An Investigation of Complex Operations with Word-Size Homomorphic Encryption

Homomorphic encryption has progressed rapidly in both efficiency and versatility since its emergence in 2009. Meanwhile, a multitude of pressing privacy needs — ranging from cloud computing to healthcare management to the handling of shared databases such as those containing genomics data — call for immediate solutions that apply fully homomorpic encryption (FHE) and somewhat homomorphic encryption (SHE) technologies. Further progress towards these ends requires new ideas for the efficient implementation of algebraic operations on wordbased (as opposed to bit-wise) encrypted data. While the arithmetic operations that are essential for cloud computing have known boolean representations, most of them require access to each bit individually, such as comparison and we lack solutions on a word domain. In this work, we tackle this challenging problem of finding ways to solve algebraic problems -including comparison and homomorphic divisionin word-based encryption via an SHE scheme. We present concrete performance figures for all proposed primitives.