Fast computation of 2-isogenies in dimension 4 and cryptographic applications

TL;DR

This paper introduces efficient algorithms for computing chains of 2-isogenies in dimension 4 using theta-coordinates, enabling practical cryptanalysis of SIDH and related schemes, which was previously computationally infeasible.

Contribution

It generalizes 2-isogeny computation algorithms to dimension 4 with theta-coordinates, facilitating cryptanalysis of SIDH and related cryptographic protocols.

Findings

Able to perform SIDH key recovery attacks in seconds on standard parameters

Developed algorithms for chains of 2-isogenies in dimension 4

Implemented practical cryptanalysis tools for SIDH and SQIsignHD

Abstract

Dimension 4 isogenies have first been introduced in cryptography for the cryptanalysis of Supersingular Isogeny Diffie-Hellman (SIDH) and have been used constructively in several schemes, including SQIsignHD, a derivative of SQIsign isogeny based signature scheme. Unlike in dimensions 2 and 3, we can no longer rely on the Jacobian model and its derivatives to compute isogenies. In dimension 4 (and higher), we can only use theta-models. Previous works by Romain Cosset, David Lubicz and Damien Robert have focused on the computation of $\ell$-isogenies in theta-models of level $n$ coprime to $\ell$ (which requires to use $n^g$ coordinates in dimension $g$). For cryptographic applications, we need to compute chains of $2$-isogenies, requiring to use $\geq 3^g$ coordinates in dimension $g$ with state of the art algorithms. In this paper, we present algorithms to compute chains of $2$-isogenies between abelian varieties of dimension $g\geq 1$ with theta-coordinates of level $n=2$, generalizing a previous work by Pierrick Dartois, Luciano Maino, Giacomo Pope and Damien Robert in dimension $g=2$. We propose an implementation of these algorithms in dimension $g=4$ to compute endomorphisms of elliptic curve products derived from Kani's lemma with applications to SQIsignHD and SIDH cryptanalysis. We are now able to run a complete key recovery attack on SIDH when the endomorphism ring of the starting curve is unknown within a few seconds on a laptop for all NIST SIKE parameters.