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@@ -0,0 +1,91 @@
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+def random_mersenne_prime():
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+ while True:
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+ p = random_prime(100, 200)
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+ m = 2^p - 1
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+ if is_prime(m):
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+ return m
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+
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+#p = random_mersenne_prime()
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+p = 8191
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+
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+# -1 should not be a quadratic residue modulo p
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+assert legendre_symbol(-1, p) == -1
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+assert p % 4 == 3
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+
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+F = GF(p)
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+R.<x> = F[]
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+K.<i> = F.extension(x^2 + 1)
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+
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+def get_point(t):
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+ x = (1 - t^2)/(1 + t^2)
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+ y = 2*t / (1 + t^2)
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+ return (x, y)
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+
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+(p1_x, p1_y) = get_point(K(3))
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+assert p1_x^2 + p1_y^2 == 1
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+z = p1_x + i*p1_y
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+
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+(p2_x, p2_y) = get_point(K(7))
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+assert p2_x^2 + p2_y^2 == 1
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+w = p2_x + i*p2_y
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+
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+def abs(z):
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+ return z[0]^2 + z[1]^2
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+
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+# z and w are now elements of F_p(i) which is a field
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+assert abs(z) == 1
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+assert abs(w) == 1
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+assert abs(z * w) == 1
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+
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+# We can also construct the inverse
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+def conjugate(z):
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+ return z[0] - i*z[1]
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+# Remember that (x + iy)(x - iy) = x^2 - i^2 y^2 = x^2 + y^2
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+assert z * conjugate(z) in F
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+# Now we find the multiplicative inverse
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+z_inv = conjugate(z) / (z * conjugate(z))
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+assert z * z_inv == 1
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+
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+# Size of K is p^2
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+assert len(K) == p^2
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+
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+# Because in sage we cannot construct a homomorphism to GF(p^2) directly
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+# we instead construct the isomorphic field extension, and use that instead.
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+Fp2.<a> = GF(p^2)
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+conway_fp2 = a.minimal_polynomial()
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+L.<j> = F.extension(conway_fp2(x=x))
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+
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+y = L.multiplicative_generator()
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+phi = K.hom([y^(y.multiplicative_order()/i.multiplicative_order())])
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+# K ≌ GF(p^2)
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+assert phi.is_injective() and phi.is_surjective()
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+
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+assert a.multiplicative_order() == p^2 - 1
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+g_K = phi.inverse()(y)
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+g1 = g_K^int((p^2 - 1)/(p + 1))
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+assert abs(g1) == 1
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+
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+g2 = K.multiplicative_generator()
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+g2 = g2^(p - 1)
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+assert abs(g2) == 1
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+
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+# bug in sage where .unit_group is missing
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+# https://ask.sagemath.org/question/62822/make-morphism-from-gfp2s-multiplicative-group-to-gfps-multiplicative-group/
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+C = AbelianGroup([p + 1])
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+g, = C.gens()
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+while True:
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+ print(f"Group of order = {g.order()}")
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+ for C2 in C.subgroups():
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+ print(f" {C2}")
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+ print()
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+
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+ if g.order() == 1:
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+ break
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+
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+ # For some annoying reason, I cannot iterate on subgroups
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+ # C = C.subgroup([g^2])
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+ # Trying to get the subgroup of this will give me some error.
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+ # Lets just construct it manually.
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+ C = AbelianGroup([(g^2).order()])
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+ g, = C.gens()
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+
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darkfi