Paper 2013/198

On Evaluating Circuits with Inputs Encrypted by Different Fully Homomorphic Encryption Schemes

Zhizhou Li and Ten H. Lai

Abstract

We consider the problem of evaluating circuits whose inputs are encrypted with possibly different encryption schemes. Let $\mathcal{C}$ be any circuit with input $x_1, \dots, x_t \in \{0,1\}$, and let $\mathcal{E}_i$, $1 \le i \le t$, be (possibly) different fully homomorphic encryption schemes, whose encryption algorithms are $\Enc_i$. Suppose $x_i$ is encrypted with $\mathcal{E}_i$ under a public key $pk_i$, say $c_i \leftarrow \Enc_i({pk_i}, x_i)$. Is there any algorithm $\Evaluate$ such that $\Evaluate(\mathcal{C}, \langle \mathcal{E}_1, pk_1, c_1\rangle, \dots, \langle \mathcal{E}_t, pk_t, c_t\rangle)$ returns a ciphertext $c$ that, once decrypted, equals $\mathcal{C}(x_1, \dots, x_t)$? We propose a solution to this seemingly impossible problem with the number of different schemes and/or keys limited to a small value. Our result also provides a partial solution to the open problem of converting any FHE scheme to a multikey FHE scheme.