@article {IOPORT.05876216,
    author       = {Lison\v ek, Petr and Moisio, Marko},
    title        = {On zeros of Kloosterman sums.},
    year         = {2011},
    journal      = {Designs, Codes and Cryptography},
    volume       = {59},
    number       = {1-3},
    issn         = {0925-1022},
    pages        = {223-230},
    publisher    = {Springer, Norwell, MA},
    doi          = {10.1007/s10623-010-9457-x},
    abstract     = {Let $p$ be a prime number, $m$ a positive integer, $\Bbb F_q$ the finite field of order $q=p^m$ and $\Bbb F^*= \Bbb F_q \setminus \{0 \}$. The Kloosterman sum on $\Bbb F$ is the map $K_q: \Bbb F_q \to \Bbb R $ defined by $$ K_q(a): = 1+ \sum_{x \in \Bbb F_{q}^{*}} e^{2\pi i \text{tr}(x^{-1}+ax)/p}, $$ where $\text{tr}: \Bbb F_q \to \Bbb F_p$ is the absolute trace. Any $a \in \Bbb F_{q}^{*}$ such that $K_q(a)=0$ is called a Kloosterman zero. These zeros play a significant role in the construction of highly nonlinear functions that are used in Cryptography. In particular, Kloosterman zeros can be used to construct monomial hyperbent (bent) functions in even (odd) characteristic, respectively. The authors give an elementary and nearly self-contained proof of the fact that for characteristic $2$ and $3$, no Kloosterman zero in $\Bbb F_q$ belongs to a proper subfield of $\Bbb F_q$ with the only exception that occurs at $q=16$. This (and even more general) result was earlier obtained by {\it M. Moisio} [IEEE Trans. Inf. Theory 55, 3563--3564 (2009)], but the proof there is more involved and not self-contained. Moreover, quite recently it was proved by {\it K. Kononen, M. Rinta-aho} and {\it K. V\"a\"an\"anen} [IEEE Trans. Inf. Theory 58, No. 8, 4011--4013 (2010)] that no Kloosterman zero exists in a field of characteristic greater than $3$. In conclusion, the authors also characterize those binary Kloosterman sums that are divisible by 16 as well as those ternary Kloosterman sums that are divisible by 9.},
    reviewer     = {Sergey Stepanov (Moskva)},
    identifier   = {05876216},
}