CCZ-equivalence and Boolean functions

We study further CCZ-equivalence of (n,m)-functions. We prove that for Boolean functions (that is, for m = 1), CCZ-equivalence coincides with EA-equivalence. On the contrary, we show that for (n,m)functions, CCZ-equivalence is strictly more general than EAequivalence when n ≥ 5 and m is greater or equal to the smallest positive divisor of n different from 1. Our result on Boolean functions allows us to study the natural generalization of CCZ-equivalence corresponding to the CCZ-equivalence of the indicators of the graphs of the functions. We show that it coincides with CCZ-equivalence.