aboutsummaryrefslogtreecommitdiff
path: root/src/secp256k1/src/group_impl.h
blob: a5fbc91a0f88d2e9eddcb13072b95d9f5938c258 (plain)
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
/**********************************************************************
 * Copyright (c) 2013, 2014 Pieter Wuille                             *
 * Distributed under the MIT software license, see the accompanying   *
 * file COPYING or http://www.opensource.org/licenses/mit-license.php.*
 **********************************************************************/

#ifndef SECP256K1_GROUP_IMPL_H
#define SECP256K1_GROUP_IMPL_H

#include "num.h"
#include "field.h"
#include "group.h"

/* These exhaustive group test orders and generators are chosen such that:
 * - The field size is equal to that of secp256k1, so field code is the same.
 * - The curve equation is of the form y^2=x^3+B for some constant B.
 * - The subgroup has a generator 2*P, where P.x=1.
 * - The subgroup has size less than 1000 to permit exhaustive testing.
 * - The subgroup admits an endomorphism of the form lambda*(x,y) == (beta*x,y).
 *
 * These parameters are generated using sage/gen_exhaustive_groups.sage.
 */
#if defined(EXHAUSTIVE_TEST_ORDER)
#  if EXHAUSTIVE_TEST_ORDER == 13
static const secp256k1_ge secp256k1_ge_const_g = SECP256K1_GE_CONST(
    0xc3459c3d, 0x35326167, 0xcd86cce8, 0x07a2417f,
    0x5b8bd567, 0xde8538ee, 0x0d507b0c, 0xd128f5bb,
    0x8e467fec, 0xcd30000a, 0x6cc1184e, 0x25d382c2,
    0xa2f4494e, 0x2fbe9abc, 0x8b64abac, 0xd005fb24
);
static const secp256k1_fe secp256k1_fe_const_b = SECP256K1_FE_CONST(
    0x3d3486b2, 0x159a9ca5, 0xc75638be, 0xb23a69bc,
    0x946a45ab, 0x24801247, 0xb4ed2b8e, 0x26b6a417
);
#  elif EXHAUSTIVE_TEST_ORDER == 199
static const secp256k1_ge secp256k1_ge_const_g = SECP256K1_GE_CONST(
    0x226e653f, 0xc8df7744, 0x9bacbf12, 0x7d1dcbf9,
    0x87f05b2a, 0xe7edbd28, 0x1f564575, 0xc48dcf18,
    0xa13872c2, 0xe933bb17, 0x5d9ffd5b, 0xb5b6e10c,
    0x57fe3c00, 0xbaaaa15a, 0xe003ec3e, 0x9c269bae
);
static const secp256k1_fe secp256k1_fe_const_b = SECP256K1_FE_CONST(
    0x2cca28fa, 0xfc614b80, 0x2a3db42b, 0x00ba00b1,
    0xbea8d943, 0xdace9ab2, 0x9536daea, 0x0074defb
);
#  else
#    error No known generator for the specified exhaustive test group order.
#  endif
#else
/** Generator for secp256k1, value 'g' defined in
 *  "Standards for Efficient Cryptography" (SEC2) 2.7.1.
 */
static const secp256k1_ge secp256k1_ge_const_g = SECP256K1_GE_CONST(
    0x79BE667EUL, 0xF9DCBBACUL, 0x55A06295UL, 0xCE870B07UL,
    0x029BFCDBUL, 0x2DCE28D9UL, 0x59F2815BUL, 0x16F81798UL,
    0x483ADA77UL, 0x26A3C465UL, 0x5DA4FBFCUL, 0x0E1108A8UL,
    0xFD17B448UL, 0xA6855419UL, 0x9C47D08FUL, 0xFB10D4B8UL
);

static const secp256k1_fe secp256k1_fe_const_b = SECP256K1_FE_CONST(0, 0, 0, 0, 0, 0, 0, 7);
#endif

static void secp256k1_ge_set_gej_zinv(secp256k1_ge *r, const secp256k1_gej *a, const secp256k1_fe *zi) {
    secp256k1_fe zi2;
    secp256k1_fe zi3;
    secp256k1_fe_sqr(&zi2, zi);
    secp256k1_fe_mul(&zi3, &zi2, zi);
    secp256k1_fe_mul(&r->x, &a->x, &zi2);
    secp256k1_fe_mul(&r->y, &a->y, &zi3);
    r->infinity = a->infinity;
}

static void secp256k1_ge_set_xy(secp256k1_ge *r, const secp256k1_fe *x, const secp256k1_fe *y) {
    r->infinity = 0;
    r->x = *x;
    r->y = *y;
}

static int secp256k1_ge_is_infinity(const secp256k1_ge *a) {
    return a->infinity;
}

