src/group_impl.h

   1 /**********************************************************************
   2  * Copyright (c) 2013, 2014 Pieter Wuille                             *
   3  * Distributed under the MIT software license, see the accompanying   *
   4  * file COPYING or http://www.opensource.org/licenses/mit-license.php.*
   5  **********************************************************************/
   6
   7 #ifndef SECP256K1_GROUP_IMPL_H
   8 #define SECP256K1_GROUP_IMPL_H
   9
  10 #include "num.h"
  11 #include "field.h"
  12 #include "group.h"
  13
  14 /* These points can be generated in sage as follows:
  15  *
  16  * 0. Setup a worksheet with the following parameters.
  17  *   b = 4  # whatever CURVE_B will be set to
  18  *   F = FiniteField (0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC2F)
  19  *   C = EllipticCurve ([F (0), F (b)])
  20  *
  21  * 1. Determine all the small orders available to you. (If there are
  22  *    no satisfactory ones, go back and change b.)
  23  *   print C.order().factor(limit=1000)
  24  *
  25  * 2. Choose an order as one of the prime factors listed in the above step.
  26  *    (You can also multiply some to get a composite order, though the
  27  *    tests will crash trying to invert scalars during signing.) We take a
  28  *    random point and scale it to drop its order to the desired value.
  29  *    There is some probability this won't work; just try again.
  30  *   order = 199
  31  *   P = C.random_point()
  32  *   P = (int(P.order()) / int(order)) * P
  33  *   assert(P.order() == order)
  34  *
  35  * 3. Print the values. You'll need to use a vim macro or something to
  36  *    split the hex output into 4-byte chunks.
  37  *   print "%x %x" % P.xy()
  38  */
  39 #if defined(EXHAUSTIVE_TEST_ORDER)
  40 #  if EXHAUSTIVE_TEST_ORDER == 199
  41 const secp256k1_ge secp256k1_ge_const_g = SECP256K1_GE_CONST(
  42     0xFA7CC9A7, 0x0737F2DB, 0xA749DD39, 0x2B4FB069,
  43     0x3B017A7D, 0xA808C2F1, 0xFB12940C, 0x9EA66C18,
  44     0x78AC123A, 0x5ED8AEF3, 0x8732BC91, 0x1F3A2868,
  45     0x48DF246C, 0x808DAE72, 0xCFE52572, 0x7F0501ED
  46 );
  47
  48 const int CURVE_B = 4;
  49 #  elif EXHAUSTIVE_TEST_ORDER == 13
  50 const secp256k1_ge secp256k1_ge_const_g = SECP256K1_GE_CONST(
  51     0xedc60018, 0xa51a786b, 0x2ea91f4d, 0x4c9416c0,
  52     0x9de54c3b, 0xa1316554, 0x6cf4345c, 0x7277ef15,
  53     0x54cb1b6b, 0xdc8c1273, 0x087844ea, 0x43f4603e,
  54     0x0eaf9a43, 0xf6effe55, 0x939f806d, 0x37adf8ac
  55 );
  56 const int CURVE_B = 2;
  57 #  else
  58 #    error No known generator for the specified exhaustive test group order.
  59 #  endif
  60 #else
  61 /** Generator for secp256k1, value 'g' defined in
  62  *  "Standards for Efficient Cryptography" (SEC2) 2.7.1.
  63  */
  64 static const secp256k1_ge secp256k1_ge_const_g = SECP256K1_GE_CONST(
  65     0x79BE667EUL, 0xF9DCBBACUL, 0x55A06295UL, 0xCE870B07UL,
  66     0x029BFCDBUL, 0x2DCE28D9UL, 0x59F2815BUL, 0x16F81798UL,
  67     0x483ADA77UL, 0x26A3C465UL, 0x5DA4FBFCUL, 0x0E1108A8UL,
  68     0xFD17B448UL, 0xA6855419UL, 0x9C47D08FUL, 0xFB10D4B8UL
  69 );
  70
  71 const int CURVE_B = 7;
  72 #endif
  73
  74 static void secp256k1_ge_set_gej_zinv(secp256k1_ge *r, const secp256k1_gej *a, const secp256k1_fe *zi) {
  75     secp256k1_fe zi2;
  76     secp256k1_fe zi3;
  77     secp256k1_fe_sqr(&zi2, zi);
  78     secp256k1_fe_mul(&zi3, &zi2, zi);
  79     secp256k1_fe_mul(&r->x, &a->x, &zi2);
  80     secp256k1_fe_mul(&r->y, &a->y, &zi3);
  81     r->infinity = a->infinity;
  82 }
  83
  84 static void secp256k1_ge_set_xy(secp256k1_ge *r, const secp256k1_fe *x, const secp256k1_fe *y) {
  85     r->infinity = 0;
  86     r->x = *x;
  87     r->y = *y;
  88 }
  89
  90 static int secp256k1_ge_is_infinity(const secp256k1_ge *a) {
  91     return a->infinity;
  92 }
  93
