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ecmult_const_impl.h
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1 /***********************************************************************
2  * Copyright (c) 2015, 2022 Pieter Wuille, Andrew Poelstra *
3  * Distributed under the MIT software license, see the accompanying *
4  * file COPYING or https://www.opensource.org/licenses/mit-license.php.*
5  ***********************************************************************/
6 
7 #ifndef SECP256K1_ECMULT_CONST_IMPL_H
8 #define SECP256K1_ECMULT_CONST_IMPL_H
9 
10 #include "scalar.h"
11 #include "group.h"
12 #include "ecmult_const.h"
13 #include "ecmult_impl.h"
14 
15 #if defined(EXHAUSTIVE_TEST_ORDER)
16 /* We need 2^ECMULT_CONST_GROUP_SIZE - 1 to be less than EXHAUSTIVE_TEST_ORDER, because
17  * the tables cannot have infinities in them (this breaks the effective-affine technique's
18  * z-ratio tracking) */
19 # if EXHAUSTIVE_TEST_ORDER == 199
20 # define ECMULT_CONST_GROUP_SIZE 4
21 # elif EXHAUSTIVE_TEST_ORDER == 13
22 # define ECMULT_CONST_GROUP_SIZE 3
23 # elif EXHAUSTIVE_TEST_ORDER == 7
24 # define ECMULT_CONST_GROUP_SIZE 2
25 # else
26 # error "Unknown EXHAUSTIVE_TEST_ORDER"
27 # endif
28 #else
29 /* Group size 4 or 5 appears optimal. */
30 # define ECMULT_CONST_GROUP_SIZE 5
31 #endif
32 
33 #define ECMULT_CONST_TABLE_SIZE (1L << (ECMULT_CONST_GROUP_SIZE - 1))
34 #define ECMULT_CONST_GROUPS ((129 + ECMULT_CONST_GROUP_SIZE - 1) / ECMULT_CONST_GROUP_SIZE)
35 #define ECMULT_CONST_BITS (ECMULT_CONST_GROUPS * ECMULT_CONST_GROUP_SIZE)
36 
46 
49 }
50 
51 /* Given a table 'pre' with odd multiples of a point, put in r the signed-bit multiplication of n with that point.
52  *
53  * For example, if ECMULT_CONST_GROUP_SIZE is 4, then pre is expected to contain 8 entries:
54  * [1*P, 3*P, 5*P, 7*P, 9*P, 11*P, 13*P, 15*P]. n is then expected to be a 4-bit integer (range 0-15), and its
55  * bits are interpreted as signs of powers of two to look up.
56  *
57  * For example, if n=4, which is 0100 in binary, which is interpreted as [- + - -], so the looked up value is
58  * [ -(2^3) + (2^2) - (2^1) - (2^0) ]*P = -7*P. Every valid n translates to an odd number in range [-15,15],
59  * which means we just need to look up one of the precomputed values, and optionally negate it.
60  */
61 #define ECMULT_CONST_TABLE_GET_GE(r,pre,n) do { \
62  unsigned int m = 0; \
63  /* If the top bit of n is 0, we want the negation. */ \
64  volatile unsigned int negative = ((n) >> (ECMULT_CONST_GROUP_SIZE - 1)) ^ 1; \
65  /* Let n[i] be the i-th bit of n, then the index is
66  * sum(cnot(n[i]) * 2^i, i=0..l-2)
67  * where cnot(b) = b if n[l-1] = 1 and 1 - b otherwise.
68  * For example, if n = 4, in binary 0100, the index is 3, in binary 011.
69  *
70  * Proof:
71  * Let
72  * x = sum((2*n[i] - 1)*2^i, i=0..l-1)
73  * = 2*sum(n[i] * 2^i, i=0..l-1) - 2^l + 1
74  * be the value represented by n.
75  * The index is (x - 1)/2 if x > 0 and -(x + 1)/2 otherwise.
