| /*********************************************************************** | |
| * Copyright (c) Pieter Wuille, Gregory Maxwell, Peter Dettman * | |
| * Distributed under the MIT software license, see the accompanying * | |
| * file COPYING or https://www.opensource.org/licenses/mit-license.php.* | |
| ***********************************************************************/ | |
| | |
| #ifndef SECP256K1_ECMULT_GEN_IMPL_H | |
| #define SECP256K1_ECMULT_GEN_IMPL_H | |
| | |
| #include "util.h" | |
| #include "scalar.h" | |
| #include "group.h" | |
| #include "ecmult_gen.h" | |
| #include "hash_impl.h" | |
| #include "precomputed_ecmult_gen.h" | |
| | |
| static void secp256k1_ecmult_gen_context_build(secp256k1_ecmult_gen_context *ctx) { | |
| secp256k1_ecmult_gen_blind(ctx, NULL); | |
| ctx->built = 1; | |
| } | |
| | |
| static int secp256k1_ecmult_gen_context_is_built(const secp256k1_ecmult_gen_context* ctx) { | |
| return ctx->built; | |
| } | |
| | |
| static void secp256k1_ecmult_gen_context_clear(secp256k1_ecmult_gen_context *ctx) { | |
| ctx->built = 0; | |
| secp256k1_scalar_clear(&ctx->scalar_offset); | |
| secp256k1_ge_clear(&ctx->ge_offset); | |
| secp256k1_fe_clear(&ctx->proj_blind); | |
| } | |
| | |
| /* Compute the scalar (2^COMB_BITS - 1) / 2, the difference between the gn argument to | |
| * secp256k1_ecmult_gen, and the scalar whose encoding the table lookup bits are drawn | |
| * from (before applying blinding). */ | |
| static void secp256k1_ecmult_gen_scalar_diff(secp256k1_scalar* diff) { | |
| int i; | |
| | |
| /* Compute scalar -1/2. */ | |
| secp256k1_scalar neghalf; | |
| secp256k1_scalar_half(&neghalf, &secp256k1_scalar_one); | |
| secp256k1_scalar_negate(&neghalf, &neghalf); | |
| | |
| /* Compute offset = 2^(COMB_BITS - 1). */ | |
| *diff = secp256k1_scalar_one; | |
| for (i = 0; i < COMB_BITS - 1; ++i) { | |
| secp256k1_scalar_add(diff, diff, diff); | |
| } | |
| | |
| /* The result is the sum 2^(COMB_BITS - 1) + (-1/2). */ | |
| secp256k1_scalar_add(diff, diff, &neghalf); | |
| } | |
| | |
| static void secp256k1_ecmult_gen(const secp256k1_ecmult_gen_context *ctx, secp256k1_gej *r, const secp256k1_scalar *gn) { | |
| uint32_t comb_off; | |
| secp256k1_ge add; | |
| secp256k1_fe neg; | |
| secp256k1_ge_storage adds; | |
| secp256k1_scalar d; | |
| /* Array of uint32_t values large enough to store COMB_BITS bits. Only the bottom | |
| * 8 are ever nonzero, but having the zero padding at the end if COMB_BITS>256 | |
| * avoids the need to deal with out-of-bounds reads from a scalar. */ | |
| uint32_t recoded[(COMB_BITS + 31) >> 5] = {0}; | |
| int first = 1, i; | |
| | |
| memset(&adds, 0, sizeof(adds)); | |
| | |
| /* We want to compute R = gn*G. | |
| * | |
| * To blind the scalar used in the computation, we rewrite this to be | |
| * R = (gn - b)*G + b*G, with a blinding value b determined by the context. | |
| * | |
| * The multiplication (gn-b)*G will be performed using a signed-digit multi-comb (see Section | |
| * 3.3 of "Fast and compact elliptic-curve cryptography" by Mike Hamburg, | |
| * https://eprint.iacr.org/2012/309). | |
| * | |
| * Let comb(s, P) = sum((2*s[i]-1)*2^i*P for i=0..COMB_BITS-1), where s[i] is the i'th bit of | |
