An interesting problem in the field of nonlinear Boolean functions is the existence of APN (almost perfect nonlinear) functions with non-classical Walsh spectra. A quadratic APN function F : GF(2^{2m}) -> GF(2^{2m}) is said to have a classical spectrum if the Walsh values comprises 0, +- 2^m, +- 2^{m+1}. The only known quadratic function with non-classical spectrum was given in [1] for m = 3 and the problem of finding another example on larger extension degrees (or an infinite class of functions) became an interesting open problem. In this thesis, we use the observation that the example in [1] can be written in a specific way which may help finding another example by a computer search after applying some theoretical ideas.
The student should apply the ideas in the case m = 4. Finding another APN function with a non-classical spectrum would be an excellent result. Showing that the function cannot be generalized by this method is another very good result. But what we expect minimally is to apply the ideas, write computer programs, and explaining the method in a nice way.
Seznam odborné literatury
[1] Browning, K. A.; Dillon, J. F.; McQuistan, M. T.; Wolfe, A. J. An APN permutation in dimension six. Finite fields: theory and applications, 33–42, Contemp. Math., 518, Amer. Math. Soc., Providence, RI, 2010.
- zadáno a potvrzeno stud. odd.