diff --git a/sage/gen_exhaustive_groups.sage b/sage/gen_exhaustive_groups.sage index 01d15dc..6026580 100644 --- a/sage/gen_exhaustive_groups.sage +++ b/sage/gen_exhaustive_groups.sage @@ -1,124 +1,161 @@ load("secp256k1_params.sage") +MAX_ORDER = 1000 + +# Set of (curve) orders we have encountered so far. orders_done = set() -results = {} -first = True + +# Map from (subgroup) orders to [b, int(gen.x), int(gen.y), gen, lambda] for those subgroups. +solutions = {} + +# Iterate over curves of the form y^2 = x^3 + B. 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. + # 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") + print() continue orders_done.add(n) + # Skip curves isomorphic to the real secp256k1 if n.is_pseudoprime(): - print(" - Isomorphic to secp256k1") + assert E.is_isomorphic(C) + print("- Isomorphic to secp256k1") + print() continue - print("- Finding subgroups") + print("- Finding prime 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 + # Map from group_order to a set of independent generators for that order. + curve_gens = {} - # 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): + for g in E.gens(): + # Find what prime subgroups of group generated by g exist. + g_order = g.order() + for f, _ in g.order().factor(): + # Skip subgroups that have bad size. + if f < 4: + print(f" - Subgroup of size {f}: too small") 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") + if f > MAX_ORDER: + print(f" - Subgroup of size {f}: too large") continue - G = G * (G.order() // f) + + # Construct a generator for that subgroup. + gen = g * (g_order // f) + assert(gen.order() == f) + + # Add to set the minimal multiple of gen. + curve_gens.setdefault(f, set()).add(min([j*gen for j in range(1, f)])) + print(f" - Subgroup of size {f}: ok") + + for f in sorted(curve_gens.keys()): + print(f"- Constructing group of order {f}") + cbrts = sorted([int(c) for c in Integers(f)(1).nth_root(3, all=true) if c != 1]) + gens = list(curve_gens[f]) + sol_count = 0 + no_endo_count = 0 + + # Consider all non-zero linear combinations of the independent generators. + for j in range(1, f**len(gens)): + gen = sum(gens[k] * ((j // f**k) % f) for k in range(len(gens))) + assert not gen.is_zero() + assert (f*gen).is_zero() # 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) + for l in cbrts: + if l*gen == E(BETA*gen[0], gen[1]): + lam = l break + if lam is None: - print(" - No endomorphism for this subgroup") - break + no_endo_count += 1 + else: + sol_count += 1 + solutions.setdefault(f, []).append((b, int(gen[0]), int(gen[1]), gen, lam)) - # 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]) + print(f" - Found {sol_count} generators (plus {no_endo_count} without endomorphism)") - # 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])) + print() - # 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 +def output_generator(g, name): + print(f"#define {name} 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("") - -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(");") +def output_b(b): 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() +print("To be put in src/group_impl.h:") +print() +print("/* Begin of section generated by sage/gen_exhaustive_groups.sage. */") +for f in sorted(solutions.keys()): + # Use as generator/2 the one with lowest b, and lowest (x, y) generator (interpreted as non-negative integers). + b, _, _, HALF_G, lam = min(solutions[f]) + output_generator(2 * HALF_G, f"SECP256K1_G_ORDER_{f}") +print("/** Generator for secp256k1, value 'g' defined in") +print(" * \"Standards for Efficient Cryptography\" (SEC2) 2.7.1.") +print(" */") +output_generator(G, "SECP256K1_G") +print("/* These exhaustive group test orders and generators are chosen such that:") +print(" * - The field size is equal to that of secp256k1, so field code is the