A New Efficient Algorithm for Solving Systems of Multivariate Polynomial Equations

The security of many recently proposed cryptosystems is based on the difficulty of solving large systems of quadratic multivariate polynomial equations. The classical algorithm for solving such a system is Buchberger\u27s algorithm for constructing Gröbner bases. Another algorithm for solving such a system is XL algorithm. For sparse system, Buchberger\u27s algorithm benefits from sparsity of the system, but its complexity is impractical and hard to determine. XL could not make a good use of sparse structure of the system, since XL has no good strategy of choosing the multiply monomials. In this paper, based on Extended Dixon Resultants, a new algorithm DR is proposed to solve systems of multivariate polynomial equations. The basic idea of DR is to apply Extended Dixon Resultants method to system of multivariate polynomial equations, by taking $x_1 \ldots x_{n-1}$ as variables and $x_n$ as parameter. The time complexity of DR technique is evaluated, it seems to be polynomial when the system is sparse and $m=n$ and mixed volume is polynomial. As far as we know, it is the first algorithm which has better behavior than exhaustive search for some sparse systems over large field. Moreover, DR technique is compared with Buchberger\u27s algorithm and XL technique in this paper. It is shown that DR is far more efficient than Buchberger\u27s algorithm and XL when $m=n$. DR is a quite efficient algorithm, it makes a good use of the sparsity of the sparse system. Besides its efficiency, another advantage of DR is that its complexity is easy to determine