Paper 2012/078

Fully Homomorphic Encryption without Modulus Switching from Classical GapSVP

Zvika Brakerski


We present a new tensoring technique for LWE-based fully homomorphic encryption. While in all previous works, the ciphertext noise grows quadratically ($B \to B^2\cdot\poly(n)$) with every multiplication (before ``refreshing''), our noise only grows linearly ($B \to B\cdot\poly(n)$). We use this technique to construct a \emph{scale-invariant} fully homomorphic encryption scheme, whose properties only depend on the ratio between the modulus $q$ and the initial noise level $B$, and not on their absolute values. Our scheme has a number of advantages over previous candidates: It uses the same modulus throughout the evaluation process (no need for ``modulus switching''), and this modulus can take arbitrary form, including a power of $2$ which carries obvious advantages for implementation. In addition, security can be \emph{classically} reduced to the worst-case hardness of the GapSVP problem (with quasi-polynomial approximation factor), whereas previous constructions could only exhibit a quantum reduction to GapSVP.

Note: Revised due to typos.

Available format(s)
Public-key cryptography
Publication info
Published elsewhere. Unknown where it was published
fully homomorphic encryptionlearning with errors
Contact author(s)
zvika @ stanford edu
2012-05-18: last of 5 revisions
2012-02-23: received
See all versions
Short URL
Creative Commons Attribution


      author = {Zvika Brakerski},
      title = {Fully Homomorphic Encryption without Modulus Switching from Classical GapSVP},
      howpublished = {Cryptology ePrint Archive, Paper 2012/078},
      year = {2012},
      note = {\url{}},
      url = {}
Note: In order to protect the privacy of readers, does not use cookies or embedded third party content.