same.") +print(" * - The curve equation is of the form y^2=x^3+B for some small constant B.") +print(" * - The subgroup has a generator 2*P, where P.x is as small as possible.") +print(f" * - The subgroup has size less than {MAX_ORDER} to permit exhaustive testing.") +print(" * - The subgroup admits an endomorphism of the form lambda*(x,y) == (beta*x,y).") +print(" */") +print("#if defined(EXHAUSTIVE_TEST_ORDER)") +first = True +for f in sorted(solutions.keys()): + b, _, _, _, lam = min(solutions[f]) + print(f"# {'if' if first else 'elif'} EXHAUSTIVE_TEST_ORDER == {f}") + first = False + print() + print(f"static const secp256k1_ge secp256k1_ge_const_g = SECP256K1_G_ORDER_{f};") + print("") + output_b(b) + print() print("# else") print("# error No known generator for the specified exhaustive test group order.") print("# endif") +print("#else") +print() +print("static const secp256k1_ge secp256k1_ge_const_g = SECP256K1_G;") +print("") +output_b(7) +print() +print("#endif") +print("/* End of section generated by sage/gen_exhaustive_groups.sage. */") -print("") -print("") -print("/* To be put in src/scalar_impl.h: */") + +print() +print() +print("To be put in src/scalar_impl.h:") +print() +print("/* Begin of section generated by sage/gen_exhaustive_groups.sage. */") first = True -for f in sorted(results.keys()): - lam = results[f]["lambda"] +for f in sorted(solutions.keys()): + _, _, _, _, lam = min(solutions[f]) 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("") +print("/* End of section generated by sage/gen_exhaustive_groups.sage. */") diff --git a/src/group_impl.h b/src/group_impl.h index dfe6e32..ed2b50b 100644 --- a/src/group_impl.h +++ b/src/group_impl.h @@ -10,59 +10,83 @@ #include "field.h" #include "group.h" +/* Begin of section generated by sage/gen_exhaustive_groups.sage. */ +#define SECP256K1_G_ORDER_7 SECP256K1_GE_CONST(\ + 0x66625d13, 0x317ffe44, 0x63d32cff, 0x1ca02b9b,\ + 0xe5c6d070, 0x50b4b05e, 0x81cc30db, 0xf5166f0a,\ + 0x1e60e897, 0xa7c00c7c, 0x2df53eb6, 0x98274ff4,\ + 0x64252f42, 0x8ca44e17, 0x3b25418c, 0xff4ab0cf\ +) #define SECP256K1_G_ORDER_13 SECP256K1_GE_CONST(\ - 0xc3459c3d, 0x35326167, 0xcd86cce8, 0x07a2417f,\ - 0x5b8bd567, 0xde8538ee, 0x0d507b0c, 0xd128f5bb,\ - 0x8e467fec, 0xcd30000a, 0x6cc1184e, 0x25d382c2,\ - 0xa2f4494e, 0x2fbe9abc, 0x8b64abac, 0xd005fb24\ + 0xa2482ff8, 0x4bf34edf, 0xa51262fd, 0xe57921db,\ + 0xe0dd2cb7, 0xa5914790, 0xbc71631f, 0xc09704fb,\ + 0x942536cb, 0xa3e49492, 0x3a701cc3, 0xee3e443f,\ + 0xdf182aa9, 0x15b8aa6a, 0x166d3b19, 0xba84b045\ ) #define SECP256K1_G_ORDER_199 SECP256K1_GE_CONST(\ - 0x226e653f, 0xc8df7744, 0x9bacbf12, 0x7d1dcbf9,\ - 0x87f05b2a, 0xe7edbd28, 0x1f564575, 0xc48dcf18,\ - 0xa13872c2, 0xe933bb17, 0x5d9ffd5b, 0xb5b6e10c,\ - 0x57fe3c00, 0xbaaaa15a, 0xe003ec3e, 0x9c269bae\ + 0x7fb07b5c, 0xd07c3bda, 0x553902e2, 0x7a87ea2c,\ + 0x35108a7f, 0x051f41e5, 0xb76abad5, 0x1f2703ad,\ + 0x0a251539, 0x5b4c4438, 0x952a634f, 0xac10dd4d,\ + 0x6d6f4745, 0x98990c27, 0x3a4f3116, 0xd32ff969\ ) /** Generator for secp256k1, value 'g' defined in * "Standards for Efficient Cryptography" (SEC2) 2.7.1. */ #define SECP256K1_G SECP256K1_GE_CONST(\ - 0x79BE667EUL, 0xF9DCBBACUL, 0x55A06295UL, 0xCE870B07UL,\ - 0x029BFCDBUL, 0x2DCE28D9UL, 0x59F2815BUL, 0x16F81798UL,\ - 0x483ADA77UL, 0x26A3C465UL, 0x5DA4FBFCUL, 0x0E1108A8UL,\ - 0xFD17B448UL, 0xA6855419UL, 0x9C47D08FUL, 0xFB10D4B8UL\ + 0x79be667e, 0xf9dcbbac, 0x55a06295, 0xce870b07,\ + 0x029bfcdb, 0x2dce28d9, 0x59f2815b, 0x16f81798,\ + 0x483ada77, 0x26a3c465, 0x5da4fbfc, 0x0e1108a8,\ + 0xfd17b448, 0xa6855419, 0x9c47d08f, 0xfb10d4b8\ ) /* 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 curve equation is of the form y^2=x^3+B for some small constant B. + * - The subgroup has a generator 2*P, where P.x is as small as possible. * - 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). - * - * These