Given a database of $N$ entries of which exactly one satisfies some
easily checked condition, classically it takes $O(N)$ trials to find the
satisfying entry. Grover's algorithm is a quantum algorithm which
reduces the work to $O(\sqrt{N})$ trials. One consequence is that in the
post-quantum regime, hash functions and symmetric ciphers only provide
half the security (measured as the log of the number of trials) as
currently provided. In this talk, we will give a brief description of
Grover's algorithm, including all of the necessary background in quantum
computing.

In a series of papers, Kani studies elliptic subcovers of genus 2 curves. This talk will focus on summarizing this work and it's applications to a recent question of Galbraith concerning isogeny graphs of abelian surfaces. Let A be a Jacobian of a curve of genus 2 which is isogeneous, but not isomorphic, to a product of elliptic curves. Galbraith's question is how "far" is A from a product of ellitpic curves in the $\ell$-isogeny graph for a small prime $\ell$.

We will continue discussing the paper by Eisentraeger-Hallgren-Lauter-Morrison-Petit (available at https://eprint.iacr.org/2018/371). In this part of the talk, we will focus on the reductions between the major problems. We will also outline the supplementary algorithms used as a black box in these reductions.

This talk will be more of a discussion on some open problems we have seen so far. It should be independent of the previous talks. We will focus on a few of the problems on class groups and isogenies referred to in the Altug-Chen paper discussed on 2/22/19. The preprint is available here: https://eprint.iacr.org/2018/926 and notes from the previous talk can be found here: https://www.math.uci.edu/~schollt/multilinear_map_seminar/scholl-02-22-1.... The problems we will focus on are finding an elliptic curve with a specified endomorphism ring, and finding l-isogenies over composite rings.