On zeros of Kloosterman sums.

Designs, Codes and Cryptography 59 (2011), 223-230.

A Kloosterman zero is a non-zero element of F_{q} for which the
Kloosterman sum on F_{q} attains the value 0. Kloosterman zeros can
be used to construct monomial hyperbent (bent) functions in even
(odd) characteristic, respectively. We give an elementary
proof of the fact
that for characteristic 2 and 3, no
Kloosterman zero in F_{q} belongs to a proper subfield of F_{q} with
one exception that occurs at q=16.
It was recently proved
that no Kloosterman zero exists in a field of
characteristic greater than 3.
We also characterize
those binary Kloosterman sums that are divisible by 16
as well as
those ternary Kloosterman sums that are divisible by 9.
Hence we provide necessary conditions
that Kloosterman zeros must satisfy.