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 \F_{2^n} or \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 \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

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

An efficient deterministic test for Kloosterman sum zeros

We propose a simple deterministic test for deciding whether or not an element a ∈ F×2n or F×3n is a zero of the corresponding Kloosterman sum over these fields, and rigorously analyse its runtime. The test seems to have been overlooked in the literature. The expected cost of the test for binary fields is a single point-halving on an associated elliptic curve, while for ternary fields the expected cost is one-half of a point-thirding on an associated elliptic curve. For binary fields of practical interest, this represents an O(n) speedup over the previous fastest test. By repeatedly invoking the test on random elements of F×2n we obtain the most efficient probabilistic method to date to find nontrivial Kloosterman sum zeros. The analysis depends on the distribution of Sylow p-subgroups in the two families of associated elliptic curves, which we ascertain using a theorem due to Howe

An efficient deterministic test for Kloosterman sum zeros

We propose a simple deterministic test for deciding whether or not an element a ∈ F×2n or F×3n is a zero of the corresponding Kloosterman sum over these fields, and rigorously analyse its runtime. The test seems to have been overlooked in the literature. The expected cost of the test for binary fields is a single point-halving on an associated elliptic curve, while for ternary fields the expected cost is one-half of a point-thirding on an associated elliptic curve. For binary fields of practical interest, this represents an O(n) speedup over the previous fastest test. By repeatedly invoking the test on random elements of F×2n we obtain the most efficient probabilistic method to date to find nontrivial Kloosterman sum zeros. The analysis depends on the distribution of Sylow p-subgroups in the two families of associated elliptic curves, which we ascertain using a theorem due to Howe