Non-interactive key exchange in a generic multilinear group: An underwhelming lower bound

This paper proves a lower bound on the degree of multilinearity needed to accomplish perfectly correct non-interactive k-party key exchange in a generic multilinear group, where k is only polylogarithmic in the security parameter. We conjecture that our result can be extended to polynomial k, as well as that similar lower bounds can be proven for more complex functionalties. We were surprised at our inability to extend the result easily, and we describe here the obstruction to completing our argument for polynomial values of k.