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