A student of mine is trying to use Sage to prove the commutativity of the
group law of elliptic curves. Below is the code used when trying to show
(P + Q) + R = P + (Q + R), when R = P + Q. I let it run for almost 4 hours
with no result. The same code translated to Magma was instantaneous. Am I
doing something wrong with Sage? Or is just that the implementation in
Magma is more efficient? (We can use Magma, but it made me curious.)
def add_EC(vE, P, Q):
if P == 0:
return Q
if Q == 0:
return P
a = vE[0]
b = vE[1]
x1 = P[0]
y1 = P[1]
x2 = Q[0]
y2 = Q[1]
if P == Q:
if y1 == 0:
return 0
else:
m = (3*x1^2 + a)/(2*y1)
else:
if x1 == x2:
return 0
else:
m = (y2 - y1)/(x2 - x1)
c = y1 - m*x1
x3 = m^2 - x1 - x2
y3 = m*x3 + c
return [x3, -y3]
R.<A, B, a, b, c, d> = PolynomialRing(QQ)
eq1 = b^2 - a^3 - A*a - B
eq2 = d^2 - c^3 - A*c - B
S = R.quo([eq1, eq2])
F = FractionField(S)
vE = [F(A), F(B)]
P = [F(a), F(b)]
Q = [F(c), F(d)]
R = add_EC(vE, P, Q)
LHS = add_EC(vE, add_EC(vE, P, Q), R)
RHS = add_EC(vE, P, add_EC(vE, Q, R))
LHS == RHS
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