doing arithmetic mod p. */
MontyContext *mc;
/* Modsqrt context for point decompression. NULL if this curve was
* constructed without providing nonsquare_mod_p. */
ModsqrtContext *sc;
/* Parameters of the curve, in Montgomery-multiplication
* transformed form. */
mp_int *a, *b;
};
WeierstrassCurve *ecc_weierstrass_curve(
mp_int *p, mp_int *a, mp_int *b, mp_int *nonsquare_mod_p)
{
WeierstrassCurve *wc = snew(WeierstrassCurve);
wc->p = mp_copy(p);
wc->mc = monty_new(p);
wc->a = monty_import(wc->mc, a);
wc->b = monty_import(wc->mc, b);
if (nonsquare_mod_p)
wc->sc = modsqrt_new(p, nonsquare_mod_p);
else
wc->sc = NULL;
return wc;
}
void ecc_weierstrass_curve_free(WeierstrassCurve *wc)
{
mp_free(wc->p);
mp_free(wc->a);
mp_free(wc->b);
monty_free(wc->mc);
if (wc->sc)
modsqrt_free(wc->sc);
sfree(wc);
}
static WeierstrassPoint *ecc_weierstrass_point_new_empty(WeierstrassCurve *wc)
{
WeierstrassPoint *wp = snew(WeierstrassPoint);
wp->wc = wc;
wp->X = wp->Y = wp->Z = NULL;
return wp;
}
static WeierstrassPoint *ecc_weierstrass_point_new_imported(
WeierstrassCurve *wc, mp_int *monty_x, mp_int *monty_y)
{
WeierstrassPoint *wp = ecc_weierstrass_point_new_empty(wc);
wp->X = monty_x;
wp->Y = monty_y;
wp->Z = mp_copy(monty_identity(wc->mc));
return wp;
}
WeierstrassPoint *ecc_weierstrass_point_new(
WeierstrassCurve *wc, mp_int *x, mp_int *y)
{
return ecc_weierstrass_point_new_imported(
wc, monty_import(wc->mc, x), monty_import(wc->mc, y));
}
WeierstrassPoint *ecc_weierstrass_point_new_identity(WeierstrassCurve *wc)
{
WeierstrassPoint *wp = ecc_weierstrass_point_new_empty(wc);
size_t bits = mp_max_bits(wc->p);
wp->X = mp_new(bits);
wp->Y = mp_new(bits);
wp->Z = mp_new(bits);
return wp;
}
void ecc_weierstrass_point_copy_into(
WeierstrassPoint *dest, WeierstrassPoint *src)
{
mp_copy_into(dest->X, src->X);
mp_copy_into(dest->Y, src->Y);
mp_copy_into(dest->Z, src->Z);
}
WeierstrassPoint *ecc_weierstrass_point_copy(WeierstrassPoint *orig)
{
WeierstrassPoint *wp = ecc_weierstrass_point_new_empty(orig->wc);
wp->X = mp_copy(orig->X);
wp->Y = mp_copy(orig->Y);
wp->Z = mp_copy(orig->Z);
return wp;
}
void ecc_weierstrass_point_free(WeierstrassPoint *wp)
{
mp_free(wp->X);
mp_free(wp->Y);
mp_free(wp->Z);
smemclr(wp, sizeof(*wp));
sfree(wp);
}
WeierstrassPoint *ecc_weierstrass_point_new_from_x(
WeierstrassCurve *wc, mp_int *xorig, unsigned desired_y_parity)
{
assert(wc->sc);
/*
* The curve equation is y^2 = x^3 + ax + b, which is already
* conveniently in a form where we can compute the RHS and take
* the square root of it to get y.
*/
unsigned success;
mp_int *x = monty_import(wc->mc, xorig);
/*
* Compute the RHS of the curve equation. We don't need to take
* account of z here, because we're constructing the point from
* scratch. So it really is just x^3 + ax + b.
*/
mp_int *x2 = monty_mul(wc->mc, x, x);
mp_int *x2_plus_a = monty_add(wc->mc, x2, wc->a);
mp_int *x3_plus_ax = monty_mul(wc->mc, x2_plus_a, x);
mp_int *rhs = monty_add(wc->mc, x3_plus_ax, wc->b);
mp_free(x2);
mp_free(x2_plus_a);
mp_free(x3_plus_ax);
mp_int *y = monty_modsqrt(wc->sc, rhs, &success);
mp_free(rhs);
if (!success) {
/* Failure! x^3+ax+b worked out to be a number that has no
* square root mod p. In this situation there's no point in
* trying to be time-constant, since the protocol sequence is
* going to diverge anyway when we complain to whoever gave us
* this bogus value. */
mp_free(x);
mp_free(y);
return NULL;
}
/*
* Choose whichever of y and p-y has the specified parity (of its
* lowest positive residue mod p).
*/
mp_int *tmp = monty_export(wc->mc, y);
unsigned flip = (mp_get_bit(tmp, 0) ^ desired_y_parity) & 1;
mp_sub_into(tmp, wc->p, y);
mp_select_into(y, y, tmp, flip);
mp_free(tmp);
return ecc_weierstrass_point_new_imported(wc, x, y);
}
static void ecc_weierstrass_cond_overwrite(
WeierstrassPoint *dest, WeierstrassPoint *src, unsigned overwrite)
{
mp_select_into(dest->X, dest->X, src->X, overwrite);
mp_select_into(dest->Y, dest->Y, src->Y, overwrite);
mp_select_into(dest->Z, dest->Z, src->Z, overwrite);
}
static void ecc_weierstrass_cond_swap(
WeierstrassPoint *P, WeierstrassPoint *Q, unsigned swap)
{
mp_cond_swap(P->X, Q->X, swap);
mp_cond_swap(P->Y, Q->Y, swap);
mp_cond_swap(P->Z, Q->Z, swap);
}
/*
* Shared code between all three of the basic arithmetic functions:
* once we've determined the slope of the line that we're intersecting
* the curve with, this takes care of finding the coordinates of the
* third intersection point (given the two input x-coordinates and one
* of the y-coords) and negating it to generate the output.