  94 static void secp256k1_ge_neg(secp256k1_ge *r, const secp256k1_ge *a) {
  95     *r = *a;
  96     secp256k1_fe_normalize_weak(&r->y);
  97     secp256k1_fe_negate(&r->y, &r->y, 1);
  98 }
  99
 100 static void secp256k1_ge_set_gej(secp256k1_ge *r, secp256k1_gej *a) {
 101     secp256k1_fe z2, z3;
 102     r->infinity = a->infinity;
 103     secp256k1_fe_inv(&a->z, &a->z);
 104     secp256k1_fe_sqr(&z2, &a->z);
 105     secp256k1_fe_mul(&z3, &a->z, &z2);
 106     secp256k1_fe_mul(&a->x, &a->x, &z2);
 107     secp256k1_fe_mul(&a->y, &a->y, &z3);
 108     secp256k1_fe_set_int(&a->z, 1);
 109     r->x = a->x;
 110     r->y = a->y;
 111 }
 112
 113 static void secp256k1_ge_set_gej_var(secp256k1_ge *r, secp256k1_gej *a) {
 114     secp256k1_fe z2, z3;
 115     r->infinity = a->infinity;
 116     if (a->infinity) {
 117         return;
 118     }
 119     secp256k1_fe_inv_var(&a->z, &a->z);
 120     secp256k1_fe_sqr(&z2, &a->z);
 121     secp256k1_fe_mul(&z3, &a->z, &z2);
 122     secp256k1_fe_mul(&a->x, &a->x, &z2);
 123     secp256k1_fe_mul(&a->y, &a->y, &z3);
 124     secp256k1_fe_set_int(&a->z, 1);
 125     r->x = a->x;
 126     r->y = a->y;
 127 }
 128
 129 static void secp256k1_ge_set_all_gej_var(secp256k1_ge *r, const secp256k1_gej *a, size_t len, const secp256k1_callback *cb) {
 130     secp256k1_fe *az;
 131     secp256k1_fe *azi;
 132     size_t i;
 133     size_t count = 0;
 134     az = (secp256k1_fe *)checked_malloc(cb, sizeof(secp256k1_fe) * len);
 135     for (i = 0; i < len; i++) {
 136         if (!a[i].infinity) {
 137             az[count++] = a[i].z;
 138         }
 139     }
 140
 141     azi = (secp256k1_fe *)checked_malloc(cb, sizeof(secp256k1_fe) * count);
 142     secp256k1_fe_inv_all_var(azi, az, count);
 143     free(az);
 144
 145     count = 0;
 146     for (i = 0; i < len; i++) {
 147         r[i].infinity = a[i].infinity;
 148         if (!a[i].infinity) {
 149             secp256k1_ge_set_gej_zinv(&r[i], &a[i], &azi[count++]);
 150         }
 151     }
 152     free(azi);
 153 }
 154
 155 static void secp256k1_ge_set_table_gej_var(secp256k1_ge *r, const secp256k1_gej *a, const secp256k1_fe *zr, size_t len) {
 156     size_t i = len - 1;
 157     secp256k1_fe zi;
 158
 159     if (len > 0) {
 160         /* Compute the inverse of the last z coordinate, and use it to compute the last affine output. */
 161         secp256k1_fe_inv(&zi, &a[i].z);
 162         secp256k1_ge_set_gej_zinv(&r[i], &a[i], &zi);
 163
 164         /* Work out way backwards, using the z-ratios to scale the x/y values. */
 165         while (i > 0) {
 166             secp256k1_fe_mul(&zi, &zi, &zr[i]);
 167             i--;
 168             secp256k1_ge_set_gej_zinv(&r[i], &a[i], &zi);
 169         }
 170     }
 171 }
 172
 173 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) {
 174     size_t i = len - 1;
 175     secp256k1_fe zs;
 176
 177     if (len > 0) {
 178         /* The z of the final point gives us the "global Z" for the table. */
 179         r[i].x = a[i].x;
 180         r[i].y = a[i].y;
 181         *globalz = a[i].z;
 182         r[i].infinity = 0;
 183         zs = zr[i];
 184
 185         /* Work our way backwards, using the z-ratios to scale the x/y values. */
 186         while (i > 0) {
 187             if (i != len - 1) {
 188                 secp256k1_fe_mul(&zs, &zs, &zr[i]);
 189             }
 190             i--;
 191             secp256k1_ge_set_gej_zinv(&r[i], &a[i], &zs);
 192         }
 193     }
 194 }
 195
 196 static void secp256k1_gej_set_infinity(secp256k1_gej *r) {
 197     r->infinity = 1;
 198     secp256k1_fe_clear(&r->x);
 199     secp256k1_fe_clear(&r->y);
 200     secp256k1_fe_clear(&r->z);
 201 }
 202
 203 static void secp256k1_gej_clear(secp256k1_gej *r) {
 204     r->infinity = 0;
 205     secp256k1_fe_clear(&r->x);
 206     secp256k1_fe_clear(&r->y);
 207     secp256k1_fe_clear(&r->z);
