## Cryptology ePrint Archive: Report 2008/208

Remarks on the Attack of Fouque et al. against the {\ell}IC Scheme

Naoki Ogura and Shigenori Uchiyama

Abstract: In 2007, $\ell$-Invertible Cycles ($\ell$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 public-key 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 $\ell$IC signature scheme by using Gr\"obner basis algorithms. However, they only explicitly dealt with the odd case, i.e. $\ell$ 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 $\ell$IC encryption/signature scheme. Our proposed attack does not employ Gr\"obner 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 $\ell$IC- scheme. To the best of our knowledge, we for the first time show some experimental results of a practical attack against the $\ell$IC- scheme for the even case.

Category / Keywords: public-key cryptography / $\ell$-Invertible Cycles, Multivariate Quadratic Public Key Cryptosystem, SFLASH

Date: received 12 May 2008, last revised 20 May 2008

Contact author: ogura-naoki at ed tmu ac jp

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