Fully Homomorphic Encryption over the Integers with Shorter Public Keys [chapter]

Jean-Sébastien Coron, Avradip Mandal, David Naccache, Mehdi Tibouchi
2011 Lecture Notes in Computer Science  
At Eurocrypt 2010 van Dijk 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 inÕ(λ 10 ) which is too large for any practical system. In this paper we reduce the public key size toÕ(λ 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,
more » ... ed on a stronger variant of the approximate-GCD problem, already considered by van Dijk 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. An extended abstract of this paper will appear at crypto 2011. This is the full version. The DGHV fully homomorphic scheme over the integers. At Eurocrypt 2010, van Dijk, Gentry, Halevi and Vaikuntanathan described a fully homomorphic encryption scheme over the integers [4]. As in Gentry's scheme the authors first describe a somewhat homomorphic scheme supporting a limited number of additions and multiplications over encrypted bits. Then they apply Gentry's "squash decryption"
doi:10.1007/978-3-642-22792-9_28 fatcat:zdkmjyttz5h2jdmzbndugvhcfy