 208 }
 209
 210 static void secp256k1_ge_clear(secp256k1_ge *r) {
 211     r->infinity = 0;
 212     secp256k1_fe_clear(&r->x);
 213     secp256k1_fe_clear(&r->y);
 214 }
 215
 216 static int secp256k1_ge_set_xquad(secp256k1_ge *r, const secp256k1_fe *x) {
 217     secp256k1_fe x2, x3, c;
 218     r->x = *x;
 219     secp256k1_fe_sqr(&x2, x);
 220     secp256k1_fe_mul(&x3, x, &x2);
 221     r->infinity = 0;
 222     secp256k1_fe_set_int(&c, CURVE_B);
 223     secp256k1_fe_add(&c, &x3);
 224     return secp256k1_fe_sqrt(&r->y, &c);
 225 }
 226
 227 static int secp256k1_ge_set_xo_var(secp256k1_ge *r, const secp256k1_fe *x, int odd) {
 228     if (!secp256k1_ge_set_xquad(r, x)) {
 229         return 0;
 230     }
 231     secp256k1_fe_normalize_var(&r->y);
 232     if (secp256k1_fe_is_odd(&r->y) != odd) {
 233         secp256k1_fe_negate(&r->y, &r->y, 1);
 234     }
 235     return 1;
 236
 237 }
 238
 239 static void secp256k1_gej_set_ge(secp256k1_gej *r, const secp256k1_ge *a) {
 240    r->infinity = a->infinity;
 241    r->x = a->x;
 242    r->y = a->y;
 243    secp256k1_fe_set_int(&r->z, 1);
 244 }
 245
 246 static int secp256k1_gej_eq_x_var(const secp256k1_fe *x, const secp256k1_gej *a) {
 247     secp256k1_fe r, r2;
 248     VERIFY_CHECK(!a->infinity);
 249     secp256k1_fe_sqr(&r, &a->z); secp256k1_fe_mul(&r, &r, x);
 250     r2 = a->x; secp256k1_fe_normalize_weak(&r2);
 251     return secp256k1_fe_equal_var(&r, &r2);
 252 }
 253
 254 static void secp256k1_gej_neg(secp256k1_gej *r, const secp256k1_gej *a) {
 255     r->infinity = a->infinity;
 256     r->x = a->x;
 257     r->y = a->y;
 258     r->z = a->z;
 259     secp256k1_fe_normalize_weak(&r->y);
 260     secp256k1_fe_negate(&r->y, &r->y, 1);
 261 }
 262
 263 static int secp256k1_gej_is_infinity(const secp256k1_gej *a) {
 264     return a->infinity;
 265 }
 266
 267 static int secp256k1_gej_is_valid_var(const secp256k1_gej *a) {
 268     secp256k1_fe y2, x3, z2, z6;
 269     if (a->infinity) {
 270         return 0;
 271     }
 272     /** y^2 = x^3 + 7
 273      *  (Y/Z^3)^2 = (X/Z^2)^3 + 7
 274      *  Y^2 / Z^6 = X^3 / Z^6 + 7
 275      *  Y^2 = X^3 + 7*Z^6
 276      */
 277     secp256k1_fe_sqr(&y2, &a->y);
 278     secp256k1_fe_sqr(&x3, &a->x); secp256k1_fe_mul(&x3, &x3, &a->x);
 279     secp256k1_fe_sqr(&z2, &a->z);
 280     secp256k1_fe_sqr(&z6, &z2); secp256k1_fe_mul(&z6, &z6, &z2);
 281     secp256k1_fe_mul_int(&z6, CURVE_B);
 282     secp256k1_fe_add(&x3, &z6);
 283     secp256k1_fe_normalize_weak(&x3);
 284     return secp256k1_fe_equal_var(&y2, &x3);
 285 }
 286
 287 static int secp256k1_ge_is_valid_var(const secp256k1_ge *a) {
 288     secp256k1_fe y2, x3, c;
 289     if (a->infinity) {
 290         return 0;
 291     }
 292     /* y^2 = x^3 + 7 */
 293     secp256k1_fe_sqr(&y2, &a->y);
 294     secp256k1_fe_sqr(&x3, &a->x); secp256k1_fe_mul(&x3, &x3, &a->x);
 295     secp256k1_fe_set_int(&c, CURVE_B);
 296     secp256k1_fe_add(&x3, &c);
 297     secp256k1_fe_normalize_weak(&x3);
 298     return secp256k1_fe_equal_var(&y2, &x3);
 299 }
 300
 301 static void secp256k1_gej_double_var(secp256k1_gej *r, const secp256k1_gej *a, secp256k1_fe *rzr) {
 302     /* Operations: 3 mul, 4 sqr, 0 normalize, 12 mul_int/add/negate.
 303      *
 304      * Note that there is an implementation described at
 305      *     https://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#doubling-dbl-2009-l
 306      * which trades a multiply for a square, but in practice this is actually slower,
 307      * mainly because it requires more normalizations.
 308      */
 309     secp256k1_fe t1,t2,t3,t4;
 310     /** For secp256k1, 2Q is infinity if and only if Q is infinity. This is because if 2Q = infinity,
 311      *  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
 312      *  y=0, x^3 must be -7 mod p. However, -7 has no cube root mod p.
