Comparison-Based Attacks Against Noise-Free Fully Homomorphic Encryption Schemes

Abstract

Homomorphic Encryption provides one of the most promising means to delegate computation to the cloud while retaining data confidentiality. We present a plaintext recovery attack against fully homomorphic schemes which have a polynomial time distinguisher for a given fixed plaintext, and rely on the capability of homomorphically compare a pair of encrypted integer values. We improve by a constant factor the computational complexity of an exhaustive search strategy, which is linear in the recovered plaintext value, and show that it significantly increases the number of recoverable plaintexts. We successfully validate our attack against two noise-free fully homomorphic encryption schemes, which fulfill the mentioned requisite and were claimed to be secure against plaintext recovery attacks.

Toward Understanding Efficient Privacy-Preserving Homomorphic Comparison

“…The described properties of Lagrange interpolation allow us to use this approach to compute any function 𝜑 with 𝑚𝑖𝑛 𝜑 efficiently. The evaluation of 𝑚𝑖𝑛 𝜑 has a minimum multiplicative depth on HE schemes over a field with small characteristics (e.g., integer-based homomorphic schemes), and it includes two-variable sign and comparison functions [36], [37].…”

Plaintext recovery attacks against linearly decryptable fully homomorphic encryption schemes

“…In this section, we analyze the computational effort required by our attack for the target FHE schemes OctoM and JordanM, by estimating the constant factors (in particular, the time complexity of the 1-distinguisher T distinguisher ) in Equation 1. In addition, we show that the speed-up of our attack against the exhaustive search strategy described in Section 3 is significant for the target FHE schemes.…”

“…Such a drawback can be overcome in an effective and elegant way if a cryptographic primitive which allows to perform computations on encrypted data, retaining the correctness of the decrypted result, is available. Such a primitive goes by the name of Fully Homomorphic Encryption (FHE), and was proposed as a pioneering A preliminary version of this paper appeared under the title "Comparison-based Attacks against Noise-free Fully Homomorphic Encryption Schemes," in Proceedings of the 20th International Conference on Information and Communications Security ICICS 2018, Lille, France, Oct. 2018 [1].…”