Using double Weil sums in finding the $c$-Boomerang Connectivity Table for monomial functions on finite fields

TL;DR

This paper characterizes the $c$-Boomerang Connectivity Table for monomial functions over finite fields using Weil sums, providing a complete description for all parameters and simplifying for Gold functions.

Contribution

It offers the first comprehensive characterization of the $c$-BCT for monomial functions, including explicit formulas and simplifications for Gold functions.

Findings

Complete description of $c$-BCT for monomials using Weil sums

Simplified formulas for Gold functions $x^{p^k+1}$

First full characterization for all parameters involved

Abstract

In this paper we characterize the $c$-Boomerang Connectivity Table (BCT), $c\neq 0$ (thus, including the classical $c=1$ case), for all monomial function $x^d$ in terms of characters and Weil sums on the finite field~$\F_{p^n}$. We further simplify these expressions for the Gold functions $x^{p^k+1}$ for all $1\leq k<n$, and $p$ odd. It is the first such attempt for a complete description for the classical BCT and its relative $c$-BCT, for all parameters involved.