Paper 2011/441

Fully Homomorphic Encryption over the Integers with Shorter Public Keys

Jean-Sebastien Coron, Avradip Mandal, David Naccache, and Mehdi Tibouchi


At Eurocrypt 2010 van Dijk {\sl et al.} described a fully homomorphic encryption scheme over the integers. The main appeal of this scheme (compared to Gentry's) is its conceptual simplicity. This simplicity comes at the expense of a public key size in ${\cal \tilde O}(\lambda^{10})$ which is too large for any practical system. In this paper we reduce the public key size to ${\cal \tilde O}(\lambda^{7})$ by encrypting with a quadratic form in the public key elements, instead of a linear form. We prove that the scheme remains semantically secure, based on a stronger variant of the approximate-GCD problem, already considered by van Dijk {\sl et al}. We also describe the first implementation of the resulting fully homomorphic scheme. Borrowing some optimizations from the recent Gentry-Halevi implementation of Gentry's scheme, we obtain roughly the same level of efficiency. This shows that fully homomorphic encryption can be implemented using simple arithmetic operations.

Available format(s)
Public-key cryptography
Publication info
Published elsewhere. An extended abstract will appear at CRYPTO 2011
Fully Homomorphic Encryption
Contact author(s)
jean-sebastien coron @ uni lu
2011-08-15: received
Short URL
Creative Commons Attribution


      author = {Jean-Sebastien Coron and Avradip Mandal and David Naccache and Mehdi Tibouchi},
      title = {Fully Homomorphic Encryption over the Integers with Shorter Public Keys},
      howpublished = {Cryptology ePrint Archive, Paper 2011/441},
      year = {2011},
      note = {\url{}},
      url = {}
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