diff --git a/src/secp256k1/sage/gen_exhaustive_groups.sage b/src/secp256k1/sage/gen_exhaustive_groups.sage new file mode 100644 --- /dev/null +++ b/src/secp256k1/sage/gen_exhaustive_groups.sage @@ -0,0 +1,129 @@ +# Define field size and field +P = 2^256 - 2^32 - 977 +F = GF(P) +BETA = F(0x7ae96a2b657c07106e64479eac3434e99cf0497512f58995c1396c28719501ee) + +assert(BETA != F(1) and BETA^3 == F(1)) + +orders_done = set() +results = {} +first = True +for b in range(1, P): + # There are only 6 curves (up to isomorphism) of the form y^2=x^3+B. Stop once we have tried all. + if len(orders_done) == 6: + break + + E = EllipticCurve(F, [0, b]) + print("Analyzing curve y^2 = x^3 + %i" % b) + n = E.order() + # Skip curves with an order we've already tried + if n in orders_done: + print("- Isomorphic to earlier curve") + continue + orders_done.add(n) + # Skip curves isomorphic to the real secp256k1 + if n.is_pseudoprime(): + print(" - Isomorphic to secp256k1") + continue + + print("- Finding subgroups") + + # Find what prime subgroups exist + for f, _ in n.factor(): + print("- Analyzing subgroup of order %i" % f) + # Skip subgroups of order >1000 + if f < 4 or f > 1000: + print(" - Bad size") + continue + + # Iterate over X coordinates until we find one that is on the curve, has order f, + # and for which curve isomorphism exists that maps it to X coordinate 1. + for x in range(1, P): + # Skip X coordinates not on the curve, and construct the full point otherwise. + if not E.is_x_coord(x): + continue + G = E.lift_x(F(x)) + + print(" - Analyzing (multiples of) point with X=%i" % x) + + # Skip points whose order is not a multiple of f. Project the point to have + # order f otherwise. + if (G.order() % f): + print(" - Bad order") + continue + G = G * (G.order() // f) + + # Find lambda for endomorphism. Skip if none can be found. + lam = None + for l in Integers(f)(1).nth_root(3, all=True): + if int(l)*G == E(BETA*G[0], G[1]): + lam = int(l) + break + if lam is None: + print(" - No endomorphism for this subgroup") + break + + # Now look for an isomorphism of the curve that gives this point an X + # coordinate equal to 1. + # If (x,y) is on y^2 = x^3 + b, then (a^2*x, a^3*y) is on y^2 = x^3 + a^6*b. + # So look for m=a^2=1/x. + m = F(1)/G[0] + if not m.is_square(): + print(" - No curve isomorphism maps it to a point with X=1") + continue + a = m.sqrt() + rb = a^6*b + RE = EllipticCurve(F, [0, rb]) + + # Use as generator twice the image of G under the above isormorphism. + # This means that generator*(1/2 mod f) will have X coordinate 1. + RG = RE(1, a^3*G[1]) * 2 + # And even Y coordinate. + if int(RG[1]) % 2: + RG = -RG + assert(RG.order() == f) + assert(lam*RG == RE(BETA*RG[0], RG[1])) + + # We have found curve RE:y^2=x^3+rb with generator RG of order f. Remember it + results[f] = {"b": rb, "G": RG, "lambda": lam} + print(" - Found solution") + break + + print("") + +print("") +print("") +print("/* To be put in src/group_impl.h: */") +first = True +for f in sorted(results.keys()): + b = results[f]["b"] + G = results[f]["G"] + print("# %s EXHAUSTIVE_TEST_ORDER == %i" % ("if" if first else "elif", f)) + first = False + print("static const secp256k1_ge secp256k1_ge_const_g = SECP256K1_GE_CONST(") + print(" 0x%08x, 0x%08x, 0x%08x, 0x%08x," % tuple((int(G[0]) >> (32 * (7 - i))) & 0xffffffff for i in range(4))) + print(" 0x%08x, 0x%08x, 0x%08x, 0x%08x," % tuple((int(G[0]) >> (32 * (7 - i))) & 0xffffffff for i in range(4, 8))) + print(" 0x%08x, 0x%08x, 0x%08x, 0x%08x," % tuple((int(G[1]) >> (32 * (7 - i))) & 0xffffffff for i in range(4))) + print(" 0x%08x, 0x%08x, 0x%08x, 0x%08x" % tuple((int(G[1]) >> (32 * (7 - i))) & 0xffffffff for i in range(4, 8))) + print(");") + print("static const secp256k1_fe secp256k1_fe_const_b = SECP256K1_FE_CONST(") + print(" 0x%08x, 0x%08x, 0x%08x, 0x%08x," % tuple((int(b) >> (32 * (7 - i))) & 0xffffffff for i in range(4))) + print(" 0x%08x, 0x%08x, 0x%08x, 0x%08x" % tuple((int(b) >> (32 * (7 - i))) & 0xffffffff for i in range(4, 8))) + print(");") +print("# else") +print("# error No known generator for the specified exhaustive test group order.") +print("# endif") + +print("") +print("") +print("/* To be put in src/scalar_impl.h: */") +first = True +for f in sorted(results.keys()): + lam = results[f]["lambda"] + print("# %s EXHAUSTIVE_TEST_ORDER == %i" % ("if" if first else "elif", f)) + first = False + print("# define EXHAUSTIVE_TEST_LAMBDA %i" % lam) +print("# else") +print("# error No known lambda for the specified exhaustive test group order.") +print("# endif") +print("") diff --git a/src/secp256k1/src/group_impl.h b/src/secp256k1/src/group_impl.h --- a/src/secp256k1/src/group_impl.h +++ b/src/secp256k1/src/group_impl.h @@ -11,52 +11,38 @@ #include "field.h" #include "group.h" -/* These points can be generated in sage as follows: +/* These exhaustive group test orders and generators are chosen such that: + * - The field size is equal to that of secp256k1, so field code is the same. + * - The curve equation is of the form y^2=x^3+B for some constant B. + * - The subgroup has a generator 2*P, where P.x=1. + * - The subgroup has size less than 1000 to permit exhaustive testing. + * - The subgroup admits an endomorphism of the form lambda*(x,y) == (beta*x,y). * - * 0. Setup a worksheet with the following parameters. - * b = 4 # whatever secp256k1_fe_const_b will be set to - * F = FiniteField (0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC2F) - * C = EllipticCurve ([F (0), F (b)]) - * - * 1. Determine all the small orders available to you. (If there are - * no satisfactory ones, go back and change b.) - * print C.order().factor(limit=1000) - * - * 2. Choose an order as one of the prime factors listed in the above step. - * (You can also multiply some to get a composite order, though the - * tests will crash trying to invert scalars during signing.) We take a - * random point and scale it to drop its order to the desired value. - * There is some probability this won't work; just try again. - * order = 199 - * P = C.random_point() - * P = (int(P.order()) / int(order)) * P - * assert(P.order() == order) - * - * 3. Print the values. You'll need to use a