From 47045270fa90f81205d989f7107769bce1e71c4d Mon Sep 17 00:00:00 2001 From: Andrew Poelstra Date: Thu, 20 Sep 2018 23:34:02 +0000 Subject: [PATCH] ecmult_impl: eliminate scratch memory used when generating context --- src/ecmult_impl.h | 115 +++++++++++++++++++++++++++++++++++++++------- src/group.h | 5 -- src/group_impl.h | 18 -------- src/tests.c | 4 -- 4 files changed, 99 insertions(+), 43 deletions(-) diff --git a/src/ecmult_impl.h b/src/ecmult_impl.h index d5fb6c5b61dd2..74c350fcde902 100644 --- a/src/ecmult_impl.h +++ b/src/ecmult_impl.h @@ -137,24 +137,107 @@ static void secp256k1_ecmult_odd_multiples_table_globalz_windowa(secp256k1_ge *p 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 *pre, const secp256k1_gej *a, const secp256k1_callback *cb) { - secp256k1_gej *prej = (secp256k1_gej*)checked_malloc(cb, sizeof(secp256k1_gej) * n); - secp256k1_ge *prea = (secp256k1_ge*)checked_malloc(cb, sizeof(secp256k1_ge) * n); - secp256k1_fe *zr = (secp256k1_fe*)checked_malloc(cb, sizeof(secp256k1_fe) * n); +static void secp256k1_ecmult_odd_multiples_table_storage_var(const int n, secp256k1_ge_storage *pre, const secp256k1_gej *a) { + secp256k1_gej d; + secp256k1_ge a_ge, d_ge, p_ge; + secp256k1_ge last_ge; + secp256k1_gej pj; + secp256k1_fe zi; + secp256k1_fe zr; + secp256k1_fe dx_over_dz_squared; 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(prea, prej, zr, n); - /* Convert them to compact storage form. */ - for (i = 0; i < n; i++) { - secp256k1_ge_to_storage(&pre[i], &prea[i]); + VERIFY_CHECK(!a->infinity); + + secp256k1_gej_double_var(&d, a, NULL); + + /* First, we perform all the additions in an isomorphic curve obtained by multiplying + * all `z` coordinates by 1/`d.z`. In these coordinates `d` is affine so we can use + * `secp256k1_gej_add_ge_var` to perform the additions. For each addition, we store + * the resulting y-coordinate and the z-ratio, since we only have enough memory to + * store two field elements. These are sufficient to efficiently undo the isomorphism + * and recompute all the `x`s. + */ + d_ge.x = d.x; + d_ge.y = d.y; + d_ge.infinity = 0; + + secp256k1_ge_set_gej_zinv(&a_ge, a, &d.z); + pj.x = a_ge.x; + pj.y = a_ge.y; + pj.z = a->z; + pj.infinity = 0; + + zr = d.z; + secp256k1_fe_normalize_var(&zr); + secp256k1_fe_to_storage(&pre[0].x, &zr); + secp256k1_fe_normalize_var(&pj.y); + secp256k1_fe_to_storage(&pre[0].y, &pj.y); + + for (i = 1; i < n; i++) { + secp256k1_gej_add_ge_var(&pj, &pj, &d_ge, &zr); + secp256k1_fe_normalize_var(&zr); + secp256k1_fe_to_storage(&pre[i].x, &zr); + secp256k1_fe_normalize_var(&pj.y); + secp256k1_fe_to_storage(&pre[i].y, &pj.y); } - free(prea); - free(prej); - free(zr); + /* Map `pj` back to our curve by multiplying its z-coordinate by `d.z`. */ + secp256k1_fe_mul(&pj.z, &pj.z, &d.z); + /* Directly set `pre[n - 1]` to `pj`, saving the inverted z-coordinate so + * that we can combine it with the saved z-ratios to compute the other zs + * without any more inversions. */ + secp256k1_fe_inv_var(&zi, &pj.z); + secp256k1_ge_set_gej_zinv(&p_ge, &pj, &zi); + secp256k1_ge_from_storage(&last_ge, &pre[n - 1]); + secp256k1_ge_to_storage(&pre[n - 1], &p_ge); + + /* Compute the actual x-coordinate of D, which will be needed below. */ + secp256k1_fe_inv_var(&d.z, &d.z); + secp256k1_fe_sqr(&dx_over_dz_squared, &d.z); + secp256k1_fe_mul(&dx_over_dz_squared, &dx_over_dz_squared, &d.x); + + i = n - 1; + while (i > 0) { + secp256k1_fe zi2, zi3; + i--; + /* For the remaining points, we extract the z-ratio from the stored + * x-coordinate, compute its z^-1 from that, and compute the full + * point from that. The z-ratio for the next iteration is stored in + * the x-coordinate at the end of the loop. */ + secp256k1_fe_mul(&zi, &zi, &last_ge.x); + secp256k1_fe_sqr(&zi2, &zi); + secp256k1_fe_mul(&zi3, &zi2, &zi); + /* To compute the actual x-coordinate, we use the stored z ratio and + * y-coordinate, which we obtained from `secp256k1_gej_add_ge_var` + * in the loop above, as well as the inverse of the square of its + * z-coordinate. We store the latter in the `zi2` variable, which is + * computed iteratively starting from the overall Z inverse then + * multiplying by each z-ratio in turn. + * + * Denoting the z-ratio as `rzr` (though the actual variable binding + * is `last_ge.x`), we observe that it equal to `h` from the inside + * of the above `gej_add_ge_var` call. This satisfies + * + * rzr = d_x * z^2 - x + * + * where `d_x` is the x coordinate of `D` and `(x, z)` are Jacobian + * coordinates of our desired point. + * + * Rearranging and dividing by `z^2` to convert to affine, we get + * + * x = d_x - rzr / z^2 + * = d_x - rzr * zi2 + */ + secp256k1_fe_mul(&p_ge.x, &last_ge.x, &zi2); + secp256k1_fe_negate(&p_ge.x, &p_ge.x, 1); + secp256k1_fe_add(&p_ge.x, &dx_over_dz_squared); + /* y is stored_y/z^3, as we expect */ + secp256k1_ge_from_storage(&last_ge, &pre[i]); + secp256k1_fe_mul(&p_ge.y, &last_ge.y, &zi3); + /* Store */ + secp256k1_ge_to_storage(&pre[i], &p_ge); + } } /** The following two macro retrieves a particular odd multiple from a table @@ -202,7 +285,7 @@ static void secp256k1_ecmult_context_build(secp256k1_ecmult_context *ctx, const ctx->pre_g = (secp256k1_ge_storage (*)[])checked_malloc(cb, sizeof((*ctx->pre_g)[0]) * ECMULT_TABLE_SIZE(WINDOW_G)); /* precompute the tables with odd multiples */ - secp256k1_ecmult_odd_multiples_table_storage_var(ECMULT_TABLE_SIZE(WINDOW_G), *ctx->pre_g, &gj, cb); + secp256k1_ecmult_odd_multiples_table_storage_var(ECMULT_TABLE_SIZE(WINDOW_G), *ctx->pre_g, &gj); #ifdef USE_ENDOMORPHISM { @@ -216,7 +299,7 @@ static void secp256k1_ecmult_context_build(secp256k1_ecmult_context *ctx, const for (i = 0; i < 128; i++) { secp256k1_gej_double_var(&g_128j, &g_128j, NULL); } - secp256k1_ecmult_odd_multiples_table_storage_var(ECMULT_TABLE_SIZE(WINDOW_G), *ctx->pre_g_128, &g_128j, cb); + secp256k1_ecmult_odd_multiples_table_storage_var(ECMULT_TABLE_SIZE(WINDOW_G), *ctx->pre_g_128, &g_128j); } #endif } diff --git a/src/group.h b/src/group.h index 0911df2cb51e6..8e122ab429c56 100644 --- a/src/group.h +++ b/src/group.h @@ -67,11 +67,6 @@ static void secp256k1_ge_set_gej(secp256k1_ge *r, secp256k1_gej *a); /** Set a batch of group elements equal to the inputs given in jacobian coordinates */ static void secp256k1_ge_set_all_gej_var(secp256k1_ge *r, const secp256k1_gej *a, size_t len); -/** 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(secp256k1_ge *r, const secp256k1_gej *a, const secp256k1_fe *zr, size_t len); - /** 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 diff --git a/src/group_impl.h b/src/group_impl.h index 006a4548876a5..5caf421b5e182 100644 --- a/src/group_impl.h +++ b/src/group_impl.h @@ -167,24 +167,6 @@ static void secp256k1_ge_set_all_gej_var(secp256k1_ge *r, const secp256k1_gej *a } } -static void secp256k1_ge_set_table_gej_var(secp256k1_ge *r, const secp256k1_gej *a, const secp256k1_fe *zr, size_t len) { - size_t i = len - 1; - secp256k1_fe zi; - - if (len > 0) { - /* 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 *r, secp256k1_fe *globalz, const secp256k1_gej *a, const secp256k1_fe *zr) { size_t i = len - 1; secp256k1_fe zs; diff --git a/src/tests.c b/src/tests.c index 589cf85e1844f..3414a0f4cee3c 100644 --- a/src/tests.c +++ b/src/tests.c @@ -2095,7 +2095,6 @@ void test_ge(void) { /* Test batch gej -> ge conversion with and without known z ratios. */ { secp256k1_fe *zr = (secp256k1_fe *)checked_malloc(&ctx->error_callback, (4 * runs + 1) * sizeof(secp256k1_fe)); - secp256k1_ge *ge_set_table = (secp256k1_ge *)checked_malloc(&ctx->error_callback, (4 * runs + 1) * sizeof(secp256k1_ge)); secp256k1_ge *ge_set_all = (secp256k1_ge *)checked_malloc(&ctx->error_callback, (4 * runs + 1) * sizeof(secp256k1_ge)); for (i = 0; i < 4 * runs + 1; i++) { /* Compute gej[i + 1].z / gez[i].z (with gej[n].z taken to be 1). */ @@ -2103,16 +2102,13 @@ void test_ge(void) { secp256k1_fe_mul(&zr[i + 1], &zinv[i], &gej[i + 1].z); } } - secp256k1_ge_set_table_gej_var(ge_set_table, gej, zr, 4 * runs + 1); secp256k1_ge_set_all_gej_var(ge_set_all, gej, 4 * runs + 1); for (i = 0; i < 4 * runs + 1; i++) { secp256k1_fe 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); }