Homomorphic encryption enables various calculations while preserving the data confidentiality. In this paper, we apply the somewhat homomorphic encryption scheme proposed by Brakerski and Vaikuntanathan (CRYPTO 2011) to secure matrix multiplication between two matrices. To reduce both the ciphertext size and the computation cost, we propose a new method to pack a matrix into a single ciphertexts so that it also enables efficient matrix multiplication over the packed ciphertexts. Our packing method generalizes Yasuda et al.’s methods (Security Comm. Networks 2015 and ACISP 2015), which are for secure inner product. We also implement our methods and give a comparison with previous packing methods.

Let $G$ be a finitely generated nilpotent group. The representation zeta function $\zeta _G(s)$ of $G$ enumerates twist isoclasses of finite-dimensional irreducible complex representations of $G$ . We prove that $\zeta _G(s)$ has rational abscissa of convergence $\alpha (G)$ and may be meromorphically continued to the left of $\alpha (G)$ and that, on the line $\{s\in \mathbb {C} \mid \mathrm {Re}(s) = \alpha (G)\}$ , the continued function is holomorphic except for a pole at $s=\alpha (G)$ . A Tauberian theorem yields a precise asymptotic result on the representation growth of $G$ in terms of the position and order of this pole. We obtain these results as a consequence of a result establishing uniform analytic properties of representation zeta functions of torsion-free finitely generated nilpotent groups of the form $\mathbf {G}(\mathcal {O})$ , where $\mathbf {G}$ is a unipotent group scheme defined in terms of a nilpotent Lie lattice over the ring $\mathcal {O}$ of integers of a number field. This allows us to show, in particular, that the abscissae of convergence of the representation zeta functions of such groups and their pole orders are invariants of $\mathbf {G}$ , independent of $\mathcal {O}$ .

Homomorphic Proxy Re-Encryption (HPRE) is an extension of Proxy Re-Encryption (PRE) which combines the advantages of both Homomorphic Encryption (HE) and PRE. A HPRE scheme allows arbitrary evaluations to be performed on ciphertexts under one (the delegator's) public key and, using a re-encryption key, it transforms the resulting ciphertext to a new ciphertext under another (the delegatee's) public key. Prior HPRE schemes are either CPA-secure or CCA-secure but only support partial homomorphic operations. We propose a generic construction of single-hop HPRE scheme which supports fully homomorphic operations. The proposed scheme is proven secure in our new index-based CCA-HPRE model. Our technique is to give a generic transformation that turns any multi-identity identity-based FHE (IBFHE) scheme with key switching into Fully Homomorphic Encryption (FHE) with key switching from which we can obtain the proposed single-hop HPRE scheme. We also present a concrete instantiation of multi-identity IBFHE with key switching from learning with errors (LWE) in the standard model.