76  * Case x > 0:
77  * n[l-1] = 1
78  * index = sum(n[i] * 2^i, i=0..l-1) - 2^(l-1)
79  * = sum(n[i] * 2^i, i=0..l-2)
80  * Case x <= 0:
81  * n[l-1] = 0
82  * index = -(2*sum(n[i] * 2^i, i=0..l-1) - 2^l + 2)/2
83  * = 2^(l-1) - 1 - sum(n[i] * 2^i, i=0..l-1)
84  * = sum((1 - n[i]) * 2^i, i=0..l-2)
85  */ \
86  unsigned int index = ((unsigned int)(-negative) ^ n) & ((1U << (ECMULT_CONST_GROUP_SIZE - 1)) - 1U); \
87  secp256k1_fe neg_y; \
88  VERIFY_CHECK((n) < (1U << ECMULT_CONST_GROUP_SIZE)); \
89  VERIFY_CHECK(index < (1U << (ECMULT_CONST_GROUP_SIZE - 1))); \
90  /* Unconditionally set r->x = (pre)[m].x. r->y = (pre)[m].y. because it's either the correct one
91  * or will get replaced in the later iterations, this is needed to make sure `r` is initialized. */ \
92  (r)->x = (pre)[m].x; \
93  (r)->y = (pre)[m].y; \
94  for (m = 1; m < ECMULT_CONST_TABLE_SIZE; m++) { \
95  /* This loop is used to avoid secret data in array indices. See
96  * the comment in ecmult_gen_impl.h for rationale. */ \
97  secp256k1_fe_cmov(&(r)->x, &(pre)[m].x, m == index); \
98  secp256k1_fe_cmov(&(r)->y, &(pre)[m].y, m == index); \
99  } \
100  (r)->infinity = 0; \
101  secp256k1_fe_negate(&neg_y, &(r)->y, 1); \
102  secp256k1_fe_cmov(&(r)->y, &neg_y, negative); \
103 } while(0)
104 
105 /* For K as defined in the comment of secp256k1_ecmult_const, we have several precomputed
106  * formulas/constants.
107  * - in exhaustive test mode, we give an explicit expression to compute it at compile time: */
108 #ifdef EXHAUSTIVE_TEST_ORDER
109 static const secp256k1_scalar secp256k1_ecmult_const_K = ((SECP256K1_SCALAR_CONST(0, 0, 0, (1U << (ECMULT_CONST_BITS - 128)) - 2U, 0, 0, 0, 0) + EXHAUSTIVE_TEST_ORDER - 1U) * (1U + EXHAUSTIVE_TEST_LAMBDA)) % EXHAUSTIVE_TEST_ORDER;
110 /* - for the real secp256k1 group we have constants for various ECMULT_CONST_BITS values. */
111 #elif ECMULT_CONST_BITS == 129
112 /* For GROUP_SIZE = 1,3. */
113 static const secp256k1_scalar secp256k1_ecmult_const_K = SECP256K1_SCALAR_CONST(0xac9c52b3ul, 0x3fa3cf1ful, 0x5ad9e3fdul, 0x77ed9ba4ul, 0xa880b9fcul, 0x8ec739c2ul, 0xe0cfc810ul, 0xb51283ceul);
114 #elif ECMULT_CONST_BITS == 130
115 /* For GROUP_SIZE = 2,5. */
116 static const secp256k1_scalar secp256k1_ecmult_const_K = SECP256K1_SCALAR_CONST(0xa4e88a7dul, 0xcb13034eul, 0xc2bdd6bful, 0x7c118d6bul, 0x589ae848ul, 0x26ba29e4ul, 0xb5c2c1dcul, 0xde9798d9ul);
117 #elif ECMULT_CONST_BITS == 132
118 /* For GROUP_SIZE = 4,6 */
119 static const secp256k1_scalar secp256k1_ecmult_const_K = SECP256K1_SCALAR_CONST(0x76b1d93dul, 0x0fae3c6bul, 0x3215874bul, 0x94e93813ul, 0x7937fe0dul, 0xb66bcaaful, 0xb3749ca5ul, 0xd7b6171bul);
120 #else
121 # error "Unknown ECMULT_CONST_BITS"
122 #endif
123 
124 static void secp256k1_ecmult_const(secp256k1_gej *r, const secp256k1_ge *a, const secp256k1_scalar *q) {
125  /* The approach below combines the signed-digit logic from Mike Hamburg's
126  * "Fast and compact elliptic-curve cryptography" (https://eprint.iacr.org/2012/309)
127  * Section 3.3, with the GLV endomorphism.
