  At Eurocrypt'18, Cid, Huang, Peyrin, Sasaki, and Song introduced a new tool
called Boomerang Connectivity Table (BCT) for measuring the resistance of a
block cipher against the boomerang attack (which is an important cryptanalysis
technique introduced by Wagner in 1999 against block ciphers). Next, Boura and
Canteaut introduced an important parameter (related to the BCT) for
cryptographic Sboxes called boomerang uniformity. In this context, we present a
brief state-of-the-art on the notion of boomerang uniformity of vectorial
functions (or Sboxes) and provide new results. More specifically, we present a
slightly different (and more convenient) formulation of the boomerang
uniformity and show that the row sum and the column sum of the boomerang
connectivity table can be expressed in terms of the zeros of the second-order
derivative of the permutation or its inverse. Most importantly, we specialize
our study of boomerang uniformity to quadratic permutations in even dimension
and generalize the previous results on quadratic permutation with optimal BCT
(optimal means that the maximal value in the Boomerang Connectivity Table
equals the lowest known differential uniformity). As a consequence of our
general result, we prove that the boomerang uniformity of the binomial
differentially $4$-uniform permutations presented by Bracken, Tan, and Tan
equals $4$. This result gives rise to a new family of optimal Sboxes.
