Some Results About Linearized Permutation Polynomials And Their Applications

In this paper we present the close relationship between planar function and the construction of SHDS. Then we do some research on linearized permutation polynomials, the rank of its coefficient matrix is given. And we give another proof of the sufficient and necessary condition for a linearized polynomial to be a permutation polynomial. At last, using the result, a necessary condition for a DO polynomial to be a APN polynomial is obtained.The structure of this paper is as follows:In first chapter, we give a brief description of the research on combinatorial design and the background of the paper.In second chapter, we introduce some definitions and properties needed in the paper and then present the relationship between planar function and the construction of SHDS. Moreover, we list some SHDS that constructed from planar functions.In chapter3, we do some research on the linearized polynomials. According to the research of its coefficient matrix, the rank of its coefficient matrix is obtained. What’s more, as a special case, we get a sufficient and necessary condition for a linearized polynomial to be a permutation polynomial. In addition, the number of linearized permutation polynomials is given.In chapter4, using the result obtained in chapter3, we give a necessary condition for DO polynomial to be APN function.