Multiparty Non-Interactive Key Exchange and More From Isogenies on Elliptic Curves
This work addresses the challenge of secure key exchange for multiple parties in cryptography, but it is incremental as it builds on existing concepts without providing a complete solution.
The authors tackled the problem of constructing a non-interactive key exchange protocol for multiple parties by proposing a framework based on isogenies on elliptic curves, but they did not achieve a working protocol due to an unresolved mathematical issue. They introduced a cryptographic invariant map as a new primitive to build various cryptographic primitives, including NIKE, which were previously reliant on multilinear maps and indistinguishability obfuscation.
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.