An Algorithmic Approach to (2, 2)-isogenies in the Theta Model and Applications to Isogeny-based Cryptography

Pierrick Dartois (1, 2, 3) , Luciano Maino (4) , Giacomo Pope (5, 4) , Damien Robert (1, 3, 2, 6)

1 IMB - Institut de Mathématiques de Bordeaux
2 CANARI - Analyse cryptographique et arithmétique
3 LFANT - Lithe and fast algorithmic number theory
4 University of Bristol [Bristol]
5 NCC Group
6 Inria Bordeaux - Sud-Ouest

In this paper, we describe an algorithm to compute chains of (2, 2)-isogenies between products of elliptic curves in the theta model. The description of the algorithm is split into various subroutines to allow for a precise field operation counting. We present a constant time implementation of our algorithm in Rust and an alternative implementation in SageMath. Our work in SageMath runs ten times faster than a comparable implementation of an isogeny chain using the Richelot correspondence. The Rust implementation runs up to forty times faster than the equivalent isogeny in SageMath and has been designed to be portable for future research in higher-dimensional isogeny-based cryptography.

Mots clés: Post-Quantum Cryptography; Isogenies; Theta coordinates; Dimension 2

Domaines: Mathématiques [math]; Informatique [cs]

Pierrick Dartois, Luciano Maino, Giacomo Pope, Damien Robert. An Algorithmic Approach to (2, 2)-isogenies in the Theta Model and Applications to Isogeny-based Cryptography. Advances in Cryptology – ASIACRYPT 2024, 15486, Springer Nature Singapore, pp.304-338, 2025, Lecture Notes in Computer Science, ⟨10.1007/978-981-96-0891-1_10⟩. ⟨hal-04297088⟩