128  *
129  * The idea there is to interpret the bits of a scalar as signs (1 = +, 0 = -), and compute a
130  * point multiplication in that fashion. Let v be an n-bit non-negative integer (0 <= v < 2^n),
131  * and v[i] its i'th bit (so v = sum(v[i] * 2^i, i=0..n-1)). Then define:
132  *
133  * C_l(v, A) = sum((2*v[i] - 1) * 2^i*A, i=0..l-1)
134  *
135  * Then it holds that C_l(v, A) = sum((2*v[i] - 1) * 2^i*A, i=0..l-1)
136  * = (2*sum(v[i] * 2^i, i=0..l-1) + 1 - 2^l) * A
137  * = (2*v + 1 - 2^l) * A
138  *
139  * Thus, one can compute q*A as C_256((q + 2^256 - 1) / 2, A). This is the basis for the
140  * paper's signed-digit multi-comb algorithm for multiplication using a precomputed table.
141  *
142  * It is appealing to try to combine this with the GLV optimization: the idea that a scalar
143  * s can be written as s1 + lambda*s2, where lambda is a curve-specific constant such that
144  * lambda*A is easy to compute, and where s1 and s2 are small. In particular we have the
145  * secp256k1_scalar_split_lambda function which performs such a split with the resulting s1
146  * and s2 in range (-2^128, 2^128) mod n. This does work, but is uninteresting:
147  *
148  * To compute q*A:
149  * - Let s1, s2 = split_lambda(q)
150  * - Let R1 = C_256((s1 + 2^256 - 1) / 2, A)
151  * - Let R2 = C_256((s2 + 2^256 - 1) / 2, lambda*A)
152  * - Return R1 + R2
153  *
154  * The issue is that while s1 and s2 are small-range numbers, (s1 + 2^256 - 1) / 2 (mod n)
155  * and (s2 + 2^256 - 1) / 2 (mod n) are not, undoing the benefit of the splitting.
156  *
157  * To make it work, we want to modify the input scalar q first, before splitting, and then only
158  * add a 2^128 offset of the split results (so that they end up in the single 129-bit range
159  * [0,2^129]). A slightly smaller offset would work due to the bounds on the split, but we pick
160  * 2^128 for simplicity. Let s be the scalar fed to split_lambda, and f(q) the function to
161  * compute it from q:
162  *
163  * To compute q*A:
164  * - Compute s = f(q)
165  * - Let s1, s2 = split_lambda(s)
166  * - Let v1 = s1 + 2^128 (mod n)
167  * - Let v2 = s2 + 2^128 (mod n)
168  * - Let R1 = C_l(v1, A)
169  * - Let R2 = C_l(v2, lambda*A)
170  * - Return R1 + R2
171  *
172  * l will thus need to be at least 129, but we may overshoot by a few bits (see
173  * further), so keep it as a variable.
174  *
175  * To solve for s, we reason:
176  * q*A = R1 + R2
177  * <=> q*A = C_l(s1 + 2^128, A) + C_l(s2 + 2^128, lambda*A)
178  * <=> q*A = (2*(s1 + 2^128) + 1 - 2^l) * A + (2*(s2 + 2^128) + 1 - 2^l) * lambda*A
179  * <=> q*A = (2*(s1 + s2*lambda) + (2^129 + 1 - 2^l) * (1 + lambda)) * A
180  * <=> q = 2*(s1 + s2*lambda) + (2^129 + 1 - 2^l) * (1 + lambda) (mod n)
181  * <=> q = 2*s + (2^129 + 1 - 2^l) * (1 + lambda) (mod n)
182  * <=> s = (q + (2^l - 2^129 - 1) * (1 + lambda)) / 2 (mod n)
183  * <=> f(q) = (q + K) / 2 (mod n)
184  * where K = (2^l - 2^129 - 1)*(1 + lambda) (mod n)
185  *
186  * We will process the computation of C_l(v1, A) and C_l(v2, lambda*A) in groups of
187  * ECMULT_CONST_GROUP_SIZE, so we set l to the smallest multiple of ECMULT_CONST_GROUP_SIZE
188  * that is not less than 129; this equals ECMULT_CONST_BITS.
