An efficient deterministic test for Kloosterman sum zeros

We propose a simple deterministic test for deciding whether or not a non-zero element $a \in \mathbb{F}_{2^n}$ or $\mathbb{F}_{3^n}$ is a zero of the corresponding Kloosterman sum over these fields, and analyse its complexity. The test seems to have been overlooked in the literature. For binary fields, the test has an expected operation count dominated by just two $\mathbb{F}_{2^n}$-multiplications when $n$ is odd (with a slightly higher cost for even extension degrees), making its repeated invocation the most efficient method to date to find a non-trivial Kloosterman sum zero in these fields. The analysis depends on the distribution of Sylow $p$ subgroups in two corresponding families of elliptic curves, which we prove using a theorem due to Howe.