 313      *
 314      *  Having said this, if this function receives a point on a sextic twist, e.g. by
 315      *  a fault attack, it is possible for y to be 0. This happens for y^2 = x^3 + 6,
 316      *  since -6 does have a cube root mod p. For this point, this function will not set
 317      *  the infinity flag even though the point doubles to infinity, and the result
 318      *  point will be gibberish (z = 0 but infinity = 0).
 319      */
 320     r->infinity = a->infinity;
 321     if (r->infinity) {
 322         if (rzr != NULL) {
 323             secp256k1_fe_set_int(rzr, 1);
 324         }
 325         return;
 326     }
 327
 328     if (rzr != NULL) {
 329         *rzr = a->y;
 330         secp256k1_fe_normalize_weak(rzr);
 331         secp256k1_fe_mul_int(rzr, 2);
 332     }
 333
 334     secp256k1_fe_mul(&r->z, &a->z, &a->y);
 335     secp256k1_fe_mul_int(&r->z, 2);       /* Z' = 2*Y*Z (2) */
 336     secp256k1_fe_sqr(&t1, &a->x);
 337     secp256k1_fe_mul_int(&t1, 3);         /* T1 = 3*X^2 (3) */
 338     secp256k1_fe_sqr(&t2, &t1);           /* T2 = 9*X^4 (1) */
 339     secp256k1_fe_sqr(&t3, &a->y);
 340     secp256k1_fe_mul_int(&t3, 2);         /* T3 = 2*Y^2 (2) */
 341     secp256k1_fe_sqr(&t4, &t3);
 342     secp256k1_fe_mul_int(&t4, 2);         /* T4 = 8*Y^4 (2) */
 343     secp256k1_fe_mul(&t3, &t3, &a->x);    /* T3 = 2*X*Y^2 (1) */
 344     r->x = t3;
 345     secp256k1_fe_mul_int(&r->x, 4);       /* X' = 8*X*Y^2 (4) */
 346     secp256k1_fe_negate(&r->x, &r->x, 4); /* X' = -8*X*Y^2 (5) */
 347     secp256k1_fe_add(&r->x, &t2);         /* X' = 9*X^4 - 8*X*Y^2 (6) */
 348     secp256k1_fe_negate(&t2, &t2, 1);     /* T2 = -9*X^4 (2) */
 349     secp256k1_fe_mul_int(&t3, 6);         /* T3 = 12*X*Y^2 (6) */
 350     secp256k1_fe_add(&t3, &t2);           /* T3 = 12*X*Y^2 - 9*X^4 (8) */
 351     secp256k1_fe_mul(&r->y, &t1, &t3);    /* Y' = 36*X^3*Y^2 - 27*X^6 (1) */
 352     secp256k1_fe_negate(&t2, &t4, 2);     /* T2 = -8*Y^4 (3) */
 353     secp256k1_fe_add(&r->y, &t2);         /* Y' = 36*X^3*Y^2 - 27*X^6 - 8*Y^4 (4) */
 354 }
 355
 356 static SECP256K1_INLINE void secp256k1_gej_double_nonzero(secp256k1_gej *r, const secp256k1_gej *a, secp256k1_fe *rzr) {
 357     VERIFY_CHECK(!secp256k1_gej_is_infinity(a));
 358     secp256k1_gej_double_var(r, a, rzr);
 359 }
 360
 361 static void secp256k1_gej_add_var(secp256k1_gej *r, const secp256k1_gej *a, const secp256k1_gej *b, secp256k1_fe *rzr) {
 362     /* Operations: 12 mul, 4 sqr, 2 normalize, 12 mul_int/add/negate */
 363     secp256k1_fe z22, z12, u1, u2, s1, s2, h, i, i2, h2, h3, t;
 364
 365     if (a->infinity) {
 366         VERIFY_CHECK(rzr == NULL);
 367         *r = *b;
 368         return;
 369     }
 370
 371     if (b->infinity) {
 372         if (rzr != NULL) {
 373             secp256k1_fe_set_int(rzr, 1);
 374         }
 375         *r = *a;
 376         return;
 377     }
 378
 379     r->infinity = 0;
 380     secp256k1_fe_sqr(&z22, &b->z);
 381     secp256k1_fe_sqr(&z12, &a->z);
 382     secp256k1_fe_mul(&u1, &a->x, &z22);
 383     secp256k1_fe_mul(&u2, &b->x, &z12);
 384     secp256k1_fe_mul(&s1, &a->y, &z22); secp256k1_fe_mul(&s1, &s1, &b->z);
 385     secp256k1_fe_mul(&s2, &b->y, &z12); secp256k1_fe_mul(&s2, &s2, &a->z);
 386     secp256k1_fe_negate(&h, &u1, 1); secp256k1_fe_add(&h, &u2);
 387     secp256k1_fe_negate(&i, &s1, 1); secp256k1_fe_add(&i, &s2);
 388     if (secp256k1_fe_normalizes_to_zero_var(&h)) {
 389         if (secp256k1_fe_normalizes_to_zero_var(&i)) {
 390             secp256k1_gej_double_var(r, a, rzr);
 391         } else {
 392             if (rzr != NULL) {
 393                 secp256k1_fe_set_int(rzr, 0);