189  */
190 
191  /* The offset to add to s1 and s2 to make them non-negative. Equal to 2^128. */
192  static const secp256k1_scalar S_OFFSET = SECP256K1_SCALAR_CONST(0, 0, 0, 1, 0, 0, 0, 0);
193  secp256k1_scalar s, v1, v2;
196  secp256k1_fe global_z;
197  int group, i;
198 
199  /* We're allowed to be non-constant time in the point, and the code below (in particular,
200  * secp256k1_ecmult_const_odd_multiples_table_globalz) cannot deal with infinity in a
201  * constant-time manner anyway. */
202  if (secp256k1_ge_is_infinity(a)) {
204  return;
205  }
206 
207  /* Compute v1 and v2. */
209  secp256k1_scalar_half(&s, &s);
210  secp256k1_scalar_split_lambda(&v1, &v2, &s);
211  secp256k1_scalar_add(&v1, &v1, &S_OFFSET);
212  secp256k1_scalar_add(&v2, &v2, &S_OFFSET);
213 
214 #ifdef VERIFY
215  /* Verify that v1 and v2 are in range [0, 2^129-1]. */
216  for (i = 129; i < 256; ++i) {
217  VERIFY_CHECK(secp256k1_scalar_get_bits(&v1, i, 1) == 0);
218  VERIFY_CHECK(secp256k1_scalar_get_bits(&v2, i, 1) == 0);
219  }
220 #endif
221 
222  /* Calculate odd multiples of A and A*lambda.
223  * All multiples are brought to the same Z 'denominator', which is stored
224  * in global_z. Due to secp256k1' isomorphism we can do all operations pretending
225  * that the Z coordinate was 1, use affine addition formulae, and correct
226  * the Z coordinate of the result once at the end.
227  */
228  secp256k1_gej_set_ge(r, a);
230  for (i = 0; i < ECMULT_CONST_TABLE_SIZE; i++) {
231  secp256k1_ge_mul_lambda(&pre_a_lam[i], &pre_a[i]);
232  }
233 
234  /* Next, we compute r = C_l(v1, A) + C_l(v2, lambda*A).
235  *
236  * We proceed in groups of ECMULT_CONST_GROUP_SIZE bits, operating on that many bits
237  * at a time, from high in v1, v2 to low. Call these bits1 (from v1) and bits2 (from v2).
238  *
239  * Now note that ECMULT_CONST_TABLE_GET_GE(&t, pre_a, bits1) loads into t a point equal
240  * to C_{ECMULT_CONST_GROUP_SIZE}(bits1, A), and analogously for pre_lam_a / bits2.
241  * This means that all we need to do is add these looked up values together, multiplied
242  * by 2^(ECMULT_GROUP_SIZE * group).
