The Gowers U3 norm of five classes of power permutations

Vectorial Boolean functions from F 2 n to F 2 m are fundamental objects in theoretical computer science and mathematics. Understanding their structure, particularly how well they can be approximated by low-degree functions, is crucial in various applications, including pseudorandomness, property testing, and cryptography. The Gowers uniformity norm, introduced in additive combinatorics, provides a powerful measure for these purposes, serving as a key indicator of the approximation and has significant applications in mathematics and theoretical computer science. While the Gowers U 2 norm is well-understood, the analysis of higher-order structures, particularly the Gowers U k norm for k ≥ 3 , poses significant challenges. Indeed, the computation of the Gowers U 3 norm is intrinsically linked to the second-order differential spectrum of a function. However, determining this spectrum is a notoriously difficult problem, and to date, it has only been solved for a few specific cases, such as the inverse function over F 2 n for n = 6 , 8 and the APN permutation over F 2 6 . In this paper, we investigate the Gowers U 3 norms of vectorial Boolean functions, with specific focus on five classes of cubic highly nonlinear power permutations over finite fields. We provide a comprehensive analysis of the Gowers U 3 norm for the Welch function, the Modified Welch function, the cubic Kasami function, the Bracken-Leander function, and the Cusick-Dobbertin function. By characterizing the distribution of solutions for all second-order derivatives of these functions, we derive exact expressions for their Gowers U 3 norms. Our results reveal quantitative differences in higher-order uniformity among these functions, with the Modified Welch function exhibiting a smaller Gowers U 3 norm compared to the Welch and cubic Kasami functions. And the Bracken-Leander function and the Cusick-Dobbertin function are shown to possess larger Gowers U 3 norm compared to the inverse function. These findings contribute to a deeper understanding of these highly nonlinear power permutations.