From 62c3f55a9da1e132e4ee567fa6ce17127ed40a8b Mon Sep 17 00:00:00 2001 From: Pieter Wuille Date: Fri, 29 Aug 2014 01:47:20 +0200 Subject: [PATCH] Nothing-up-my-sleeving blinding for a*G --- README.md | 2 +- src/ecmult_impl.h | 80 +++++++++++++++++++++++++++++------------------ 2 files changed, 51 insertions(+), 31 deletions(-) diff --git a/README.md b/README.md index ec6c5475611..32750bde924 100644 --- a/README.md +++ b/README.md @@ -36,7 +36,7 @@ Implementation details * Point multiplication for signing * Use a precomputed table of multiples of powers of 16 multiplied with the generator, so general multiplication becomes a series of additions. * Slice the precomputed table in memory per byte, so memory access to the table becomes uniform. - * Not fully constant-time. + * Not fully constant-time, but the precomputed tables add and eventually subtract points for which no known scalar (private key) is known, blinding non-constant time effects even from an attacker with control over the private key used. Build steps ----------- diff --git a/src/ecmult_impl.h b/src/ecmult_impl.h index e893add4bcf..b313b6c0766 100644 --- a/src/ecmult_impl.h +++ b/src/ecmult_impl.h @@ -73,14 +73,16 @@ typedef struct { // For accelerating the computation of a*G: // To harden against timing attacks, use the following mechanism: // * Break up the multiplicand into groups of 4 bits, called n_0, n_1, n_2, ..., n_63. - // * Compute sum((n_i + 1) * 16^i * G, i=0..63). - // * Subtract sum(1 * 16^i * G, i=0..63). - // For each i, and each of the 16 possible values of n_i, ((n_i + 1) * 16^i * G) is - // precomputed (call it prec(i,n_i), as well as -sum(1 * 16^i * G) (called fin). - // The formula now becomes sum(prec(i, n_i), i=0..63) + fin. - // To make memory access uniform, the bytes of prec(i,n_i) are sliced per value of n_i. - unsigned char prec[64][sizeof(secp256k1_ge_t)][16]; // prec[j][k][i] = k'th byte of (16^j * (i+1) * G) - secp256k1_ge_t fin; // -(sum(prec[j][0], j=0..63)) + // * Compute sum(n_i * 16^i * G + U_i, i=0..63), where: + // * U_i = U * 2^i (for i=0..62) + // * U_i = U * (1-2^63) (for i=63) + // where U is a point with no known corresponding scalar. Note that sum(U_i, i=0..63) = 0. + // For each i, and each of the 16 possible values of n_i, (n_i * 16^i * G + U_i) is + // precomputed (call it prec(i, n_i)). The formula now becomes sum(prec(i, n_i), i=0..63). + // None of the resulting prec group elements have a known scalar, and neither do any of + // the intermediate sums while computing a*G. + // To make memory access uniform, the bytes of prec(i, n_i) are sliced per value of n_i. + unsigned char prec[64][sizeof(secp256k1_ge_t)][16]; // prec[j][k][i] = k'th byte of (16^j * i * G + U_i) } secp256k1_ecmult_consts_t; static const secp256k1_ecmult_consts_t *secp256k1_ecmult_consts = NULL; @@ -105,34 +107,53 @@ static void secp256k1_ecmult_start(void) { secp256k1_ecmult_table_precomp_ge(ret->pre_g, &gj, WINDOW_G); secp256k1_ecmult_table_precomp_ge(ret->pre_g_128, &g_128j, WINDOW_G); - // compute prec and fin - secp256k1_gej_t tj[961]; - int pos = 0; - secp256k1_gej_t fn; secp256k1_gej_set_infinity(&fn); - for (int j=0; j<64; j++) { - secp256k1_gej_add(&fn, &fn, &gj); - secp256k1_gej_t adj = gj; - for (int i=1; i<16; i++) { - secp256k1_gej_add(&gj, &gj, &adj); - tj[pos++] = gj; + // Construct a group element with no known corresponding scalar (nothing up my sleeve). + secp256k1_gej_t nums_gej; + { + static const unsigned char nums_b32[32] = "The scalar for this x is unknown"; + secp256k1_fe_t nums_x; + secp256k1_fe_set_b32(&nums_x, nums_b32); + secp256k1_ge_t nums_ge; + VERIFY_CHECK(secp256k1_ge_set_xo(&nums_ge, &nums_x, 0)); + secp256k1_gej_set_ge(&nums_gej, &nums_ge); + // Add G to make the bits in x uniformly distributed. + secp256k1_gej_add_ge(&nums_gej, &nums_gej, g); + } + + // compute prec. + secp256k1_ge_t prec[1024]; + { + secp256k1_gej_t precj[1024]; // Jacobian versions of prec. + int j = 0; + secp256k1_gej_t gbase; gbase = gj; // 16^j * G + secp256k1_gej_t numsbase; numsbase = nums_gej; // 2^j * nums. + for (int j=0; j<64; j++) { + // Set precj[j*16 .. j*16+15] to (numsbase, numsbase + gbase, ..., numsbase + 15*gbase). + precj[j*16] = numsbase; + for (int i=1; i<16; i++) { + secp256k1_gej_add(&precj[j*16 + i], &precj[j*16 + i - 1], &gbase); + } + // Multiply gbase by 16. + for (int i=0; i<4; i++) { + secp256k1_gej_double(&gbase, &gbase); + } + // Multiply numbase by 2. + secp256k1_gej_double(&numsbase, &numsbase); + if (j == 62) { + // In the last iteration, numsbase is (1 - 2^j) * nums instead. + secp256k1_gej_neg(&numsbase, &numsbase); + secp256k1_gej_add(&numsbase, &numsbase, &nums_gej); + } } + secp256k1_ge_set_all_gej(1024, prec, precj); } - VERIFY_CHECK(pos == 960); - tj[pos] = fn; - secp256k1_ge_t t[961]; secp256k1_ge_set_all_gej(961, t, tj); - pos = 0; - const unsigned char *raw = (const unsigned char*)g; for (int j=0; j<64; j++) { - for (int k=0; kprec[j][k][0] = raw[k]; - for (int i=1; i<16; i++) { - raw = (const unsigned char*)(&t[pos++]); + for (int i=0; i<16; i++) { + const unsigned char* raw = (const unsigned char*)(&prec[j*16 + i]); for (int k=0; kprec[j][k][i] = raw[k]; } } - VERIFY_CHECK(pos == 960); - secp256k1_ge_neg(&ret->fin, &t[pos]); } static void secp256k1_ecmult_stop(void) { @@ -202,7 +223,6 @@ void static secp256k1_ecmult_gen(secp256k1_gej_t *r, const secp256k1_num_t *gn) secp256k1_ge_clear(&add); secp256k1_num_clear(&n); secp256k1_num_free(&n); - secp256k1_gej_add_ge(r, r, &c->fin); } void static secp256k1_ecmult(secp256k1_gej_t *r, const secp256k1_gej_t *a, const secp256k1_num_t *na, const secp256k1_num_t *ng) {