On the spectra of quadratic functions

Kaşıkcı, Canan (2015) On the spectra of quadratic functions. [Thesis]

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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.
Item Type: Thesis
Uncontrolled Keywords: Quadratic functions. -- Walsh transform. -- Expected value. -- Variance. -- Nonlinearity. Discrete Fourier transform. -- Kuadratik fonksiyonlar. -- Walsh dönüşümü. -- Beklenen değer. -- Varyans. -- Doğrusalsızlık. -- Kesikli Fourier dönüşümü.
Subjects: Q Science > QA Mathematics
Divisions: Faculty of Engineering and Natural Sciences > Basic Sciences > Mathematics
Faculty of Engineering and Natural Sciences
Depositing User: IC-Cataloging
Date Deposited: 12 Jun 2017 15:24
Last Modified: 26 Apr 2022 10:10
URI: https://research.sabanciuniv.edu/id/eprint/32335

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