vim macro or something to - * split the hex output into 4-byte chunks. - * print "%x %x" % P.xy() + * These parameters are generated using sage/gen_exhaustive_groups.sage. */ #if defined(EXHAUSTIVE_TEST_ORDER) -# if EXHAUSTIVE_TEST_ORDER == 199 +# if EXHAUSTIVE_TEST_ORDER == 13 static const secp256k1_ge secp256k1_ge_const_g = SECP256K1_GE_CONST( - 0xFA7CC9A7, 0x0737F2DB, 0xA749DD39, 0x2B4FB069, - 0x3B017A7D, 0xA808C2F1, 0xFB12940C, 0x9EA66C18, - 0x78AC123A, 0x5ED8AEF3, 0x8732BC91, 0x1F3A2868, - 0x48DF246C, 0x808DAE72, 0xCFE52572, 0x7F0501ED + 0xc3459c3d, 0x35326167, 0xcd86cce8, 0x07a2417f, + 0x5b8bd567, 0xde8538ee, 0x0d507b0c, 0xd128f5bb, + 0x8e467fec, 0xcd30000a, 0x6cc1184e, 0x25d382c2, + 0xa2f4494e, 0x2fbe9abc, 0x8b64abac, 0xd005fb24 ); - -static const secp256k1_fe secp256k1_fe_const_b = SECP256K1_FE_CONST(0, 0, 0, 0, 0, 0, 0, 4); - -# elif EXHAUSTIVE_TEST_ORDER == 13 +static const secp256k1_fe secp256k1_fe_const_b = SECP256K1_FE_CONST( + 0x3d3486b2, 0x159a9ca5, 0xc75638be, 0xb23a69bc, + 0x946a45ab, 0x24801247, 0xb4ed2b8e, 0x26b6a417 +); +# elif EXHAUSTIVE_TEST_ORDER == 199 static const secp256k1_ge secp256k1_ge_const_g = SECP256K1_GE_CONST( - 0xedc60018, 0xa51a786b, 0x2ea91f4d, 0x4c9416c0, - 0x9de54c3b, 0xa1316554, 0x6cf4345c, 0x7277ef15, - 0x54cb1b6b, 0xdc8c1273, 0x087844ea, 0x43f4603e, - 0x0eaf9a43, 0xf6effe55, 0x939f806d, 0x37adf8ac + 0x226e653f, 0xc8df7744, 0x9bacbf12, 0x7d1dcbf9, + 0x87f05b2a, 0xe7edbd28, 0x1f564575, 0xc48dcf18, + 0xa13872c2, 0xe933bb17, 0x5d9ffd5b, 0xb5b6e10c, + 0x57fe3c00, 0xbaaaa15a, 0xe003ec3e, 0x9c269bae +); +static const secp256k1_fe secp256k1_fe_const_b = SECP256K1_FE_CONST( + 0x2cca28fa, 0xfc614b80, 0x2a3db42b, 0x00ba00b1, + 0xbea8d943, 0xdace9ab2, 0x9536daea, 0x0074defb ); - -static const secp256k1_fe secp256k1_fe_const_b = SECP256K1_FE_CONST(0, 0, 0, 0, 0, 0, 0, 2); - # else # error No known generator for the specified exhaustive test group order. # endif diff --git a/src/secp256k1/src/modules/recovery/tests_exhaustive_impl.h b/src/secp256k1/src/modules/recovery/tests_exhaustive_impl.h --- a/src/secp256k1/src/modules/recovery/tests_exhaustive_impl.h +++ b/src/secp256k1/src/modules/recovery/tests_exhaustive_impl.h @@ -25,6 +25,7 @@ unsigned char sk32[32], msg32[32]; int expected_recid; int recid; + int overflow; secp256k1_scalar_set_int(&msg, i); secp256k1_scalar_set_int(&sk, j); secp256k1_scalar_get_b32(sk32, &sk); @@ -34,17 +35,18 @@ /* Check directly */ secp256k1_ecdsa_recoverable_signature_load(ctx, &r, &s, &recid, &rsig); - r_from_k(&expected_r, group, k); + r_from_k(&expected_r, group, k, &overflow); CHECK(r == expected_r); CHECK((k * s) % EXHAUSTIVE_TEST_ORDER == (i + r * j) % EXHAUSTIVE_TEST_ORDER || (k * (EXHAUSTIVE_TEST_ORDER - s)) % EXHAUSTIVE_TEST_ORDER == (i + r * j) % EXHAUSTIVE_TEST_ORDER); /* The recid's second bit is for conveying overflow (R.x value >= group order). * In the actual secp256k1 this is an astronomically unlikely event, but in the - * small group used here, it will always be the case. + * small group used here, it will be the case for all points except the ones where + * R.x=1 (which the group is specifically selected to have). * Note that this isn't actually useful; full recovery would need to convey * floor(R.x / group_order), but only one bit is used as that is sufficient * in the real group. */ - expected_recid = 2; + expected_recid = overflow ? 