A degree bound for the c-boomerang uniformity of permutation monomials

Let \(\mathbb{F}_q\) ${\mathbb{F}}_q$ be a finite field of characteristic p. In this paper we prove that the c-Boomerang Uniformity, \(c \ne 0\) $c\ne 0$ , for all permutation monomials \(x^d\) $x^d$ , where \(d > 1\) $d>1$ and \(p \not \mid d\) $p\nmid d$ , is bounded by \(\left\{ \begin{array}{ll} d^2, & c^2 \ne 1, \\ d \cdot (d - 1), & c = - 1, \\ d \cdot (d - 2), & c = 1 \end{array} \right\} .\) $\left\{\begin{array}{cc}{d}^2,\hfill & c^2\ne 1,\hfill \\ d·(d-1),\hfill & c=-1,\hfill \\ d·(d-2),\hfill & c=1\hfill \end{array}\right\}.$ Further, we utilize this bound to estimate the c-boomerang uniformity of a large class of generalized triangular dynamical systems, a polynomial-based approach to describe cryptographic permutations of \(\mathbb{F}_{q}^{n}\) ${\mathbb{F}}_q^n$ , including the well-known substitution–permutation network.