Paper 2025/543 Original: eprint.iacr.org/2025/543 ↗

Models of Kummer lines and Galois representations

Abstract

In order to compute a multiple of a point on an elliptic curve in Weierstrass form one can use formulas in only one of the two coordinates of the points. These $x$-only formulas can be seen as an arithmetic on the Kummer line associated to the curve. In this paper, we look at models of Kummer lines, and define an intrinsic notion of isomorphisms of Kummer lines. This allows us to give conversion formulas between Kummer models in a unified manner. When there is one rational point $T$ of $2$-torsion on the curve, we also use Mumford's theory of theta groups to show that there are two type of models: the “symmetric” ones with respect to $T$ and the “anti-symmetric“ ones. We show how this recovers the Montgomery model and various variants of the theta model. We also classify when curves admit these different models via Galois representations and modular curves. When an elliptic curve is viewed inside a $2$-isogeny volcano, we give a criteria to say if it has a given Kummer model based solely on its position in the volcano. We also give applications to the ECM factorization algorithm.