| * the binary representation of scalar s. So the s[i] values determine whether -2^i*P (s[i]=0) | |
| * or +2^i*P (s[i]=1) are added together. COMB_BITS is at least 256, so all bits of s are | |
| * covered. By manipulating: | |
| * | |
| * comb(s, P) = sum((2*s[i]-1)*2^i*P for i=0..COMB_BITS-1) | |
| * <=> comb(s, P) = sum((2*s[i]-1)*2^i for i=0..COMB_BITS-1) * P | |
| * <=> comb(s, P) = (2*sum(s[i]*2^i for i=0..COMB_BITS-1) - sum(2^i for i=0..COMB_BITS-1)) * P | |
| * <=> comb(s, P) = (2*s - (2^COMB_BITS - 1)) * P | |
| * | |
| * If we wanted to compute (gn-b)*G as comb(s, G), it would need to hold that | |
| * | |
| * (gn - b) * G = (2*s - (2^COMB_BITS - 1)) * G | |
| * <=> s = (gn - b + (2^COMB_BITS - 1))/2 (mod order) | |
| * | |
| * We use an alternative here that avoids the modular division by two: instead we compute | |
| * (gn-b)*G as comb(d, G/2). For that to hold it must be the case that | |
| * | |
| * (gn - b) * G = (2*d - (2^COMB_BITS - 1)) * (G/2) | |
| * <=> d = gn - b + (2^COMB_BITS - 1)/2 (mod order) | |
| * | |
| * Adding precomputation, our final equations become: | |
| * | |
| * ctx->scalar_offset = (2^COMB_BITS - 1)/2 - b (mod order) | |
| * ctx->ge_offset = b*G | |
| * d = gn + ctx->scalar_offset (mod order) | |
| * R = comb(d, G/2) + ctx->ge_offset | |
| * | |
| * comb(d, G/2) function is then computed by summing + or - 2^(i-1)*G, for i=0..COMB_BITS-1, | |
| * depending on the value of the bits d[i] of the binary representation of scalar d. | |
| */ | |
| | |
| /* Compute the scalar d = (gn + ctx->scalar_offset). */ | |
| secp256k1_scalar_add(&d, &ctx->scalar_offset, gn); | |
| /* Convert to recoded array. */ | |
| for (i = 0; i < 8 && i < ((COMB_BITS + 31) >> 5); ++i) { | |
| recoded[i] = secp256k1_scalar_get_bits_limb32(&d, 32 * i, 32); | |
| } | |
| secp256k1_scalar_clear(&d); | |
| | |
| /* In secp256k1_ecmult_gen_prec_table we have precomputed sums of the | |
| * (2*d[i]-1) * 2^(i-1) * G points, for various combinations of i positions. | |
| * We rewrite our equation in terms of these table entries. | |
| * | |
| * Let mask(b) = sum(2^((b*COMB_TEETH + t)*COMB_SPACING) for t=0..COMB_TEETH-1), | |
| * with b ranging from 0 to COMB_BLOCKS-1. So for example with COMB_BLOCKS=11, | |
| * COMB_TEETH=6, COMB_SPACING=4, we would have: | |
| * mask(0) = 2^0 + 2^4 + 2^8 + 2^12 + 2^16 + 2^20, | |
| * mask(1) = 2^24 + 2^28 + 2^32 + 2^36 + 2^40 + 2^44, | |
| * mask(2) = 2^48 + 2^52 + 2^56 + 2^60 + 2^64 + 2^68, | |
| * ... | |
| * mask(10) = 2^240 + 2^244 + 2^248 + 2^252 + 2^256 + 2^260 | |
| * | |
| * We will split up the bits d[i] using these masks. Specifically, each mask is | |
| * used COMB_SPACING times, with different shifts: | |
| * | |
| * d = (d & mask(0)<<0) + (d & mask(1)<<0) + ... + (d & mask(COMB_BLOCKS-1)<<0) + | |
| * (d & mask(0)<<1) + (d & mask(1)<<1) + ... + (d & mask(COMB_BLOCKS-1)<<1) + | |
| * ... | |
| * (d & mask(0)<<(COMB_SPACING-1)) + ... | |
| * | |
| * Now define table(b, m) = (m - mask(b)/2) * G, and we will precompute these values for | |
| * b=0..COMB_BLOCKS-1, and for all values m which (d & mask(b)) can take (so m can take on | |
| * 2^COMB_TEETH distinct values). | |
| * | |
| * If m=(d & mask(b)), then table(b, m) is the sum of 2^i * (2*d[i]-1) * G/2, with i | |
