Cryptanalysis of factoring-based fully homomorphic encryption

Abstract

This paper deals with fully homomorphic cryptosystems exploiting the problem of big integers factoring. We give a short review of them and highlight two main types of such fully homomorphic cryptosystems (FHCs): polynomial-based and matrix-based. The main focus of the discussion is placed on one recently proposed polynomial-based FHC. Its construction is recalled, but mainly we concentrate on security issues. And here our contribution is twofold. First, we review a known-plaintext attack (KPA) proposed in literature on this FHC. We give the general idea of KPA, the probability of its success and the number of pairs (plaintext, ciphertext) necessary to break the FHC. Second, we discuss how the reviewed KPA may be extended in order to decrease the necessary number of pairs. On a high level the proposed extension of KPA may be applied not only to this concrete FHC, but to all reviewed here FHCs. Our KPA essentially uses non-uniformity of probabilistic distribution over plaintexts to obtain a high probability of success. And instead of missing pairs it requires an additional sequence of ciphertexts produced on the same key.