New Binomial Bent Function over the Finite Fields of Odd Characteristic

TL;DR

This paper introduces a new binomial bent function over finite fields of odd characteristic, proving its weak regularity and calculating its Walsh transform coefficients, with novel results in exponential sums.

Contribution

The paper presents a new binomial bent function over GF(p^{4k}) and provides exact Walsh transform values, advancing understanding of bent functions in finite field theory.

Findings

Proves the function is weakly regular bent.

Calculates exact Walsh transform coefficients.

Introduces new results in exponential sums over finite fields.

Abstract

The $p$-ary function $f(x)$ mapping $\mathrm{GF}(p^{4k})$ to $\mathrm{GF}(p)$ given by $f(x)={\rm Tr}_{4k}\big(x^{p^{3k}+p^{2k}-p^k+1}+x^2\big)$ is proven to be a weakly regular bent function and the exact values of its Walsh transform coefficients are found. The proof is based on a few new results in the area of exponential sums and polynomials over finite fields that may also be interesting as independent problems.