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Effective affine addition in EC multiplication 

* Make secp256k1_gej_add_var and secp256k1_gej_double return the
  Z ratio to go from a.z to r.z.
* Use these Z ratios to speed up batch point conversion to affine
  coordinates, and to speed up batch conversion of points to a
  common Z coordinate.
* Add a point addition function that takes a point with a known
  Z inverse.
* Due to secp256k1's endomorphism, all additions in the EC
  multiplication code can work on affine coordinate (with an
  implicit common Z coordinate), correcting the Z coordinate of
  the result afterwards.

Refactoring by Pieter Wuille:
* Move more global-z logic into the group code.
* Separate code for computing the odd multiples from the code to bring it
  to either storage or globalz format.
* Rename functions.
* Make all addition operations return Z ratios, and test them.
* Make the zr table format compatible with future batch chaining
  (the first entry in zr becomes the ratio between the input and the
  first output).

Original idea and code by Peter Dettman.
This commit is contained in:
Peter Dettman 2014-11-04 19:16:55 +07:00 committed by Pieter Wuille
parent 22f60a6280
commit 4f9791abba
6 changed files with 339 additions and 95 deletions
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4
src/bench_internal.c
View File

@ -193,7 +193,7 @@ void bench_group_double_var(void* arg) {
bench_inv_t *data = (bench_inv_t*)arg;
for (i = 0; i < 200000; i++) {
secp256k1_gej_double_var(&data->gej_x, &data->gej_x);
secp256k1_gej_double_var(&data->gej_x, &data->gej_x, NULL);
}
}
@ -202,7 +202,7 @@ void bench_group_add_var(void* arg) {
bench_inv_t *data = (bench_inv_t*)arg;
for (i = 0; i < 200000; i++) {
secp256k1_gej_add_var(&data->gej_x, &data->gej_x, &data->gej_y);
secp256k1_gej_add_var(&data->gej_x, &data->gej_x, &data->gej_y, NULL);
}
}

8
src/ecmult_gen_impl.h
View File

@ -54,18 +54,18 @@ static void secp256k1_ecmult_gen_context_build(secp256k1_ecmult_gen_context_t *c
/* Set precj[j*16 .. j*16+15] to (numsbase, numsbase + gbase, ..., numsbase + 15*gbase). */
precj[j*16] = numsbase;
for (i = 1; i < 16; i++) {
secp256k1_gej_add_var(&precj[j*16 + i], &precj[j*16 + i - 1], &gbase);
secp256k1_gej_add_var(&precj[j*16 + i], &precj[j*16 + i - 1], &gbase, NULL);
}
/* Multiply gbase by 16. */
for (i = 0; i < 4; i++) {
secp256k1_gej_double_var(&gbase, &gbase);
secp256k1_gej_double_var(&gbase, &gbase, NULL);
}
/* Multiply numbase by 2. */
secp256k1_gej_double_var(&numsbase, &numsbase);
secp256k1_gej_double_var(&numsbase, &numsbase, NULL);
if (j == 62) {
/* In the last iteration, numsbase is (1 - 2^j) * nums instead. */
secp256k1_gej_neg(&numsbase, &numsbase);
secp256k1_gej_add_var(&numsbase, &numsbase, &nums_gej);
secp256k1_gej_add_var(&numsbase, &numsbase, &nums_gej, NULL);
}
}
secp256k1_ge_set_all_gej_var(1024, prec, precj);

162
src/ecmult_impl.h
View File

@ -24,62 +24,85 @@
#define WINDOW_G 16
#endif
/** Fill a table 'pre' with precomputed odd multiples of a. W determines the size of the table.
* pre will contains the values [1*a,3*a,5*a,...,(2^(w-1)-1)*a], so it needs place for
* 2^(w-2) entries.
*
* There are two versions of this function:
* - secp256k1_ecmult_precomp_wnaf_gej, which operates on group elements in jacobian notation,
* fast to precompute, but slower to use in later additions.
* - secp256k1_ecmult_precomp_wnaf_ge, which operates on group elements in affine notations,
* (much) slower to precompute, but a bit faster to use in later additions.
* To compute a*P + b*G, we use the jacobian version for P, and the affine version for G, as
* G is constant, so it only needs to be done once in advance.
