Fast computation of 2-isogenies in dimension 4 and cryptographic applications
<|reference_start|>Fast computation of 2-isogenies in dimension 4 and cryptographic applications: 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.<|reference_end|>
@article{dartois2024fast,
    title={Fast computation of 2-isogenies in dimension 4 and cryptographic
  applications},
    author={Pierrick Dartois},
    journal={arXiv preprint arXiv:2407.15492},
    year={2024},
    archivePrefix={arXiv},
    eprint={2407.15492},
    primaryClass={cs.CR math.AG}
}