add working ordp function

script/research/ec/ordp.sage

@@ -0,0 +1,68 @@
+from tabulate import tabulate
+# This is a more usable version of valuate.sage, less instructional
+
+K.<x, y> = GF(11)[]
+Px, Py = K(2), K(4)
+S = K.quotient(y^2 - x^3 - 4*x).fraction_field()
+X, Y = S(x), S(y)
+
+EC_A = 4
+EC_B = 0
+EC = y^2 - x^3 - EC_A*x - EC_B
+
+original_f = (y - 2*x)^2
+b0, b1, b2 = [(x - Px), (y - Py), 1]
+
+# Return components for basis
+def decomp(f):
+    a0, r = f.quo_rem(b0)
+    a1, r = r.quo_rem(b1)
+    a2, r = r.quo_rem(b2)
+    assert r == 0
+    return [a0, a1, a2]
+
+def comp(comps):
+    return sum(a*b for a, b in zip(comps, (b0, b1, b2)))
+
+assert comp(decomp(original_f)) == original_f
+
+# so we can replace (y - Py) with this
+Ef = b0^2 + binomial(3,2)*Px*b0^1 + (3*Px^2 + EC_A)
+Eg = (y + Py)
+assert EC == b1*Eg - b0*Ef
+
+def apply_reduction(a, g):
+    assert a[2] == 0
+    a[0] = a[0]*Eg + a[1]*Ef
+    a[1] = 0
+    g[0] *= Eg
+
+k = 0
+a = [original_f, 0, 0]
+g = [1]
+
+table = []
+table.append(("", "a", "g", "k"))
+
+def log(step_name, a, g, k):
+    table.append((step_name, str(a), str(g), k))
+
+log("start", a, g, k)
+
+while True:
+    f = a[0]
+    a = decomp(f)
+    log("decomp", a, g, k)
+
+    # Check remainder
+    if a[2] != 0:
+        break
+
+    # We can apply a reduction
+    k += 1
+
+    apply_reduction(a, g)
+    log("reduce", a, g, k)
+
+print(tabulate(table))
+print(f"k = {k}")

script/research/ec/valuate.sage

@@ -42,9 +42,10 @@ assert comp(decomp(original_f, basis), basis) == original_f
 EC_A = 4
 EC_B = 0
 
-EC = y^2 - x^3 - A*x - B
+EC = y^2 - x^3 - EC_A*x - EC_B
 
 # so we can replace (y - Py) with this
+b0, b1, b2 = basis
 Ef = b0^2 + binomial(3,2)*Px*b0^1 + (3*Px^2 + EC_A)
 Eg = (y + Py)
 assert EC == b1*Eg - b0*Ef