*/
static void secp256k1_ecmult_table_precomp_gej_var(secp256k1_gej_t *pre, const secp256k1_gej_t *a, int w) {
secp256k1_gej_t d;
int i;
pre[0] = *a;
secp256k1_gej_double_var(&d, &pre[0]);
for (i = 1; i < (1 << (w-2)); i++) {
secp256k1_gej_add_var(&pre[i], &d, &pre[i-1]);
}
}
static void secp256k1_ecmult_table_precomp_ge_storage_var(secp256k1_ge_storage_t *pre, const secp256k1_gej_t *a, int w) {
secp256k1_gej_t d;
int i;
const int table_size = 1 << (w-2);
secp256k1_gej_t *prej = (secp256k1_gej_t *)checked_malloc(sizeof(secp256k1_gej_t) * table_size);
secp256k1_ge_t *prea = (secp256k1_ge_t *)checked_malloc(sizeof(secp256k1_ge_t) * table_size);
prej[0] = *a;
secp256k1_gej_double_var(&d, a);
for (i = 1; i < table_size; i++) {
secp256k1_gej_add_var(&prej[i], &d, &prej[i-1]);
}
secp256k1_ge_set_all_gej_var(table_size, prea, prej);
for (i = 0; i < table_size; i++) {
secp256k1_ge_to_storage(&pre[i], &prea[i]);
}
free(prej);
free(prea);
}
/** The number of entries a table with precomputed multiples needs to have. */
#define ECMULT_TABLE_SIZE(w) (1 << ((w)-2))
/** Fill a table 'prej' with precomputed odd multiples of a. Prej will contain
* the values [1*a,3*a,...,(2*n-1)*a], so it space for n values. zr[0] will
* contain prej[0].z / a.z. The other zr[i] values = prej[i].z / prej[i-1].z.
*/
static void secp256k1_ecmult_odd_multiples_table(int n, secp256k1_gej_t *prej, secp256k1_fe_t *zr, const secp256k1_gej_t *a) {
secp256k1_gej_t d;
int i;
VERIFY_CHECK(!a->infinity);
prej[0] = *a;
secp256k1_gej_double_var(&d, &prej[0], NULL);
secp256k1_fe_set_int(zr, 1);
for (i = 1; i < n; i++) {
secp256k1_gej_add_var(&prej[i], &prej[i-1], &d, &zr[i]);
}
}
/** Fill a table 'pre' with precomputed odd multiples of a.
*
* There are two versions of this function:
* - secp256k1_ecmult_odd_multiples_table_globalz_windowa which brings its
* resulting point set to a single constant Z denominator, stores the X and Y
* coordinates as ge_storage points in pre, and stores the global Z in rz.
* It only operates on tables sized for WINDOW_A wnaf multiples.
* - secp256k1_ecmult_odd_multiples_table_storage_var, which converts its
* resulting point set to actually affine points, and stores those in pre.
* It operates on tables of any size, but uses heap-allocated temporaries.
*
* To compute a*P + b*G, we compute a table for P using the first function,
* and for G using the second (which requires an inverse, but it only needs to
* happen once).
*/
static void secp256k1_ecmult_odd_multiples_table_globalz_windowa(secp256k1_ge_t *pre, secp256k1_fe_t *globalz, const secp256k1_gej_t *a) {
secp256k1_gej_t prej[ECMULT_TABLE_SIZE(WINDOW_A)];
secp256k1_fe_t zr[ECMULT_TABLE_SIZE(WINDOW_A)];
/* Compute the odd multiples in Jacobian form. */
secp256k1_ecmult_odd_multiples_table(ECMULT_TABLE_SIZE(WINDOW_A), prej, zr, a);
/* Bring them to the same Z denominator. */
secp256k1_ge_globalz_set_table_gej(ECMULT_TABLE_SIZE(WINDOW_A), pre, globalz, prej, zr);
}
static void secp256k1_ecmult_odd_multiples_table_storage_var(int n, secp256k1_ge_storage_t *pre, const secp256k1_gej_t *a) {
secp256k1_gej_t *prej = checked_malloc(sizeof(secp256k1_gej_t) * n);
secp256k1_ge_t *prea = checked_malloc(sizeof(secp256k1_ge_t) * n);
secp256k1_fe_t *zr = checked_malloc(sizeof(secp256k1_fe_t) * n);
int i;
/* Compute the odd multiples in Jacobian form. */
