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 this paper we present a new variant of the Zhuang-Zi algorithm, which solves multivariate polynomial equations over a finite field by converting it into a single variable problem over a large extension field. The improvement is based on the newly developed concept of mutant in solving multivariate equations.

We show all the existing TTM implementation schemes have a defect that there exist linearization equations \sum_{i=1,j=1}^{n,m} a_{ij} x_i y_j(x_1,...,x_n) + \sum_{i=1}^n b_i x_i + \sum_{j=1}^m c_j y_j(x_1,...,x_n) + d = 0, which are satisfied by the components y_i(x_1,...,x_n) of the ciphers of the TTM schemes. We further demonstrate that, for the case of the most recent two implementation schemes in two versions of the paper \cite{CM}, where the inventor of TTM used them to refute a claim in \cite{CG}, if we do a linear substitution with the linear equations derived from the linearization equations for a given ciphertext, we can find the plaintext easily by an iteration of the procedure of first search for linear equations by linear combinations and then linear substitution. The computation complexity of the attack on these two schemes is less than 2^{35} over a finite field of size 2^8.

Recent developments in multivariate polynomial solving algorithms have made algebraic cryptanalysis a plausible threat to many cryptosystems. However, theoretical complexity estimates have shown this kind of attack unfeasible for most realistic applications. In this paper we present a strategy for computing Gröbner basis that challenges those complexity estimates. It uses a flexible partial enlargement technique together with reduced row echelon forms to generate lower degree elements–mutants. This new strategy surpasses old boundaries and obligates us to think of new paradigms for estimating complexity of Gröbner basis computation. The new proposed algorithm computed a Gröbner basis of a degree 2 random system with 32 variables and 32 equations using 30 GB which was never done before by any known Gröbner bases solver.

The problem is concerned about how large (e.g. the Hausdorff dimension) is Whitney's critical set contained in a given fractal. For this, we prove that the Moran arc, an arc containing a Moran set, is a Whitney's critical set. The excellent open set condition is defined, when the condition holds, the associated self-similar set contains a Whitney's critical subset of full dimension. As its application, the Sierpinski gasket and Koch curve have Whitney's critical subset of full dimension. Finally, we provide a self-similar tree which never contains any Whitney's critical set.