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.

Though the Perturbed Matsumoto-Imai (PMI) cryptosystem is considered insecure due to the recent differential attack of Fouque, Granboulan, and Stern, even more recently Ding and Gower showed that PMI can be repaired with the Plus (+) method of externally adding as few as 10 randomly chosen quadratic polynomials. Since relatively few extra polynomials are added, the attack complexity of a Gröbner basis attack on PMI+ will be roughly equal to that of PMI. Using Magma’s implementation of the F 4 Gröbner basis algorithm, we attack PMI with parameters q = 2, 0 ≤ r ≤ 10, and 14 ≤ n ≤ 59. Here, q is the number of field elements, n the number of equations/variables, and r the perturbation dimension. Based on our experimental results, we give estimates for the running time for such an attack. We use these estimates to judge the security of some proposed schemes, and we suggest more efficient schemes. In particular, we estimate that an attack using F 4 against the parameters q = 2, r = 5, n = 96 (suggested in [7]) has a time complexity of less than 250 3-DES computations, which would be considered insecure for practical applications.

Hidden field equation (HFE) multivariable cryptosystems were first suggested by Patarin. Kipnis and Shamir showed that to make the cryptosystem secure, a special parameter D of any HFE cryptosystem can not be too small. Consequently Kipnis, Patarin and Goubin proposed an enhanced variant of the HFE cryptosystem by combining the idea of Oil and Vinegar construction with the HFE construction. Essentially they “perturb” the HFE system with some external variables. In this paper, we will first present a new cryptanalysis method for the HFEv schemes. We then use the idea of internal perturbation to build a new cryptosystem, an internally perturbed HFE cryptosystem (IPHFE).