Remarks on the Attack of Fouque et al. against the ℓIC Scheme

Abstract

Abstract. In 2007, -Invertible Cycles ( IC) was proposed by Ding et al. This is one of the most efficient trapdoors for encryption/signature schemes, and of the mixed field type for multivariate quadratic publickey cryptosystems. Such schemes fit on the implementation over low cost smart cards or PDAs. In 2008, Fouque et al. proposed an efficient attack against the IC signature scheme by using Gröbner basis algorithms. However, they only explicitly dealt with the odd case, i.e. is odd, but the even case; they only implemented their proposed attack in the odd case. In this paper, we propose an another practical attack against the IC encryption/signature scheme. Our proposed attack does not employ Gröbner basis algorithms, and can be applied to the both even and odd cases. We show the efficiency of the attack by using some experimental results. Furthermore, the attack can be also applied to the IC-scheme. To the best of our knowledge, we for the first time show some experimental results of a practical attack against the IC-scheme for the even case.

Practical-Sized Instances of Multivariate PKCs: Rainbow, TTS, and ℓIC-Derivatives

“…Similarly, [24] and the related [29] shows a similar attack (with a more complex backend) almost surely breaks 3IC − and any other IC − . For the IC case, the point is the differential expose the symmetry for a linear map (…”

“…− and discover that again the attacks from [24,29] are prevented. We call this setup 3IC − p(q, k, s).…”