Paper 2016/675

Accelerating Homomorphic Computations on Rational Numbers

Angela Jäschke and Frederik Armknecht

Abstract

Fully Homomorphic Encryption (FHE) schemes are conceptually very powerful tools for outsourcing computations on confidential data. However, experience shows that FHE-based solutions are not sufficiently efficient for practical applications yet. Hence, there is a huge interest in improving the performance of applying FHE to concrete use cases. What has been mainly overlooked so far is that not only the FHE schemes themselves contribute to the slowdown, but also the choice of data encoding. While FHE schemes usually allow for homomorphic executions of algebraic operations over finite fields (often $\mathbb{Z}_2$), many applications call for different algebraic structures like signed rational numbers. Thus, before an FHE scheme can be used at all, the data needs to be mapped into the structure supported by the FHE scheme. We show that the choice of the encoding can already incur a significant slowdown of the overall process, which is independent of the efficiency of the employed FHE scheme. We compare different methods for representing signed rational numbers and investigate their impact on the effort needed for processing encrypted values. In addition to forming a new encoding technique which is superior under some circumstances, we also present further techniques to speed up computations on encrypted data under certain conditions, each of independent interest. We confirm our results by experiments.