On zeros of Kloosterman sums

Abstract

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.

On the enumeration of irreducible polynomials over GF(q) with prescribed coefficients

“…, and hence there are four cases to consider when computing the zeta functions of each of the curves (38). In order to express F 2 (n, t 1 , t 2 , t 3 , t 4 ) compactly, we further define the following polynomials:…”

“…There are two other polynomials which occur as factors of the characteristic polynomial of Frobenius of the above curves, but they are even polynomials and hence can be ignored for n odd. Using Magma to compute the zeta functions of the curves (38) and applying (36) gives the following result.…”

“…Computing Kloosterman sum zeros is generally regarded as being difficult, currently taking exponential time (in n) to find a single non-trivial (a = 0) zero. Besides the deterministic test due to Ahmadi and Granger [4], which computes the cardinality of the Sylow 2-subgroup of any E 2 n (a) via point-halving, and thus by Lemma 11 the maximum power of 2 dividing K 2 n (a), research has focused on characterising Kloosterman sums modulo small integers [41,37,18,26,38,21,22,20]. In order to analyse the expected running time of the algorithm of Ahmadi and Granger, it is necessary to know the distribution of Kloosterman sums which are divisible by successive powers of 2.…”

“…[51]), hence T 2 n (3) = 2 n−1 − 1. Finally, Lisoněk and Moisio proved that T (n, 4) = (2 n − (−1 + i) n − (−1 − i) n )/4 − 1, connecting it with the number of points on a supersingular elliptic curve [38,Theorem 3.6].…”

Binary Kloosterman sums using Stickelberger's theorem and the Gross–Koblitz formula

“…However determining a zero of a Kloosterman sum is not easy. A recent result in this direction is the following: a binary or ternary Kloosterman sum K p n (a) is not zero if a is in a proper subfield of F p n except when p = 2, n = 4, a = 1, see [14]. Given the difficulty of the problem of finding zeros (or explicit values) of Kloosterman sums, and that they sometimes do not exist, one is generally satisfied with divisibility results and characterisation of Kloosterman sums modulo some integer (see [15,13,3,1,14]).…”