Fully Homomorphic Encryption over the Integers

We describe a very simple “somewhat homomorphic” encryption scheme using only elementary modular arithmetic, and use Gentry’s techniques to convert it into a fully homomorphic scheme. Compared to Gentry’s construction, our somewhat homomorphic scheme merely uses addition and multiplication over the integers rather than working with ideal lattices over a polynomial ring. The main appeal of our approach is the conceptual simplicity. We reduce the security of our somewhat homomorphic scheme to finding an approximate integer gcd – i.e., given a list of integers that are near-multiples of a hidden integer, output that hidden integer. We investigate the hardness of this task, building on earlier work of HowgraveGraham.

• Type: Conference
• Year: 2010
• Where: EUROCRYPT
• Authors: Marten van Dijk, Craig Gentry, Shai Halevi, Vinod Vaikuntanathan
• Keywords: Approximate Integer Gcd, Cryptology, EUROCRYPT 2010, Hidden Integer, Homomorphic Scheme