*/
static inline void ecc_weierstrass_epilogue(
mp_int *Px, mp_int *Qx, mp_int *Py, mp_int *common_Z,
mp_int *lambda_n, mp_int *lambda_d, WeierstrassPoint *out)
{
WeierstrassCurve *wc = out->wc;
/* Powers of the numerator and denominator of the slope lambda */
mp_int *lambda_n2 = monty_mul(wc->mc, lambda_n, lambda_n);
mp_int *lambda_d2 = monty_mul(wc->mc, lambda_d, lambda_d);
mp_int *lambda_d3 = monty_mul(wc->mc, lambda_d, lambda_d2);
/* Make the output x-coordinate */
mp_int *xsum = monty_add(wc->mc, Px, Qx);
mp_int *lambda_d2_xsum = monty_mul(wc->mc, lambda_d2, xsum);
out->X = monty_sub(wc->mc, lambda_n2, lambda_d2_xsum);
/* Make the output y-coordinate */
mp_int *lambda_d2_Px = monty_mul(wc->mc, lambda_d2, Px);
mp_int *xdiff = monty_sub(wc->mc, lambda_d2_Px, out->X);
mp_int *lambda_n_xdiff = monty_mul(wc->mc, lambda_n, xdiff);
mp_int *lambda_d3_Py = monty_mul(wc->mc, lambda_d3, Py);
out->Y = monty_sub(wc->mc, lambda_n_xdiff, lambda_d3_Py);
/* Make the output z-coordinate */
out->Z = monty_mul(wc->mc, common_Z, lambda_d);
mp_free(lambda_n2);
mp_free(lambda_d2);
mp_free(lambda_d3);
mp_free(xsum);
mp_free(xdiff);
mp_free(lambda_d2_xsum);
mp_free(lambda_n_xdiff);
mp_free(lambda_d2_Px);
mp_free(lambda_d3_Py);
}
/*
* Shared code between add and add_general: put the two input points
* over a common denominator, and determine the slope lambda of the
* line through both of them. If the points have the same
* x-coordinate, then the slope will be returned with a zero
* denominator.
*/
static inline void ecc_weierstrass_add_prologue(
WeierstrassPoint *P, WeierstrassPoint *Q,
mp_int **Px, mp_int **Py, mp_int **Qx, mp_int **denom,
mp_int **lambda_n, mp_int **lambda_d)
{
WeierstrassCurve *wc = P->wc;
/* Powers of the points' denominators */
mp_int *Pz2 = monty_mul(wc->mc, P->Z, P->Z);
mp_int *Pz3 = monty_mul(wc->mc, Pz2, P->Z);
mp_int *Qz2 = monty_mul(wc->mc, Q->Z, Q->Z);
mp_int *Qz3 = monty_mul(wc->mc, Qz2, Q->Z);
/* Points' x,y coordinates scaled by the other one's denominator
* (raised to the appropriate power) */
*Px = monty_mul(wc->mc, P->X, Qz2);
*Py = monty_mul(wc->mc, P->Y, Qz3);
*Qx = monty_mul(wc->mc, Q->X, Pz2);
mp_int *Qy = monty_mul(wc->mc, Q->Y, Pz3);
/* Common denominator */
*denom = monty_mul(wc->mc, P->Z, Q->Z);
/* Slope of the line through the two points, if P != Q */
*lambda_n = monty_sub(wc->mc, Qy, *Py);
*lambda_d = monty_sub(wc->mc, *Qx, *Px);
mp_free(Pz2);
mp_free(Pz3);
mp_free(Qz2);
mp_free(Qz3);
mp_free(Qy);
}
WeierstrassPoint *ecc_weierstrass_add(WeierstrassPoint *P, WeierstrassPoint *Q)
{
WeierstrassCurve *wc = P->wc;
assert(Q->wc == wc);
WeierstrassPoint *S = ecc_weierstrass_point_new_empty(wc);
mp_int *Px, *Py, *Qx, *denom, *lambda_n, *lambda_d;
ecc_weierstrass_add_prologue(
P, Q, &Px, &Py, &Qx, &denom, &lambda_n, &lambda_d);
/* Never expect to have received two mutually inverse inputs, or
* two identical ones (which would make this a doubling). In other
* words, the two input x-coordinates (after putting over a common
* denominator) should never have been equal. */
assert(!mp_eq_integer(lambda_n, 0));
/* Now go to the common epilogue code. */
ecc_weierstrass_epilogue(Px, Qx, Py, denom, lambda_n, lambda_d, S);
mp_free(Px);
mp_free(Py);
mp_free(Qx);
mp_free(denom);
mp_free(lambda_n);
mp_free(lambda_d);
return S;
}
/*
* Code to determine the slope of the line you need to intersect with
* the curve in the case where you're adding a point to itself. In
* this situation you can't just say "the line through both input
* points" because that's under-determined; instead, you have to take
* the _tangent_ to the curve at the given point, by differentiating
* the curve equation y^2=x^3+ax+b to get 2y dy/dx = 3x^2+a.
