A note on the moments of Kloosterman sums
This paper considers several types of Kloosterman sums and proves identities between such sums. Three main types are considered: Kloosterman sums \(K_n(a)\) of degree \(n\), \(m\)-dimensional Kloosterman sums \(K^{(m)}(a)\), and \(m\)-dimensional Kloosterman sums \(K_n^{(m)}(a)\) of degree \(n\). In order to define these, we recall some notation: \(p\) is a prime, \(q=p^r\) for some fixed power \(r\), \({\text{tr}}\) denotes the trace from \(\mathbb{F}_q\) to \(\mathbb{F}_p\), namely \({\text{tr}}(x) = x+ x^p + x^{p^2} + \cdots + x^{p^{r-1}}\) for \(x \in \mathbb{F}_q\), and \({\text{Tr}}\) denotes the trace from \(\mathbb{F}_{q^m}\) to \(\mathbb{F}_q\), namely \({\text{Tr}}(x) = x+ x^q + x^{q^2} + \cdots + x^{q^{m-1}}\) for \(x \in \mathbb{F}_{q^m}\). Let \(\omega\) be a primitive \(p\)-th root of unity. The relevant Kloosterman sums are defined by: \[ K_n(a) = \sum_{x_1,\ldots, x_n \in \mathbb{F}_q^*} \omega ^{{\text{tr}} (x_1 + \cdots + x_n + \frac{a}{x_1 \cdots x_n})}, \] \[ K^{(m)}(a) = \sum_{x \in \mathbb{F}^*_{q^m}} \omega^{{\text{tr}}( {\text{Tr}}(x + ax^{-1}))}, \] \[ K_n^{(m)}(a) = \sum_{x_1,\ldots, x_n \in \mathbb{F}_{q^m}^*} \omega^{{\text{tr}}( {\text{Tr}}(x_1+ \cdots + x_n + \frac{a}{x_1\cdots x_n}))}. \] For a Kloosterman sum \(K(a)\) (where \(K\) may be any of the above three types), \(\sum_a (K(a))^d\) is the \(d\)-th moment. The main results of the paper pertain to the case \(q=2^r\), for which the author computes \(d\)-th moments in the following cases: for \(K_2\), the \(d\)-th moment for \(d=1,2,3,4\); for \(K^{(4)}\), \(d=2\); for \(K^{(2)},\) \(d=4\). A key theorem generalizes a result of \textit{L. Carlitz} [Math. Scand. 42, 39--48 (1978; Zbl 0388.12007)] and proves that in the case \(q=2^r\), for every positive integer \(m\), \[ K_2^{(m)}(a) = (K^{(m)}(a))^2 -q^m, \] for all \(a \in \mathbb{F}_{q^m}^*\). From this, the author is able to compute the following \(d\)-th moments: for \(K_2^{(2)},\) \(d=1,2\); for \(K_2^{(3)}\), \(d=1\); for \(K_2^{(4)}\), \(d=1\). The author also considers the correlation of pairs of sequences constructed from Kloosterman sums, as well as generalized Kloosterman sums.
scientific article; zbMATH DE number 5659686