On the Equivalence of Nonlinear Functions

Recently, many new almost perfect nonlinear (APN) and almost bent (AB) functions have been constructed. These functions F-2(n) -> F-2(n) play an important role in cryptography. In this article, we will summarize different concepts of equivalence between these functions, and discuss some invariants. Two codes can be associated with APN and AB functions. This is useful to distinguish functions up to equivalence. We give a short proof about the dimension of one of these codes. We slightly extend the known concepts of equivalence to the more general case of functions F-2(n) -> F-2(m). Moreover, we show that CCZ equivalence is the same as extended affine equivalence if F is a vectorial bent function.