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## An efficient deterministic test for Kloosterman sum zeros

### Abstract

We propose a simple deterministic test for deciding whether or not an element $a \in \F_{2^n}^{\times}$ or $\F_{3^n}^{\times}$ 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 $\F_{2^n}^{\times}$ we obtain the most efficient probabilistic method to date to find non-trivial 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.Comment: 17 pages, accepted to Mathematics of Computation. This is the final version with many improvements upon the first version. We also include new records for the computation of Kloosterman zero

Topics: Mathematics - Number Theory, Computer Science - Cryptography and Security, 11L05, 14H52, 20D20
Year: 2012
OAI identifier: oai:arXiv.org:1104.3882