static void secp256k1_ge_neg(secp256k1_ge *r, const secp256k1_ge *a) {
    *r = *a;
    secp256k1_fe_normalize_weak(&r->y);
    secp256k1_fe_negate(&r->y, &r->y, 1);
}

static void secp256k1_ge_set_gej(secp256k1_ge *r, secp256k1_gej *a) {
    secp256k1_fe z2, z3;
    r->infinity = a->infinity;
    secp256k1_fe_inv(&a->z, &a->z);
    secp256k1_fe_sqr(&z2, &a->z);
    secp256k1_fe_mul(&z3, &a->z, &z2);
    secp256k1_fe_mul(&a->x, &a->x, &z2);
    secp256k1_fe_mul(&a->y, &a->y, &z3);
    secp256k1_fe_set_int(&a->z, 1);
    r->x = a->x;
    r->y = a->y;
}

static void secp256k1_ge_set_gej_var(secp256k1_ge *r, secp256k1_gej *a) {
    secp256k1_fe z2, z3;
    r->infinity = a->infinity;
    if (a->infinity) {
        return;
    }
    secp256k1_fe_inv_var(&a->z, &a->z);
    secp256k1_fe_sqr(&z2, &a->z);
    secp256k1_fe_mul(&z3, &a->z, &z2);
    secp256k1_fe_mul(&a->x, &a->x, &z2);
    secp256k1_fe_mul(&a->y, &a->y, &z3);
    secp256k1_fe_set_int(&a->z, 1);
    r->x = a->x;
    r->y = a->y;
}

static void secp256k1_ge_set_all_gej_var(secp256k1_ge *r, const secp256k1_gej *a, size_t len) {
    secp256k1_fe u;
    size_t i;
    size_t last_i = SIZE_MAX;

    for (i = 0; i < len; i++) {
        if (!a[i].infinity) {
            /* Use destination's x coordinates as scratch space */
            if (last_i == SIZE_MAX) {
                r[i].x = a[i].z;
            } else {
                secp256k1_fe_mul(&r[i].x, &r[last_i].x, &a[i].z);
            }
            last_i = i;
        }
    }
    if (last_i == SIZE_MAX) {
        return;
    }
    secp256k1_fe_inv_var(&u, &r[last_i].x);

    i = last_i;
    while (i > 0) {
        i--;
        if (!a[i].infinity) {
            secp256k1_fe_mul(&r[last_i].x, &r[i].x, &u);
            secp256k1_fe_mul(&u, &u, &a[last_i].z);
            last_i = i;
        }
    }
    VERIFY_CHECK(!a[last_i].infinity);
    r[last_i].x = u;

    for (i = 0; i < len; i++) {
        r[i].infinity = a[i].infinity;
        if (!a[i].infinity) {
            secp256k1_ge_set_gej_zinv(&r[i], &a[i], &r[i].x);
        }
    }
}

static void secp256k1_ge_globalz_set_table_gej(size_t len, secp256k1_ge *r, secp256k1_fe *globalz, const secp256k1_gej *a, const secp256k1_fe *zr) {
    size_t i = len - 1;
    secp256k1_fe zs;

    if (len > 0) {
        /* The z of the final point gives us the "global Z" for the table. */
        r[i].x = a[i].x;
        r[i].y = a[i].y;
        /* Ensure all y values are in weak normal form for fast negation of points */
        secp256k1_fe_normalize_weak(&r[i].y);
        *globalz = a[i].z;
        r[i].infinity = 0;
        zs = zr[i];

        /* Work our way backwards, using the z-ratios to scale the x/y values. */
        while (i > 0) {
            if (i != len - 1) {
                secp256k1_fe_mul(&zs, &zs, &zr[i]);
            }
            i--;
            secp256k1_ge_set_gej_zinv(&r[i], &a[i], &zs);
        }
    }
}

static void secp256k1_gej_set_infinity(secp256k1_gej *r) {
    r->infinity = 1;
    secp256k1_fe_clear(&r->x);
    secp256k1_fe_clear(&r->y);
    secp256k1_fe_clear(&r->z);
}

static void secp256k1_ge_set_infinity(secp256k1_ge *r) {
    r->infinity = 1;
    secp256k1_fe_clear(&r->x);
    secp256k1_fe_clear(&r->y);
}

static void secp256k1_gej_clear(secp256k1_gej *r) {
    r->infinity = 0;
    secp256k1_fe_clear(&r->x);
    secp256k1_fe_clear(&r->y);
    secp256k1_fe_clear(&r->z);
}

static void secp256k1_ge_clear(secp256k1_ge *r) {
    r->infinity = 0;
    secp256k1_fe_clear(&r->x);
    secp256k1_fe_clear(&r->y);
}

