A New Somewhat Homomorphic Encryption Scheme over Integers

At Eurocrypt 2010 van Dijk et al. presented a very simple somewhat homomorphic encryption scheme over the integers. However, this simplicity came at the cost of a public key size in Õ(λ¹⁰). Although at Crypto 2011 Coron et al. reduced the public key size to Õ(λ⁷), it was still too large for practical applications. In this paper we further reduce the public key size to Õ(λ³) by encrypting with a new form. The semantic security of our scheme is based on approximate-GCD problem of two integers. By using Gentry´s techniques, we can easily convert the somewhat scheme into a practical fully homomorphic encryption scheme available in cloud computing.