Cryptanalysis of GiophantusTM Schemes against Hybrid Attack

Abstract

The hybrid attack was proposed by Howgrave-Graham in CRYPTO2007, which was originally designed for the cryptanalysis of NTRU cryptosystems. In this paper, based on Howgrave-Graham's attack model, we propose a simulator of hybrid attack to evaluate the hardness of the unique shortest vector problem. By a dynamical computation, our algorithm can trade off the cost between reduction and MitM, while both of them run in exponential time. Further, we adapt our simulator to Giophantus™, Giophantus+ and Giophantus-cryptosystems, proposed by Akiyama et al. in SAC2017, SCIS2019 and SCIS2020, respectively. Our analysis shows that by the hybrid attack, the security levels can be reduced by at most 19 bits for Giophantus' parameters proposed in NIST Post Quantum Cryptography (PQC) standardization 1st round submission. Meanwhile, the parameter sets of Giophantus+ and Giophantus-are secure against the hybrid attack.