Generalized Learning Problems and Applications to Non-commutative Cryptography

Abstract. We propose a generalization of the learning parity with noise (LPN) and learning with errors (LWE) problems to an abstract class of group-theoretic learning problems that we term learning homomorphisms with noise (LHN). This class of problems contains LPN and LWE as spe-cial cases, but is much more general. It allows, for example, instantiations based on non-abelian groups, resulting in a new avenue for the applica-tion of combinatorial group theory to the development of cryptographic primitives. We then study a particular instantiation using relatively free groups and construct a symmetric cryptosystem based upon it