Fixed-Point Arithmetic in SHE Schemes

The purpose of this paper is to investigate fixed point arithmetic in ring-based Somewhat Homomorphic Encryption (SHE) schemes. We provide three main contributions: Firstly, we investigate the representation of fixed point numbers. We analyse the two representations from Dowlin et al, representing a fixed point number as a large integer (encoded as a scaled polynomial) versus a polynomial-based fractional representation. We show that these two are, in fact, isomorphic by presenting an explicit isomorphism between the two that enables us to map the parameters from one representation to another. Secondly, given a computation and a bound on the fixed point numbers used as inputs and scalars within the computation, we achieve a way of producing lower bounds on the plaintext modulus p and the degree of the ring d needed to support complex homomorphic operations. Finally, we investigate an application in homomorphic image processing. We have an image given in encrypted form and are required to perform the standard image processing pipeline of Fourier Transform–Hadamard Product–Inverse Fourier Transform. In particular we examine applications in which the specific matrices involved in the Hadamard multiplication are also encrypted. We propose a mixed Fourier Transform Algorithm which aims to strike a compromise between the number of homomorphic multiplications and the parameter sizes of the underlying SHE scheme.