static int secp256k1_ge_set_xquad(secp256k1_ge *r, const secp256k1_fe *x) {
    secp256k1_fe x2, x3;
    r->x = *x;
    secp256k1_fe_sqr(&x2, x);
    secp256k1_fe_mul(&x3, x, &x2);
    r->infinity = 0;
    secp256k1_fe_add(&x3, &secp256k1_fe_const_b);
    return secp256k1_fe_sqrt(&r->y, &x3);
}

static int secp256k1_ge_set_xo_var(secp256k1_ge *r, const secp256k1_fe *x, int odd) {
    if (!secp256k1_ge_set_xquad(r, x)) {
        return 0;
    }
    secp256k1_fe_normalize_var(&r->y);
    if (secp256k1_fe_is_odd(&r->y) != odd) {
        secp256k1_fe_negate(&r->y, &r->y, 1);
    }
    return 1;

}

static void secp256k1_gej_set_ge(secp256k1_gej *r, const secp256k1_ge *a) {
   r->infinity = a->infinity;
   r->x = a->x;
   r->y = a->y;
   secp256k1_fe_set_int(&r->z, 1);
}

static int secp256k1_gej_eq_x_var(const secp256k1_fe *x, const secp256k1_gej *a) {
    secp256k1_fe r, r2;
    VERIFY_CHECK(!a->infinity);
    secp256k1_fe_sqr(&r, &a->z); secp256k1_fe_mul(&r, &r, x);
    r2 = a->x; secp256k1_fe_normalize_weak(&r2);
    return secp256k1_fe_equal_var(&r, &r2);
}

static void secp256k1_gej_neg(secp256k1_gej *r, const secp256k1_gej *a) {
    r->infinity = a->infinity;
    r->x = a->x;
    r->y = a->y;
    r->z = a->z;
    secp256k1_fe_normalize_weak(&r->y);
    secp256k1_fe_negate(&r->y, &r->y, 1);
}

static int secp256k1_gej_is_infinity(const secp256k1_gej *a) {
    return a->infinity;
}

static int secp256k1_ge_is_valid_var(const secp256k1_ge *a) {
    secp256k1_fe y2, x3;
    if (a->infinity) {
        return 0;
    }
    /* y^2 = x^3 + 7 */
    secp256k1_fe_sqr(&y2, &a->y);
    secp256k1_fe_sqr(&x3, &a->x); secp256k1_fe_mul(&x3, &x3, &a->x);
    secp256k1_fe_add(&x3, &secp256k1_fe_const_b);
    secp256k1_fe_normalize_weak(&x3);
    return secp256k1_fe_equal_var(&y2, &x3);
}

static SECP256K1_INLINE void secp256k1_gej_double(secp256k1_gej *r, const secp256k1_gej *a) {
    /* Operations: 3 mul, 4 sqr, 0 normalize, 12 mul_int/add/negate.
     *
     * Note that there is an implementation described at
     *     https://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#doubling-dbl-2009-l
     * which trades a multiply for a square, but in practice this is actually slower,
     * mainly because it requires more normalizations.
     */
    secp256k1_fe t1,t2,t3,t4;

    r->infinity = a->infinity;

    secp256k1_fe_mul(&r->z, &a->z, &a->y);
    secp256k1_fe_mul_int(&r->z, 2);       /* Z' = 2*Y*Z (2) */
    secp256k1_fe_sqr(&t1, &a->x);
    secp256k1_fe_mul_int(&t1, 3);         /* T1 = 3*X^2 (3) */
    secp256k1_fe_sqr(&t2, &t1);           /* T2 = 9*X^4 (1) */
    secp256k1_fe_sqr(&t3, &a->y);
    secp256k1_fe_mul_int(&t3, 2);         /* T3 = 2*Y^2 (2) */
    secp256k1_fe_sqr(&t4, &t3);
    secp256k1_fe_mul_int(&t4, 2);         /* T4 = 8*Y^4 (2) */
    secp256k1_fe_mul(&t3, &t3, &a->x);    /* T3 = 2*X*Y^2 (1) */
    r->x = t3;
    secp256k1_fe_mul_int(&r->x, 4);       /* X' = 8*X*Y^2 (4) */
    secp256k1_fe_negate(&r->x, &r->x, 4); /* X' = -8*X*Y^2 (5) */
    secp256k1_fe_add(&r->x, &t2);         /* X' = 9*X^4 - 8*X*Y^2 (6) */
    secp256k1_fe_negate(&t2, &t2, 1);     /* T2 = -9*X^4 (2) */
    secp256k1_fe_mul_int(&t3, 6);         /* T3 = 12*X*Y^2 (6) */
    secp256k1_fe_add(&t3, &t2);           /* T3 = 12*X*Y^2 - 9*X^4 (8) */
    secp256k1_fe_mul(&r->y, &t1, &t3);    /* Y' = 36*X^3*Y^2 - 27*X^6 (1) */
    secp256k1_fe_negate(&t2, &t4, 2);     /* T2 = -8*Y^4 (3) */
    secp256k1_fe_add(&r->y, &t2);         /* Y' = 36*X^3*Y^2 - 27*X^6 - 8*Y^4 (4) */
}