*/
static inline void ecc_weierstrass_tangent_slope(
WeierstrassPoint *P, mp_int **lambda_n, mp_int **lambda_d)
{
WeierstrassCurve *wc = P->wc;
mp_int *X2 = monty_mul(wc->mc, P->X, P->X);
mp_int *twoX2 = monty_add(wc->mc, X2, X2);
mp_int *threeX2 = monty_add(wc->mc, twoX2, X2);
mp_int *Z2 = monty_mul(wc->mc, P->Z, P->Z);
mp_int *Z4 = monty_mul(wc->mc, Z2, Z2);
mp_int *aZ4 = monty_mul(wc->mc, wc->a, Z4);
*lambda_n = monty_add(wc->mc, threeX2, aZ4);
*lambda_d = monty_add(wc->mc, P->Y, P->Y);
mp_free(X2);
mp_free(twoX2);
mp_free(threeX2);
mp_free(Z2);
mp_free(Z4);
mp_free(aZ4);
}
WeierstrassPoint *ecc_weierstrass_double(WeierstrassPoint *P)
{
WeierstrassCurve *wc = P->wc;
WeierstrassPoint *D = ecc_weierstrass_point_new_empty(wc);
mp_int *lambda_n, *lambda_d;
ecc_weierstrass_tangent_slope(P, &lambda_n, &lambda_d);
ecc_weierstrass_epilogue(P->X, P->X, P->Y, P->Z, lambda_n, lambda_d, D);
mp_free(lambda_n);
mp_free(lambda_d);
return D;
}
static inline void ecc_weierstrass_select_into(
WeierstrassPoint *dest, WeierstrassPoint *P, WeierstrassPoint *Q,
unsigned choose_Q)
{
mp_select_into(dest->X, P->X, Q->X, choose_Q);
mp_select_into(dest->Y, P->Y, Q->Y, choose_Q);
mp_select_into(dest->Z, P->Z, Q->Z, choose_Q);
}
WeierstrassPoint *ecc_weierstrass_add_general(
WeierstrassPoint *P, WeierstrassPoint *Q)
{
WeierstrassCurve *wc = P->wc;
assert(Q->wc == wc);
WeierstrassPoint *S = ecc_weierstrass_point_new_empty(wc);
/* Parameters for the epilogue, and slope of the line if P != Q */
mp_int *Px, *Py, *Qx, *denom, *lambda_n, *lambda_d;
ecc_weierstrass_add_prologue(
P, Q, &Px, &Py, &Qx, &denom, &lambda_n, &lambda_d);
/* Slope if P == Q */
mp_int *lambda_n_tangent, *lambda_d_tangent;
ecc_weierstrass_tangent_slope(P, &lambda_n_tangent, &lambda_d_tangent);
/* Select between those slopes depending on whether P == Q */
unsigned same_x_coord = mp_eq_integer(lambda_d, 0);
unsigned same_y_coord = mp_eq_integer(lambda_n, 0);
unsigned equality = same_x_coord & same_y_coord;
mp_select_into(lambda_n, lambda_n, lambda_n_tangent, equality);
mp_select_into(lambda_d, lambda_d, lambda_d_tangent, equality);
/* Now go to the common code between addition and doubling */
ecc_weierstrass_epilogue(Px, Qx, Py, denom, lambda_n, lambda_d, S);
/* Check for the input identity cases, and overwrite the output if
* necessary. */
ecc_weierstrass_select_into(S, S, Q, mp_eq_integer(P->Z, 0));
ecc_weierstrass_select_into(S, S, P, mp_eq_integer(Q->Z, 0));
/*
* In the case where P == -Q and so the output is the identity,
* we'll have calculated lambda_d = 0 and so the output will have
* z==0 already. Detect that and use it to normalise the other two
* coordinates to zero.
*/
unsigned output_id = mp_eq_integer(S->Z, 0);
mp_cond_clear(S->X, output_id);
mp_cond_clear(S->Y, output_id);
mp_free(Px);
mp_free(Py);
mp_free(Qx);
mp_free(denom);
mp_free(lambda_n);
mp_free(lambda_d);
mp_free(lambda_n_tangent);
mp_free(lambda_d_tangent);
return S;
}
WeierstrassPoint *ecc_weierstrass_multiply(WeierstrassPoint *B, mp_int *n)
{
WeierstrassPoint *two_B = ecc_weierstrass_double(B);
WeierstrassPoint *k_B = ecc_weierstrass_point_copy(B);
WeierstrassPoint *kplus1_B = ecc_weierstrass_point_copy(two_B);
/*
* This multiply routine more or less follows the shape of the
* 'Montgomery ladder' technique that you have to use under the
* extra constraint on addition in Montgomery curves, because it
* was fresh in my mind and easier to just do it the same way. See
* the comment in ecc_montgomery_multiply.
