A method of evaluation of exponential sum of binary quadratic functions

作者：Zhang, Xiyong^*; Cao, Xiwang; Feng, Rongquan

来源：Finite Fields and Their Applications, 2012, 18(6): 1089-1103.

DOI：10.1016/j.ffa.2012.09.007

• Binary quadratic function
• Exponential sum
• Companion polynomial
• Reciprocal polynomial

In this paper, by using the factorization of the companion polynomial of the binary quadratic function f(x) = Sigma(1 <= i <= k)aix(1+2ai) +aox, x is an element of F(2)n, a(i) is an element of F(2)m m vertical bar n we give a method to compute the exponential sum S(f, n) = Sigma(x is an element of)F(2)n (-1)(Tr(f(x))) for the quadratic functions f (x), where Tr(.) is the trace function from F(2)n to F-2. 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 f'(z) equivalent to g(z) g*(z) (mod (2. z(2s) + 1)) 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, 2(s)) for most binary quadratic functions f (x) is an element of F-2[x] can be computed.

• 出版日期2012-11
• 单位中国科学院信息工程研究所; 北京大学; 南京航空航天大学; 中国科学技术信息研究所