243  */
244  for (group = ECMULT_CONST_GROUPS - 1; group >= 0; --group) {
245  /* Using the _var get_bits function is ok here, since it's only variable in offset and count, not in the scalar. */
249  int j;
250 
251  ECMULT_CONST_TABLE_GET_GE(&t, pre_a, bits1);
252  if (group == ECMULT_CONST_GROUPS - 1) {
253  /* Directly set r in the first iteration. */
254  secp256k1_gej_set_ge(r, &t);
255  } else {
256  /* Shift the result so far up. */
257  for (j = 0; j < ECMULT_CONST_GROUP_SIZE; ++j) {
258  secp256k1_gej_double(r, r);
259  }
260  secp256k1_gej_add_ge(r, r, &t);
261  }
262  ECMULT_CONST_TABLE_GET_GE(&t, pre_a_lam, bits2);
263  secp256k1_gej_add_ge(r, r, &t);
264  }
265 
266  /* Map the result back to the secp256k1 curve from the isomorphic curve. */
267  secp256k1_fe_mul(&r->z, &r->z, &global_z);
268 }
269 
270 static int secp256k1_ecmult_const_xonly(secp256k1_fe* r, const secp256k1_fe *n, const secp256k1_fe *d, const secp256k1_scalar *q, int known_on_curve) {
271 
272  /* This algorithm is a generalization of Peter Dettman's technique for
273  * avoiding the square root in a random-basepoint x-only multiplication
274  * on a Weierstrass curve:
275  * https://mailarchive.ietf.org/arch/msg/cfrg/7DyYY6gg32wDgHAhgSb6XxMDlJA/
276  *
277  *
278  * === Background: the effective affine technique ===
279  *
280  * Let phi_u be the isomorphism that maps (x, y) on secp256k1 curve y^2 = x^3 + 7 to
281  * x' = u^2*x, y' = u^3*y on curve y'^2 = x'^3 + u^6*7. This new curve has the same order as
282  * the original (it is isomorphic), but moreover, has the same addition/doubling formulas, as
283  * the curve b=7 coefficient does not appear in those formulas (or at least does not appear in
284  * the formulas implemented in this codebase, both affine and Jacobian). See also Example 9.5.2
285  * in https://www.math.auckland.ac.nz/~sgal018/crypto-book/ch9.pdf.
286  *
287  * This means any linear combination of secp256k1 points can be computed by applying phi_u
288  * (with non-zero u) on all input points (including the generator, if used), computing the
289  * linear combination on the isomorphic curve (using the same group laws), and then applying
290  * phi_u^{-1} to get back to secp256k1.
291  *
292  * Switching to Jacobian coordinates, note that phi_u applied to (X, Y, Z) is simply
293  * (X, Y, Z/u). Thus, if we want to compute (X1, Y1, Z) + (X2, Y2, Z), with identical Z
294  * coordinates, we can use phi_Z to transform it to (X1, Y1, 1) + (X2, Y2, 1) on an isomorphic
295  * curve where the affine addition formula can be used instead.
296  * If (X3, Y3, Z3) = (X1, Y1) + (X2, Y2) on that curve, then our answer on secp256k1 is
297  * (X3, Y3, Z3*Z).
298  *
299  * This is the effective affine technique: if we have a linear combination of group elements
300  * to compute, and all those group elements have the same Z coordinate, we can simply pretend
301  * that all those Z coordinates are 1, perform the computation that way, and then multiply the
302  * original Z coordinate back in.
303  *
304  * The technique works on any a=0 short Weierstrass curve. It is possible to generalize it to
305  * other curves too, but there the isomorphic curves will have different 'a' coefficients,
306  * which typically does affect the group laws.
307  *
308  *
309  * === Avoiding the square root for x-only point multiplication ===
310  *
311  * In this function, we want to compute the X coordinate of q*(n/d, y), for
312  * y = sqrt((n/d)^3 + 7). Its negation would also be a valid Y coordinate, but by convention
313  * we pick whatever sqrt returns (which we assume to be a deterministic function).
314  *
315  * Let g = y^2*d^3 = n^3 + 7*d^3. This also means y = sqrt(g/d^3).
316  * Further let v = sqrt(d*g), which must exist as d*g = y^2*d^4 = (y*d^2)^2.
317  *
318  * The input point (n/d, y) also has Jacobian coordinates:
319  *
320  * (n/d, y, 1)
321  * = (n/d * v^2, y * v^3, v)
322  * = (n/d * d*g, y * sqrt(d^3*g^3), v)
323  * = (n/d * d*g, sqrt(y^2 * d^3*g^3), v)
324  * = (n*g, sqrt(g/d^3 * d^3*g^3), v)
325  * = (n*g, sqrt(g^4), v)
326  * = (n*g, g^2, v)
327  *
328  * It is easy to verify that both (n*g, g^2, v) and its negation (n*g, -g^2, v) have affine X
329  * coordinate n/d, and this holds even when the square root function doesn't have a
330  * deterministic sign. We choose the (n*g, g^2, v) version.
