On the spectra of quadratic functions

Official URL: http://risc01.sabanciuniv.edu/record=b1606666 (Table of Contents)

Study of quadratic forms goes back to the 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. The Walsh transform f of a quadratic function f : Fpn - Fp satisfies ∣f (y)∣ ∈ 2 {0, p n+s/2 for all y∈ 2 F pn and for an integer 0 ≤ s < n. In other words quadratic functions form a subclass of the so-called plateaued functions. The value of s is 0 for example, in the case of the well-known bent functions, hence bent functions are 0-plateued. In this thesis we study quadratic functions F p, n = ∑k i=0 T rn (ai x p i+1) given in trace form with the restriction that a i ∈ Fp, 0 ≤i ≤ k. Extensive work on quadratic functions with such restrictions on coefficients shows that they have many interesting features. In this work we determine the expected value for the parameter s for such quadratic functions, for many classes of integers n. Our exact formulas con rm that on average the value of s is small, and hence the average nonlinearity of this class of quadratic functions is high when p = 2.