A method of evaluation of exponential sum of binary quadratic functions

In this paper, by using the factorization of the companion polynomial of the binary quadratic function f(x)=∑1⩽i⩽kaix1+2αi+a0x, x∈F2n, ai∈F2m, m|n, we give a method to compute the exponential sum S(f,n)=∑x∈F2n(−1)Tr(f(x)) for the quadratic functions f(x), where Tr(⋅) is the trace function from F2n to F2. The computation of the exponential sum of quadratic functions with many terms can be transformed to that of some quadratic functions that can be explicitly evaluated by present results. Moreover, the necessary and sufficient condition for is given, where g⁎(z) is the generalized reciprocal polynomial of g(z) and f′(z) is the companion polynomial of f(x). As a consequence, the exponential sums S(f,2s) for most binary quadratic functions f(x)∈F2[x] can be computed.