*/
unsigned not_started_yet = 1;
for (size_t bitindex = mp_max_bits(n); bitindex-- > 0 ;) {
unsigned nbit = mp_get_bit(n, bitindex);
WeierstrassPoint *sum = ecc_weierstrass_add(k_B, kplus1_B);
ecc_weierstrass_cond_swap(k_B, kplus1_B, nbit);
WeierstrassPoint *other = ecc_weierstrass_double(k_B);
ecc_weierstrass_point_free(k_B);
ecc_weierstrass_point_free(kplus1_B);
k_B = other;
kplus1_B = sum;
ecc_weierstrass_cond_swap(k_B, kplus1_B, nbit);
ecc_weierstrass_cond_overwrite(k_B, B, not_started_yet);
ecc_weierstrass_cond_overwrite(kplus1_B, two_B, not_started_yet);
not_started_yet &= ~nbit;
}
ecc_weierstrass_point_free(two_B);
ecc_weierstrass_point_free(kplus1_B);
return k_B;
}
unsigned ecc_weierstrass_is_identity(WeierstrassPoint *wp)
{
return mp_eq_integer(wp->Z, 0);
}
/*
* Normalise a point by scaling its Jacobian coordinates so that Z=1.
* This doesn't change what point is represented by the triple, but it
* means the affine x,y can now be easily recovered from X and Y.
*/
static void ecc_weierstrass_normalise(WeierstrassPoint *wp)
{
WeierstrassCurve *wc = wp->wc;
mp_int *zinv = monty_invert(wc->mc, wp->Z);
mp_int *zinv2 = monty_mul(wc->mc, zinv, zinv);
mp_int *zinv3 = monty_mul(wc->mc, zinv2, zinv);
monty_mul_into(wc->mc, wp->X, wp->X, zinv2);
monty_mul_into(wc->mc, wp->Y, wp->Y, zinv3);
monty_mul_into(wc->mc, wp->Z, wp->Z, zinv);
mp_free(zinv);
mp_free(zinv2);
mp_free(zinv3);
}
void ecc_weierstrass_get_affine(
WeierstrassPoint *wp, mp_int **x, mp_int **y)
{
WeierstrassCurve *wc = wp->wc;
ecc_weierstrass_normalise(wp);
if (x)
*x = monty_export(wc->mc, wp->X);
if (y)
*y = monty_export(wc->mc, wp->Y);
}
unsigned ecc_weierstrass_point_valid(WeierstrassPoint *P)
{
WeierstrassCurve *wc = P->wc;
/*
* The projective version of the curve equation is
* Y^2 = X^3 + a X Z^4 + b Z^6
*/
mp_int *lhs = monty_mul(P->wc->mc, P->Y, P->Y);
mp_int *x2 = monty_mul(wc->mc, P->X, P->X);
mp_int *x3 = monty_mul(wc->mc, x2, P->X);
mp_int *z2 = monty_mul(wc->mc, P->Z, P->Z);
mp_int *z4 = monty_mul(wc->mc, z2, z2);
mp_int *az4 = monty_mul(wc->mc, wc->a, z4);
mp_int *axz4 = monty_mul(wc->mc, az4, P->X);
mp_int *x3_plus_axz4 = monty_add(wc->mc, x3, axz4);
mp_int *z6 = monty_mul(wc->mc, z2, z4);
mp_int *bz6 = monty_mul(wc->mc, wc->b, z6);
mp_int *rhs = monty_add(wc->mc, x3_plus_axz4, bz6);
unsigned valid = mp_cmp_eq(lhs, rhs);
mp_free(lhs);
mp_free(x2);
mp_free(x3);
mp_free(z2);
mp_free(z4);
mp_free(az4);
mp_free(axz4);
mp_free(x3_plus_axz4);
mp_free(z6);
mp_free(bz6);
mp_free(rhs);
return valid;
}
/* ----------------------------------------------------------------------
* Montgomery curves.
*/
struct MontgomeryPoint {
/* XZ coordinates. These represent the affine x coordinate by the
* relationship x = X/Z. */
mp_int *X, *Z;
MontgomeryCurve *mc;
};
struct MontgomeryCurve {
/* Prime modulus of the finite field. */
mp_int *p;
/* Montgomery context for arithmetic mod p. */
MontyContext *mc;
/* Parameters of the curve, in Montgomery-multiplication
* transformed form. */
mp_int *a, *b;
/* (a+2)/4, also in Montgomery-multiplication form. */
mp_int *aplus2over4;
};
MontgomeryCurve *ecc_montgomery_curve(
mp_int *p, mp_int *a, mp_int *b)
{
MontgomeryCurve *mc = snew(MontgomeryCurve);
mc->p = mp_copy(p);
mc->mc = monty_new(p);
mc->a = monty_import(mc->mc, a);
mc->b = monty_import(mc->mc, b);
mp_int *four = mp_from_integer(4);
mp_int *fourinverse = mp_invert(four, mc->p);
mp_int *aplus2 = mp_copy(a);
mp_add_integer_into(aplus2, aplus2, 2);
mp_int *aplus2over4 = mp_modmul(aplus2, fourinverse, mc->p);
mc->aplus2over4 = monty_import(mc->mc, aplus2over4);
mp_free(four);
mp_free(fourinverse);
mp_free(aplus2);
mp_free(aplus2over4);
return mc;
}
void ecc_montgomery_curve_free(MontgomeryCurve *mc)
{
mp_free(mc->p);
