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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.

In 2008, Hashimoto and Sakurai proposed a new efficient signature scheme, which is a noncommutative version of Shamir's birational permutation signature scheme. Shamir's scheme is a generalization of the Ong-Schnorr-Shamir scheme and was broken by Coppersmith et al. using its linearity and commutativity. The HS (Hashimoto-Sakurai) scheme is expected to be secure against the attack from its non-commutative structure. In this paper, we propose an attack against the HS scheme, which is practical under the condition that its step size and the number of steps are small. We discuss its efficiency by using some experimental results.

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