secp256k1_ecmult_odd_multiples_table(n, prej, zr, a);
/* Convert them in batch to affine coordinates. */
secp256k1_ge_set_table_gej_var(n, prea, prej, zr);
/* Convert them to compact storage form. */
for (i = 0; i < n; i++) {
secp256k1_ge_to_storage(&pre[i], &prea[i]);
}
free(prea);
free(prej);
free(zr);
}
/** The following two macro retrieves a particular odd multiple from a table
* of precomputed multiples. */
#define ECMULT_TABLE_GET_GEJ(r,pre,n,w) do { \
#define ECMULT_TABLE_GET_GE(r,pre,n,w) do { \
VERIFY_CHECK(((n) & 1) == 1); \
VERIFY_CHECK((n) >= -((1 << ((w)-1)) - 1)); \
VERIFY_CHECK((n) <= ((1 << ((w)-1)) - 1)); \
if ((n) > 0) { \
*(r) = (pre)[((n)-1)/2]; \
} else { \
secp256k1_gej_neg((r), &(pre)[(-(n)-1)/2]); \
secp256k1_ge_neg((r), &(pre)[(-(n)-1)/2]); \
} \
} while(0)
#define ECMULT_TABLE_GET_GE_STORAGE(r,pre,n,w) do { \
VERIFY_CHECK(((n) & 1) == 1); \
VERIFY_CHECK((n) >= -((1 << ((w)-1)) - 1)); \
@ -112,7 +135,7 @@ static void secp256k1_ecmult_context_build(secp256k1_ecmult_context_t *ctx) {
ctx->pre_g = (secp256k1_ge_storage_t (*)[])checked_malloc(sizeof((*ctx->pre_g)[0]) * ECMULT_TABLE_SIZE(WINDOW_G));
/* precompute the tables with odd multiples */
secp256k1_ecmult_table_precomp_ge_storage_var(*ctx->pre_g, &gj, WINDOW_G);
secp256k1_ecmult_odd_multiples_table_storage_var(ECMULT_TABLE_SIZE(WINDOW_G), *ctx->pre_g, &gj);
#ifdef USE_ENDOMORPHISM
{
@ -124,9 +147,9 @@ static void secp256k1_ecmult_context_build(secp256k1_ecmult_context_t *ctx) {
/* calculate 2^128*generator */
g_128j = gj;
for (i = 0; i < 128; i++) {
secp256k1_gej_double_var(&g_128j, &g_128j);
secp256k1_gej_double_var(&g_128j, &g_128j, NULL);
}
secp256k1_ecmult_table_precomp_ge_storage_var(*ctx->pre_g_128, &g_128j, WINDOW_G);
secp256k1_ecmult_odd_multiples_table_storage_var(ECMULT_TABLE_SIZE(WINDOW_G), *ctx->pre_g_128, &g_128j);
}
#endif
}
@ -208,11 +231,11 @@ static int secp256k1_ecmult_wnaf(int *wnaf, const secp256k1_scalar_t *a, int w)
}
static void secp256k1_ecmult(const secp256k1_ecmult_context_t *ctx, secp256k1_gej_t *r, const secp256k1_gej_t *a, const secp256k1_scalar_t *na, const secp256k1_scalar_t *ng) {
secp256k1_gej_t tmpj;
secp256k1_gej_t pre_a[ECMULT_TABLE_SIZE(WINDOW_A)];
secp256k1_ge_t pre_a[ECMULT_TABLE_SIZE(WINDOW_A)];
secp256k1_ge_t tmpa;
secp256k1_fe_t Z;
#ifdef USE_ENDOMORPHISM
secp256k1_gej_t pre_a_lam[ECMULT_TABLE_SIZE(WINDOW_A)];
secp256k1_ge_t pre_a_lam[ECMULT_TABLE_SIZE(WINDOW_A)];
secp256k1_scalar_t na_1, na_lam;
/* Splitted G factors. */
secp256k1_scalar_t ng_1, ng_128;
@ -252,12 +275,21 @@ static void secp256k1_ecmult(const secp256k1_ecmult_context_t *ctx, secp256k1_ge
bits = bits_na;
#endif
/* calculate odd multiples of a */
secp256k1_ecmult_table_precomp_gej_var(pre_a, a, WINDOW_A);
/* Calculate odd multiples of a.
* All multiples are brought to the same Z 'denominator', which is stored
* in Z. Due to secp256k1' isomorphism we can do all operations pretending
* that the Z coordinate was 1, use affine addition formulae, and correct
* the Z coordinate of the result once at the end.
* The exception is the precomputed G table points, which are actually
* affine. Compared to the base used for other points, they have a Z ratio
* of 1/Z, so we can use secp256k1_gej_add_zinv_var, which uses the same
* isomorphism to efficiently add with a known Z inverse.