mp_free(mc->a);
mp_free(mc->b);
mp_free(mc->aplus2over4);
monty_free(mc->mc);
sfree(mc);
}
static MontgomeryPoint *ecc_montgomery_point_new_empty(MontgomeryCurve *mc)
{
MontgomeryPoint *mp = snew(MontgomeryPoint);
mp->mc = mc;
mp->X = mp->Z = NULL;
return mp;
}
MontgomeryPoint *ecc_montgomery_point_new(MontgomeryCurve *mc, mp_int *x)
{
MontgomeryPoint *mp = ecc_montgomery_point_new_empty(mc);
mp->X = monty_import(mc->mc, x);
mp->Z = mp_copy(monty_identity(mc->mc));
return mp;
}
void ecc_montgomery_point_copy_into(
MontgomeryPoint *dest, MontgomeryPoint *src)
{
mp_copy_into(dest->X, src->X);
mp_copy_into(dest->Z, src->Z);
}
MontgomeryPoint *ecc_montgomery_point_copy(MontgomeryPoint *orig)
{
MontgomeryPoint *mp = ecc_montgomery_point_new_empty(orig->mc);
mp->X = mp_copy(orig->X);
mp->Z = mp_copy(orig->Z);
return mp;
}
void ecc_montgomery_point_free(MontgomeryPoint *mp)
{
mp_free(mp->X);
mp_free(mp->Z);
smemclr(mp, sizeof(*mp));
sfree(mp);
}
static void ecc_montgomery_cond_overwrite(
MontgomeryPoint *dest, MontgomeryPoint *src, unsigned overwrite)
{
mp_select_into(dest->X, dest->X, src->X, overwrite);
mp_select_into(dest->Z, dest->Z, src->Z, overwrite);
}
static void ecc_montgomery_cond_swap(
MontgomeryPoint *P, MontgomeryPoint *Q, unsigned swap)
{
mp_cond_swap(P->X, Q->X, swap);
mp_cond_swap(P->Z, Q->Z, swap);
}
MontgomeryPoint *ecc_montgomery_diff_add(
MontgomeryPoint *P, MontgomeryPoint *Q, MontgomeryPoint *PminusQ)
{
MontgomeryCurve *mc = P->mc;
assert(Q->mc == mc);
assert(PminusQ->mc == mc);
/*
* Differential addition is achieved using the following formula
* that relates the affine x-coordinates of P, Q, P+Q and P-Q:
*
* x(P+Q) x(P-Q) (x(Q)-x(P))^2 = (x(P)x(Q) - 1)^2
*
* As with the Weierstrass coordinates, the code below transforms
* that affine relation into a projective one to avoid having to
* do a division during the main arithmetic.
*/
MontgomeryPoint *S = ecc_montgomery_point_new_empty(mc);
mp_int *Px_m_Pz = monty_sub(mc->mc, P->X, P->Z);
mp_int *Px_p_Pz = monty_add(mc->mc, P->X, P->Z);
mp_int *Qx_m_Qz = monty_sub(mc->mc, Q->X, Q->Z);
mp_int *Qx_p_Qz = monty_add(mc->mc, Q->X, Q->Z);
mp_int *PmQp = monty_mul(mc->mc, Px_m_Pz, Qx_p_Qz);
mp_int *PpQm = monty_mul(mc->mc, Px_p_Pz, Qx_m_Qz);
mp_int *Xpre = monty_add(mc->mc, PmQp, PpQm);
mp_int *Zpre = monty_sub(mc->mc, PmQp, PpQm);
mp_int *Xpre2 = monty_mul(mc->mc, Xpre, Xpre);
mp_int *Zpre2 = monty_mul(mc->mc, Zpre, Zpre);
S->X = monty_mul(mc->mc, Xpre2, PminusQ->Z);
S->Z = monty_mul(mc->mc, Zpre2, PminusQ->X);
mp_free(Px_m_Pz);
mp_free(Px_p_Pz);
mp_free(Qx_m_Qz);
mp_free(Qx_p_Qz);
mp_free(PmQp);
mp_free(PpQm);
mp_free(Xpre);
mp_free(Zpre);
mp_free(Xpre2);
mp_free(Zpre2);
return S;
}
MontgomeryPoint *ecc_montgomery_double(MontgomeryPoint *P)
{
MontgomeryCurve *mc = P->mc;
MontgomeryPoint *D = ecc_montgomery_point_new_empty(mc);
/*
* To double a point in affine coordinates, in principle you can
* use the same technique as for Weierstrass: differentiate the
* curve equation to get the tangent line at the input point, use
* that to get an expression for y which you substitute back into
* the curve equation, and subtract the known two roots (in this
* case both the same) from the x^2 coefficient of the resulting
* cubic.
*
* In this case, we don't have an input y-coordinate, so you have
* to do a bit of extra transformation to find a formula that can
* work without it. The tangent formula is (3x^2 + 2ax + 1)/(2y),
* and when that appears in the final formula it will be squared -
* so we can substitute the y^2 in the denominator for the RHS of
* the curve equation. Put together, that gives
*
* x_out = (x+1)^2 (x-1)^2 / 4(x^3+ax^2+x)
*
* and, as usual, the code below transforms that into projective
* form to avoid the division.
