Isogeny Club - Season 4 - Talk #6:Using Kummer surfaces in SQIsign verification

This is the discussion thread for the sixth and last session of season four of The Isogeny Club, where I presented a talk on using Kummer surfaces in SQIsign verification.

The use of Kummer surfaces in isogeny-based cryptography dates back to a suggestion by the Chudnovsky brothers in the 1980s. Later, Gaudry showed that Kummer surfaces can be used for high-speed curve cryptography, and subsequent works improved the usage of Kummer surfaces in cryptography. With the recent interest in higher-dimensional isogenies, Kummer surfaces have also appeared as tools for efficient (2,2)-isogenies between Jacobians of genus-2 curves, and similarly, Costello suggested the usage of Kummer surfaces for SIDH already in 2018.

In this talk, we go over these developments to give an introduction to Kummer surfaces and their use in cryptography, and their connection to theta structures. We then apply Scholten’s construction to SQIsign verification, which allows us to perform this verification on Kummer surfaces over’ 𝔽p, instead of elliptic curves over 𝔽p2. This requires us to develop a list of non-trivial computational tools on Kummer surfaces, which have general applications for higher-dimensional isogeny-based cryptography. In particular, we describe how pairings can be used to decompose Jacobians, which allows for efficient sampling of torsion points.


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