*/
secp256k1_ecmult_odd_multiples_table_globalz_windowa(pre_a, &Z, a);
#ifdef USE_ENDOMORPHISM
for (i = 0; i < ECMULT_TABLE_SIZE(WINDOW_A); i++) {
secp256k1_gej_mul_lambda(&pre_a_lam[i], &pre_a[i]);
secp256k1_ge_mul_lambda(&pre_a_lam[i], &pre_a[i]);
}
/* split ng into ng_1 and ng_128 (where gn = gn_1 + gn_128*2^128, and gn_1 and gn_128 are ~128 bit) */
@ -281,37 +313,41 @@ static void secp256k1_ecmult(const secp256k1_ecmult_context_t *ctx, secp256k1_ge
secp256k1_gej_set_infinity(r);
for (i = bits-1; i >= 0; i--) {
for (i = bits - 1; i >= 0; i--) {
int n;
secp256k1_gej_double_var(r, r);
secp256k1_gej_double_var(r, r, NULL);
#ifdef USE_ENDOMORPHISM
if (i < bits_na_1 && (n = wnaf_na_1[i])) {
ECMULT_TABLE_GET_GEJ(&tmpj, pre_a, n, WINDOW_A);
secp256k1_gej_add_var(r, r, &tmpj);
ECMULT_TABLE_GET_GE(&tmpa, pre_a, n, WINDOW_A);
secp256k1_gej_add_ge_var(r, r, &tmpa);
}
if (i < bits_na_lam && (n = wnaf_na_lam[i])) {
ECMULT_TABLE_GET_GEJ(&tmpj, pre_a_lam, n, WINDOW_A);
secp256k1_gej_add_var(r, r, &tmpj);
ECMULT_TABLE_GET_GE(&tmpa, pre_a_lam, n, WINDOW_A);
secp256k1_gej_add_ge_var(r, r, &tmpa);
}
if (i < bits_ng_1 && (n = wnaf_ng_1[i])) {
ECMULT_TABLE_GET_GE_STORAGE(&tmpa, *ctx->pre_g, n, WINDOW_G);
secp256k1_gej_add_ge_var(r, r, &tmpa);
secp256k1_gej_add_zinv_var(r, r, &tmpa, &Z);
}
if (i < bits_ng_128 && (n = wnaf_ng_128[i])) {
ECMULT_TABLE_GET_GE_STORAGE(&tmpa, *ctx->pre_g_128, n, WINDOW_G);
secp256k1_gej_add_ge_var(r, r, &tmpa);
secp256k1_gej_add_zinv_var(r, r, &tmpa, &Z);
}
#else
if (i < bits_na && (n = wnaf_na[i])) {
ECMULT_TABLE_GET_GEJ(&tmpj, pre_a, n, WINDOW_A);
secp256k1_gej_add_var(r, r, &tmpj);
ECMULT_TABLE_GET_GE(&tmpa, pre_a, n, WINDOW_A);
secp256k1_gej_add_ge_var(r, r, &tmpa);
}
if (i < bits_ng && (n = wnaf_ng[i])) {
ECMULT_TABLE_GET_GE_STORAGE(&tmpa, *ctx->pre_g, n, WINDOW_G);
secp256k1_gej_add_ge_var(r, r, &tmpa);
secp256k1_gej_add_zinv_var(r, r, &tmpa, &Z);
}
#endif
}
if (!r->infinity) {
secp256k1_fe_mul(&r->z, &r->z, &Z);
}
}
#endif

24
src/group.h
View File

@ -62,6 +62,17 @@ static void secp256k1_ge_set_gej(secp256k1_ge_t *r, secp256k1_gej_t *a);
/** Set a batch of group elements equal to the inputs given in jacobian coordinates */
static void secp256k1_ge_set_all_gej_var(size_t len, secp256k1_ge_t *r, const secp256k1_gej_t *a);
/** Set a batch of group elements equal to the inputs given in jacobian
* coordinates (with known z-ratios). zr must contain the known z-ratios such
* that mul(a[i].z, zr[i+1]) == a[i+1].z. zr[0] is ignored. */
static void secp256k1_ge_set_table_gej_var(size_t len, secp256k1_ge_t *r, const secp256k1_gej_t *a, const secp256k1_fe_t *zr);
/** Bring a batch inputs given in jacobian coordinates (with known z-ratios) to
* the same global z "denominator". zr must contain the known z-ratios such
* that mul(a[i].z, zr[i+1]) == a[i+1].z. zr[0] is ignored. The x and y
* coordinates of the result are stored in r, the common z coordinate is
* stored in globalz. */
static void secp256k1_ge_globalz_set_table_gej(size_t len, secp256k1_ge_t *r, secp256k1_fe_t *globalz, const secp256k1_gej_t *a, const secp256k1_fe_t *zr);
/** Set a group element (jacobian) equal to the point at infinity. */
static void secp256k1_gej_set_infinity(secp256k1_gej_t *r);
