Homomorphic Encryption within Lattice-Based Encryption System

Abstract In 2009 a system of fully homomorphic encryption was constructed, in the future, many works were done based on it. In this work, will be performed an analysis of the possibility to use the ideal lattices for constructing homomorphic operations over ciphertexts. This paper represents the analysis of an encryption system based on the primitive of a union in ideal lattices space. The advantage of this approach consists in the possibility of segregated analysis of encryption security and homomorphic properties of the system. The work will be based on the method of analyzing generalized operations over ciphertext using the concept of the base reducing element. It will be shown that some encryption systems can be supplemented by homomorphism between opentext and ciphertext. Thus such systems can be decomposed by encryption and homomorphic parts which would affect each other but although can be analyzed separately. Different systems (probably within one class) can be represented via an identical transform of sets. Separated from the cryptographic scheme the underlying math can be used to analyze only the homomorphic part, particularly under some simplifications. The building of ideal-based ciphertext laying on the assumption that ideals can be extracted further, it will be shown in the paper what the ”remainder theorem” can be one of the principal ways to do this.