Closed formulas for exponential sums of symmetric polynomials over Galois fields

TL;DR

This paper derives closed-form formulas for exponential sums of symmetric polynomials over Galois fields, extending previous binary field results and revealing their linear recursive nature over any finite field.

Contribution

It generalizes known formulas from Boolean functions to all Galois fields, providing explicit linear recurrences and linking to multinomial coefficient problems.

Findings

Exponential sums are linear recursive over any finite field.

Closed formulas for these sums are explicitly derived.

A connection to multinomial coefficient problems is established.

Abstract

Exponential sums have applications to a variety of scientific fields, including, but not limited to, cryptography, coding theory and information theory. Closed formulas for exponential sums of symmetric Boolean functions were found by Cai, Green and Thierauf in the late 1990's. Their closed formulas imply that these exponential sums are linear recursive. The linear recursivity of these sums has been exploited in numerous papers and has been used to compute the asymptotic behavior of such sequences. In this article, we extend the result of Cai, Green and Thierauf, that is, we find closed formulas for exponential sums of symmetric polynomials over any Galois fields. Our result also implies that the recursive nature of these sequences is not unique to the binary field, as they are also linear recursive over any finite field. In fact, we provide explicit linear recurrences with integer coefficients for such sequences. As a byproduct of our results, we discover a link between exponential sums of symmetric polynomials over Galois fields and a problem for multinomial coefficients which similar to the problem of bisecting binomial coefficients.