*/
mp_int *Px_m_Pz = monty_sub(mc->mc, P->X, P->Z);
mp_int *Px_p_Pz = monty_add(mc->mc, P->X, P->Z);
mp_int *Px_m_Pz_2 = monty_mul(mc->mc, Px_m_Pz, Px_m_Pz);
mp_int *Px_p_Pz_2 = monty_mul(mc->mc, Px_p_Pz, Px_p_Pz);
D->X = monty_mul(mc->mc, Px_m_Pz_2, Px_p_Pz_2);
mp_int *XZ = monty_mul(mc->mc, P->X, P->Z);
mp_int *twoXZ = monty_add(mc->mc, XZ, XZ);
mp_int *fourXZ = monty_add(mc->mc, twoXZ, twoXZ);
mp_int *fourXZ_scaled = monty_mul(mc->mc, fourXZ, mc->aplus2over4);
mp_int *Zpre = monty_add(mc->mc, Px_m_Pz_2, fourXZ_scaled);
D->Z = monty_mul(mc->mc, fourXZ, Zpre);
mp_free(Px_m_Pz);
mp_free(Px_p_Pz);
mp_free(Px_m_Pz_2);
mp_free(Px_p_Pz_2);
mp_free(XZ);
mp_free(twoXZ);
mp_free(fourXZ);
mp_free(fourXZ_scaled);
mp_free(Zpre);
return D;
}
static void ecc_montgomery_normalise(MontgomeryPoint *mp)
{
MontgomeryCurve *mc = mp->mc;
mp_int *zinv = monty_invert(mc->mc, mp->Z);
monty_mul_into(mc->mc, mp->X, mp->X, zinv);
monty_mul_into(mc->mc, mp->Z, mp->Z, zinv);
mp_free(zinv);
}
MontgomeryPoint *ecc_montgomery_multiply(MontgomeryPoint *B, mp_int *n)
{
/*
* 'Montgomery ladder' technique, to compute an arbitrary integer
* multiple of B under the constraint that you can only add two
* unequal points if you also know their difference.
*
* The setup is that you maintain two curve points one of which is
* always the other one plus B. Call them kB and (k+1)B, where k
* is some integer that evolves as we go along. We begin by
* doubling the input B, to initialise those points to B and 2B,
* so that k=1.
*
* At each stage, we add kB and (k+1)B together - which we can do
* under the differential-addition constraint because we know
* their difference is always just B - to give us (2k+1)B. Then we
* double one of kB or (k+1)B, and depending on which one we
* choose, we end up with (2k)B or (2k+2)B. Either way, that
* differs by B from the other value we've just computed. So in
* each iteration, we do one diff-add and one doubling, plus a
* couple of conditional swaps to choose which value we double and
* which way round we put the output points, and the effect is to
* replace k with either 2k or 2k+1, which we choose based on the
* appropriate bit of the desired exponent.
*
* This routine doesn't assume we know the exact location of the
* topmost set bit of the exponent. So to maintain constant time
* it does an iteration for every _potential_ bit, starting from
* the top downwards; after each iteration in which we haven't
* seen a set exponent bit yet, we just overwrite the two points
* with B and 2B again,
*/
MontgomeryPoint *two_B = ecc_montgomery_double(B);
MontgomeryPoint *k_B = ecc_montgomery_point_copy(B);
MontgomeryPoint *kplus1_B = ecc_montgomery_point_copy(two_B);
unsigned not_started_yet = 1;
for (size_t bitindex = mp_max_bits(n); bitindex-- > 0 ;) {
unsigned nbit = mp_get_bit(n, bitindex);
MontgomeryPoint *sum = ecc_montgomery_diff_add(k_B, kplus1_B, B);
ecc_montgomery_cond_swap(k_B, kplus1_B, nbit);
MontgomeryPoint *other = ecc_montgomery_double(k_B);
ecc_montgomery_point_free(k_B);
ecc_montgomery_point_free(kplus1_B);
k_B = other;
kplus1_B = sum;
ecc_montgomery_cond_swap(k_B, kplus1_B, nbit);
ecc_montgomery_cond_overwrite(k_B, B, not_started_yet);
ecc_montgomery_cond_overwrite(kplus1_B, two_B, not_started_yet);
not_started_yet &= ~nbit;
}
ecc_montgomery_point_free(two_B);
ecc_montgomery_point_free(kplus1_B);
return k_B;
}
void ecc_montgomery_get_affine(MontgomeryPoint *mp, mp_int **x)
{
MontgomeryCurve *mc = mp->mc;
ecc_montgomery_normalise(mp);
if (x)
*x = monty_export(mc->mc, mp->X);
}
unsigned ecc_montgomery_is_identity(MontgomeryPoint *mp)
{
return mp_eq_integer(mp->Z, 0);
}
/* ----------------------------------------------------------------------
* Twisted Edwards curves.
*/
struct EdwardsPoint {
/*
* We represent an Edwards curve point in 'extended coordinates'.
* There's more than one coordinate system going by that name,
* unfortunately. These ones have the semantics that X,Y,Z are
* ordinary projective coordinates (so x=X/Z and y=Y/Z), but also,
* we store the extra value T = xyZ = XY/Z.
