Closed formulas for exponential sums of symmetric polynomials over Galois fields

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.

Value distribution of elementary symmetric polynomials and its perturbations over finite fields

“…Furthermore, in this article we use the term exponential sums to refer to both (1.1) and (1.2). In [11], closed formulas for exponential sums of type (1.1) of elementary symmetric polynomials were found (extending the results of [3] to every finite field). There is a natural connection between the formulas presented in [11] and the value distribution of elementary symmetric polynomials over F q .…”

“…Recently, some cryptographic applications when the characteristic of the field is different than 2 has been found. This has prompted new research in exponential sums of the type (1.1) and many of the results available for the binary field have been extended to other finite fields [10,11,21,22,23]. Let L : F q → F q be a linear function and X an indeterminate.…”

“…Suppose that x ∈ A n and that a j appears m j times in x. Following [11], the value of e k (x) will be denoted by Λ a1,...,as (k, m 1 , . .…”

“…As mentioned before, one of the main results of [11] are closed formulas for exponential sums of elementary symmetric polynomials over any finite field. For convenience, we include their result next.…”

“…Theorem 1.1 can be generalized without too much effort to linear combinations of elementary symmetric polynomials. See [11] for more details. Theorem 1.1 is a generalization of the results presented in [3] for the binary field.…”