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 easy to generalize the somewhat homomorphic versions of the corresponding schemes to support arithmetic operations modulo Q for any Q > 2, but bootstrapping those generalized variants into fully homomorphic schemes is not easy. Thus, Nuida and Kurosawa settled a significant 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, we now have two different ways of homomorphically evaluating a mod-Q arithmetic circuit with an FHE scheme over the integers: 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 evaluate that converted circuit using an FHE scheme with binary message space. In this paper, we compare both approaches and show that the latter is often preferable to the former.