Characteristic set algorithms for equation solving in finite fields

Efficient characteristic set methods for computing solutions of a polynomial equation system in a finite field is proposed. We introduce the concept of proper triangular sets and prove that proper triangular sets are square-free in certain sense. We present an improved algorithm which can be used to reduce the zero set of an equation system in general form as the union of zero sets of proper triangular sets. As a consequence, we can give an explicit formula for the number of solutions of an equation system. We also give a characteristic set method for equation solving in F2 with better complexity bounds. The methods are implemented and extensive experiments show that they are quite efficient for solving a class of equations raised in analyzing stream ciphers.