parameters are generated using sage/gen_exhaustive_groups.sage. */ #if defined(EXHAUSTIVE_TEST_ORDER) -# if EXHAUSTIVE_TEST_ORDER == 13 +# if EXHAUSTIVE_TEST_ORDER == 7 + +static const secp256k1_ge secp256k1_ge_const_g = SECP256K1_G_ORDER_7; + +static const secp256k1_fe secp256k1_fe_const_b = SECP256K1_FE_CONST( + 0x00000000, 0x00000000, 0x00000000, 0x00000000, + 0x00000000, 0x00000000, 0x00000000, 0x00000006 +); + +# elif EXHAUSTIVE_TEST_ORDER == 13 + static const secp256k1_ge secp256k1_ge_const_g = SECP256K1_G_ORDER_13; static const secp256k1_fe secp256k1_fe_const_b = SECP256K1_FE_CONST( - 0x3d3486b2, 0x159a9ca5, 0xc75638be, 0xb23a69bc, - 0x946a45ab, 0x24801247, 0xb4ed2b8e, 0x26b6a417 + 0x00000000, 0x00000000, 0x00000000, 0x00000000, + 0x00000000, 0x00000000, 0x00000000, 0x00000002 ); + # elif EXHAUSTIVE_TEST_ORDER == 199 + static const secp256k1_ge secp256k1_ge_const_g = SECP256K1_G_ORDER_199; static const secp256k1_fe secp256k1_fe_const_b = SECP256K1_FE_CONST( - 0x2cca28fa, 0xfc614b80, 0x2a3db42b, 0x00ba00b1, - 0xbea8d943, 0xdace9ab2, 0x9536daea, 0x0074defb + 0x00000000, 0x00000000, 0x00000000, 0x00000000, + 0x00000000, 0x00000000, 0x00000000, 0x00000004 ); + # else # error No known generator for the specified exhaustive test group order. # endif #else + static const secp256k1_ge secp256k1_ge_const_g = SECP256K1_G; -static const secp256k1_fe secp256k1_fe_const_b = SECP256K1_FE_CONST(0, 0, 0, 0, 0, 0, 0, 7); +static const secp256k1_fe secp256k1_fe_const_b = SECP256K1_FE_CONST( + 0x00000000, 0x00000000, 0x00000000, 0x00000000, + 0x00000000, 0x00000000, 0x00000000, 0x00000007 +); + #endif +/* End of section generated by sage/gen_exhaustive_groups.sage. */ static void secp256k1_ge_set_gej_zinv(secp256k1_ge *r, const secp256k1_gej *a, const secp256k1_fe *zi) { secp256k1_fe zi2; diff --git a/src/modules/recovery/tests_exhaustive_impl.h b/src/modules/recovery/tests_exhaustive_impl.h index ed9386b..142f713 100644 --- a/src/modules/recovery/tests_exhaustive_impl.h +++ b/src/modules/recovery/tests_exhaustive_impl.h @@ -43,8 +43,7 @@ void test_exhaustive_recovery_sign(const secp256k1_context *ctx, const secp256k1 (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 be the case for all points except the ones where - * R.x=1 (which the group is specifically selected to have). + * small group used here, it will almost certainly be the case for all points. * 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. */ diff --git a/src/precomputed_ecmult.h b/src/precomputed_ecmult.h index 949b62c..a4aa83e 100644 --- a/src/precomputed_ecmult.h +++ b/src/precomputed_ecmult.h @@ -13,7 +13,9 @@ extern "C" { #include "group.h" #if defined(EXHAUSTIVE_TEST_ORDER) -#if EXHAUSTIVE_TEST_ORDER == 13 +# if EXHAUSTIVE_TEST_ORDER == 7 +# define WINDOW_G 3 +# elif EXHAUSTIVE_TEST_ORDER == 13 # define WINDOW_G 4 # elif EXHAUSTIVE_TEST_ORDER == 199 # define WINDOW_G 8 diff --git a/src/scalar_impl.h b/src/scalar_impl.h index 3a57f56..9e72df2 100644 --- a/src/scalar_impl.h +++ b/src/scalar_impl.h @@ -33,15 +33,18 @@ static int secp256k1_scalar_set_b32_seckey(secp256k1_scalar *r, const unsigned c return (!overflow) & (!secp256k1_scalar_is_zero(r)); } -/* These parameters are generated using sage/gen_exhaustive_groups.sage. */ #if defined(EXHAUSTIVE_TEST_ORDER) -# if EXHAUSTIVE_TEST_ORDER == 13 +/* Begin of section generated by sage/gen_exhaustive_groups.sage. */ +# if EXHAUSTIVE_TEST_ORDER == 7 +# define EXHAUSTIVE_TEST_LAMBDA 2 +# elif EXHAUSTIVE_TEST_ORDER == 13 # define EXHAUSTIVE_TEST_LAMBDA 9 # elif EXHAUSTIVE_TEST_ORDER == 199 # define EXHAUSTIVE_TEST_LAMBDA 92 # else # error No known lambda for the specified exhaustive test group order. # endif +/* End of section generated by sage/gen_exhaustive_groups.sage. */ /** * Find r1 and r2 given k, such that r1 + r2 * lambda == k mod n; unlike in the