static void secp256k1_gej_double_var(secp256k1_gej *r, const secp256k1_gej *a, secp256k1_fe *rzr) {
    /** For secp256k1, 2Q is infinity if and only if Q is infinity. This is because if 2Q = infinity,
     *  Q must equal -Q, or that Q.y == -(Q.y), or Q.y is 0. For a point on y^2 = x^3 + 7 to have
     *  y=0, x^3 must be -7 mod p. However, -7 has no cube root mod p.
     *
     *  Having said this, if this function receives a point on a sextic twist, e.g. by
     *  a fault attack, it is possible for y to be 0. This happens for y^2 = x^3 + 6,
     *  since -6 does have a cube root mod p. For this point, this function will not set
     *  the infinity flag even though the point doubles to infinity, and the result
     *  point will be gibberish (z = 0 but infinity = 0).
     */
    if (a->infinity) {
        r->infinity = 1;
        if (rzr != NULL) {
            secp256k1_fe_set_int(rzr, 1);
        }
        return;
    }

    if (rzr != NULL) {
        *rzr = a->y;
        secp256k1_fe_normalize_weak(rzr);
        secp256k1_fe_mul_int(rzr, 2);
    }

    secp256k1_gej_double(r, a);
}

static void secp256k1_gej_add_var(secp256k1_gej *r, const secp256k1_gej *a, const secp256k1_gej *b, secp256k1_fe *rzr) {
    /* Operations: 12 mul, 4 sqr, 2 normalize, 12 mul_int/add/negate */
    secp256k1_fe z22, z12, u1, u2, s1, s2, h, i, i2, h2, h3, t;

    if (a->infinity) {
        VERIFY_CHECK(rzr == NULL);
        *r = *b;
        return;
    }

    if (b->infinity) {
        if (rzr != NULL) {
            secp256k1_fe_set_int(rzr, 1);
        }
        *r = *a;
        return;
    }

    r->infinity = 0;
    secp256k1_fe_sqr(&z22, &b->z);
    secp256k1_fe_sqr(&z12, &a->z);
    secp256k1_fe_mul(&u1, &a->x, &z22);
    secp256k1_fe_mul(&u2, &b->x, &z12);
    secp256k1_fe_mul(&s1, &a->y, &z22); secp256k1_fe_mul(&s1, &s1, &b->z);
    secp256k1_fe_mul(&s2, &b->y, &z12); secp256k1_fe_mul(&s2, &s2, &a->z);
    secp256k1_fe_negate(&h, &u1, 1); secp256k1_fe_add(&h, &u2);
    secp256k1_fe_negate(&i, &s1, 1); secp256k1_fe_add(&i, &s2);
    if (secp256k1_fe_normalizes_to_zero_var(&h)) {
        if (secp256k1_fe_normalizes_to_zero_var(&i)) {
            secp256k1_gej_double_var(r, a, rzr);
        } else {
            if (rzr != NULL) {
                secp256k1_fe_set_int(rzr, 0);
            }
            secp256k1_gej_set_infinity(r);
        }
        return;
    }
    secp256k1_fe_sqr(&i2, &i);
    secp256k1_fe_sqr(&h2, &h);
    secp256k1_fe_mul(&h3, &h, &h2);
    secp256k1_fe_mul(&h, &h, &b->z);
    if (rzr != NULL) {
        *rzr = h;
    }
    secp256k1_fe_mul(&r->z, &a->z, &h);
    secp256k1_fe_mul(&t, &u1, &h2);
    r->x = t; secp256k1_fe_mul_int(&r->x, 2); secp256k1_fe_add(&r->x, &h3); secp256k1_fe_negate(&r->x, &r->x, 3); secp256k1_fe_add(&r->x, &i2);
    secp256k1_fe_negate(&r->y, &r->x, 5); secp256k1_fe_add(&r->y, &t); secp256k1_fe_mul(&r->y, &r->y, &i);
    secp256k1_fe_mul(&h3, &h3, &s1); secp256k1_fe_negate(&h3, &h3, 1);
    secp256k1_fe_add(&r->y, &h3);
}

