  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.
