Cryptographic applications of multilinear maps beyond bilinear pairings were first proposed in the work of Boneh and Silverberg. However the existence of cryptographically interesting $n$-multilinear maps for $n > 2$ remains an open problem. Very recently Lin and Tessaro showed that trilinear maps are sufficient for the purpose of achieving indistinguishability obfuscation. This striking result put spotlight on the following question: can a cryptographically interesting algebraic trilinear map be constructed? In this talk we discuss a method for constructing such a trilinear map, and present concrete candidate trilinear maps which involve Weil descent and the Jacobian varieties of hyperelliptic curves. A preprint is available here: https://arxiv.org/abs/1810.03646.

The McEliece cryptosystem is the first code-based public key cryptosystem proposed by Robert McEliece in 1978 a few years after the appearance of RSA. The original McEliece cryptosystem uses binary Goppa codes which are a subclass of Algebraic Geometric Codes and it is still unbroken under quantum attack. In this talk, we introduce basic facts about coding theory and discuss various code-based public key cryptosystems including our new cryptosystem McNie, which is a combination of the McEliece cryptosystem and the Niederreiter cryptosystem.

In this talk, we will summarize a candidate multilinear map construction due to Gentry-Gorbunov-Halevi (GGH15). While the original key exchange protocol using GGH15 is broken, new safeguards have recently been proposed which aim to prevent this and other ``zeroizing'' attacks. Our goal will be to understand the construction, attack, and proposed safeguards.

In this talk we will introduce the Weil pairing on an elliptic curve and give several cryptographic applications. We will review the argument of Boneh and Silverberg which suggests that this kind of pairing does not exist naturally on higher dimensional varieties. We will also look at some constructions of pairing-friendly elliptic curves.