| * iterating over the set bits in mask(b). In our example, table(2, 2^48 + 2^56 + 2^68) | |
| * would equal (2^48 - 2^52 + 2^56 - 2^60 - 2^64 + 2^68) * G/2. | |
| * | |
| * With that, we can rewrite comb(d, G/2) as: | |
| * | |
| * 2^0 * (table(0, d>>0 & mask(0)) + ... + table(COMB_BLOCKS-1, d>>0 & mask(COMP_BLOCKS-1))) | |
| * + 2^1 * (table(0, d>>1 & mask(0)) + ... + table(COMB_BLOCKS-1, d>>1 & mask(COMP_BLOCKS-1))) | |
| * + 2^2 * (table(0, d>>2 & mask(0)) + ... + table(COMB_BLOCKS-1, d>>2 & mask(COMP_BLOCKS-1))) | |
| * + ... | |
| * + 2^(COMB_SPACING-1) * (table(0, d>>(COMB_SPACING-1) & mask(0)) + ...) | |
| * | |
| * Or more generically as | |
| * | |
| * sum(2^i * sum(table(b, d>>i & mask(b)), b=0..COMB_BLOCKS-1), i=0..COMB_SPACING-1) | |
| * | |
| * This is implemented using an outer loop that runs in reverse order over the lines of this | |
| * equation, which in each iteration runs an inner loop that adds the terms of that line and | |
| * then doubles the result before proceeding to the next line. | |
| * | |
| * In pseudocode: | |
| * c = infinity | |
| * for comb_off in range(COMB_SPACING - 1, -1, -1): | |
| * for block in range(COMB_BLOCKS): | |
| * c += table(block, (d >> comb_off) & mask(block)) | |
| * if comb_off > 0: | |
| * c = 2*c | |
| * return c | |
| * | |
| * This computes c = comb(d, G/2), and thus finally R = c + ctx->ge_offset. Note that it would | |
| * be possible to apply an initial offset instead of a final offset (moving ge_offset to take | |
| * the place of infinity above), but the chosen approach allows using (in a future improvement) | |
| * an incomplete addition formula for most of the multiplication. | |
| * | |
| * The last question is how to implement the table(b, m) function. For any value of b, | |
| * m=(d & mask(b)) can only take on at most 2^COMB_TEETH possible values (the last one may have | |
| * fewer as there mask(b) may exceed the curve order). So we could create COMB_BLOCK tables | |
| * which contain a value for each such m value. | |
| * | |
| * Now note that if m=(d & mask(b)), then flipping the relevant bits of m results in negating | |
| * the result of table(b, m). This is because table(b,m XOR mask(b)) = table(b, mask(b) - m) = | |
| * (mask(b) - m - mask(b)/2)*G = (-m + mask(b)/2)*G = -(m - mask(b)/2)*G = -table(b, m). | |
| * Because of this it suffices to only store the first half of the m values for every b. If an | |
| * entry from the second half is needed, we look up its bit-flipped version instead, and negate | |
| * it. | |
| * | |
| * secp256k1_ecmult_gen_prec_table[b][index] stores the table(b, m) entries. Index | |
| * is the relevant mask(b) bits of m packed together without gaps. */ | |
| | |
| /* Outer loop: iterate over comb_off from COMB_SPACING - 1 down to 0. */ | |
| comb_off = COMB_SPACING - 1; | |
| while (1) { | |
| uint32_t block; | |
| uint32_t bit_pos = comb_off; | |
| /* Inner loop: for each block, add table entries to the result. */ | |
| for (block = 0; block < COMB_BLOCKS; ++block) { | |
| /* Gather the mask(block)-selected bits of d into bits. They're packed: | |
| * bits[tooth] = d[(block*COMB_TEETH + tooth)*COMB_SPACING + comb_off]. */ | |
| uint32_t bits = 0, sign, abs, index, tooth; | |
| /* Instead of reading individual bits here to construct the bits variable, | |
