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**2024/948**

**Title:**

Return of the Kummer: a toolbox for genus 2 cryptography

**Authors:**
Maria Corte-Real Santos, Krijn Reijnders

**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.

**ePrint:**
https://eprint.iacr.org/2024/948

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