### Fully Homomorphic Encryption without Bootstrapping

Zvika Brakerski, Craig Gentry, and Vinod Vaikuntanathan

##### Abstract

We present a radically new approach to fully homomorphic encryption (FHE) that dramatically improves performance and bases security on weaker assumptions. A central conceptual contribution in our work is a new way of constructing leveled fully homomorphic encryption schemes (capable of evaluating arbitrary polynomial-size circuits), {\em without Gentry's bootstrapping procedure}. Specifically, we offer a choice of FHE schemes based on the learning with error (LWE) or ring-LWE (RLWE) problems that have $2^\secparam$ security against known attacks. For RLWE, we have: 1. A leveled FHE scheme that can evaluate $L$-level arithmetic circuits with $\tilde{O}(\secparam \cdot L^3)$ per-gate computation -- i.e., computation {\em quasi-linear} in the security parameter. Security is based on RLWE for an approximation factor exponential in $L$. This construction does not use the bootstrapping procedure. 2. A leveled FHE scheme that uses bootstrapping {\em as an optimization}, where the per-gate computation (which includes the bootstrapping procedure) is $\tilde{O}(\secparam^2)$, {\em independent of $L$}. Security is based on the hardness of RLWE for {\em quasi-polynomial} factors (as opposed to the sub-exponential factors needed in previous schemes). We obtain similar results for LWE, but with worse performance. We introduce a number of further optimizations to our schemes. As an example, for circuits of large width -- e.g., where a constant fraction of levels have width at least $\secparam$ -- we can reduce the per-gate computation of the bootstrapped version to $\tilde{O}(\secparam)$, independent of $L$, by {\em batching the bootstrapping operation}. Previous FHE schemes all required $\tilde{\Omega}(\secparam^{3.5})$ computation per gate. At the core of our construction is a much more effective approach for managing the noise level of lattice-based ciphertexts as homomorphic operations are performed, using some new techniques recently introduced by Brakerski and Vaikuntanathan (FOCS 2011).

Available format(s)
Publication info
Published elsewhere. Unknown where it was published
Keywords
fully homomorphic encryptioncryptography
Contact author(s)
craigbgentry @ gmail com
History
2011-08-11: last of 3 revisions
See all versions
Short URL
https://ia.cr/2011/277

CC BY

BibTeX

@misc{cryptoeprint:2011/277,
author = {Zvika Brakerski and Craig Gentry and Vinod Vaikuntanathan},
title = {Fully Homomorphic Encryption without Bootstrapping},
howpublished = {Cryptology ePrint Archive, Paper 2011/277},
year = {2011},
note = {\url{https://eprint.iacr.org/2011/277}},
url = {https://eprint.iacr.org/2011/277}
}

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