A new characterization of almost bent functions

Abstract

We study the functions from Fm2 into Fm2 for odd m which oppose an optimal resistance to linear cryptanalysis. These functions are called almost bent. It is known that almost bent functions are also almost perfect nonlinear, i.e. they also ensure an optimal resistance to di_eren- tial cryptanalysis but the converse is not true. We here give a necessary and sufficient condition for an almost perfect nonlinear function to be almost bent. This notably enables us to exhibit some infinite families of power functions which are not almost bent.