 394             }
 395             r->infinity = 1;
 396         }
 397         return;
 398     }
 399     secp256k1_fe_sqr(&i2, &i);
 400     secp256k1_fe_sqr(&h2, &h);
 401     secp256k1_fe_mul(&h3, &h, &h2);
 402     secp256k1_fe_mul(&h, &h, &b->z);
 403     if (rzr != NULL) {
 404         *rzr = h;
 405     }
 406     secp256k1_fe_mul(&r->z, &a->z, &h);
 407     secp256k1_fe_mul(&t, &u1, &h2);
 408     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);
 409     secp256k1_fe_negate(&r->y, &r->x, 5); secp256k1_fe_add(&r->y, &t); secp256k1_fe_mul(&r->y, &r->y, &i);
 410     secp256k1_fe_mul(&h3, &h3, &s1); secp256k1_fe_negate(&h3, &h3, 1);
 411     secp256k1_fe_add(&r->y, &h3);
 412 }
 413
 414 static void secp256k1_gej_add_ge_var(secp256k1_gej *r, const secp256k1_gej *a, const secp256k1_ge *b, secp256k1_fe *rzr) {
 415     /* 8 mul, 3 sqr, 4 normalize, 12 mul_int/add/negate */
 416     secp256k1_fe z12, u1, u2, s1, s2, h, i, i2, h2, h3, t;
 417     if (a->infinity) {
 418         VERIFY_CHECK(rzr == NULL);
 419         secp256k1_gej_set_ge(r, b);
 420         return;
 421     }
 422     if (b->infinity) {
 423         if (rzr != NULL) {
 424             secp256k1_fe_set_int(rzr, 1);
 425         }
 426         *r = *a;
 427         return;
 428     }
 429     r->infinity = 0;
 430
 431     secp256k1_fe_sqr(&z12, &a->z);
 432     u1 = a->x; secp256k1_fe_normalize_weak(&u1);
 433     secp256k1_fe_mul(&u2, &b->x, &z12);
 434     s1 = a->y; secp256k1_fe_normalize_weak(&s1);
 435     secp256k1_fe_mul(&s2, &b->y, &z12); secp256k1_fe_mul(&s2, &s2, &a->z);
 436     secp256k1_fe_negate(&h, &u1, 1); secp256k1_fe_add(&h, &u2);
 437     secp256k1_fe_negate(&i, &s1, 1); secp256k1_fe_add(&i, &s2);
 438     if (secp256k1_fe_normalizes_to_zero_var(&h)) {
 439         if (secp256k1_fe_normalizes_to_zero_var(&i)) {
 440             secp256k1_gej_double_var(r, a, rzr);
 441         } else {
 442             if (rzr != NULL) {
 443                 secp256k1_fe_set_int(rzr, 0);
 444             }
 445             r->infinity = 1;
 446         }
 447         return;
 448     }
 449     secp256k1_fe_sqr(&i2, &i);
 450     secp256k1_fe_sqr(&h2, &h);
 451     secp256k1_fe_mul(&h3, &h, &h2);
 452     if (rzr != NULL) {
 453         *rzr = h;
 454     }
 455     secp256k1_fe_mul(&r->z, &a->z, &h);
 456     secp256k1_fe_mul(&t, &u1, &h2);
 457     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);
 458     secp256k1_fe_negate(&r->y, &r->x, 5); secp256k1_fe_add(&r->y, &t); secp256k1_fe_mul(&r->y, &r->y, &i);
 459     secp256k1_fe_mul(&h3, &h3, &s1); secp256k1_fe_negate(&h3, &h3, 1);
 460     secp256k1_fe_add(&r->y, &h3);
 461 }
 462
 463 static void secp256k1_gej_add_zinv_var(secp256k1_gej *r, const secp256k1_gej *a, const secp256k1_ge *b, const secp256k1_fe *bzinv) {
 464     /* 9 mul, 3 sqr, 4 normalize, 12 mul_int/add/negate */
 465     secp256k1_fe az, z12, u1, u2, s1, s2, h, i, i2, h2, h3, t;
 466
 467     if (b->infinity) {
 468         *r = *a;
 469         return;
 470     }
 471     if (a->infinity) {
 472         secp256k1_fe bzinv2, bzinv3;
 473         r->infinity = b->infinity;
 474         secp256k1_fe_sqr(&bzinv2, bzinv);
 475         secp256k1_fe_mul(&bzinv3, &bzinv2, bzinv);
 476         secp256k1_fe_mul(&r->x, &b->x, &bzinv2);
 477         secp256k1_fe_mul(&r->y, &b->y, &bzinv3);
 478         secp256k1_fe_set_int(&r->z, 1);
 479         return;
 480     }
 481     r->infinity = 0;
 482
 483     /** We need to calculate (rx,ry,rz) = (ax,ay,az) + (bx,by,1/bzinv). Due to
 484      *  secp256k1's isomorphism we can multiply the Z coordinates on both sides
 485      *  by bzinv, and get: (rx,ry,rz*bzinv) = (ax,ay,az*bzinv) + (bx,by,1).
