Paper 2024/778

Ideal-to-isogeny algorithm using 2-dimensional isogenies and its application to SQIsign

Hiroshi Onuki, The University of Tokyo
Kohei Nakagawa, NTT Social Informatics Laboratories
Abstract

The Deuring correspondence is a correspondence between supersingular elliptic curves and quaternion orders. Under this correspondence, an isogeny between elliptic curves corresponds to a quaternion ideal. This correspondence plays an important role in isogeny-based cryptography and several algorithms to compute an isogeny corresponding to a quaternion ideal (ideal-to-isogeny algorithms) have been proposed. In particular, SQIsign is a signature scheme based on the Deuring correspondence and uses an ideal-to-isogeny algorithm. In this paper, we propose a novel ideal-to-isogeny algorithm using isogenies of dimension $2$. Our algorithm is based on Kani's reducibility theorem, which gives a connection between isogenies of dimension $1$ and $2$. By using the characteristic $p$ of the base field of the form $2^fg - 1$ for a small odd integer $g$, our algorithm works by only $2$-isogenies and $(2, 2)$-isogenies in the operations in $\mathbb{F}_{p^2}$. We apply our algorithm to SQIsign and compare the efficiency of the new algorithm with the existing one. Our analysis shows that the key generation and the signing in our algorithm are at least twice as fast as those in the existing algorithm at the NIST security level 1. This advantage becomes more significant at higher security levels. In addition, our algorithm also improves the efficiency of the verification in SQIsign.

Metadata
Available format(s)
PDF
Category
Public-key cryptography
Publication info
Preprint.
Keywords
post-quantum cryptographySQIsignthe Deuring correspondenceKani's theorem
Contact author(s)
onuki @ mist i u-tokyo ac jp
kohei nakagawa @ ntt com
History
2024-06-06: revised
2024-05-21: received
See all versions
Short URL
https://ia.cr/2024/778
License
Creative Commons Attribution
CC BY

BibTeX

@misc{cryptoeprint:2024/778,
      author = {Hiroshi Onuki and Kohei Nakagawa},
      title = {Ideal-to-isogeny algorithm using 2-dimensional isogenies and its application to {SQIsign}},
      howpublished = {Cryptology ePrint Archive, Paper 2024/778},
      year = {2024},
      note = {\url{https://eprint.iacr.org/2024/778}},
      url = {https://eprint.iacr.org/2024/778}
}
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