FHE over the Integers

This is a fully homomorphic encryption relying only on modular arithmetic. The authors first create an somewhat homomorphic scheme which is bootstrappable and then apply Gentry's technique to construct a fully homomorphic scheme.

One of the main advantages of this scheme represents its conceptual simplicity. The security of the scheme is reduced to the approximate gcd problem over the integers, that is, given a list of integers that are near-multiples of an unknown integer d , find d .

The bootstrappable encryption scheme

Using a theorem of Gentry, the authors are able to construct a homomorphic encryption scheme that can handle circuits of any depth from a scheme that is capable of evaluating just a little more than its own decryption circuit.

Definition. Let ${\mathcal {E}}=(Keygen,Encrypt,Decrypt,Evaluate)$ be an encryption scheme where $Decrypt$ is implemented by a circuit that depends only on the security parameter. For a given value $\lambda$ of the security parameter, the set of augmented decryption circuits consists of two circuits, both take as input a secret key and two ciphertexts. One circuit decrypts both ciphertexts and adds the resulting plaintext bits modulo $2$, the other decrypts both ciphertexts and multiplies the resulting plaintext bits modulo $2$. The authors denote this set by $D_{\mathcal {E}}(\lambda )$.