Fully Homomorphic Encryption from the Finite Field Isomorphism Problem

If q is a prime and n is a positive integer then any two finite fields of order \(q^n\) are isomorphic. Elements of these fields can be thought of as polynomials with coefficients chosen modulo q, and a notion of length can be associated to these polynomials. A non-trivial isomorphism between the fields, in general, does not preserve this length, and a short element in one field will usually have an image in the other field with coefficients appearing to be randomly and uniformly distributed modulo q. This key feature allows us to create a new family of cryptographic constructions based on the difficulty of recovering a secret isomorphism between two finite fields. In this paper we describe a fully homomorphic encryption scheme based on this new hard problem.