static void secp256k1_gej_add_ge_var(secp256k1_gej *r, const secp256k1_gej *a, const secp256k1_ge *b, secp256k1_fe *rzr) {
    /* 8 mul, 3 sqr, 4 normalize, 12 mul_int/add/negate */
    secp256k1_fe z12, u1, u2, s1, s2, h, i, i2, h2, h3, t;
    if (a->infinity) {
        VERIFY_CHECK(rzr == NULL);
        secp256k1_gej_set_ge(r, b);
        return;
    }
    if (b->infinity) {
        if (rzr != NULL) {
            secp256k1_fe_set_int(rzr, 1);
        }
        *r = *a;
        return;
    }
    r->infinity = 0;

    secp256k1_fe_sqr(&z12, &a->z);
    u1 = a->x; secp256k1_fe_normalize_weak(&u1);
    secp256k1_fe_mul(&u2, &b->x, &z12);
    s1 = a->y; secp256k1_fe_normalize_weak(&s1);
    secp256k1_fe_mul(&s2, &b->y, &z12); secp256k1_fe_mul(&s2, &s2, &a->z);
    secp256k1_fe_negate(&h, &u1, 1); secp256k1_fe_add(&h, &u2);
    secp256k1_fe_negate(&i, &s1, 1); secp256k1_fe_add(&i, &s2);
    if (secp256k1_fe_normalizes_to_zero_var(&h)) {
        if (secp256k1_fe_normalizes_to_zero_var(&i)) {
            secp256k1_gej_double_var(r, a, rzr);
        } else {
            if (rzr != NULL) {
                secp256k1_fe_set_int(rzr, 0);
            }
            secp256k1_gej_set_infinity(r);
        }
        return;
    }
    secp256k1_fe_sqr(&i2, &i);
    secp256k1_fe_sqr(&h2, &h);
    secp256k1_fe_mul(&h3, &h, &h2);
    if (rzr != NULL) {
        *rzr = h;
    }
    secp256k1_fe_mul(&r->z, &a->z, &h);
    secp256k1_fe_mul(&t, &u1, &h2);
    r->x = t; secp256k1_fe_mul_int(&r->x, 2); secp256k1_fe_add(&r->x, &h3); secp256k1_fe_negate(&r->x, &r->x, 3); secp256k1_fe_add(&r->x, &i2);
    secp256k1_fe_negate(&r->y, &r->x, 5); secp256k1_fe_add(&r->y, &t); secp256k1_fe_mul(&r->y, &r->y, &i);
    secp256k1_fe_mul(&h3, &h3, &s1); secp256k1_fe_negate(&h3, &h3, 1);
    secp256k1_fe_add(&r->y, &h3);
}

static void secp256k1_gej_add_zinv_var(secp256k1_gej *r, const secp256k1_gej *a, const secp256k1_ge *b, const secp256k1_fe *bzinv) {
    /* 9 mul, 3 sqr, 4 normalize, 12 mul_int/add/negate */
    secp256k1_fe az, z12, u1, u2, s1, s2, h, i, i2, h2, h3, t;

    if (b->infinity) {
        *r = *a;
        return;
    }
    if (a->infinity) {
        secp256k1_fe bzinv2, bzinv3;
        r->infinity = b->infinity;
        secp256k1_fe_sqr(&bzinv2, bzinv);
        secp256k1_fe_mul(&bzinv3, &bzinv2, bzinv);
        secp256k1_fe_mul(&r->x, &b->x, &bzinv2);
        secp256k1_fe_mul(&r->y, &b->y, &bzinv3);
        secp256k1_fe_set_int(&r->z, 1);
        return;
    }
    r->infinity = 0;

    /** We need to calculate (rx,ry,rz) = (ax,ay,az) + (bx,by,1/bzinv). Due to
     *  secp256k1's isomorphism we can multiply the Z coordinates on both sides
     *  by bzinv, and get: (rx,ry,rz*bzinv) = (ax,ay,az*bzinv) + (bx,by,1).
     *  This means that (rx,ry,rz) can be calculated as
     *  (ax,ay,az*bzinv) + (bx,by,1), when not applying the bzinv factor to rz.
     *  The variable az below holds the modified Z coordinate for a, which is used
     *  for the computation of rx and ry, but not for rz.
     */
    secp256k1_fe_mul(&az, &a->z, bzinv);

