Multiparty Non-Interactive Key Exchange and More From Isogenies on Elliptic Curves

TL;DR

This paper proposes a framework for non-interactive key exchange using isogenies of elliptic curves, introducing a new cryptographic primitive called invariant maps, but the protocol remains incomplete due to an open mathematical problem.

Contribution

It introduces a novel framework for NIKE based on isogenies and cryptographic invariant maps, extending cryptographic primitives beyond multilinear maps.

Findings

Framework for n-party NIKE using isogenies

Introduction of cryptographic invariant maps as a primitive

Potential to build cryptographic primitives without multilinear maps

Abstract

We describe a framework for constructing an efficient non-interactive key exchange (NIKE) protocol for n parties for any n >= 2. Our approach is based on the problem of computing isogenies between isogenous elliptic curves, which is believed to be difficult. We do not obtain a working protocol because of a missing step that is currently an open mathematical problem. What we need to complete our protocol is an efficient algorithm that takes as input an abelian variety presented as a product of isogenous elliptic curves, and outputs an isomorphism invariant of the abelian variety. Our framework builds a cryptographic invariant map, which is a new primitive closely related to a cryptographic multilinear map, but whose range does not necessarily have a group structure. Nevertheless, we show that a cryptographic invariant map can be used to build several cryptographic primitives, including NIKE, that were previously constructed from multilinear maps and indistinguishability obfuscation.