@ -81,11 +92,11 @@ static void secp256k1_gej_neg(secp256k1_gej_t *r, const secp256k1_gej_t *a);
/** Check whether a group element is the point at infinity. */
static int secp256k1_gej_is_infinity(const secp256k1_gej_t *a);
/** Set r equal to the double of a. */
static void secp256k1_gej_double_var(secp256k1_gej_t *r, const secp256k1_gej_t *a);
/** Set r equal to the double of a. If rzr is not-NULL, r->z = a->z * *rzr (where infinity means an implicit z = 0). */
static void secp256k1_gej_double_var(secp256k1_gej_t *r, const secp256k1_gej_t *a, secp256k1_fe_t *rzr);
/** Set r equal to the sum of a and b. */
static void secp256k1_gej_add_var(secp256k1_gej_t *r, const secp256k1_gej_t *a, const secp256k1_gej_t *b);
/** Set r equal to the sum of a and b. If rzr is non-NULL, r->z = a->z * *rzr (a cannot be infinity in that case). */
static void secp256k1_gej_add_var(secp256k1_gej_t *r, const secp256k1_gej_t *a, const secp256k1_gej_t *b, secp256k1_fe_t *rzr);
/** Set r equal to the sum of a and b (with b given in affine coordinates, and not infinity). */
static void secp256k1_gej_add_ge(secp256k1_gej_t *r, const secp256k1_gej_t *a, const secp256k1_ge_t *b);
@ -95,9 +106,12 @@ static void secp256k1_gej_add_ge(secp256k1_gej_t *r, const secp256k1_gej_t *a, c
guarantee, and b is allowed to be infinity. */
static void secp256k1_gej_add_ge_var(secp256k1_gej_t *r, const secp256k1_gej_t *a, const secp256k1_ge_t *b);
/** Set r equal to the sum of a and b (with the inverse of b's Z coordinate passed as bzinv). */
static void secp256k1_gej_add_zinv_var(secp256k1_gej_t *r, const secp256k1_gej_t *a, const secp256k1_ge_t *b, const secp256k1_fe_t *bzinv);
#ifdef USE_ENDOMORPHISM
/** 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_mul_lambda(secp256k1_gej_t *r, const secp256k1_gej_t *a);
static void secp256k1_ge_mul_lambda(secp256k1_ge_t *r, const secp256k1_ge_t *a);
#endif
/** Clear a secp256k1_gej_t to prevent leaking sensitive information. */

153
src/group_impl.h
View File

@ -23,6 +23,16 @@ static const secp256k1_ge_t secp256k1_ge_const_g = SECP256K1_GE_CONST(
0xFD17B448UL, 0xA6855419UL, 0x9C47D08FUL, 0xFB10D4B8UL
);
static void secp256k1_ge_set_gej_zinv(secp256k1_ge_t *r, const secp256k1_gej_t *a, const secp256k1_fe_t *zi) {
secp256k1_fe_t zi2;
secp256k1_fe_t 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_infinity(secp256k1_ge_t *r) {
r->infinity = 1;
}
@ -92,17 +102,55 @@ static void secp256k1_ge_set_all_gej_var(size_t len, secp256k1_ge_t *r, const se
for (i = 0; i < len; i++) {
r[i].infinity = a[i].infinity;
if (!a[i].infinity) {
secp256k1_fe_t zi2, zi3;
secp256k1_fe_t *zi = &azi[count++];
secp256k1_fe_sqr(&zi2, zi);
secp256k1_fe_mul(&zi3, &zi2, zi);
secp256k1_fe_mul(&r[i].x, &a[i].x, &zi2);
secp256k1_fe_mul(&r[i].y, &a[i].y, &zi3);
secp256k1_ge_set_gej_zinv(&r[i], &a[i], &azi[count++]);
}
}
free(azi);
}
static void secp256k1_ge_set_table_gej_var(size_t len, secp256k1_ge_t *r, const secp256k1_gej_t *a, const secp256k1_fe_t *zr) {
size_t i = len - 1;
secp256k1_fe_t zi;
if (len < 1)
return;
/* Compute the inverse of the last z coordinate, and use it to compute the last affine output. */
secp256k1_fe_inv(&zi, &a[i].z);
secp256k1_ge_set_gej_zinv(&r[i], &a[i], &zi);
/* Work out way backwards, using the z-ratios to scale the x/y values. */
