Partitioning CCZ classes into EA classes

EA equivalence classes and the coarser CCZ equivalence classes of functions over $GF(p^n)$ each preserve measures of nonlinearity desirable in cryptographic functions. We identify very precisely the condition on a linear permutation defining a CCZ isomorphism between functions which ensures that the CCZ isomorphism can be rewritten as EA isomorphism. We introduce new algebraic invariants $n(f)$ of the EA isomorphism class of $f$ and $s(f)$ of the CCZ isomorphism class of $f$, with $n(f) < s(f)$, and relate them to the differential uniformity of $f$. We formulate three questions about partitioning CCZ classes into EA classes and relate these to a conjecture of Edel's about quadratic APN functions.