The c-boomerang uniformity and c-boomerang spectrum of two classes of permutation polynomials over the finite field F2n

The boomerang attack developed by Wagner is a cryptanalysis technique against block ciphers. A new theoretical tool, the Boomerang Connectivity Table (BCT) and the corresponding boomerang uniformity were introduced to evaluate the resistance of a block cipher against the boomerang attack. Using a multiplier differential, Stănică (2021) [28] extended the notion of boomerang uniformity to c-boomerang uniformity. In this paper, we focus on two classes of permutation polynomials over $F_{2^n}$. For one of these, we show that the c-boomerang uniformity of this function is equal to 1. For the second type of function, we first consider the c-BCT entries. We then explicitly determine the c-boomerang spectrum of this function by means of characters and some techniques in solving equations over $F_{2^n}$.