    secp256k1_fe_sqr(&z12, &az);
    u1 = a->x; secp256k1_fe_normalize_weak(&u1);
    secp256k1_fe_mul(&u2, &b->x, &z12);
    s1 = a->y; secp256k1_fe_normalize_weak(&s1);
    secp256k1_fe_mul(&s2, &b->y, &z12); secp256k1_fe_mul(&s2, &s2, &az);
    secp256k1_fe_negate(&h, &u1, 1); secp256k1_fe_add(&h, &u2);
    secp256k1_fe_negate(&i, &s1, 1); secp256k1_fe_add(&i, &s2);
    if (secp256k1_fe_normalizes_to_zero_var(&h)) {
        if (secp256k1_fe_normalizes_to_zero_var(&i)) {
            secp256k1_gej_double_var(r, a, NULL);
        } else {
            secp256k1_gej_set_infinity(r);
        }
        return;
    }
    secp256k1_fe_sqr(&i2, &i);
    secp256k1_fe_sqr(&h2, &h);
    secp256k1_fe_mul(&h3, &h, &h2);
    r->z = a->z; secp256k1_fe_mul(&r->z, &r->z, &h);
    secp256k1_fe_mul(&t, &u1, &h2);
    r->x = t; secp256k1_fe_mul_int(&r->x, 2); secp256k1_fe_add(&r->x, &h3); secp256k1_fe_negate(&r->x, &r->x, 3); secp256k1_fe_add(&r->x, &i2);
    secp256k1_fe_negate(&r->y, &r->x, 5); secp256k1_fe_add(&r->y, &t); secp256k1_fe_mul(&r->y, &r->y, &i);
    secp256k1_fe_mul(&h3, &h3, &s1); secp256k1_fe_negate(&h3, &h3, 1);
    secp256k1_fe_add(&r->y, &h3);
}


static void secp256k1_gej_add_ge(secp256k1_gej *r, const secp256k1_gej *a, const secp256k1_ge *b) {
    /* Operations: 7 mul, 5 sqr, 4 normalize, 21 mul_int/add/negate/cmov */
    static const secp256k1_fe fe_1 = SECP256K1_FE_CONST(0, 0, 0, 0, 0, 0, 0, 1);
    secp256k1_fe zz, u1, u2, s1, s2, t, tt, m, n, q, rr;
    secp256k1_fe m_alt, rr_alt;
    int infinity, degenerate;
    VERIFY_CHECK(!b->infinity);
    VERIFY_CHECK(a->infinity == 0 || a->infinity == 1);

    /** In:
     *    Eric Brier and Marc Joye, Weierstrass Elliptic Curves and Side-Channel Attacks.
     *    In D. Naccache and P. Paillier, Eds., Public Key Cryptography, vol. 2274 of Lecture Notes in Computer Science, pages 335-345. Springer-Verlag, 2002.
     *  we find as solution for a unified addition/doubling formula:
     *    lambda = ((x1 + x2)^2 - x1 * x2 + a) / (y1 + y2), with a = 0 for secp256k1's curve equation.
     *    x3 = lambda^2 - (x1 + x2)
     *    2*y3 = lambda * (x1 + x2 - 2 * x3) - (y1 + y2).
     *
     *  Substituting x_i = Xi / Zi^2 and yi = Yi / Zi^3, for i=1,2,3, gives:
     *    U1 = X1*Z2^2, U2 = X2*Z1^2
     *    S1 = Y1*Z2^3, S2 = Y2*Z1^3
     *    Z = Z1*Z2
     *    T = U1+U2
     *    M = S1+S2
     *    Q = T*M^2
     *    R = T^2-U1*U2
     *    X3 = 4*(R^2-Q)
     *    Y3 = 4*(R*(3*Q-2*R^2)-M^4)
     *    Z3 = 2*M*Z
     *  (Note that the paper uses xi = Xi / Zi and yi = Yi / Zi instead.)
     *
     *  This formula has the benefit of being the same for both addition
     *  of distinct points and doubling. However, it breaks down in the
     *  case that either point is infinity, or that y1 = -y2. We handle
     *  these cases in the following ways:
     *
     *    - If b is infinity we simply bail by means of a VERIFY_CHECK.
     *
     *    - If a is infinity, we detect this, and at the end of the
     *      computation replace the result (which will be meaningless,
     *      but we compute to be constant-time) with b.x : b.y : 1.
     *
     *    - If a = -b, we have y1 = -y2, which is a degenerate case.
     *      But here the answer is infinity, so we simply set the
     *      infinity flag of the result, overriding the computed values
     *      without even needing to cmov.
     *
     *    - If y1 = -y2 but x1 != x2, which does occur thanks to certain
     *      properties of our curve (specifically, 1 has nontrivial cube
     *      roots in our field, and the curve equation has no x coefficient)
     *      then the answer is not infinity but also not given by the above
     *      equation. In this case, we cmov in place an alternate expression
     *      for lambda. Specifically (y1 - y2)/(x1 - x2). Where both these
     *      expressions for lambda are defined, they are equal, and can be
     *      obtained from each other by multiplication by (y1 + y2)/(y1 + y2)
     *      then substitution of x^3 + 7 for y^2 (using the curve equation).
     *      For all pairs of nonzero points (a, b) at least one is defined,
     *      so this covers everything.
     */

