The Differential and Boomerang Properties of a Class of Binomials

Let <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</i> be an odd prime power with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q\equiv 3\ ({\mathrm {mod}}\,4)$ </tex-math></inline-formula>. In this paper, we study the differential and boomerang properties of the function <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$F_{2,u}(x)=x^{2}\big (1+u\eta (x)\big)$ </tex-math></inline-formula> over <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbb {F}_{q}$ </tex-math></inline-formula>, where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$u\in \mathbb {F}_{q}^{*}$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\eta $ </tex-math></inline-formula> is the quadratic character of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbb {F}_{q}$ </tex-math></inline-formula>. We determine the differential uniformity of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$F_{2,u}$ </tex-math></inline-formula> for any <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$u\in \mathbb {F}_{q}^{*}$ </tex-math></inline-formula>, as well as the differential spectra and boomerang uniformity of the locally-APN functions <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$F_{2,\pm 1}$ </tex-math></inline-formula>, thereby disproving a conjecture proposed in Budaghyan and Pal (2024), which states that there exist infinitely many values of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</i> and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">u</i> such that <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$F_{2,u}$ </tex-math></inline-formula> is an APN function.