while (i > 0) {
secp256k1_fe_mul(&zi, &zi, &zr[i]);
i--;
secp256k1_ge_set_gej_zinv(&r[i], &a[i], &zi);
}
}
static void secp256k1_ge_globalz_set_table_gej(size_t len, secp256k1_ge_t *r, secp256k1_fe_t *globalz, const secp256k1_gej_t *a, const secp256k1_fe_t *zr) {
size_t i = len - 1;
secp256k1_fe_t zs;
if (len < 1)
return;
/* 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;
*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_t *r) {
r->infinity = 1;
secp256k1_fe_set_int(&r->x, 0);
@ -210,7 +258,7 @@ static int secp256k1_ge_is_valid_var(const secp256k1_ge_t *a) {
return secp256k1_fe_equal_var(&y2, &x3);
}
static void secp256k1_gej_double_var(secp256k1_gej_t *r, const secp256k1_gej_t *a) {
static void secp256k1_gej_double_var(secp256k1_gej_t *r, const secp256k1_gej_t *a, secp256k1_fe_t *rzr) {
/* Operations: 3 mul, 4 sqr, 0 normalize, 12 mul_int/add/negate */
secp256k1_fe_t t1,t2,t3,t4;
/** For secp256k1, 2Q is infinity if and only if Q is infinity. This is because if 2Q = infinity,
@ -219,9 +267,18 @@ static void secp256k1_gej_double_var(secp256k1_gej_t *r, const secp256k1_gej_t *
*/
r->infinity = a->infinity;
if (r->infinity) {
if (rzr) {
secp256k1_fe_set_int(rzr, 1);
}
return;
}
if (rzr) {
*rzr = a->y;
secp256k1_fe_normalize_weak(rzr);
secp256k1_fe_mul_int(rzr, 2);
}
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);
@ -244,17 +301,24 @@ static void secp256k1_gej_double_var(secp256k1_gej_t *r, const secp256k1_gej_t *
secp256k1_fe_add(&r->y, &t2); /* Y' = 36*X^3*Y^2 - 27*X^6 - 8*Y^4 (4) */
}
static void secp256k1_gej_add_var(secp256k1_gej_t *r, const secp256k1_gej_t *a, const secp256k1_gej_t *b) {
static void secp256k1_gej_add_var(secp256k1_gej_t *r, const secp256k1_gej_t *a, const secp256k1_gej_t *b, secp256k1_fe_t *rzr) {
/* Operations: 12 mul, 4 sqr, 2 normalize, 12 mul_int/add/negate */
secp256k1_fe_t 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) {
secp256k1_fe_set_int(rzr, 1);
}
*r = *a;
return;
}
r->infinity = 0;
secp256k1_fe_sqr(&z22, &b->z);
secp256k1_fe_sqr(&z12, &a->z);
@ -266,8 +330,11 @@ static void secp256k1_gej_add_var(secp256k1_gej_t *r, const secp256k1_gej_t *a,
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);
secp256k1_gej_double_var(r, a, rzr);
} else {
if (rzr) {
secp256k1_fe_set_int(rzr, 0);
}
r->infinity = 1;
}
return;
@ -275,7 +342,11 @@ static void secp256k1_gej_add_var(secp256k1_gej_t *r, const secp256k1_gej_t *a,
secp256k1_fe_sqr(&i2, &i);
secp256k1_fe_sqr(&h2, &h);
secp256k1_fe_mul(&h3, &h, &h2);
secp256k1_fe_mul(&r->z, &a->z, &b->z); secp256k1_fe_mul(&r->z, &r->z, &h);
secp256k1_fe_mul(&h, &h, &b->z);
if (rzr) {
*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);
@ -298,6 +369,7 @@ static void secp256k1_gej_add_ge_var(secp256k1_gej_t *r, const secp256k1_gej_t *
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);
@ -307,7 +379,7 @@ static void secp256k1_gej_add_ge_var(secp256k1_gej_t *r, const secp256k1_gej_t *
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);
secp256k1_gej_double_var(r, a, NULL);
} else {
r->infinity = 1;
}
@ -324,6 +396,63 @@ static void secp256k1_gej_add_ge_var(secp256k1_gej_t *r, const secp256k1_gej_t *
secp256k1_fe_add(&r->y, &h3);
}