2 : 0; r_dot_y_normalized = group[k].y; secp256k1_fe_normalize(&r_dot_y_normalized); /* Also the recovery id is flipped depending if we hit the low-s branch */ @@ -61,7 +63,7 @@ /* Note that we compute expected_r *after* signing -- this is important * because our nonce-computing function function might change k during * signing. */ - r_from_k(&expected_r, group, k); + r_from_k(&expected_r, group, k, NULL); CHECK(r == expected_r); CHECK((k * s) % EXHAUSTIVE_TEST_ORDER == (i + r * j) % EXHAUSTIVE_TEST_ORDER || (k * (EXHAUSTIVE_TEST_ORDER - s)) % EXHAUSTIVE_TEST_ORDER == (i + r * j) % EXHAUSTIVE_TEST_ORDER); @@ -104,7 +106,7 @@ should_verify = 0; for (k = 0; k < EXHAUSTIVE_TEST_ORDER; k++) { secp256k1_scalar check_x_s; - r_from_k(&check_x_s, group, k); + r_from_k(&check_x_s, group, k, NULL); if (r_s == check_x_s) { secp256k1_scalar_set_int(&s_times_k_s, k); secp256k1_scalar_mul(&s_times_k_s, &s_times_k_s, &s_s); diff --git a/src/secp256k1/src/scalar_impl.h b/src/secp256k1/src/scalar_impl.h --- a/src/secp256k1/src/scalar_impl.h +++ b/src/secp256k1/src/scalar_impl.h @@ -253,6 +253,7 @@ } #ifdef USE_ENDOMORPHISM +/* These parameters are generated using sage/gen_exhaustive_groups.sage. */ #if defined(EXHAUSTIVE_TEST_ORDER) # if EXHAUSTIVE_TEST_ORDER == 13 # define EXHAUSTIVE_TEST_LAMBDA 9 diff --git a/src/secp256k1/src/tests_exhaustive.c b/src/secp256k1/src/tests_exhaustive.c --- a/src/secp256k1/src/tests_exhaustive.c +++ b/src/secp256k1/src/tests_exhaustive.c @@ -215,14 +215,14 @@ secp256k1_scratch_destroy(&ctx->error_callback, scratch); } -void r_from_k(secp256k1_scalar *r, const secp256k1_ge *group, int k) { +void r_from_k(secp256k1_scalar *r, const secp256k1_ge *group, int k, int* overflow) { secp256k1_fe x; unsigned char x_bin[32]; k %= EXHAUSTIVE_TEST_ORDER; x = group[k].x; secp256k1_fe_normalize(&x); secp256k1_fe_get_b32(x_bin, &x); - secp256k1_scalar_set_b32(r, x_bin, NULL); + secp256k1_scalar_set_b32(r, x_bin, overflow); } void test_exhaustive_verify(const secp256k1_context *ctx, const secp256k1_ge *group) { @@ -250,7 +250,7 @@ should_verify = 0; for (k = 0; k < EXHAUSTIVE_TEST_ORDER; k++) { secp256k1_scalar check_x_s; - r_from_k(&check_x_s, group, k); + r_from_k(&check_x_s, group, k, NULL); if (r_s == check_x_s) { secp256k1_scalar_set_int(&s_times_k_s, k); secp256k1_scalar_mul(&s_times_k_s, &s_times_k_s, &s_s); @@ -297,7 +297,7 @@ /* Note that we compute expected_r *after* signing -- this is important * because our nonce-computing function function might change k during * signing. */ - r_from_k(&expected_r, group, k); + r_from_k(&expected_r, group, k, NULL); CHECK(r == expected_r); CHECK((k * s) % EXHAUSTIVE_TEST_ORDER == (i + r * j) % EXHAUSTIVE_TEST_ORDER || (k * (EXHAUSTIVE_TEST_ORDER - s)) % EXHAUSTIVE_TEST_ORDER == (i + r * j) % EXHAUSTIVE_TEST_ORDER);