We present the Zhuang-Zi algorithm, a new method for solving multivariate polynomial equations over a finite field. We describe the algorithm and present examples, some of which cannot be solved with the fastest known algorithms.

In the CT-track of the 2006 RSA conference, a new multivariate public key cryptosystem, which is called the Medium Field Equation (MFE) multivariate public key cryptosystem, is proposed by Wang, Yang, Hu and Lai. We use the second order linearization equation attack method by Patarin to break MFE. Given a ciphertext, we can derive the plaintext within 2^{23} F_{2^16}-multiplications, after performing once for any given public key a computation of complexity less than 2^{52}. We also propose a high order linearization equation (HOLE) attack on multivariate public key cryptosystems, which is a further generalization of the (first and second order) linearization equation (LE). This method can be used to attack extensions of the current MFE.

Balanced Oil and Vinegar signature schemes and the unbalanced Oil and Vinegar signature schemes are public key signature schemes based on multivariable polynomials. In this paper, we suggest a new signature scheme, which is a generalization of the Oil-Vinegar construction to improve the efficiency of the unbalanced Oil and Vinegar signature scheme. The basic idea can be described as a construction of multi-layer Oil-Vinegar construction and its generalization. We call our system a Rainbow signature scheme. We propose and implement a practical scheme, which works better than Sflash^{v_2}, in particular, in terms of signature generating time.

We apply internal perturbation [3] to the matrix-type cryptosystems [C_n] and HM constructed in [9]. Using small instances of these variants, we investigate the existence of linearization equations and degree 2 equations that could be used in a XL attack. Our results indicate that these new variants may be suitable for use in practical implementations. We propose a specific instance for practical implementation, and estimate its performance and security.

In ACISP ’00, Wu et al. proposed attacks to break the Polynomial Authentication and Signature Scheme (PASS), in particular, they are able to generate valid authentication transcripts and digital signatures without knowing the private key and any previous transcripts/ signatures. They showed that PASS can be broken with around 238.3 trials. In this paper, we analyze the security of the improved versions of PASS; viz. PASS II and MiniPASS, and extend the Wu et al.’s attacks to PASS II and MiniPASS to break them. Furthermore, we discuss why and how these schemes are broken from the view point of the structure of cryptosystems and point out the fundamental weakness behind.