Special values of Kloosterman sums and binomial bent functions

Let p ≥ 7, q = p. Kq(a) = ∑ x∈Fpm ζ m 1 (x m−2+ax) is the Kloosterman sum of a on Fpm , where ζ = e 2π √ −1 p . The value 1− 2 ζ+ζ−1 of Kq(a) and its conjugate have close relationship with a class of binomial function with Dillon exponent. This paper first presents some necessary conditions for a such that Kq(a) = 1− 2 ζ+ζ−1 . Further, we prove that if p = 11, for any a, Kq(a) 6= 1− 2 ζ+ζ−1 . And for p ≥ 13, if a ∈ Fps and s = gcd(2,m), Kq(a) 6= 1− 2 ζ+ζ−1 . In application, these results explains some class of binomial regular bent functions does not exits. Index Terms Regular bent function, Walsh transform, Kloosterman sums, π-adic expansion, cyclotomic fields