Wikipedia:Reference desk/Archives/Mathematics/2024 November 21

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November 21

Is it possible to adapt Nigel’s Smart algorithm for establshing an isomorphism when the curve is only partially anomalous ?

An anomalous elliptic curve is a curve for which $\#E(\mathbf {F} _{q})=q$. But in my case, the curve has order j×q and the underlying field has order i×q. In the situation I’m thinking about, I do have 2 points such as both G∈q and P∈q subgroup and where P=s×G.

So since the scalar $s$ lies in a common part of the additive group from both the curve along it’s underlying base field, is it possible to transfer the discrete logarithm to the underlying finite field ? Or does anomalous curves requires the whole embedding field’s order to match the one of the curve even if the discrete logarithm solution lies into a common smaller group ?

If yes, how to adapt the Nigel’s smart algorithm used for solving the discrete logarithm inside anomalous curves ? The aim is to etablish an isomorphism between the common subgroup generated by E and $F_{p}$ 82.66.26.199 (talk) 19:47, 21 November 2024 (UTC)