Fourier Spectra of Binomial APN Functions

In this paper we compute the Fourier spectra of some recently discovered binomial APN functions. One consequence of this is the determination of the nonlinearity of the functions, which measures their resistance to linear cryptanalysis. Another consequence is that certain error-correcting codes related to these functions have the same weight distribution as the 2-error-correcting BCH code. Furthermore, for field extensions of F2 of odd degree, our results provide an alternative proof of the APN property of the functions. School of Mathematical Sciences, University College Dublin, Ireland. ([email protected]) Research supported by Irish Research Council for Science, Engineering and Technology Postdoctoral Fellowship. School of Mathematical Sciences, University College Dublin, Ireland. ([email protected]) Research supported by the Claude Shannon Institute, Science Foundation Ireland Grant 06/MI/006. School of Mathematical Sciences, University College Dublin, Ireland. ([email protected]) Postdoctoral Fellow supported by the Claude Shannon Institute, Science Foundation Ireland Grant 06/MI/006. School of Mathematical Sciences, University College Dublin, Ireland. ([email protected]) Research supported by the Claude Shannon Institute, Science Foundation Ireland Grant 06/MI/006.