*/
mp_int *X, *Y, *Z, *T;
EdwardsCurve *ec;
};
struct EdwardsCurve {
/* Prime modulus of the finite field. */
mp_int *p;
/* Montgomery context for arithmetic mod p. */
MontyContext *mc;
/* Modsqrt context for point decompression. */
ModsqrtContext *sc;
/* Parameters of the curve, in Montgomery-multiplication
* transformed form. */
mp_int *d, *a;
};
EdwardsCurve *ecc_edwards_curve(mp_int *p, mp_int *d, mp_int *a,
mp_int *nonsquare_mod_p)
{
EdwardsCurve *ec = snew(EdwardsCurve);
ec->p = mp_copy(p);
ec->mc = monty_new(p);
ec->d = monty_import(ec->mc, d);
ec->a = monty_import(ec->mc, a);
if (nonsquare_mod_p)
ec->sc = modsqrt_new(p, nonsquare_mod_p);
else
ec->sc = NULL;
return ec;
}
void ecc_edwards_curve_free(EdwardsCurve *ec)
{
mp_free(ec->p);
mp_free(ec->d);
mp_free(ec->a);
monty_free(ec->mc);
if (ec->sc)
modsqrt_free(ec->sc);
sfree(ec);
}
static EdwardsPoint *ecc_edwards_point_new_empty(EdwardsCurve *ec)
{
EdwardsPoint *ep = snew(EdwardsPoint);
ep->ec = ec;
ep->X = ep->Y = ep->Z = ep->T = NULL;
return ep;
}
static EdwardsPoint *ecc_edwards_point_new_imported(
EdwardsCurve *ec, mp_int *monty_x, mp_int *monty_y)
{
EdwardsPoint *ep = ecc_edwards_point_new_empty(ec);
ep->X = monty_x;
ep->Y = monty_y;
ep->T = monty_mul(ec->mc, ep->X, ep->Y);
ep->Z = mp_copy(monty_identity(ec->mc));
return ep;
}
EdwardsPoint *ecc_edwards_point_new(
EdwardsCurve *ec, mp_int *x, mp_int *y)
{
return ecc_edwards_point_new_imported(
ec, monty_import(ec->mc, x), monty_import(ec->mc, y));
}
void ecc_edwards_point_copy_into(EdwardsPoint *dest, EdwardsPoint *src)
{
mp_copy_into(dest->X, src->X);
mp_copy_into(dest->Y, src->Y);
mp_copy_into(dest->Z, src->Z);
mp_copy_into(dest->T, src->T);
}
EdwardsPoint *ecc_edwards_point_copy(EdwardsPoint *orig)
{
EdwardsPoint *ep = ecc_edwards_point_new_empty(orig->ec);
ep->X = mp_copy(orig->X);
ep->Y = mp_copy(orig->Y);
ep->Z = mp_copy(orig->Z);
ep->T = mp_copy(orig->T);
return ep;
}
void ecc_edwards_point_free(EdwardsPoint *ep)
{
mp_free(ep->X);
mp_free(ep->Y);
mp_free(ep->Z);
mp_free(ep->T);
smemclr(ep, sizeof(*ep));
sfree(ep);
}
EdwardsPoint *ecc_edwards_point_new_from_y(
EdwardsCurve *ec, mp_int *yorig, unsigned desired_x_parity)
{
assert(ec->sc);
/*
* The curve equation is ax^2 + y^2 = 1 + dx^2y^2, which
* rearranges to x^2(dy^2-a) = y^2-1. So we compute
* (y^2-1)/(dy^2-a) and take its square root.
*/
unsigned success;
mp_int *y = monty_import(ec->mc, yorig);
mp_int *y2 = monty_mul(ec->mc, y, y);
mp_int *dy2 = monty_mul(ec->mc, ec->d, y2);
mp_int *dy2ma = monty_sub(ec->mc, dy2, ec->a);
mp_int *y2m1 = monty_sub(ec->mc, y2, monty_identity(ec->mc));
mp_int *recip_denominator = monty_invert(ec->mc, dy2ma);
mp_int *radicand = monty_mul(ec->mc, y2m1, recip_denominator);
mp_int *x = monty_modsqrt(ec->sc, radicand, &success);
mp_free(y2);
mp_free(dy2);
mp_free(dy2ma);
mp_free(y2m1);
mp_free(recip_denominator);
mp_free(radicand);
if (!success) {
/* Failure! x^2 worked out to be a number that has no square
* root mod p. In this situation there's no point in trying to
* be time-constant, since the protocol sequence is going to
* diverge anyway when we complain to whoever gave us this
* bogus value. */
mp_free(x);
mp_free(y);
return NULL;
}
/*
* Choose whichever of x and p-x has the specified parity (of its
* lowest positive residue mod p).
*/
mp_int *tmp = monty_export(ec->mc, x);
unsigned flip = (mp_get_bit(tmp, 0) ^ desired_x_parity) & 1;
mp_sub_into(tmp, ec->p, x);
mp_select_into(x, x, tmp, flip);
mp_free(tmp);
return ecc_edwards_point_new_imported(ec, x, y);
}
static void ecc_edwards_cond_overwrite(
EdwardsPoint *dest, EdwardsPoint *src, unsigned overwrite)
{
mp_select_into(dest->X, dest->X, src->X, overwrite);
mp_select_into(dest->Y, dest->Y, src->Y, overwrite);
mp_select_into(dest->Z, dest->Z, src->Z, overwrite);
mp_select_into(dest->T, dest->T, src->T, overwrite);
}
static void ecc_edwards_cond_swap(
EdwardsPoint *P, EdwardsPoint *Q, unsigned swap)
{
mp_cond_swap(P->X, Q->X, swap);
mp_cond_swap(P->Y, Q->Y, swap);
