Wikipedia:Reference desk/Archives/Mathematics/2024 July 7

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July 7

Using sagemath or an other language, how to exactly find out what the order of the base point of an elliptic curve in Edwards Form is ?

This kind of code will do it for the usual Weirestrass form :

a = 1
b = 3141592653589793238462643383279502884197169399375105820974944592307816406665
p = 2^251 + 17*2^192 +1

E = EllipticCurve(GF(p), [0,0,0,a,b])
print(E)
print(E.abelian_group())

card = E.cardinality()
print("cardinality =",card)
factor(card)

G = E(874739451078007766457464989774322083649278607533249481151382481072868806602,152666792071518830868575557812948353041420400780739481342941381225525861407)
print("Generator order q=", G.order())

But how to do it for a curve in the twisted Edwards form ? Because I suppose converting the curve and the point to the Weirestrass form would change the resulting order being computed right ? 2A01:E0A:401:A7C0:DD6F:EA1B:CCA4:2633 (talk) 21:12, 7 July 2024 (UTC)

I'm not an expert, but I'd think that the group is isomorphic to the Weierstrass group by which it is induced.  --Lambiam 11:07, 8 July 2024 (UTC)