## Efficient Fully Homomorphic Encryption from (Standard) LWE

by Zvika Brakerski and Vinod Vaikuntanathan.

This work increases the confidence in the feasibility of constructing fully homomorphic encryption schemes, let alone efficient ones. Unlike prior works, it rely on an intractability assumption that does not seem to enjoy a structure that supports addition and multiplication.

#### The original abstract

In fully homomorphic encryption, it is possible to transform an encryption of a message, $m$, into an encryption of any (efficient) function of that message, $f(m)$, without knowing the secret key. This property makes it into a very useful cryptographic building block.

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. As icing on the cake, our scheme is quite efficient, and has very short ciphertexts.

Our construction improves upon previous works in two aspects:

1. We show that somewhat homomorphic'' encryption can be based on LWE, using a new {\it re-linearization} technique. In contrast, all previous schemes relied on complexity assumptions related to ideals in various rings.
2. We deviate from the squashing paradigm'' used in all previous works. We introduce a new {\it dimension reduction} technique, which shortens the ciphertexts and reduces the decryption complexity of our scheme, without introducing additional assumptions. In contrast, all previous works required an additional, very strong assumption (namely, the sparse subset sum assumption).
Since our scheme has very short ciphertexts, we 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 $k \cdot {\rm polylog}\,k+\log |DB|$ bits per single-bit query, which is better than any known scheme. Previously, it was not known how to achieve a communication complexity of even ${\rm poly}(k, \log|DB|)$ based on LWE.

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