Given a vectorial Boolean function , it is called differentially -uniform if the equation admits at most solutions for every non-zero and .
This definition can be generalized to the case of functions . Functions with the smallest value for contribute an optimal resistance to the differential attack.
The smallest possible value is , such functions are called perfect nonlinear (PN) and they exist only for odd and . (see also planar functions)
For and the smallest value is and such optimal funtions are called almost perfect nonlinear (APN).
Differential uniformity is invariant under affine, EA- and CCZ-equivalence.