This talk outlines an ongoing research effort towards a probabilistic framework for the application of infinite groups to cryptography.

We start by analyzing a classical group-theoretic construction for public-key cryptosystems from a complexity-theoretic perspective. We then suggest a way of casting some of the standard computational problems from group theory in terms of probabilistic cryptographic assumptions---an essential ingredient for a formal security analysis. Next, we outline a new approach for finding cryptographically-suitable group-theoretic assumptions, inspired by recent advances in lattice-based cryptography. The framework relies on a new problem that we term ``Learning Homomorphisms from Images with Errors'' (LHIE), which can be viewed as a generalization of the ``Learning With Errors'' (LWE) problem from the setting of vector spaces and linear transformations to the setting of groups and homomorphisms. We conclude by discussing how this assumption yields group-theoretic public-key cryptosystems, and describe some of the remaining challenges in this effort.