特征为2的有限域上二次函数指数和计算的新方法

A new method to evaluate the exponential sums of quadratic functions on finite field with character 2

设二次函数f(x)=∑1≤i≤kaix1+2αi,k<n/2,x∈F2nai∈F2m,m|n。若f(x)的伴随多项式f'(z)有f'(z)≡g(z)·g*(z)(modzn+1)的分解形式,则可给出一种计算二次函数指数和S(f(x),n)=∑x∈F2n(-1)Tr(f(x))的新方法,其中Tr(·)是从F2n到F2的迹函数,g*(z)为g(z)的互反多项式。通过这种方法,可将一类二次函数指数和简化为可计算函数的指数和。

Let f（x）=1≤i≤k∑aix1＋2ai,k〈n/2,x∈F2nai∈F2m, m｜n be a quadratic function. If the companion polynomial off（x） has the particular factorization of f＇（z）有f＇（z）≡g（z）·g＊（z）（modzn＋1）, a new method to compute the exponential sum S（f（x）,n）=x∈F2n∑（-1）Tr（f（x）） was obtained, where Tr（ ·） is the trace function from F2n to F2, and g＊ （z） is the reciprocal polynomial of g（z）. By this method, the computation of the exponential sums of a large class of quadratic function can be converted to the present result which can be explicitly evaluated.