aboutsummaryrefslogtreecommitdiff
path: root/src/secp256k1/sage
diff options
context:
space:
mode:
Diffstat (limited to 'src/secp256k1/sage')
-rw-r--r--src/secp256k1/sage/gen_exhaustive_groups.sage129
1 files changed, 129 insertions, 0 deletions
diff --git a/src/secp256k1/sage/gen_exhaustive_groups.sage b/src/secp256k1/sage/gen_exhaustive_groups.sage
new file mode 100644
index 0000000000..3c3c984811
--- /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("")