A Study of APN Functions in Dimension 7 Using Antiderivatives

Almost perfect nonlinear (APN) functions yield the best possible resistance to differential attacks when used as a substitution box in the design of a block cipher. Constructing APN functions is a non-trivial problem and so far there is only one sporadic example that is not equivalent to either a monomial or a quadratic function. This is the Brinkmann-Leander-Edel-Pott function that is defined in 6 variables. In this paper, we consider in detail an original construction approach of this function suggested by Suder, which is based on the application of antiderivatives in conjunction with fast point spaces. We generalize this approach to higher dimensions and show that it does not yield any cubic APN function in 7 variables.