Paper 2014/032

Scale-Invariant Fully Homomorphic Encryption over the Integers

Jean-Sébastien Coron, Tancrède Lepoint, and Mehdi Tibouchi


At Crypto 2012, Brakerski constructed a scale-invariant fully homomorphic encryption scheme based on the LWE problem, in which the same modulus is used throughout the evaluation process, instead of a ladder of moduli when doing “modulus switching”. In this paper we describe a variant of the van Dijk et al. FHE scheme over the integers with the same scale-invariant property. Our scheme has a single secret modulus whose size is linear in the multiplicative depth of the circuit to be homomorphically evaluated, instead of exponential; we therefore construct a leveled fully homomorphic encryption scheme. This scheme can be transformed into a pure fully homomorphic encryption scheme using bootstrapping, and its security is still based on the Approximate-GCD problem. We also describe an implementation of the homomorphic evaluation of the full AES encryption circuit, and obtain significantly improved performance compared to previous implementations: about 23 seconds (resp. 3 minutes) per AES block at the 72-bit (resp. 80-bit) security level on a mid-range workstation. Finally, we prove the equivalence between the (error-free) decisional Approximate-GCD problem introduced by Cheon et al. (Eurocrypt 2013) and the classical computational Approximate-GCD problem. This equivalence allows to get rid of the additional noise in all the integer-based FHE schemes described so far, and therefore to simplify their security proof.

Available format(s)
Public-key cryptography
Publication info
A major revision of an IACR publication in PKC 2014
Fully Homomorphic EncryptionApproximage-GCDHomomorphic AES
Contact author(s)
jean-sebastien coron @ uni lu
2014-01-12: received
Short URL
Creative Commons Attribution


      author = {Jean-Sébastien Coron and Tancrède Lepoint and Mehdi Tibouchi},
      title = {Scale-Invariant Fully Homomorphic Encryption over the Integers},
      howpublished = {Cryptology ePrint Archive, Paper 2014/032},
      year = {2014},
      note = {\url{}},
      url = {}
Note: In order to protect the privacy of readers, does not use cookies or embedded third party content.