Cryptanalysis of a homomorphic encryption scheme from ISIT 2008

At ISIT 2008, Aguilar Melchor, Castagnos and Gaborit presented a lattice-based homomorphic encryption scheme (abbreviated as MCG). Its security is based on the Computational Knapsack Vector Problem. In this paper, we explore a secret linear relationship between the public keys and the secret keys, which can be used to construct a reduced-dimension lattice, and then we obtain a group of equivalent private keys by solving the Closest Vector Problem of the lattice. Moreover, our attack is practical on all the three settings of recommended parameters, and the running time to recover the equivalent private keys is only several hours on a single PC.