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Switch to exhaustive groups with small B coefficient
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| Original file line number | Diff line number | Diff line change |
|---|---|---|
| @@ -1,124 +1,161 @@ | ||
| load("secp256k1_params.sage") | ||
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| MAX_ORDER = 1000 | ||
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| # Set of (curve) orders we have encountered so far. | ||
| orders_done = set() | ||
| results = {} | ||
| first = True | ||
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| # Map from (subgroup) orders to [b, int(gen.x), int(gen.y), gen, lambda] for those subgroups. | ||
| solutions = {} | ||
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| # 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 | ||
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| E = EllipticCurve(F, [0, b]) | ||
| print("Analyzing curve y^2 = x^3 + %i" % b) | ||
| n = E.order() | ||
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| # 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) | ||
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| # 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 | ||
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| print("- Finding subgroups") | ||
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| # 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 | ||
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| # 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("- Finding prime subgroups") | ||
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| print(" - Analyzing (multiples of) point with X=%i" % x) | ||
| # Map from group_order to a set of independent generators for that order. | ||
| curve_gens = {} | ||
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| # 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") | ||
| 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 | ||
| if f > MAX_ORDER: | ||
| print(f" - Subgroup of size {f}: too large") | ||
| continue | ||
| G = G * (G.order() // f) | ||
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| # Construct a generator for that subgroup. | ||
| gen = g * (g_order // f) | ||
| assert(gen.order() == f) | ||
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| # 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") | ||
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| 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 | ||
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| # 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() | ||
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| # 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 | ||
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| if lam is None: | ||
| print(" - No endomorphism for this subgroup") | ||
| break | ||
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| # 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]) | ||
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| # 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])) | ||
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| # 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 | ||
| no_endo_count += 1 | ||
| else: | ||
| sol_count += 1 | ||
| solutions.setdefault(f, []).append((b, int(gen[0]), int(gen[1]), gen, lam)) | ||
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| print("") | ||
| print(f" - Found {sol_count} generators (plus {no_endo_count} without endomorphism)") | ||
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| 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() | ||
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| 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(")") | ||
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| 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(");") | ||
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| 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") | ||
|
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||
| print("#else") | ||
| print() | ||
| print("static const secp256k1_ge secp256k1_ge_const_g = SECP256K1_G;") | ||
| print("") | ||
| print("") | ||
| print("/* To be put in src/scalar_impl.h: */") | ||
| output_b(7) | ||
| print() | ||
| print("#endif") | ||
| print("/* End of section generated by sage/gen_exhaustive_groups.sage. */") | ||
|
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|
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| 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. */") |
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