Special values of Kloosterman sums and binomial bent functions

Let p≥7p≥7 and q=pmq=pm. Kq(a)=∑x∈FpmζTr1m(xpm−2+ax) is the Kloosterman sum of a   on FpmFpm, where ζ=e2π−1p. The value 1−2ζ+ζ−1 of Kq(a)Kq(a) and its conjugate have close relationship with a class of binomial functions with Dillon exponent. This paper first presents some necessary conditions for a   such that Kq(a)=1−2ζ+ζ−1. Further, we prove that if p=11p=11, for any a  , Kq(a)≠1−2ζ+ζ−1. And for p≥13p≥13, if a∈Fpsa∈Fps and s=gcd(2,m)s=gcd(2,m), Kq(a)≠1−2ζ+ζ−1. In application, these results explain that some class of binomial regular bent functions does not exist.