Constructions of APN permutations

APN functions defined on finite fields of characteristic two provide the best protection against differential cryptanalysis. They are used extensively in modern symmetric block ciphers. It is beneficial when APN functions are permutations. EA-equivalence and more generally CCZ-equivalence preserves the APN property. Only one example of APN permutations is known in even dimensions and its generalizations are called Kim-type functions. Our first result proves that all Kim-type APN functions in even dimensions greater than six are EA-equivalent to Gold functions. Combined with a previous result this shows that Kim-type APN functions are never CCZ-equivalent to permutations, except for dimension six. Our second result provides several theoretical constructions of Walsh zero spaces for Gold APN functions in odd dimensions. This allows one to construct new APN permutations that are CCZ-equivalent to Gold functions, but they are not EA-equivalent to them or their inverses.