static void secp256k1_gej_add_zinv_var(secp256k1_gej_t *r, const secp256k1_gej_t *a, const secp256k1_ge_t *b, const secp256k1_fe_t *bzinv) {
/* 9 mul, 3 sqr, 4 normalize, 12 mul_int/add/negate */
secp256k1_fe_t az, z12, u1, u2, s1, s2, h, i, i2, h2, h3, t;
if (b->infinity) {
*r = *a;
return;
}
if (a->infinity) {
secp256k1_fe_t 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 {
r->infinity = 1;
}
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_t *r, const secp256k1_gej_t *a, const secp256k1_ge_t *b) {
/* Operations: 7 mul, 5 sqr, 5 normalize, 17 mul_int/add/negate/cmov */
static const secp256k1_fe_t fe_1 = SECP256K1_FE_CONST(0, 0, 0, 0, 0, 0, 0, 1);
@ -430,7 +559,7 @@ static SECP256K1_INLINE void secp256k1_ge_storage_cmov(secp256k1_ge_storage_t *r
}
#ifdef USE_ENDOMORPHISM
static void secp256k1_gej_mul_lambda(secp256k1_gej_t *r, const secp256k1_gej_t *a) {
static void secp256k1_ge_mul_lambda(secp256k1_ge_t *r, const secp256k1_ge_t *a) {
static const secp256k1_fe_t beta = SECP256K1_FE_CONST(
0x7ae96a2bul, 0x657c0710ul, 0x6e64479eul, 0xac3434e9ul,
0x9cf04975ul, 0x12f58995ul, 0xc1396c28ul, 0x719501eeul

83
src/tests.c
View File

@ -966,6 +966,10 @@ void test_ge(void) {
*/
secp256k1_ge_t *ge = (secp256k1_ge_t *)malloc(sizeof(secp256k1_ge_t) * (1 + 4 * runs));
secp256k1_gej_t *gej = (secp256k1_gej_t *)malloc(sizeof(secp256k1_gej_t) * (1 + 4 * runs));
secp256k1_fe_t *zinv = (secp256k1_fe_t *)malloc(sizeof(secp256k1_fe_t) * (1 + 4 * runs));
secp256k1_fe_t zf;
secp256k1_fe_t zfi2, zfi3;
secp256k1_gej_set_infinity(&gej[0]);
secp256k1_ge_clear(&ge[0]);
secp256k1_ge_set_gej_var(&ge[0], &gej[0]);
@ -990,19 +994,62 @@ void test_ge(void) {
}
}
/* Compute z inverses. */
{
secp256k1_fe_t *zs = malloc(sizeof(secp256k1_fe_t) * (1 + 4 * runs));
for (i = 0; i < 4 * runs + 1; i++) {
if (i == 0) {
/* The point at infinity does not have a meaningful z inverse. Any should do. */
do {
random_field_element_test(&zs[i]);
} while(secp256k1_fe_is_zero(&zs[i]));
} else {
zs[i] = gej[i].z;
}
}
secp256k1_fe_inv_all_var(4 * runs + 1, zinv, zs);
free(zs);
}
/* Generate random zf, and zfi2 = 1/zf^2, zfi3 = 1/zf^3 */
do {
random_field_element_test(&zf);
} while(secp256k1_fe_is_zero(&zf));
random_field_element_magnitude(&zf);
secp256k1_fe_inv_var(&zfi3, &zf);
secp256k1_fe_sqr(&zfi2, &zfi3);
secp256k1_fe_mul(&zfi3, &zfi3, &zfi2);
for (i1 = 0; i1 < 1 + 4 * runs; i1++) {
int i2;
for (i2 = 0; i2 < 1 + 4 * runs; i2++) {
/* Compute reference result using gej + gej (var). */
secp256k1_gej_t refj, resj;
secp256k1_ge_t ref;
secp256k1_gej_add_var(&refj, &gej[i1], &gej[i2]);
secp256k1_fe_t zr;
secp256k1_gej_add_var(&refj, &gej[i1], &gej[i2], secp256k1_gej_is_infinity(&gej[i1]) ? NULL : &zr);
/* Check Z ratio. */
if (!secp256k1_gej_is_infinity(&gej[i1]) && !secp256k1_gej_is_infinity(&refj)) {
secp256k1_fe_t zrz; secp256k1_fe_mul(&zrz, &zr, &gej[i1].z);
CHECK(secp256k1_fe_equal_var(&zrz, &refj.z));
}
secp256k1_ge_set_gej_var(&ref, &refj);
/* Test gej + ge (var). */
secp256k1_gej_add_ge_var(&resj, &gej[i1], &ge[i2]);
ge_equals_gej(&ref, &resj);
/* Test gej + ge (var, with additional Z factor). */
{
secp256k1_ge_t ge2_zfi = ge[i2]; /* the second term with x and y rescaled for z = 1/zf */
secp256k1_fe_mul(&ge2_zfi.x, &ge2_zfi.x, &zfi2);
secp256k1_fe_mul(&ge2_zfi.y, &ge2_zfi.y, &zfi3);
