Paper 2024/948
Return of the Kummer: a toolbox for genus 2 cryptography
Abstract
This work expands the machinery we have for isogeny-based cryptography in genus 2 by developing a toolbox of several essential algorithms for Kummer surfaces, the dimension 2 analogue of x-only arithmetic on elliptic curves. Kummer surfaces have been suggested in (hyper-)elliptic curve cryptography since at least the 1980s and recently these surfaces have reappeared to efficiently compute (2,2)-isogenies. We construct several essential analogues of techniques used in one-dimensional isogeny-based cryptography, such as pairings, deterministic point sampling and point compression and give an overview of (2,2)-isogenies on Kummer surfaces. We furthermore show how Scholten's construction can be used to transform isogeny-based cryptography over elliptic curves over $\mathbb{F}_{p^2}$ into protocols over Kummer surfaces over $\mathbb{F}_p$. As an example of this approach, we demonstrate that SQIsign verification can be performed completely on Kummer surfaces, and, therefore, that one-dimensional SQIsign verification can be viewed as a two-dimensional isogeny between products of elliptic curves. Curiously, the isogeny is then defined over $\mathbb{F}_p$ rather than $\mathbb{F}_{p^2}$. Contrary to expectation, the cost of SQIsign verification using Kummer surfaces does not explode: verification costs only 1.5 times more in terms of finite field operations than the SQIsign variant AprèsSQI, optimised for fast verification. Furthermore, as Kummer surfaces allow a much higher degree of parallelization, Kummer-based protocols over $\mathbb{F}_p$ could potentially outperform elliptic curve analogues over $\mathbb{F}_{p^2}$ in terms of clock cycles and actual performance.
Metadata
- Available format(s)
- Category
- Public-key cryptography
- Publication info
- Preprint.
- Keywords
- post-quantum cryptographyisogeniesKummer surfaceSQIsigngenus 2
- Contact author(s)
-
maria santos 20 @ ucl ac uk
krijn @ cs ru nl - History
- 2024-06-13: revised
- 2024-06-13: received
- See all versions
- Short URL
- https://ia.cr/2024/948
- License
-
CC0
BibTeX
@misc{cryptoeprint:2024/948, author = {Maria Corte-Real Santos and Krijn Reijnders}, title = {Return of the Kummer: a toolbox for genus 2 cryptography}, howpublished = {Cryptology ePrint Archive, Paper 2024/948}, year = {2024}, note = {\url{https://eprint.iacr.org/2024/948}}, url = {https://eprint.iacr.org/2024/948} }