On the efficiency of solving Boolean polynomial systems with the characteristic set method

Abstract

An improved characteristic set algorithm for solving Boolean polynomial systems is proposed. This algorithm is based on the idea of converting all the polynomials into monic ones by zero decomposition, and using additions to obtain pseudo-remainders. Three important techniques are applied in the algorithm. The first one is eliminating variables by new generated linear polynomials. The second one is optimizing the strategy of choosing polynomial for zero decomposition. The third one is to compute add-remainders to eliminate the leading variable of new generated monic polynomials. By analyzing the depth of the zero decomposition tree, we present some complexity bounds of this algorithm, which are lower than the complexity bounds of previous characteristic set algorithms. Extensive experimental results show that this new algorithm is more efficient than previous characteristic set algorithms for solving Boolean polynomial systems.

Solving Boolean polynomial systems by parallelizing characteristic set method for cyber‐physical systems

“…Moreover, Huang et al 21 propose BCS method to further develop and improve the performance of MFCS. It adds a simplification process into algorithm of MFCS and uses an alternative choose function to determine the order of choosing polynomial for zero decomposition.…”

“…Finding efficient algorithms for solving such systems is not only a challenging problem in mathematics, but also in computer sciences. In recent decades, a number of methods have been proposed and developed, including the characteristic set (CS) method, 15‐21 the Gröbner bases (GB) method (F4, F5), 22‐25 the XL method, 26,27 the fast exhaustive search algorithm, 28 and other hybrid algorithms 29,30 . Moreover, many works are conducted about the asymptotic complexity of solving Boolean polynomial systems, especially the complexity of solving Boolean quadratic polynomial systems.…”

“…Previous work has demonstrated that these methods have excellent performance in solving a certain type of Boolean polynomial systems. In particular, CS method is testified to be very efficient for solving Boolean polynomial system which generated from some special cryptanalysis problems, such as stream ciphers 18,19,21 . The main idea of CS method is reducing equation systems in general form to equation systems in the form of triangular sets.…”

“…After this transformation, the process of solving an equation system can be reduced to solving univariate equations in cascaded form. There are many other scenarios where the CS method can be used, such as to compute the degree, the order, and the dimension for an polynomial system, to prove theorems from elementary and differential geometries, even to solve the radical ideal membership problem 21 . Solving the polynomial systems in finite fields, especially solving Boolean polynomial systems, is the most useful application of CS.…”

“…Solving the polynomial systems in finite fields, especially solving Boolean polynomial systems, is the most useful application of CS. Many efficient CS algorithms are proposed for solving polynomial systems, such as Multiplication‐free Characteristic Set method (MFCS), 19 Boolean Characteristic Set method (BCS) 21 …”