A Hybrid Method for Spectral Translation Equivalent Boolean Functions The equivalence of Boolean functions with respect to five invariance (aka translation) operations has been well considered with respect to the RademacherWalsh spectral domain. In this paper, we introduce a hybrid approach that uses both the ReedMuller and the RademacherWalsh spectra. A novel hybrid algorithm that maps a Boolean function to a representative function for the equivalence class containing the original function is presented. The algorithm can be used to determine a sequence of translations that maps one function to an equivalent function. We present experimental results that show the hybrid algorithm can determine the equivalence classes for 5 variables much more efficiently than before. We also show that for 6 variables where there are 150,357 equivalence classes, 8 are very difficult, a further 58 are difficult and the remainder are straightforward in terms of the CPU time required by the hybrid algorithm.

An Update on Known Invariants of Vectorial Boolean Functions Almost Perfect Nonlinear functions are of great practical importance for cryptography as they provide optimum resistance to differential cryptanalysis. Due to the large number of such functions, they are classified up to CCZequivalence, which leaves this resistance to differential cryptanalysis invariant. Deciding the equivalence of two given functions from the definition is computationally difficult. We summarize computational results on some properties that remain invariant under CCZequivalence for the known APN functions, including more than 8000 new CCZinequivalent functions that have previously never been considered with respect to these invariants. Knowing these invariants greatly facilitates the classification of APN functions, since different values of these invariants immediately imply that two functions are CCZinequivalent.