International Association for Cryptologic Research

International Association
for Cryptologic Research


Naoki Ogura


Cryptographic Pairings Based on Elliptic Nets
In 2007, Stange proposed a novel method of computing the Tate pairing on an elliptic curve over a finite field. This method is based on elliptic nets, which are maps from $\mathbb{Z}^n$ to a ring that satisfy a certain recurrence relation. In this paper, we explicitly give formulae for computing some variants of the Tate pairing: Ate, Ate$_i$, R-Ate and Optimal pairings, based on elliptic nets. We also discuss their efficiency by using some experimental results.
Remarks on the Attack of Fouque et al. against the {\ell}IC Scheme
Naoki Ogura Shigenori Uchiyama
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.