    secp256k1_fe_sqr(&zz, &a->z);                       /* z = Z1^2 */
    u1 = a->x; secp256k1_fe_normalize_weak(&u1);        /* u1 = U1 = X1*Z2^2 (1) */
    secp256k1_fe_mul(&u2, &b->x, &zz);                  /* u2 = U2 = X2*Z1^2 (1) */
    s1 = a->y; secp256k1_fe_normalize_weak(&s1);        /* s1 = S1 = Y1*Z2^3 (1) */
    secp256k1_fe_mul(&s2, &b->y, &zz);                  /* s2 = Y2*Z1^2 (1) */
    secp256k1_fe_mul(&s2, &s2, &a->z);                  /* s2 = S2 = Y2*Z1^3 (1) */
    t = u1; secp256k1_fe_add(&t, &u2);                  /* t = T = U1+U2 (2) */
    m = s1; secp256k1_fe_add(&m, &s2);                  /* m = M = S1+S2 (2) */
    secp256k1_fe_sqr(&rr, &t);                          /* rr = T^2 (1) */
    secp256k1_fe_negate(&m_alt, &u2, 1);                /* Malt = -X2*Z1^2 */
    secp256k1_fe_mul(&tt, &u1, &m_alt);                 /* tt = -U1*U2 (2) */
    secp256k1_fe_add(&rr, &tt);                         /* rr = R = T^2-U1*U2 (3) */
    /** If lambda = R/M = 0/0 we have a problem (except in the "trivial"
     *  case that Z = z1z2 = 0, and this is special-cased later on). */
    degenerate = secp256k1_fe_normalizes_to_zero(&m) &
                 secp256k1_fe_normalizes_to_zero(&rr);
    /* This only occurs when y1 == -y2 and x1^3 == x2^3, but x1 != x2.
     * This means either x1 == beta*x2 or beta*x1 == x2, where beta is
     * a nontrivial cube root of one. In either case, an alternate
     * non-indeterminate expression for lambda is (y1 - y2)/(x1 - x2),
     * so we set R/M equal to this. */
    rr_alt = s1;
    secp256k1_fe_mul_int(&rr_alt, 2);       /* rr = Y1*Z2^3 - Y2*Z1^3 (2) */
    secp256k1_fe_add(&m_alt, &u1);          /* Malt = X1*Z2^2 - X2*Z1^2 */

    secp256k1_fe_cmov(&rr_alt, &rr, !degenerate);
    secp256k1_fe_cmov(&m_alt, &m, !degenerate);
    /* Now Ralt / Malt = lambda and is guaranteed not to be 0/0.
     * From here on out Ralt and Malt represent the numerator
     * and denominator of lambda; R and M represent the explicit
     * expressions x1^2 + x2^2 + x1x2 and y1 + y2. */
    secp256k1_fe_sqr(&n, &m_alt);                       /* n = Malt^2 (1) */
    secp256k1_fe_mul(&q, &n, &t);                       /* q = Q = T*Malt^2 (1) */
    /* These two lines use the observation that either M == Malt or M == 0,
     * so M^3 * Malt is either Malt^4 (which is computed by squaring), or
     * zero (which is "computed" by cmov). So the cost is one squaring
     * versus two multiplications. */
    secp256k1_fe_sqr(&n, &n);
    secp256k1_fe_cmov(&n, &m, degenerate);              /* n = M^3 * Malt (2) */
    secp256k1_fe_sqr(&t, &rr_alt);                      /* t = Ralt^2 (1) */
    secp256k1_fe_mul(&r->z, &a->z, &m_alt);             /* r->z = Malt*Z (1) */
    infinity = secp256k1_fe_normalizes_to_zero(&r->z) * (1 - a->infinity);
    secp256k1_fe_mul_int(&r->z, 2);                     /* r->z = Z3 = 2*Malt*Z (2) */
    secp256k1_fe_negate(&q, &q, 1);                     /* q = -Q (2) */
    secp256k1_fe_add(&t, &q);                           /* t = Ralt^2-Q (3) */
    secp256k1_fe_normalize_weak(&t);
    r->x = t;                                           /* r->x = Ralt^2-Q (1) */
    secp256k1_fe_mul_int(&t, 2);                        /* t = 2*x3 (2) */
    secp256k1_fe_add(&t, &q);                           /* t = 2*x3 - Q: (4) */
    secp256k1_fe_mul(&t, &t, &rr_alt);                  /* t = Ralt*(2*x3 - Q) (1) */
    secp256k1_fe_add(&t, &n);                           /* t = Ralt*(2*x3 - Q) + M^3*Malt (3) */
    secp256k1_fe_negate(&r->y, &t, 3);                  /* r->y = Ralt*(Q - 2x3) - M^3*Malt (4) */
    secp256k1_fe_normalize_weak(&r->y);
    secp256k1_fe_mul_int(&r->x, 4);                     /* r->x = X3 = 4*(Ralt^2-Q) */
    secp256k1_fe_mul_int(&r->y, 4);                     /* r->y = Y3 = 4*Ralt*(Q - 2x3) - 4*M^3*Malt (4) */

