Paper 2010/560

On CCA-Secure Fully Homomorphic Encryption

J. Loftus, A. May, N. P. Smart, and F. Vercauteren


It is well known that any encryption scheme which supports any form of homomorphic operation cannot be secure against adaptive chosen ciphertext attacks. The question then arises as to what is the most stringent security definition which is achievable by homomorphic encryption schemes. Prior work has shown that various schemes which support a single homomorphic encryption scheme can be shown to be IND-CCA1, i.e. secure against lunchtime attacks. In this paper we extend this analysis to the recent fully homomorphic encryption scheme proposed by Gentry, as refined by Gentry, Halevi, Smart and Vercauteren. We show that the basic Gentry scheme is not IND-CCA1; indeed a trivial lunchtime attack allows one to recover the secret key. We then show that a minor modification to the variant of the somewhat homomorphic encryption scheme of Smart and Vercauteren will allow one to achieve IND-CCA1, indeed PA-1, in the standard model assuming a lattice based knowledge assumption. We also examine the security of the scheme against another security notion, namely security in the presence of ciphertext validity checking oracles; and show why CCA-like notions are important in applications in which multiple parties submit encrypted data to the ``cloud'' for secure processing.

Note: Added an appendix on embedding an FHE scheme within a non-FFHE CCA scheme.

Available format(s)
Public-key cryptography
Publication info
Published elsewhere. Unknown where it was published
Fully Homomorphic Encryption
Contact author(s)
nigel @ cs bris ac uk
2011-07-15: last of 2 revisions
2010-11-03: received
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      author = {J.  Loftus and A.  May and N. P.  Smart and F.  Vercauteren},
      title = {On CCA-Secure Fully Homomorphic Encryption},
      howpublished = {Cryptology ePrint Archive, Paper 2010/560},
      year = {2010},
      note = {\url{}},
      url = {}
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