The potential applications of homomorphic operations over encryption functions
were recognized and appreciated almost about the same time as the first publickey cryptographic algorithm RSA was invented. The RSA algorithm is multiplicatively homomorphic. The ensuing 30 years have brought on several additively or
multiplicatively homomorphic encryption functions with increasing algorithmic inventiveness and complicated mathematics. However, until 2009, it was not clear
whether a fully homomorphic encryption algorithm, one that allows both additive
and multiplicative homomorphisms, would exist. This was resolved by Craig Gentry, and followed up by several authors who have proposed fully homomorphic encryption algorithms and addressed issues related to their formulation, arithmetic,
efficiency and security. While formidable efficiency barriers remain, we now have a
variety of fully homomorphic encryption algorithms that can be applied to various
private computation problems in healthcare, finance and national security.
This book started with the desire and motivation to describe and implement partially homomorphic encryption functions using a unified mathematical notation. We
hope that we accomplished our projected goal, and you will find our exposition
helpful to understand and implement these algorithms. If a highly efficient fully homomorphic encryption function for multiple applications were to become available,
the usefulness of partially homomorphic encryption functions would be in question. But this lofty end is not yet in sight. Studying partially homomorphic encryption functions may help us to understand the difficulties ahead and perhaps to avoid
blind alleys and dead ends. Moreover, partially homomorphic encryption algorithms
may have certain perhaps limited applications for which significantly more efficient
implementations can be obtained. If not, it is always enjoyable to learn interesting
algorithms and their underlying beautiful mathematics.
The authors thank their families and dedicate this book to them