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.
Multilinear maps is a new hot topic in cryptography because they offer a significant number of applications. The main open problem in this area is constructing a secure and efficiently computable multilinear map. In this talk, we introduce cryptographic multilinear maps, go through several applications, and then discuss some possible obstructions to constructing one. The main reference for this talk is the paper "Applications of Multilinear Forms to Cryptography" by Dan Boneh and Alice Silverberg.
Despite widespread interest in cryptographic multilinear maps since
Boneh-Silverberg's 2003 paper, very few candidate maps have been
discovered. The first serious candidate was a scheme of
Garg-Gentry-Halevi (GGH), which is based on ideal lattices in cyclotomic
number rings. While the scheme was later shown to be broken, the only
other candidate schemes are hardened variants of GGH. We give a
relatively detailed description of the GGH multilinear map.
Elliptic curve cryptography (ECC) is a widely used public key cryptosystem. The security of ECC relies on the difficulty of the elliptic curve discrete log problem (ECDLP). Isogenies are morphisms of curves that can be used to transfer instances of ECDLP between elliptic curves. Suppose that we suspect that some proportion of curves are "weak" in the sense that the ECDLP can be solved quickly. To avoid an attacker moving the ECDLP to a weak curve, we would want to use curves for which it difficult to transfer the ECDLP. In this talk we will introduce the notion of an "isolated" curve. These are curves which do not admit many computable isogenies which obstructs the transferring of the ECDLP.