331  *
332  * Now switch to the effective affine curve using phi_v, where the input point has coordinates
333  * (n*g, g^2). Compute (X, Y, Z) = q * (n*g, g^2) there.
334  *
335  * Back on secp256k1, that means q * (n*g, g^2, v) = (X, Y, v*Z). This last point has affine X
336  * coordinate X / (v^2*Z^2) = X / (d*g*Z^2). Determining the affine Y coordinate would involve
337  * a square root, but as long as we only care about the resulting X coordinate, no square root
338  * is needed anywhere in this computation.
339  */
340 
341  secp256k1_fe g, i;
342  secp256k1_ge p;
343  secp256k1_gej rj;
344 
345  /* Compute g = (n^3 + B*d^3). */
346  secp256k1_fe_sqr(&g, n);
347  secp256k1_fe_mul(&g, &g, n);
348  if (d) {
349  secp256k1_fe b;
351  secp256k1_fe_sqr(&b, d);
352  VERIFY_CHECK(SECP256K1_B <= 8); /* magnitude of b will be <= 8 after the next call */
354  secp256k1_fe_mul(&b, &b, d);
355  secp256k1_fe_add(&g, &b);
356  if (!known_on_curve) {
357  /* We need to determine whether (n/d)^3 + 7 is square.
358  *
359  * is_square((n/d)^3 + 7)
360  * <=> is_square(((n/d)^3 + 7) * d^4)
361  * <=> is_square((n^3 + 7*d^3) * d)
362  * <=> is_square(g * d)
363  */
364  secp256k1_fe c;
365  secp256k1_fe_mul(&c, &g, d);
366  if (!secp256k1_fe_is_square_var(&c)) return 0;
367  }
368  } else {
370  if (!known_on_curve) {
371  /* g at this point equals x^3 + 7. Test if it is square. */
372  if (!secp256k1_fe_is_square_var(&g)) return 0;
373  }
374  }
375 
376  /* Compute base point P = (n*g, g^2), the effective affine version of (n*g, g^2, v), which has
377  * corresponding affine X coordinate n/d. */
378  secp256k1_fe_mul(&p.x, &g, n);
379  secp256k1_fe_sqr(&p.y, &g);
380  p.infinity = 0;
381 
382  /* Perform x-only EC multiplication of P with q. */
384  secp256k1_ecmult_const(&rj, &p, q);
386 
387  /* The resulting (X, Y, Z) point on the effective-affine isomorphic curve corresponds to
388  * (X, Y, Z*v) on the secp256k1 curve. The affine version of that has X coordinate
389  * (X / (Z^2*d*g)). */
390  secp256k1_fe_sqr(&i, &rj.z);
391  secp256k1_fe_mul(&i, &i, &g);
392  if (d) secp256k1_fe_mul(&i, &i, d);
393  secp256k1_fe_inv(&i, &i);
394  secp256k1_fe_mul(r, &rj.x, &i);
395 
396  return 1;
397 }
398 
399 #endif /* SECP256K1_ECMULT_CONST_IMPL_H */
static int secp256k1_ecmult_const_xonly(secp256k1_fe *r, const secp256k1_fe *n, const secp256k1_fe *d, const secp256k1_scalar *q, int known_on_curve)
static void secp256k1_ecmult_const_odd_multiples_table_globalz(secp256k1_ge *pre, secp256k1_fe *globalz, const secp256k1_gej *a)
Fill a table 'pre' with precomputed odd multiples of a.
static const secp256k1_scalar secp256k1_ecmult_const_K
#define ECMULT_CONST_GROUP_SIZE
static void secp256k1_ecmult_const(secp256k1_gej *r, const secp256k1_ge *a, const secp256k1_scalar *q)
#define ECMULT_CONST_TABLE_GET_GE(r, pre, n)
#define ECMULT_CONST_TABLE_SIZE
#define ECMULT_CONST_GROUPS
#define ECMULT_CONST_BITS
static void secp256k1_ecmult_odd_multiples_table(int n, secp256k1_ge *pre_a, secp256k1_fe *zr, secp256k1_fe *z, const secp256k1_gej *a)
Fill a table 'pre_a' with precomputed odd multiples of a.
