New Results About the Boomerang Uniformity of Permutation Polynomials

Abstract

In EUROCRYPT 2018, Cid et al. introduced a new concept on the cryptographic property of S-boxes: boomerang connectivity table (BCT for short) for evaluating the subtleties of boomerang-style attacks. Very recently, BCT and the boomerang uniformity, the maximum value in BCT, were further studied by Boura and Canteaut. In this paper, aiming at providing new insights, we show some new results about BCT and the boomerang uniformity of permutations in terms of theory and experiment. First, we present an equivalent technique to compute BCT and the boomerang uniformity, which seems to be much simpler than the original definition by Cid et al. Second, thanks to Carlet's idea, we give a characterization of functions f from F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> to itself with boomerang uniformity δ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f</sub> by means of the Walsh transform. Third, by our method, we consider boomerang uniformities of some specific permutations, mainly the ones with low differential uniformity. Finally, we obtain another class of 4-uniform BCT permutation polynomials over F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> n.