  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.
