Homomorphic Evaluation Of Lattice-Based Symmetric Encryption Schemes

Optimizing performance of Fully Homomorphic Encryption (FHE) is nowadays an active trend of research in cryptography. One way of improvement is to use a hybrid construction with a classical symmetric encryption scheme to transfer encrypted data to the Cloud. This allows to reduce the bandwidth since the expansion factor of symmetric schemes (the ratio between the ciphertext and the plaintext length) is close to one, whereas for FHE schemes it is in the order of 1,000 to 1,000,000. However, such a construction requires the decryption circuit of the symmetric scheme to be easy to evaluate homomorphically. Several works have studied the cost of homomorphically evaluating classical block ciphers, and some recent works have suggested new homomorphic oriented constructions of block ciphers or stream ciphers. Since the multiplication gate of FHE schemes usually squares the noise of the ciphertext, we cannot afford too many multiplication stages in the decryption circuit. Consequently, FHE-friendly symmetric encryption schemes have a decryption circuit with small multiplication depth.We aim at minimizing the cost of the homomorphic evaluation of the decryption of symmetric encryption schemes. To do so, we focus on schemes based on learning problems: Learning With Errors (LWE), Learning Parity with Noise (LPN) and Learning With Rounding (LWR). We show that they have lower multiplicative depth than usual block ciphers, and hence allow more FHE operations before a heavy bootstrapping becomes necessary. Moreover, some of them come with a security proof. Finally, we implement our schemes in HElib. Experimental evidence shows that they achieve lower amortized and total running time than previous performance from the literature: our schemes are from 10 to 10,000 more efficient for the time per bit and the total running time is also reduced by a factor between 20 to 10,000. Of independent interest, the security of our LWR-based scheme is related to LWE and we provide an efficient security proof that allows to take smaller parameters.