### Isogeny graphs of abelian varieties and applications to the discrete logarithm problem

Heilbronn Number Theory Seminar

13th December 2017, 4:00 pm – 5:00 pm

Howard House, 4th Floor Seminar Room

An isogeny graph is a graph whose vertices correspond to abelian varieties (with some structure) and whose edges correspond to isogenies of a certain degree. The structure of isogeny graphs for elliptic curves was first studied by David Kohel in his PhD thesis and has since been an essential tool in algorithmic number theory (for example to compute the endomorphism ring of an ordinary elliptic curve over a finite field) and cryptography (for example in the post-quantum cryptographic protocol SIDH). We present a structure theorem for isogeny graphs of ordinary abelian varieties, and explain how, under some heuristic assumptions, this can be applied to breaking the discrete logarithm problem for genus 3 curves and possibly some families of elliptic curves.

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