Extending class group action attacks via sesquilinear pairings

TL;DR

This paper introduces sesquilinear pairings as a new tool to analyze the security of class group action-based cryptographic problems involving elliptic curves with complex multiplication, extending prior research.

Contribution

It presents a novel sesquilinear pairing framework for studying isogeny-based cryptography, expanding the scope of security analysis for class group actions on elliptic curves.

Findings

Sesquilinear pairings provide new insights into elliptic curve security.

Extended analysis of class group action problems with these pairings.

Connections established with prior work on isogeny-based cryptography.

Abstract

We introduce a new tool for the study of isogeny-based cryptography, namely pairings which are sesquilinear (conjugate linear) with respect to the $\mathcal{O}$-module structure of an elliptic curve with CM by an imaginary quadratic order $\mathcal{O}$. We use these pairings to study the security of problems based on the class group action on collections of oriented ordinary or supersingular elliptic curves. This extends work of both (Castryck, Houben, Merz, Mula, Buuren, Vercauteren, 2023) and (De Feo, Fouotsa, Panny, 2024).