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The authors 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 they term Learning Homomorphisms with Noise (LHN). This class of problems contains LPN and LWE as special cases, but is much more general. It allows, for example, instantiations based on non-abelian groups, resulting in a new avenue for the application of combinatorial group theory to the development of cryptographic primitives. They then study a particular instantiation using relatively free groups and construct a symmetric cryptosystem based upon it.
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