A Secure Computation Scheme of Inner Product Based on Fully Homomorphic Encryption

Joppe W. Bos, Kristin Lauter, Jake Loftus +1 more

Sai Hu, Ruwei Huang, Liang Zhou

Abstract Traditional fully homomorphic encryption(FHE) schemes allow computation only on data encrypted under the same public key. Multi-Key Fully Homomorphic Encryption (MKFHE) enables arbitrary operations on data encrypted with different public keys, allowing all participating users jointly decrypting the final ciphertext. The multi-key BFV FHE scheme inherits BFV’s advantages in ring element encryption and scale invariance. Nonetheless, it also has some disadvantages, such as additional noise generated during the relinearization process, the need for costly transformations during the external product process, and the requirement for a Common Reference String (CRS). In this paper, we investigate the MKFHE scheme for RLWE-based BFV. Firstly, we improve the modulus size of the evaluation key and the public key to construct a modulus enchancement relinearization method, which can significantly reduce the noise generated during the relinearization process. Secondly, we propose to use an inner product via Gadget decomposition in the relinearization based on the MK-BFV scheme instead of the original outer product operation, which can reduce the complexity of the NTT operation to $$\left( {d + 2\tilde{d}} \right)r^{\prime } /\left( {d + 2} \right)\tilde{l}$$ d + 2 d ~ r ′ / d + 2 l ~ of the original one. Finally, we propose a MK-BFV without CRS on the basis of the previous ones, which enhances the user's control over his own key.

Pratish Datta, Ratna Dutta, Sourav Mukhopadhyay

Masaya Yasuda, Jun Yajima, Shimoyama Takeshi +1 more

Pratish Datta, Ratna Dutta, Sourav Mukhopadhyay

Carlos Aguilar Melchor, Slim Bettaieb, Philippe Gaborit +1 more

Abstract. In CRYPTO 2010, Aguilar et al. proposed a somewhat homomorphic encryption scheme, i.e. an encryption scheme allowing to compute a limited amount of sums and products over encrypted data, with a security reduction from LWE over general lattices. General lattices (as opposed to ideal lattices) do not have an inherent multiplicative structure but, using a tensorial product, Aguilar et al. managed to obtain a scheme allowing to compute products with a polylogarithmic amount of operands. In this paper we present an alternative construction allowing to compute products with polynomiallymany operands while preserving the security reductions of the initial scheme. Unfortunately, despite this improvement our construction seems to be incompatible with Gentry’s seminal transformation allowing to obtain fully-homomorphic encryption schemes. Recently, Brakerski et al. used the tensorial product approach introduced by Aguilar et al. in a new alternative way which allows to radically improve the performance of the obtained scheme. Based on this approach, and using two nice optimizations, their scheme is able to evaluate products with exponentially-many operands and can be transformed into an efficient fully-homomorphic encryption scheme while being based on general lattice problems. However, even if these results outperform the construction presented here, we believe the modifications we suggest for Aguilar et al.’s schemes are of independent interest.