In 2006, Nie et al proposed an attack to break an instance of TTM cryptosystems. However, the inventor of TTM disputed this attack and he proposed two new instances of TTM to support his viewpoint. At this time, he did not give the detail of key construction --- the construction of the lock polynomials in these instances which would be used in decryption. The two instances are claimed to achieve a security of 2^{109} against Nie et al attack. In this paper, we show that these instances are both still insecure, and in fact, they do not achieve a better design in the sense that we can find a ciphertext-only attack utilizing the First Order Linearization Equations while for the previous version of TTM, only Second Order Linearization Equations can be used in the beginning stage of the previous attack. Different from previous attacks, we use an iterated linearization method to break these two instances. For any given valid ciphertext, we can find its corresponding plaintext within 2^{31} F_{2^8}-computations after performing once for any public key a computation of complexity less than 2^{44} . Our experiment result shows we have unlocked the lock polynomials after several iterations, though we do not know the detailed construction of lock polynomials.

We prove that a random map drawn from any class \mathfrak{C} of polynomial maps from (F_q)^n to (F_q)^{n+r} that is (i) totally random in the affine terms, and (ii) has a negligible chance of being not strongly one-way, provides a secure PRNG (hence a secure stream cipher) for any q. Plausible choices for are semi-sparse (i.e., the affine terms are truly random) systems and other systems that are easy to evaluate from a small (compared to a generic map) number of parameters. To our knowledge, there are no other positive results for provable security of specialized polynomial systems, in particular sparse ones (which are natural candidates to investigate for speed). We can build a family of provably secure stream ciphers a rough implementation of which at the same security level can be more than twice faster than an optimized QUAD (and any other provably secure stream ciphers proposed so far), and uses much less storage. This may also help build faster provably secure hashes. We also examine the effects of recent results on specialization on security, e.g., Aumasson-Meier (ICISC 2007), which precludes Merkle-Damgaard compression using polynomials systems {uniformly very sparse in every degree} from being universally collision-free. We conclude that our ideas are consistent with and complements these new results. We think that we can build secure primitives based on specialized (versus generic) polynomial maps which are more efficient.

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.

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).

The Matsumoto-Imai (MI) cryptosystem was the first multivariate public key cryptosystem proposed for practical use. Though MI is now considered insecure due to Patarin’s linearization attack, the core idea of MI has been used to construct many variants such as Sflash, which has recently been accepted for use in the New European Schemes for Signatures, Integrity, and Encryption project. Linearization attacks take advantage of the algebraic structure of MI to produce a set of equations that can be used to recover the plaintext from a given ciphertext. In our paper, we present a solution to the problem of finding the dimension of the space of linearization equations, a measure of how much work the attack will require.

We demonstrate how to prevent differential attacks on multivariate public key cryptosystems using the Plus (+) method of external perturbation. In particular, we prescribe adding as few as 10 Plus polynomials to the Perturbed Matsumoto-Imai (PMI) cryptosystem when g=1 and r=6, where θ is the Matsumoto-Imai exponent, n is the message length, g = gcd(θ,n), and r is the internal perturbation dimension; or as few as g+10 when g ≠ 1. The external perturbation does not significantly decrease the efficiency of the system, and in fact has the additional benefit of resolving the problem of finding the true plaintext among several preimages of a given ciphertext. We call this new scheme the Perturbed Matsumoto-Imai-Plus (PMI+) cryptosystem.