mp_cond_swap(P->Z, Q->Z, swap);
mp_cond_swap(P->T, Q->T, swap);
}
EdwardsPoint *ecc_edwards_add(EdwardsPoint *P, EdwardsPoint *Q)
{
EdwardsCurve *ec = P->ec;
assert(Q->ec == ec);
EdwardsPoint *S = ecc_edwards_point_new_empty(ec);
/*
* The affine rule for Edwards addition of (x1,y1) and (x2,y2) is
*
* x_out = (x1 y2 + y1 x2) / (1 + d x1 x2 y1 y2)
* y_out = (y1 y2 - a x1 x2) / (1 - d x1 x2 y1 y2)
*
* The formulae below are listed as 'add-2008-hwcd' in
* https://hyperelliptic.org/EFD/g1p/auto-twisted-extended.html
*
* and if you undo the careful optimisation to find out what
* they're actually computing, it comes out to
*
* X_out = (X1 Y2 + Y1 X2) (Z1 Z2 - d T1 T2)
* Y_out = (Y1 Y2 - a X1 X2) (Z1 Z2 + d T1 T2)
* Z_out = (Z1 Z2 - d T1 T2) (Z1 Z2 + d T1 T2)
* T_out = (X1 Y2 + Y1 X2) (Y1 Y2 - a X1 X2)
*/
mp_int *PxQx = monty_mul(ec->mc, P->X, Q->X);
mp_int *PyQy = monty_mul(ec->mc, P->Y, Q->Y);
mp_int *PtQt = monty_mul(ec->mc, P->T, Q->T);
mp_int *PzQz = monty_mul(ec->mc, P->Z, Q->Z);
mp_int *Psum = monty_add(ec->mc, P->X, P->Y);
mp_int *Qsum = monty_add(ec->mc, Q->X, Q->Y);
mp_int *aPxQx = monty_mul(ec->mc, ec->a, PxQx);
mp_int *dPtQt = monty_mul(ec->mc, ec->d, PtQt);
mp_int *sumprod = monty_mul(ec->mc, Psum, Qsum);
mp_int *xx_p_yy = monty_add(ec->mc, PxQx, PyQy);
mp_int *E = monty_sub(ec->mc, sumprod, xx_p_yy);
mp_int *F = monty_sub(ec->mc, PzQz, dPtQt);
mp_int *G = monty_add(ec->mc, PzQz, dPtQt);
mp_int *H = monty_sub(ec->mc, PyQy, aPxQx);
S->X = monty_mul(ec->mc, E, F);
S->Z = monty_mul(ec->mc, F, G);
S->Y = monty_mul(ec->mc, G, H);
S->T = monty_mul(ec->mc, H, E);
mp_free(PxQx);
mp_free(PyQy);
mp_free(PtQt);
mp_free(PzQz);
mp_free(Psum);
mp_free(Qsum);
mp_free(aPxQx);
mp_free(dPtQt);
mp_free(sumprod);
mp_free(xx_p_yy);
mp_free(E);
mp_free(F);
mp_free(G);
mp_free(H);
return S;
}
static void ecc_edwards_normalise(EdwardsPoint *ep)
{
EdwardsCurve *ec = ep->ec;
mp_int *zinv = monty_invert(ec->mc, ep->Z);
monty_mul_into(ec->mc, ep->X, ep->X, zinv);
monty_mul_into(ec->mc, ep->Y, ep->Y, zinv);
monty_mul_into(ec->mc, ep->Z, ep->Z, zinv);
mp_free(zinv);
monty_mul_into(ec->mc, ep->T, ep->X, ep->Y);
}
EdwardsPoint *ecc_edwards_multiply(EdwardsPoint *B, mp_int *n)
{
EdwardsPoint *two_B = ecc_edwards_add(B, B);
EdwardsPoint *k_B = ecc_edwards_point_copy(B);
EdwardsPoint *kplus1_B = ecc_edwards_point_copy(two_B);
/*
* Another copy of the same exponentiation routine following the
* pattern of the Montgomery ladder, because it works as well as
* any other technique and this way I didn't have to debug two of
* them.
*/
unsigned not_started_yet = 1;
for (size_t bitindex = mp_max_bits(n); bitindex-- > 0 ;) {
unsigned nbit = mp_get_bit(n, bitindex);
EdwardsPoint *sum = ecc_edwards_add(k_B, kplus1_B);
ecc_edwards_cond_swap(k_B, kplus1_B, nbit);
EdwardsPoint *other = ecc_edwards_add(k_B, k_B);
ecc_edwards_point_free(k_B);
ecc_edwards_point_free(kplus1_B);
k_B = other;
kplus1_B = sum;
ecc_edwards_cond_swap(k_B, kplus1_B, nbit);
ecc_edwards_cond_overwrite(k_B, B, not_started_yet);
ecc_edwards_cond_overwrite(kplus1_B, two_B, not_started_yet);
not_started_yet &= ~nbit;
}
ecc_edwards_point_free(two_B);
ecc_edwards_point_free(kplus1_B);
return k_B;
}
/*
* Helper routine to determine whether two values each given as a pair
* of projective coordinates represent the same affine value.
*/
static inline unsigned projective_eq(
MontyContext *mc, mp_int *An, mp_int *Ad,
mp_int *Bn, mp_int *Bd)
{
mp_int *AnBd = monty_mul(mc, An, Bd);
mp_int *BnAd = monty_mul(mc, Bn, Ad);
unsigned toret = mp_cmp_eq(AnBd, BnAd);
mp_free(AnBd);
mp_free(BnAd);
return toret;
}
unsigned ecc_edwards_eq(EdwardsPoint *P, EdwardsPoint *Q)
{
EdwardsCurve *ec = P->ec;
assert(Q->ec == ec);
return (projective_eq(ec->mc, P->X, P->Z, Q->X, Q->Z) &
projective_eq(ec->mc, P->Y, P->Z, Q->Y, Q->Z));
}
void ecc_edwards_get_affine(EdwardsPoint *ep, mp_int **x, mp_int **y)
{
EdwardsCurve *ec = ep->ec;
ecc_edwards_normalise(ep);
if (x)
*x = monty_export(ec->mc, ep->X);
if (y)
*y = monty_export(ec->mc, ep->Y);
}
|