Isogenies of Abelian Varieties and Post-Quantum Cryptography
This is a group of researchers in mathematics and computer science interested in Isogenies of Abelian Varieties. We have organized several conferences and are focused on exploring the isogeny based post-quantum cryptography. Â
Isogeny based cryptography is based on computing isogenies among Abelian varieties. More isogenies an Abelian variety has the harder it is to "guess" another Abelian variety isogenous to the first one. So naturally one is interested on Abelian varieties with large endomorphism rings. The most basic case is that of elliptic curves which gives rise to SIDH (Supersingular Isogeny Diffee Hellman) and SIKE (Supersingular Isogeny Key Exchange). SIKE was broken in the Summer 2022, using a construction of E. Kani, glueing of two elliptic curves E_1 and E_2 into a genus 2 curve C and then checking if this genus 2 curve has (3,3)-split Jacobian by pluging the Igusa invariants of the curve into the equation of the locus of genus 2 curves with (3, 3)-split Jacobians computed by T. Shaska in his thesis (2001). This determines if there is a degree n=3 isogeny between E_1 and E_2. However, isogeny based cryptography is not dead and provides one of the most promising methods for post-quantum cryptography.
The goal of this conference is explore the isogeny based cryptography for Abelian varieties of dimension > 1. One of the main reasons that it took over ten years to break SIKE was because there was a disconnect between people involved in cryptography and the mathematicians who had studied genus 2 decomposable Jacobians and had expertise in computing with genus 2 curves. The mathematical results were available in 2012 when SIKE was first introduced and it should have been broken then instead of spending 10 years later.  The aim of this conference is to avoid that for higher dimensional Abelian varieties. We like to bring together mathematicians who are experts in the area of Abelian varieties and cryptographers who are focused on isogeny based cryptography.Â
Literature
We expect all of the students who will attend the conference to familiar with some of the basics of Abelian varieties and their use in cryptography. You can read the following, among other works, as a preparation for the conference
Frey, Gerhard; Shaska, Tony; Curves, Jacobians, and cryptography. Algebraic curves and their applications, 279–344, Contemp. Math., 724, Amer. Math. Soc., [Providence], RI, 2019.