Motivated by the success of using separable isogenies between supersingular elliptic curves to build (conjecturally) secure cryptography, we look at generalising this to arbitrary dimension. In particular, we study superspecial principally polarised abelian varieties of dimension g and the polarised isogenies between them.
In this talk, we first detail the isogeny problem in dimension two: given two principally polarised abelian surfaces defined over a finite field, find a polarised isogeny connecting them. We show how this problem relates to the problem of finding Jacobians of genus 2 curves which are (N,N)-split, and present an algorithm to solve it.
To construct efficient cryptographic protocols from this dimension 2 problem, we work instead with the Kummer surface associated to the Jacobian of a genus 2 curve, and develop algorithms to compute (N,N)-isogenies between Kummer surfaces given their kernel.
This is based on joint work with Costello, Flynn, Frengley and Smith.
