125 lines
4.7 KiB
Python
125 lines
4.7 KiB
Python
load("secp256k1_params.sage")
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orders_done = set()
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results = {}
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first = True
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for b in range(1, P):
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# There are only 6 curves (up to isomorphism) of the form y^2=x^3+B. Stop once we have tried all.
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if len(orders_done) == 6:
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break
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E = EllipticCurve(F, [0, b])
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print("Analyzing curve y^2 = x^3 + %i" % b)
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n = E.order()
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# Skip curves with an order we've already tried
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if n in orders_done:
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print("- Isomorphic to earlier curve")
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continue
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orders_done.add(n)
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# Skip curves isomorphic to the real secp256k1
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if n.is_pseudoprime():
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print(" - Isomorphic to secp256k1")
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continue
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print("- Finding subgroups")
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# Find what prime subgroups exist
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for f, _ in n.factor():
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print("- Analyzing subgroup of order %i" % f)
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# Skip subgroups of order >1000
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if f < 4 or f > 1000:
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print(" - Bad size")
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continue
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# Iterate over X coordinates until we find one that is on the curve, has order f,
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# and for which curve isomorphism exists that maps it to X coordinate 1.
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for x in range(1, P):
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# Skip X coordinates not on the curve, and construct the full point otherwise.
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if not E.is_x_coord(x):
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continue
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G = E.lift_x(F(x))
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print(" - Analyzing (multiples of) point with X=%i" % x)
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# Skip points whose order is not a multiple of f. Project the point to have
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# order f otherwise.
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if (G.order() % f):
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print(" - Bad order")
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continue
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G = G * (G.order() // f)
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# Find lambda for endomorphism. Skip if none can be found.
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lam = None
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for l in Integers(f)(1).nth_root(3, all=True):
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if int(l)*G == E(BETA*G[0], G[1]):
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lam = int(l)
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break
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if lam is None:
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print(" - No endomorphism for this subgroup")
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break
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# Now look for an isomorphism of the curve that gives this point an X
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# coordinate equal to 1.
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# 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.
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# So look for m=a^2=1/x.
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m = F(1)/G[0]
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if not m.is_square():
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print(" - No curve isomorphism maps it to a point with X=1")
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continue
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a = m.sqrt()
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rb = a^6*b
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RE = EllipticCurve(F, [0, rb])
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# Use as generator twice the image of G under the above isormorphism.
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# This means that generator*(1/2 mod f) will have X coordinate 1.
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RG = RE(1, a^3*G[1]) * 2
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# And even Y coordinate.
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if int(RG[1]) % 2:
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RG = -RG
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assert(RG.order() == f)
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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
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results[f] = {"b": rb, "G": RG, "lambda": lam}
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print(" - Found solution")
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break
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print("")
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print("")
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print("")
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print("/* To be put in src/group_impl.h: */")
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first = True
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for f in sorted(results.keys()):
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b = results[f]["b"]
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G = results[f]["G"]
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print("# %s EXHAUSTIVE_TEST_ORDER == %i" % ("if" if first else "elif", f))
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first = False
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print("static const secp256k1_ge secp256k1_ge_const_g = SECP256K1_GE_CONST(")
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print(" 0x%08x, 0x%08x, 0x%08x, 0x%08x," % tuple((int(G[0]) >> (32 * (7 - i))) & 0xffffffff for i in range(4)))
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print(" 0x%08x, 0x%08x, 0x%08x, 0x%08x," % tuple((int(G[0]) >> (32 * (7 - i))) & 0xffffffff for i in range(4, 8)))
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print(" 0x%08x, 0x%08x, 0x%08x, 0x%08x," % tuple((int(G[1]) >> (32 * (7 - i))) & 0xffffffff for i in range(4)))
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print(" 0x%08x, 0x%08x, 0x%08x, 0x%08x" % tuple((int(G[1]) >> (32 * (7 - i))) & 0xffffffff for i in range(4, 8)))
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print(");")
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print("static const secp256k1_fe secp256k1_fe_const_b = SECP256K1_FE_CONST(")
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print(" 0x%08x, 0x%08x, 0x%08x, 0x%08x," % tuple((int(b) >> (32 * (7 - i))) & 0xffffffff for i in range(4)))
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print(" 0x%08x, 0x%08x, 0x%08x, 0x%08x" % tuple((int(b) >> (32 * (7 - i))) & 0xffffffff for i in range(4, 8)))
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print(");")
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print("# else")
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print("# error No known generator for the specified exhaustive test group order.")
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print("# endif")
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print("")
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print("")
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print("/* To be put in src/scalar_impl.h: */")
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first = True
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for f in sorted(results.keys()):
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lam = results[f]["lambda"]
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print("# %s EXHAUSTIVE_TEST_ORDER == %i" % ("if" if first else "elif", f))
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first = False
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print("# define EXHAUSTIVE_TEST_LAMBDA %i" % lam)
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print("# else")
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print("# error No known lambda for the specified exhaustive test group order.")
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print("# endif")
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print("")
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