 486      *  This means that (rx,ry,rz) can be calculated as
 487      *  (ax,ay,az*bzinv) + (bx,by,1), when not applying the bzinv factor to rz.
 488      *  The variable az below holds the modified Z coordinate for a, which is used
 489      *  for the computation of rx and ry, but not for rz.
 490      */
 491     secp256k1_fe_mul(&az, &a->z, bzinv);
 492
 493     secp256k1_fe_sqr(&z12, &az);
 494     u1 = a->x; secp256k1_fe_normalize_weak(&u1);
 495     secp256k1_fe_mul(&u2, &b->x, &z12);
 496     s1 = a->y; secp256k1_fe_normalize_weak(&s1);
 497     secp256k1_fe_mul(&s2, &b->y, &z12); secp256k1_fe_mul(&s2, &s2, &az);
 498     secp256k1_fe_negate(&h, &u1, 1); secp256k1_fe_add(&h, &u2);
 499     secp256k1_fe_negate(&i, &s1, 1); secp256k1_fe_add(&i, &s2);
 500     if (secp256k1_fe_normalizes_to_zero_var(&h)) {
 501         if (secp256k1_fe_normalizes_to_zero_var(&i)) {
 502             secp256k1_gej_double_var(r, a, NULL);
 503         } else {
 504             r->infinity = 1;
 505         }
 506         return;
 507     }
 508     secp256k1_fe_sqr(&i2, &i);
 509     secp256k1_fe_sqr(&h2, &h);
 510     secp256k1_fe_mul(&h3, &h, &h2);
 511     r->z = a->z; secp256k1_fe_mul(&r->z, &r->z, &h);
 512     secp256k1_fe_mul(&t, &u1, &h2);
 513     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);
 514     secp256k1_fe_negate(&r->y, &r->x, 5); secp256k1_fe_add(&r->y, &t); secp256k1_fe_mul(&r->y, &r->y, &i);
 515     secp256k1_fe_mul(&h3, &h3, &s1); secp256k1_fe_negate(&h3, &h3, 1);
 516     secp256k1_fe_add(&r->y, &h3);
 517 }
 518
 519
 520 static void secp256k1_gej_add_ge(secp256k1_gej *r, const secp256k1_gej *a, const secp256k1_ge *b) {
 521     /* Operations: 7 mul, 5 sqr, 4 normalize, 21 mul_int/add/negate/cmov */
 522     static const secp256k1_fe fe_1 = SECP256K1_FE_CONST(0, 0, 0, 0, 0, 0, 0, 1);
 523     secp256k1_fe zz, u1, u2, s1, s2, t, tt, m, n, q, rr;
 524     secp256k1_fe m_alt, rr_alt;
 525     int infinity, degenerate;
 526     VERIFY_CHECK(!b->infinity);
 527     VERIFY_CHECK(a->infinity == 0 || a->infinity == 1);
 528
 529     /** In:
 530      *    Eric Brier and Marc Joye, Weierstrass Elliptic Curves and Side-Channel Attacks.
 531      *    In D. Naccache and P. Paillier, Eds., Public Key Cryptography, vol. 2274 of Lecture Notes in Computer Science, pages 335-345. Springer-Verlag, 2002.
 532      *  we find as solution for a unified addition/doubling formula:
 533      *    lambda = ((x1 + x2)^2 - x1 * x2 + a) / (y1 + y2), with a = 0 for secp256k1's curve equation.
 534      *    x3 = lambda^2 - (x1 + x2)
 535      *    2*y3 = lambda * (x1 + x2 - 2 * x3) - (y1 + y2).
 536      *
 537      *  Substituting x_i = Xi / Zi^2 and yi = Yi / Zi^3, for i=1,2,3, gives:
 538      *    U1 = X1*Z2^2, U2 = X2*Z1^2
 539      *    S1 = Y1*Z2^3, S2 = Y2*Z1^3
 540      *    Z = Z1*Z2
 541      *    T = U1+U2
 542      *    M = S1+S2
 543      *    Q = T*M^2
 544      *    R = T^2-U1*U2
 545      *    X3 = 4*(R^2-Q)
 546      *    Y3 = 4*(R*(3*Q-2*R^2)-M^4)
 547      *    Z3 = 2*M*Z
 548      *  (Note that the paper uses xi = Xi / Zi and yi = Yi / Zi instead.)
 549      *
 550      *  This formula has the benefit of being the same for both addition
 551      *  of distinct points and doubling. However, it breaks down in the
 552      *  case that either point is infinity, or that y1 = -y2. We handle
 553      *  these cases in the following ways:
 554      *
 555      *    - If b is infinity we simply bail by means of a VERIFY_CHECK.
 556      *
 557      *    - If a is infinity, we detect this, and at the end of the
 558      *      computation replace the result (which will be meaningless,
 559      *      but we compute to be constant-time) with b.x : b.y : 1.