| * build up the result by xoring rotated reads together. In every iteration, | |
| * one additional bit is made correct, starting at the bottom. The bits | |
| * above that contain junk. This reduces leakage by avoiding computations | |
| * on variables that can have only a low number of possible values (e.g., | |
| * just two values when reading a single bit into a variable.) See: | |
| * https://www.usenix.org/system/files/conference/usenixsecurity18/sec18-alam.pdf | |
| */ | |
| for (tooth = 0; tooth < COMB_TEETH; ++tooth) { | |
| /* Construct bitdata s.t. the bottom bit is the bit we'd like to read. | |
| * | |
| * We could just set bitdata = recoded[bit_pos >> 5] >> (bit_pos & 0x1f) | |
| * but this would simply discard the bits that fall off at the bottom, | |
| * and thus, for example, bitdata could still have only two values if we | |
| * happen to shift by exactly 31 positions. We use a rotation instead, | |
| * which ensures that bitdata doesn't loose entropy. This relies on the | |
| * rotation being atomic, i.e., the compiler emitting an actual rot | |
| * instruction. */ | |
| uint32_t bitdata = secp256k1_rotr32(recoded[bit_pos >> 5], bit_pos & 0x1f); | |
| | |
| /* Clear the bit at position tooth, but sssh, don't tell clang. */ | |
| uint32_t volatile vmask = ~(1 << tooth); | |
| bits &= vmask; | |
| | |
| /* Write the bit into position tooth (and junk into higher bits). */ | |
| bits ^= bitdata << tooth; | |
| bit_pos += COMB_SPACING; | |
| } | |
| | |
| /* If the top bit of bits is 1, flip them all (corresponding to looking up | |
| * the negated table value), and remember to negate the result in sign. */ | |
| sign = (bits >> (COMB_TEETH - 1)) & 1; | |
| abs = (bits ^ -sign) & (COMB_POINTS - 1); | |
| VERIFY_CHECK(sign == 0 || sign == 1); | |
| VERIFY_CHECK(abs < COMB_POINTS); | |
| | |
| /** This uses a conditional move to avoid any secret data in array indexes. | |
| * _Any_ use of secret indexes has been demonstrated to result in timing | |
| * sidechannels, even when the cache-line access patterns are uniform. | |
| * See also: | |
| * "A word of warning", CHES 2013 Rump Session, by Daniel J. Bernstein and Peter Schwabe | |
| * (https://cryptojedi.org/peter/data/chesrump-20130822.pdf) and | |
| * "Cache Attacks and Countermeasures: the Case of AES", RSA 2006, | |
| * by Dag Arne Osvik, Adi Shamir, and Eran Tromer | |
| * (https://www.tau.ac.il/~tromer/papers/cache.pdf) | |
| */ | |
| for (index = 0; index < COMB_POINTS; ++index) { | |
| secp256k1_ge_storage_cmov(&adds, &secp256k1_ecmult_gen_prec_table[block][index], index == abs); | |
| } | |
| | |
| /* Set add=adds or add=-adds, in constant time, based on sign. */ | |
| secp256k1_ge_from_storage(&add, &adds); | |
| secp256k1_fe_negate(&neg, &add.y, 1); | |
| secp256k1_fe_cmov(&add.y, &neg, sign); | |
| | |
| /* Add the looked up and conditionally negated value to r. */ | |
| if (EXPECT(first, 0)) { | |
| /* If this is the first table lookup, we can skip addition. */ | |
| secp256k1_gej_set_ge(r, &add); | |
| /* Give the entry a random Z coordinate to blind intermediary results. */ | |
| secp256k1_gej_rescale(r, &ctx->proj_blind); | |
| first = 0; | |
| } else { | |
| secp256k1_gej_add_ge(r, r, &add); | |
