Efficient Fully Homomorphic Encryption from (Standard) LWE

Abstract

We present a fully homomorphic encryption scheme that is based solely on the(standard) learning with errors (LWE) assumption. Applying known results on LWE, the security of our scheme is based on the worst-case hardness of ``short vector problems'' on arbitrary lattices. Our construction improves on previous works in two aspects:\begin{enumerate}\item We show that ``somewhat homomorphic'' encryption can be based on LWE, using a new {\em re-linearization} technique. In contrast, all previous schemes relied on complexity assumptions related to ideals in various rings. \item We deviate from the "squashing paradigm'' used in all previous works. We introduce a new {\em dimension-modulus reduction} technique, which shortens the cipher texts and reduces the decryption complexity of our scheme, {\em without introducing additional assumptions}. \end{enumerate}Our scheme has very short cipher texts and we therefore use it to construct an asymptotically efficient LWE-based single-server private information retrieval (PIR) protocol. The communication complexity of our protocol (in the public-key model) is $k \cdot \polylog(k)+\log \dbs$ bits per single-bit query (here, $k$ is a security parameter).