 560      *
 561      *    - If a = -b, we have y1 = -y2, which is a degenerate case.
 562      *      But here the answer is infinity, so we simply set the
 563      *      infinity flag of the result, overriding the computed values
 564      *      without even needing to cmov.
 565      *
 566      *    - If y1 = -y2 but x1 != x2, which does occur thanks to certain
 567      *      properties of our curve (specifically, 1 has nontrivial cube
 568      *      roots in our field, and the curve equation has no x coefficient)
 569      *      then the answer is not infinity but also not given by the above
 570      *      equation. In this case, we cmov in place an alternate expression
 571      *      for lambda. Specifically (y1 - y2)/(x1 - x2). Where both these
 572      *      expressions for lambda are defined, they are equal, and can be
 573      *      obtained from each other by multiplication by (y1 + y2)/(y1 + y2)
 574      *      then substitution of x^3 + 7 for y^2 (using the curve equation).
 575      *      For all pairs of nonzero points (a, b) at least one is defined,
 576      *      so this covers everything.
 577      */
 578
 579     secp256k1_fe_sqr(&zz, &a->z);                       /* z = Z1^2 */
 580     u1 = a->x; secp256k1_fe_normalize_weak(&u1);        /* u1 = U1 = X1*Z2^2 (1) */
 581     secp256k1_fe_mul(&u2, &b->x, &zz);                  /* u2 = U2 = X2*Z1^2 (1) */
 582     s1 = a->y; secp256k1_fe_normalize_weak(&s1);        /* s1 = S1 = Y1*Z2^3 (1) */
 583     secp256k1_fe_mul(&s2, &b->y, &zz);                  /* s2 = Y2*Z1^2 (1) */
 584     secp256k1_fe_mul(&s2, &s2, &a->z);                  /* s2 = S2 = Y2*Z1^3 (1) */
 585     t = u1; secp256k1_fe_add(&t, &u2);                  /* t = T = U1+U2 (2) */
 586     m = s1; secp256k1_fe_add(&m, &s2);                  /* m = M = S1+S2 (2) */
 587     secp256k1_fe_sqr(&rr, &t);                          /* rr = T^2 (1) */
 588     secp256k1_fe_negate(&m_alt, &u2, 1);                /* Malt = -X2*Z1^2 */
 589     secp256k1_fe_mul(&tt, &u1, &m_alt);                 /* tt = -U1*U2 (2) */
 590     secp256k1_fe_add(&rr, &tt);                         /* rr = R = T^2-U1*U2 (3) */
 591     /** If lambda = R/M = 0/0 we have a problem (except in the "trivial"
 592      *  case that Z = z1z2 = 0, and this is special-cased later on). */
 593     degenerate = secp256k1_fe_normalizes_to_zero(&m) &
 594                  secp256k1_fe_normalizes_to_zero(&rr);
 595     /* This only occurs when y1 == -y2 and x1^3 == x2^3, but x1 != x2.
 596      * This means either x1 == beta*x2 or beta*x1 == x2, where beta is
 597      * a nontrivial cube root of one. In either case, an alternate
 598      * non-indeterminate expression for lambda is (y1 - y2)/(x1 - x2),
 599      * so we set R/M equal to this. */
 600     rr_alt = s1;
 601     secp256k1_fe_mul_int(&rr_alt, 2);       /* rr = Y1*Z2^3 - Y2*Z1^3 (2) */
 602     secp256k1_fe_add(&m_alt, &u1);          /* Malt = X1*Z2^2 - X2*Z1^2 */
 603
 604     secp256k1_fe_cmov(&rr_alt, &rr, !degenerate);
 605     secp256k1_fe_cmov(&m_alt, &m, !degenerate);
 606     /* Now Ralt / Malt = lambda and is guaranteed not to be 0/0.
