Properties And Constructions Of Highly Nonlinear Functions Over Finite Fields

Functions with high nonlinearity have important applications in cryptography, sequences andcoding theory. In this thesis, it is firstly presented that the summary of all important known resultsabout perfect nonlinear functions, almost perfect nonlinear functions and almost bent functions overfinite fields. Then, the nonlinearity of a family of trinomial APN functions is given. Finally, theconstructions of two family of Dembowski-Ostrom type perfect nonlinear functions are presented.Meanwhile, the equivalences between them and the known ones are analyzed, respectively. The maincontributes and fruits of this thesis are as follows:(1) The extend Walsh spectrum of a family of trinomial APN functions f is calculated, and thenonlinearity of f is determined. Meanwhile, a binary linear code Cf and its extended code C_f areconstructed from f, and the minimum distance, dimension and characteristic set of Cf and C_f arepresented,respectively;(2) A new family of Dembowski-Ostrom type perfect nonlinear functions is presented over (?)_p^2k,and it is proved to be inequivalent to any known ones;(3) A family of Dembowski-Ostrom type perfect nonlinear functions is presented over (?)_p^2k.Computer program shows that this function is equivalent to x², for k 2,and prime p 11.Furthermore, the proof of the equivalence between one of its subfamily and x² is given. And it isproved that this subfamily contains (p^k+1)(p^k-3)/2elements.