    /** In case a->infinity == 1, replace r with (b->x, b->y, 1). */
    secp256k1_fe_cmov(&r->x, &b->x, a->infinity);
    secp256k1_fe_cmov(&r->y, &b->y, a->infinity);
    secp256k1_fe_cmov(&r->z, &fe_1, a->infinity);
    r->infinity = infinity;
}

static void secp256k1_gej_rescale(secp256k1_gej *r, const secp256k1_fe *s) {
    /* Operations: 4 mul, 1 sqr */
    secp256k1_fe zz;
    VERIFY_CHECK(!secp256k1_fe_is_zero(s));
    secp256k1_fe_sqr(&zz, s);
    secp256k1_fe_mul(&r->x, &r->x, &zz);                /* r->x *= s^2 */
    secp256k1_fe_mul(&r->y, &r->y, &zz);
    secp256k1_fe_mul(&r->y, &r->y, s);                  /* r->y *= s^3 */
    secp256k1_fe_mul(&r->z, &r->z, s);                  /* r->z *= s   */
}

static void secp256k1_ge_to_storage(secp256k1_ge_storage *r, const secp256k1_ge *a) {
    secp256k1_fe x, y;
    VERIFY_CHECK(!a->infinity);
    x = a->x;
    secp256k1_fe_normalize(&x);
    y = a->y;
    secp256k1_fe_normalize(&y);
    secp256k1_fe_to_storage(&r->x, &x);
    secp256k1_fe_to_storage(&r->y, &y);
}

static void secp256k1_ge_from_storage(secp256k1_ge *r, const secp256k1_ge_storage *a) {
    secp256k1_fe_from_storage(&r->x, &a->x);
    secp256k1_fe_from_storage(&r->y, &a->y);
    r->infinity = 0;
}

static SECP256K1_INLINE void secp256k1_ge_storage_cmov(secp256k1_ge_storage *r, const secp256k1_ge_storage *a, int flag) {
    secp256k1_fe_storage_cmov(&r->x, &a->x, flag);
    secp256k1_fe_storage_cmov(&r->y, &a->y, flag);
}

static void secp256k1_ge_mul_lambda(secp256k1_ge *r, const secp256k1_ge *a) {
    static const secp256k1_fe beta = SECP256K1_FE_CONST(
        0x7ae96a2bul, 0x657c0710ul, 0x6e64479eul, 0xac3434e9ul,
        0x9cf04975ul, 0x12f58995ul, 0xc1396c28ul, 0x719501eeul
    );
    *r = *a;
    secp256k1_fe_mul(&r->x, &r->x, &beta);
}

static int secp256k1_gej_has_quad_y_var(const secp256k1_gej *a) {
    secp256k1_fe yz;

    if (a->infinity) {
        return 0;
    }

    /* We rely on the fact that the Jacobi symbol of 1 / a->z^3 is the same as
     * that of a->z. Thus a->y / a->z^3 is a quadratic residue iff a->y * a->z
       is */
    secp256k1_fe_mul(&yz, &a->y, &a->z);
    return secp256k1_fe_is_quad_var(&yz);
}

static int secp256k1_ge_is_in_correct_subgroup(const secp256k1_ge* ge) {
#ifdef EXHAUSTIVE_TEST_ORDER
    secp256k1_gej out;
    int i;

    /* A very simple EC multiplication ladder that avoids a dependecy on ecmult. */
    secp256k1_gej_set_infinity(&out);
    for (i = 0; i < 32; ++i) {
        secp256k1_gej_double_var(&out, &out, NULL);
        if ((((uint32_t)EXHAUSTIVE_TEST_ORDER) >> (31 - i)) & 1) {
            secp256k1_gej_add_ge_var(&out, &out, ge, NULL);
        }
    }
    return secp256k1_gej_is_infinity(&out);
#else
    (void)ge;
    /* The real secp256k1 group has cofactor 1, so the subgroup is the entire curve. */
    return 1;
#endif
}

#endif /* SECP256K1_GROUP_IMPL_H */