 607      * From here on out Ralt and Malt represent the numerator
 608      * and denominator of lambda; R and M represent the explicit
 609      * expressions x1^2 + x2^2 + x1x2 and y1 + y2. */
 610     secp256k1_fe_sqr(&n, &m_alt);                       /* n = Malt^2 (1) */
 611     secp256k1_fe_mul(&q, &n, &t);                       /* q = Q = T*Malt^2 (1) */
 612     /* These two lines use the observation that either M == Malt or M == 0,
 613      * so M^3 * Malt is either Malt^4 (which is computed by squaring), or
 614      * zero (which is "computed" by cmov). So the cost is one squaring
 615      * versus two multiplications. */
 616     secp256k1_fe_sqr(&n, &n);
 617     secp256k1_fe_cmov(&n, &m, degenerate);              /* n = M^3 * Malt (2) */
 618     secp256k1_fe_sqr(&t, &rr_alt);                      /* t = Ralt^2 (1) */
 619     secp256k1_fe_mul(&r->z, &a->z, &m_alt);             /* r->z = Malt*Z (1) */
 620     infinity = secp256k1_fe_normalizes_to_zero(&r->z) * (1 - a->infinity);
 621     secp256k1_fe_mul_int(&r->z, 2);                     /* r->z = Z3 = 2*Malt*Z (2) */
 622     secp256k1_fe_negate(&q, &q, 1);                     /* q = -Q (2) */
 623     secp256k1_fe_add(&t, &q);                           /* t = Ralt^2-Q (3) */
 624     secp256k1_fe_normalize_weak(&t);
 625     r->x = t;                                           /* r->x = Ralt^2-Q (1) */
 626     secp256k1_fe_mul_int(&t, 2);                        /* t = 2*x3 (2) */
 627     secp256k1_fe_add(&t, &q);                           /* t = 2*x3 - Q: (4) */
 628     secp256k1_fe_mul(&t, &t, &rr_alt);                  /* t = Ralt*(2*x3 - Q) (1) */
 629     secp256k1_fe_add(&t, &n);                           /* t = Ralt*(2*x3 - Q) + M^3*Malt (3) */
 630     secp256k1_fe_negate(&r->y, &t, 3);                  /* r->y = Ralt*(Q - 2x3) - M^3*Malt (4) */
 631     secp256k1_fe_normalize_weak(&r->y);
 632     secp256k1_fe_mul_int(&r->x, 4);                     /* r->x = X3 = 4*(Ralt^2-Q) */
 633     secp256k1_fe_mul_int(&r->y, 4);                     /* r->y = Y3 = 4*Ralt*(Q - 2x3) - 4*M^3*Malt (4) */
 634
 635     /** In case a->infinity == 1, replace r with (b->x, b->y, 1). */
 636     secp256k1_fe_cmov(&r->x, &b->x, a->infinity);
 637     secp256k1_fe_cmov(&r->y, &b->y, a->infinity);
 638     secp256k1_fe_cmov(&r->z, &fe_1, a->infinity);
 639     r->infinity = infinity;
 640 }
 641
 642 static void secp256k1_gej_rescale(secp256k1_gej *r, const secp256k1_fe *s) {
 643     /* Operations: 4 mul, 1 sqr */
 644     secp256k1_fe zz;
 645     VERIFY_CHECK(!secp256k1_fe_is_zero(s));
 646     secp256k1_fe_sqr(&zz, s);
 647     secp256k1_fe_mul(&r->x, &r->x, &zz);                /* r->x *= s^2 */
 648     secp256k1_fe_mul(&r->y, &r->y, &zz);
 649     secp256k1_fe_mul(&r->y, &r->y, s);                  /* r->y *= s^3 */
 650     secp256k1_fe_mul(&r->z, &r->z, s);                  /* r->z *= s   */
 651 }
 652
 653 static void secp256k1_ge_to_storage(secp256k1_ge_storage *r, const secp256k1_ge *a) {
 654     secp256k1_fe x, y;
 655     VERIFY_CHECK(!a->infinity);
 656     x = a->x;
 657     secp256k1_fe_normalize(&x);
 658     y = a->y;
 659     secp256k1_fe_normalize(&y);
 660     secp256k1_fe_to_storage(&r->x, &x);
 661     secp256k1_fe_to_storage(&r->y, &y);
 662 }
 663
 664 static void secp256k1_ge_from_storage(secp256k1_ge *r, const secp256k1_ge_storage *a) {
 665     secp256k1_fe_from_storage(&r->x, &a->x);
 666     secp256k1_fe_from_storage(&r->y, &a->y);
 667     r->infinity = 0;
 668 }
 669
 670 static SECP256K1_INLINE void secp256k1_ge_storage_cmov(secp256k1_ge_storage *r, const secp256k1_ge_storage *a, int flag) {
 671     secp256k1_fe_storage_cmov(&r->x, &a->x, flag);
 672     secp256k1_fe_storage_cmov(&r->y, &a->y, flag);
 673 }
 674
 675 #ifdef USE_ENDOMORPHISM
 676 static void secp256k1_ge_mul_lambda(secp256k1_ge *r, const secp256k1_ge *a) {
 677     static const secp256k1_fe beta = SECP256K1_FE_CONST(
 678         0x7ae96a2bul, 0x657c0710ul, 0x6e64479eul, 0xac3434e9ul,
 679         0x9cf04975ul, 0x12f58995ul, 0xc1396c28ul, 0x719501eeul
 680     );
 681     *r = *a;
 682     secp256k1_fe_mul(&r->x, &r->x, &beta);
 683 }
 684 #endif
 685
 686 static int secp256k1_gej_has_quad_y_var(const secp256k1_gej *a) {
 687     secp256k1_fe yz;
 688
 689     if (a->infinity) {
 690         return 0;
 691     }
 692
 693     /* We rely on the fact that the Jacobi symbol of 1 / a->z^3 is the same as
 694      * that of a->z. Thus a->y / a->z^3 is a quadratic residue iff a->y * a->z
 695        is */
 696     secp256k1_fe_mul(&yz, &a->y, &a->z);
 697     return secp256k1_fe_is_quad_var(&yz);
 698 }
 699
 700 #endif /* SECP256K1_GROUP_IMPL_H */