| } | |
| } | |
| | |
| /* Double the result, except in the last iteration. */ | |
| if (comb_off-- == 0) break; | |
| secp256k1_gej_double(r, r); | |
| } | |
| | |
| /* Correct for the scalar_offset added at the start (ge_offset = b*G, while b was | |
| * subtracted from the input scalar gn). */ | |
| secp256k1_gej_add_ge(r, r, &ctx->ge_offset); | |
| | |
| /* Cleanup. */ | |
| secp256k1_fe_clear(&neg); | |
| secp256k1_ge_clear(&add); | |
| secp256k1_memclear(&adds, sizeof(adds)); | |
| secp256k1_memclear(&recoded, sizeof(recoded)); | |
| } | |
| | |
| /* Setup blinding values for secp256k1_ecmult_gen. */ | |
| static void secp256k1_ecmult_gen_blind(secp256k1_ecmult_gen_context *ctx, const unsigned char *seed32) { | |
| secp256k1_scalar b; | |
| secp256k1_scalar diff; | |
| secp256k1_gej gb; | |
| secp256k1_fe f; | |
| unsigned char nonce32[32]; | |
| secp256k1_rfc6979_hmac_sha256 rng; | |
| unsigned char keydata[64]; | |
| | |
| /* Compute the (2^COMB_BITS - 1)/2 term once. */ | |
| secp256k1_ecmult_gen_scalar_diff(&diff); | |
| | |
| if (seed32 == NULL) { | |
| /* When seed is NULL, reset the final point and blinding value. */ | |
| secp256k1_ge_neg(&ctx->ge_offset, &secp256k1_ge_const_g); | |
| secp256k1_scalar_add(&ctx->scalar_offset, &secp256k1_scalar_one, &diff); | |
| ctx->proj_blind = secp256k1_fe_one; | |
| return; | |
| } | |
| /* The prior blinding value (if not reset) is chained forward by including it in the hash. */ | |
| secp256k1_scalar_get_b32(keydata, &ctx->scalar_offset); | |
| /** Using a CSPRNG allows a failure free interface, avoids needing large amounts of random data, | |
| * and guards against weak or adversarial seeds. This is a simpler and safer interface than | |
| * asking the caller for blinding values directly and expecting them to retry on failure. | |
| */ | |
| VERIFY_CHECK(seed32 != NULL); | |
| memcpy(keydata + 32, seed32, 32); | |
| secp256k1_rfc6979_hmac_sha256_initialize(&rng, keydata, 64); | |
| secp256k1_memclear(keydata, sizeof(keydata)); | |
| | |
| /* Compute projective blinding factor (cannot be 0). */ | |
| secp256k1_rfc6979_hmac_sha256_generate(&rng, nonce32, 32); | |
| secp256k1_fe_set_b32_mod(&f, nonce32); | |
| secp256k1_fe_cmov(&f, &secp256k1_fe_one, secp256k1_fe_normalizes_to_zero(&f)); | |
| ctx->proj_blind = f; | |
| | |
| /* For a random blinding value b, set scalar_offset=diff-b, ge_offset=bG */ | |
| secp256k1_rfc6979_hmac_sha256_generate(&rng, nonce32, 32); | |
| secp256k1_scalar_set_b32(&b, nonce32, NULL); | |
| /* The blinding value cannot be zero, as that would mean ge_offset = infinity, | |
| * which secp256k1_gej_add_ge cannot handle. */ | |
| secp256k1_scalar_cmov(&b, &secp256k1_scalar_one, secp256k1_scalar_is_zero(&b)); | |
| secp256k1_rfc6979_hmac_sha256_finalize(&rng); | |
| secp256k1_ecmult_gen(ctx, &gb, &b); | |
| secp256k1_scalar_negate(&b, &b); | |
| secp256k1_scalar_add(&ctx->scalar_offset, &b, &diff); | |
| secp256k1_ge_set_gej(&ctx->ge_offset, &gb); | |
| | |
| /* Clean up. */ | |
| secp256k1_memclear(nonce32, sizeof(nonce32)); | |
| secp256k1_scalar_clear(&b); | |
| secp256k1_gej_clear(&gb); | |
| secp256k1_fe_clear(&f); | |
| secp256k1_rfc6979_hmac_sha256_clear(&rng); | |
| } | |
| | |
| #endif /* SECP256K1_ECMULT_GEN_IMPL_H */ | |