Definition: ecmult_impl.h:73
#define secp256k1_fe_mul_int(r, a)
Multiply a field element with a small integer.
Definition: field.h:238
#define secp256k1_fe_mul
Definition: field.h:94
#define secp256k1_fe_add
Definition: field.h:93
#define secp256k1_fe_is_square_var
Definition: field.h:104
#define secp256k1_fe_normalizes_to_zero
Definition: field.h:81
#define secp256k1_fe_inv
Definition: field.h:99
#define secp256k1_fe_sqr
Definition: field.h:95
#define secp256k1_fe_add_int
Definition: field.h:103
static void secp256k1_ge_mul_lambda(secp256k1_ge *r, const secp256k1_ge *a)
Set r to be equal to lambda times a, where lambda is chosen in a way such that this is very fast.
static void secp256k1_gej_set_infinity(secp256k1_gej *r)
Set a group element (jacobian) equal to the point at infinity.
static int secp256k1_gej_is_infinity(const secp256k1_gej *a)
Check whether a group element is the point at infinity.
static void secp256k1_gej_add_ge(secp256k1_gej *r, const secp256k1_gej *a, const secp256k1_ge *b)
Set r equal to the sum of a and b (with b given in affine coordinates, and not infinity).
static void secp256k1_ge_table_set_globalz(size_t len, secp256k1_ge *a, const secp256k1_fe *zr)
Bring a batch of inputs to the same global z "denominator", based on ratios between (omitted) z coord...
static int secp256k1_ge_is_infinity(const secp256k1_ge *a)
Check whether a group element is the point at infinity.
static void secp256k1_gej_double(secp256k1_gej *r, const secp256k1_gej *a)
Set r equal to the double of a.
static void secp256k1_gej_set_ge(secp256k1_gej *r, const secp256k1_ge *a)
Set a group element (jacobian) equal to another which is given in affine coordinates.
#define SECP256K1_B
Definition: group_impl.h:71
static void secp256k1_scalar_half(secp256k1_scalar *r, const secp256k1_scalar *a)
Multiply a scalar with the multiplicative inverse of 2.
static int secp256k1_scalar_is_zero(const secp256k1_scalar *a)
Check whether a scalar equals zero.
static unsigned int secp256k1_scalar_get_bits(const secp256k1_scalar *a, unsigned int offset, unsigned int count)
Access bits from a scalar.
static int secp256k1_scalar_add(secp256k1_scalar *r, const secp256k1_scalar *a, const secp256k1_scalar *b)
Add two scalars together (modulo the group order).
static void secp256k1_scalar_split_lambda(secp256k1_scalar *SECP256K1_RESTRICT r1, secp256k1_scalar *SECP256K1_RESTRICT r2, const secp256k1_scalar *SECP256K1_RESTRICT k)
Find r1 and r2 such that r1+r2*lambda = k, where r1 and r2 or their negations are maximum 128 bits lo...
static unsigned int secp256k1_scalar_get_bits_var(const secp256k1_scalar *a, unsigned int offset, unsigned int count)
Access bits from a scalar.
#define SECP256K1_SCALAR_CONST(d7, d6, d5, d4, d3, d2, d1, d0)
Definition: scalar_4x64.h:17
#define VERIFY_CHECK(cond)
Definition: util.h:139
This field implementation represents the value as 10 uint32_t limbs in base 2^26.
Definition: field_10x26.h:14
A group element in affine coordinates on the secp256k1 curve, or occasionally on an isomorphic curve ...
Definition: group.h:16
int infinity
Definition: group.h:19
secp256k1_fe x
Definition: group.h:17
secp256k1_fe y
Definition: group.h:18
A group element of the secp256k1 curve, in jacobian coordinates.
Definition: group.h:28
secp256k1_fe x
Definition: group.h:29
secp256k1_fe z
Definition: group.h:31
A scalar modulo the group order of the secp256k1 curve.
Definition: scalar_4x64.h:13
#define EXHAUSTIVE_TEST_ORDER