LAUSR

x∈Fq χ(F (x)) = 0 for all nontrivial additive characters χ of Fq. Permutation polynomials are a very important class of polynomials as they have applications in coding theory and cryptography, especially in the substitution boxes (S-boxes) of the block ciphers. The security of the S-boxes relies on certain properties of the function F (x), e.g., its differential uniformity, boomerang uniformity, nonlinearity etc. Recently, Cid et al. [4] introduced a “new tool” for analyzing the boomerang style attack proposed by Wagner [17]. This new tool is usually referred to as Boomerang Connectivity Table (BCT). Boura and Canteaut [2] further studied BCT and coined the term boomerang uniformity, which is essentially the maximum value in the BCT. Li et al. [9] provided new insights in the study of BCT and presented an equivalent technique to compute BCT, which does not require the compositional inverse of the permutation polynomial F (x) at all. In fact, Li et al. [9] also gave a characterization of BCT in terms of Walsh transform and gave a class of permutation polynomial with boomerang uniformity 4. Recently, Stănică [12] extended the notion of BCT and boomerang uniformity. In fact, he defined what he termed as c-BCT and c-boomerang uniformity for an arbitrary polynomial function F over Fq and for any c 6= 0 ∈ Fq. Let a, b ∈ Fq, then the entry of the c-Boomerang Connectivity Table (c-BCT) at (a, b) ∈ Fpn × Fpn, denoted as cBF (a, b), is the number of solutions in Fpn × Fpn of the following system