Learning with errors in the exponent

Abstract

The Snowden revelations have shown that intelligence agencies have been successful in undermining cryptography and put in question the exact security provided by the underlying intractability problem. We introduce a new class of intractability problems, called Learning with Errors in the Exponent (LWEE). We give a tight reduction from Learning with Errors (LWE) and the Representation Problem (RP) in finite groups, two seemingly unrelated problem, to LWEE. The argument holds in the classical and quantum model of computation. Furthermore, we present the very first construction of a semantically secure public-key encryption system based on LWEE in groups of composite order. The heart of our construction is an error recovery “in the exponent” technique to handle critical propagations of noise terms.