Abstract: Classical algebraic number theory, developed throughout the 1800s, has found an exciting new application in the 2000s: cryptographic systems with attractive features like high efficiency, 'post-quantum' security, and the ability to compute on encrypted data.

This talk will recap some of the foundational ideas and results in this area, like the connection between "Ring-LWE" and problems on "ideal lattices" (i.e., ideals in number rings), and how they give rise to encrypted computation. The speaker will also share some of his favorite open problems.

(A basic understanding of number fields, their rings of integers and ideals, and how these give rise to lattices, will be very helpful but not absolutely necessary.)