The Difference Distribution Table (DDT) and the differential uniformity play a major role for the design of substitution boxes in block ciphers, since they indicate the function’s resistance against differential cryptanalysis. This concept was extended recently to$c$-DDT and$c$-differential uniformity, which have the potential of extending differential cryptanalysis. Recently, a new theoretical tool, the Boomerang Connectivity Table (BCT) and the corresponding boomerang uniformity were introduced to quantify the resistance of a block cipher against boomerang-style attacks. Here we concentrate on two classes (introduced recently) of permutation polynomials over finite fields of even characteristic. For one of these, which is an involution used to construct a 4-uniform permutation, we explicitly determine the$c$-DDT entries and BCT entries. For the second type of function, which is a differentially 4-uniform function, we give bounds for its$c$-differential and boomerang uniformities.