$4$-uniform BCT permutations from generalized butterfly structure

As a generalization of Dillon's APN permutation, butterfly structure and generalizations have been of great interest since they generate permutations with the best known differential and nonlinear properties over the field of size $2^{4k+2}$. Complementary to these results, we show in this paper that butterfly structure, more precisely the closed butterfly also yields permutations with the best boomerang uniformity, a new and important parameter related to boomerang-style attacks. This is the sixth known infinite family of permutations in the literature with the best known boomerang uniformity over such fields.
Comment: This paper was originally submitted on 5 December, 2019, and as we know later, is independent of arXiv:1912.02640 which appeared at the same time and obtained essentially the same results by using quite different methods. A substantial extension of this paper was recently obtained in arXiv:2001.00418