Isogenies between Hessian curves

Abstract

Elliptic curves are used in post-quantum cryptography, where two parties can use compositions of low-degree isogenies to establish a shared secret. There are several forms for representing elliptic curves, and different forms require different isogeny formulas. This thesis is concerned with the Hessian form of elliptic curves, and explicit formulas for isogenies between these. A formula for n-isogenies between Hessian curves has recently been found for n not divisible by 3 [Bro+21]. We derive a new formula for 3-isogenies between Hessian curves. We also derive new formulas for 2- and 4-isogenies that results in a simpler formula for the latter case, compared to [Bro+21]. We find that any representative for 2-isogenies must have indeterminacies. We give a globally defined formula for morphisms that are 2-isogenies followed by translation with a 2-torsion point. In addition we describe how such morphisms can be used to construct isogenies of degree 2^e for some positive integer e, similar to how 2-isogenies are used in cryptography to construct isogenies of degree 2^e.