Attack and Improvement on a Symmetric Fully Homomorphic Encryption Scheme

Abstract

Fully homomorphic encryption (FHE) has been regarded as the holy grail of cryptography and supports meaningful computations on encrypted data. Recently, Kumer et al. proposed a symmetric FHE scheme based on the Euler theorem. However, we illustrate in the paper that the secret key can be recovered from the homomorphic computation key and only a pair of known plaintext/ciphertext by computing the greatest common divisor with the Euclidean algorithm. Our observations are well supported by the numerical experiments on the suggested parameters. Furthermore, an improvement in the scheme is presented. We mimic Dijk et al.'s idea to illustrate that the improvement can be reduced to the approximate greatest common divisor (AGCD) problem. Several known attacks, including the proposal in this paper, are also examined.

Keywords

• Symmetric homomorphic encryption
• cryptanalysis
• known plaintext attack
• greatest common divisor