Abstract: S-box is important in symmetric cryptosystem.One method is proposed to constructed perfect nonlinear S-box over finite field.Compared to the construction over vector space，it is easier to study the properties of permutation polynomials，like algebraic degree.By choosing different parameters，different types of S-boxes can be constructed.For example，by choosing α^x as the permutation polynomial，we obtain the S-box once constructed by Satoh et al.By selecting permutation of form x^d，we obtain another type of S-boxes.The algebraic degree of any nonzero linear combination of its output functions is highest.

摘要： 完全非线性S-盒在对称密码中有着重要的运用。给出有限域上完全非线性S-盒的一种构造方法。与在向量空间上构造的方法比，有限域上置换多项式的代数次数等性质更容易研究。该方法可以构造多类完全非线性S-盒，例如，通过选择幂函数形式的置换α^x，得到Satoh等人构造的S-盒；通过选取指数形式的置换x^d，所得完全非线性S-盒的分量函数的任意非零线性组合的代数次数达到最高。