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

Let $$\mathbb{F}_q$$ F q be a finite field of characteristic p . In this paper we prove that the c -Boomerang Uniformity, $$c \ne 0$$ c ≠ 0 , for all permutation monomials $$x^d$$ x d , where $$d > 1$$ d > 1 and $$p \not \mid d$$ p ∤ d , is bounded by $$\left\{ \begin{array}{ll} d^2, & c^2 \ne 1, \\ d \cdot (d - 1), & c = - 1, \\ d \cdot (d - 2), & c = 1 \end{array} \right\} .$$ d 2 , c 2 ≠ 1 , d · ( d - 1 ) , c = - 1 , d · ( d - 2 ) , c = 1 . 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 $$\mathbb{F}_{q}^{n}$$ F q n , including the well-known substitution–permutation network.