FHE over the integers and modular arithmetic circuits

Abstract

Fully homomorphic encryption (FHE) over the integers, as proposed by van Dijk et al. in 2010 and developed in a number of papers afterwards, originally supported the evaluation of Boolean circuits (i.e. mod-2 arithmetic circuits) only. It is easily generalised to the somewhat homomorphic versions of the corresponding schemes to support arithmetic operations modulo Q for any Q > 2, but bootstrapping those generalised variants into fully homomorphic schemes is not easy. Thus, Nuida and Kurosawa settled an interesting open problem in 2015 by showing that one could in fact construct FHE over the integers with message space ℤ/Qℤ for any constant prime Q. As a result of their work, the authors can homomorphically evaluate a mod-Q arithmetic circuit with an FHE scheme over the integers in two different ways: one could either use their scheme with message space ℤ/Qℤ directly, or one could first convert the arithmetic circuit to a Boolean one, and then evaluate that converted circuit using an FHE scheme with binary message space. In this study, they compare both approaches and show that the latter is often preferable to the former.