The Construction Of Semi-bent Functions With Polynomial Trace Form

Semi-bent functions have been widely studied for their superior properties.But,it is difficult to give a precise classification of semi-bent functions under the current development level of the Mathematics.Hence,the most important thing for us to study the semi-bent functions is to look for the methods to construct them.Many researchers have made a success in studying the semi-bent functions and they made a conclusion that almost all families of semi-bent functions have been derived from the power polynomial Tr1n（xd）.The semi-bent functions that we want to construct it in this paper which are also comes from this kind of power polynomial.This paper is organized as follows.1.Let ga,b,c,d（r,s）（x）=Tr1n（axr（2m-1））+Tr14（bx（2n-1）/5）+Tr1n（cx（2m-1）/2+1）+Tr1n（dx（2m-1）s+1）be a Boolean function on the finite fields F2m,where n=2m with m≡2（mod4）,r be a positive integer, s∈{0,1/4,1/6,3),a∈F2n*,b∈F16*,c∈F2n and d∈F2. We study the semi-bent properties by Walsh transform.We can discuss the equation to detemine whether the Boolean functions ga,b,c,d（r,s）（x） on F2n. could be semi-bent functions,where the function fa,b（r）（x）=Tr1n（axr（2m-1））+Tr14（bx（2n-1）/5） with a∈F2n*and b∈F16*.Thus,it can be divided into three parts:（1）r=1,then we look for the values of a and b such that the equation holds true.（2）r=5, then we look for the proper values of a and b such that the equation （3）We assume r∈Z and make a generalization of the above situations.2.We will make a generalization of the class of semi-bent functions with the trace polynomial form ga,b,c,d（r,s）（x）=Tr1n（axr（2m-1））+Tr12（bx（2n-1）/3）+Tr1n（cx（2m-1）/2+1）+Tr1n（dx（2m-1）s+1）,where n=2n with m is an odd integer, r is a positive integer, s∈{0,1/4,1/6,3) a∈F2n*,,b∈F16*,c∈F2n and d∈F2. 3. We need use Kloosterman sums, cubic sums and Weil sums to study the two above questions. Infact, the question we must discuss is to calculate the exponential sums ofa∈F_2m on the cyclicgroupU.