A New Ring-Based SPHF and PAKE Protocol On Ideal Lattices

emph{ Smooth Projective Hash Functions } ( SPHFs ) as a specific pattern of zero knowledge proof system are fundamental tools to build many efficient cryptographic schemes and protocols. As an application of SPHFs, emph { Password - Based Authenticated Key Exchange } ( PAKE ) protocol is well-studied area in the last few years. In 2009, Katz and Vaikuntanathan described the first lattice-based PAKE using the Learning With Errors ( LWE ) problem. In this work, we present a new efficient emph { ring-based } smooth projectice hash function `` ( Ring - SPHF ) " using Lyubashevsky, Peikert, and Regev's dual-style cryptosystem based on the Learning With Errors over Rings ( Ring - LWE ) problem. Then, using our ring-SPHF, we propose the first efficient password-based authenticated key exchange ` ` ( Ring - PAKE ) " protocol over emph{ rings } whose security relies on ideal lattice assumptions.