Efficient Fully Homomorphic Encryption from (Standard) LWE
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Author(s)
Brakerski, Zvika
Vaikuntanathan, Vinod
Date Issued
April 2014
Journal
SIAM Journal on Computing
Publisher
Society for Industrial & Applied Mathematics (SIAM)
Citation
Brakerski, Zvika, and Vinod Vaikuntanathan. “Efficient Fully Homomorphic Encryption from (Standard) LWE.” SIAM Journal on Computing, vol. 43, no. 2, Jan. 2014, pp. 831–71. © The Authors
Version
Final published version
Abstract
A fully homomorphic encryption (FHE) scheme allows anyone 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. We present a leveled FHE scheme that is based solely on the (standard) learning with errors (LWE) assumption. (Leveled FHE schemes are initialized with a bound on the maximal evaluation depth. However, this restriction can be removed by assuming “weak circular security.”) 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: 1. We show that “somewhat homomorphic” encryption can be based on LWE, using a new relinearization 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 dimension-modulus reduction technique, which shortens the ciphertexts and reduces the decryption complexity of our scheme, without introducing additional
assumptions. Our scheme has very short ciphertexts, 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·polylog(k)+log |DB| bits per
single-bit query, in order to achieve security against 2k-time adversaries (based on the best known attacks against our underlying assumptions). Key words. cryptology, public-key encryption, fully homomorphic encryption, learning with errors, private information retrieval
MIT Department
Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
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DOI of Published Version
https://doi.org/10.1137/120868669
