The Gowers U3 norm of one family of cubic power permutations

Abstract

The Gowers uniformity norm has emerged as a significant metric in the evaluation of Boolean functions employed in symmetric-key encryptions, particularly in assessing their resilience against low degree approximation attacks. Beyond cryptography, this norm plays a pivotal role in theoretical computer science, including pseudorandomness and property testing of Boolean functions. However, the determination of the Gowers $U_k$ norm for $k\ge 3$ for general Boolean functions presents substantial computational and theoretical challenges. Recently, the relationship between the Gowers uniformity norm and the higher-order differential spectrum of Boolean functions has been derived. Furthermore, the fact that the Gowers uniformity norm of a power permutation can be determined by the Gowers uniformity norm of its any component function. In this paper, we focus on determining the Gowers $U_3$ norm of one family of power permutations, thereby assessing their resistance to quadratic approximation attacks. The family contains five classes of power permutations, which are $x^{2^{\frac{n}{2}}+3}$ with even $n\ge 4$, $x^{2^{2s}+2^s+1}$ with $1\le s\le \lfloor \frac{n}{2}\rfloor$ and $n\ge 3$, $x^{2^{n-1}+2^{n-2}+1}$ with $n\ge 3$, $x^{2^{n-1}+2^{\frac{n}{2}}+1}$ with even $n\ge 4$, and $x^{2^{n-1}+2^{\frac{n-1}{2}}+1}$ with odd $n\ge 3$. We establish the second-order differential spectrum of these power permutations, by determining the number of solutions of certain linearized polynomials. Thus, we are able to derive the Gowers $U_3$ norms of these power permutations. At last we present a comparison of the Gowers $U_3$ norms of these power permutations. Our analysis yielded a significant finding that the power permutation $x^{2^{n-1}+2^{n-2}+1}$ with $n\ge 3$ exhibits a better resistance against quadratic approximation attacks than the other four classes of power permutations.