Fully Homomorphic Encryption over the Integers

We construct a simple fully homomorphic encryption scheme, using only elementary modular arithmetic. We use Gentry’s technique to construct fully homomorphic scheme from a “bootstrappable” somewhat homomorphic scheme. However, instead of using ideal lattices over a polynomial ring, our bootstrappable encryption scheme merely uses addition and multiplication over the integers. The main appeal of our scheme is the conceptual simplicity. We reduce the security of our 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 Howgrave-Graham.