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gen_exhaustive_groups.sage
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gen_exhaustive_groups.sage
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load("secp256k1_params.sage")
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("")

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