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Extending class group action attacks via sesquilinear pairings

Joseph Macula, Katherine E. Stange

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).

Extending class group action attacks via sesquilinear pairings

Abstract

We introduce a new tool for the study of isogeny-based cryptography, namely pairings which are sesquilinear (conjugate linear) with respect to the -module structure of an elliptic curve with CM by an imaginary quadratic order . 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).
Paper Structure (9 sections, 4 theorems, 8 equations)

This paper contains 9 sections, 4 theorems, 8 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Notations.
  4. Orientations.
  5. Cyclic self-pairings
  6. Level Structure
  7. Computational assumptions
  8. The Tate-Lichtenbaum Frey-Rück Pairing.
  9. Structure of $E[\alpha]$

Key Result

proposition thmcounterproposition

Definition defn: tate is well-defined, and has the following properties:

Theorems & Definitions (7)

  • definition thmcounterdefinition: Level
  • definition thmcounterdefinition
  • proposition thmcounterproposition
  • theorem thmcountertheorem: Lenstra
  • theorem thmcountertheorem: Lenstra
  • theorem thmcountertheorem
  • proof