ON THE SPECTRA OF QUADRATIC FUNCTIONS
Mathematics, PhD Dissertation, 2015
Prof. Dr Alev Topuzoğlu (Thesis Advisor), Prof. Dr. Henning Stichtehoth, Assoc. Prof. Cem Güneri, Assoc. Prof. Albert Levi, Assoc. Prof. Dr. Wilfried Meidl (Co-Thesis Advısor)
Date & Time: January 5th, 2015 – 10:00 AM
Place: FENS 2008
Keywords : Quadratic functions, Walsh Transform, expected value, variance, nonlinearity,
discrete Fourier transform
Study of quadratic forms goes back to 18th century. They attracted particular interest in the last decades also because of their applications. Indeed, there is an interaction between quadratic functions, cryptography and coding theory via their relation with Boolean bent/semi-bent functions,sequences, and various types of codes.
Quadratic functions form a subclass of the so-called s-plateaued. The value of s is 0 for example, in the case of the well-known bent functions, hence bent functions are 0-plateaued.
In this thesis quadratic functions with coefficients in the prime subfield given in trace form are studied. Extensive work on quadratic functions with such restrictions on coefficients shows that they have many interesting futures. Recently Meidl, Topuzoğlu, Roy obtained results on the number of s-plateaued functions with prescribed s.
In this work we determine the expected value for the parameter s for such quadratic unctions, or many classes of n. Our exact formulas confirm that on average the value of s is small, and hence the average nonlinearity o this class of quadratic functions is high when p=2.