random_field_element_magnitude(&ge2_zfi.x);
random_field_element_magnitude(&ge2_zfi.y);
secp256k1_gej_add_zinv_var(&resj, &gej[i1], &ge2_zfi, &zf);
ge_equals_gej(&ref, &resj);
}
/* Test gej + ge (const). */
if (i2 != 0) {
/* secp256k1_gej_add_ge does not support its second argument being infinity. */
@ -1012,10 +1059,15 @@ void test_ge(void) {
/* Test doubling (var). */
if ((i1 == 0 && i2 == 0) || ((i1 + 3)/4 == (i2 + 3)/4 && ((i1 + 3)%4)/2 == ((i2 + 3)%4)/2)) {
/* Normal doubling. */
secp256k1_gej_double_var(&resj, &gej[i1]);
secp256k1_fe_t zr2;
/* Normal doubling with Z ratio result. */
secp256k1_gej_double_var(&resj, &gej[i1], &zr2);
ge_equals_gej(&ref, &resj);
secp256k1_gej_double_var(&resj, &gej[i2]);
/* Check Z ratio. */
secp256k1_fe_mul(&zr2, &zr2, &gej[i1].z);
CHECK(secp256k1_fe_equal_var(&zr2, &resj.z));
/* Normal doubling. */
secp256k1_gej_double_var(&resj, &gej[i2], NULL);
ge_equals_gej(&ref, &resj);
}
@ -1054,27 +1106,40 @@ void test_ge(void) {
}
}
for (i = 0; i < 4 * runs + 1; i++) {
secp256k1_gej_add_var(&sum, &sum, &gej_shuffled[i]);
secp256k1_gej_add_var(&sum, &sum, &gej_shuffled[i], NULL);
}
CHECK(secp256k1_gej_is_infinity(&sum));
free(gej_shuffled);
}
/* Test batch gej -> ge conversion. */
/* Test batch gej -> ge conversion with and without known z ratios. */
{
secp256k1_fe_t *zr = (secp256k1_fe_t *)malloc((4 * runs + 1) * sizeof(secp256k1_fe_t));
secp256k1_ge_t *ge_set_table = (secp256k1_ge_t *)malloc((4 * runs + 1) * sizeof(secp256k1_ge_t));
secp256k1_ge_t *ge_set_all = (secp256k1_ge_t *)malloc((4 * runs + 1) * sizeof(secp256k1_ge_t));
for (i = 0; i < 4 * runs + 1; i++) {
/* Compute gej[i + 1].z / gez[i].z (with gej[n].z taken to be 1). */
if (i < 4 * runs) {
secp256k1_fe_mul(&zr[i + 1], &zinv[i], &gej[i + 1].z);
}
}
secp256k1_ge_set_table_gej_var(4 * runs + 1, ge_set_table, gej, zr);
secp256k1_ge_set_all_gej_var(4 * runs + 1, ge_set_all, gej);
for (i = 0; i < 4 * runs + 1; i++) {
secp256k1_fe_t s;
random_fe_non_zero(&s);
secp256k1_gej_rescale(&gej[i], &s);
ge_equals_gej(&ge_set_table[i], &gej[i]);
ge_equals_gej(&ge_set_all[i], &gej[i]);
}
free(ge_set_table);
free(ge_set_all);
free(zr);
}
free(ge);
free(gej);
free(zinv);
}
void run_ge(void) {
@ -1139,14 +1204,14 @@ void run_ecmult_chain(void) {
);
secp256k1_gej_neg(&rp, &rp);
secp256k1_gej_add_var(&rp, &rp, &x);
secp256k1_gej_add_var(&rp, &rp, &x, NULL);
CHECK(secp256k1_gej_is_infinity(&rp));
}
}
/* redo the computation, but directly with the resulting ae and ge coefficients: */
secp256k1_ecmult(&ctx->ecmult_ctx, &x2, &a, &ae, &ge);
secp256k1_gej_neg(&x2, &x2);
secp256k1_gej_add_var(&x2, &x2, &x);
secp256k1_gej_add_var(&x2, &x2, &x, NULL);
CHECK(secp256k1_gej_is_infinity(&x2));
}
@ -1162,7 +1227,7 @@ void test_point_times_order(const secp256k1_gej_t *point) {
secp256k1_scalar_negate(&nx, &x);
secp256k1_ecmult(&ctx->ecmult_ctx, &res1, point, &x, &x); /* calc res1 = x * point + x * G; */
secp256k1_ecmult(&ctx->ecmult_ctx, &res2, point, &nx, &nx); /* calc res2 = (order - x) * point + (order - x) * G; */
secp256k1_gej_add_var(&res1, &res1, &res2);
secp256k1_gej_add_var(&res1, &res1, &res2, NULL);
CHECK(secp256k1_gej_is_infinity(&res1));
CHECK(secp256k1_gej_is_valid_var(&res1) == 0);
secp256k1_ge_set_gej(&res3, &res1);