Zero Knowledge Authentication Protocols With Algebraic Geometry Techniques

Several cryptographic methods have been developed based on the difficulty to determine the set of solutions of a polynomial system over a given field. We build a polynomial ideal whose algebraic set is related to the set of isomorphisms between two graphs. The problem isomorphism, posed in the context of Graph Theory, has been extensively used in zero knowledge authentication protocols. Thus, any cryptographic method based on isomorphism can be translated into an equivalent method based on the problem of finding rational points in algebraic sets associated to polynomial ideals. Keywords— zero knowledge procedures, graph isomorphism problem, multivariate polynomial system.