On Plateaued Functions, Linear Structures, Permutation Polynomials and c-Differential Uniformity

Date

2023-8-31

Author

Kaytancı, Kübra

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A desired goal in designing good cryptosystems is to construct boolean functions with good cryptographic properties, such as having high nonlinearity, balancedness, high correlation immunity, and high algebraic immunity. In this thesis, we obtain concrete upper bounds on the algebraic immunity of a class of highly nonlinear plateaued functions without linear structures than the one given recently in 2017 by Cusick. Moreover, we extend Cusick's class to a much bigger explicit class, and we show that our class has better algebraic immunity by an explicit example. We also give a new notion of the linear translator, which includes the Frobenius linear translator given in 2018, Cepak, Pasalic, and Muratovic-Ribic as a particular case. We find some applications of our new notion of linear translator to the construction of permutation polynomials. Furthermore, we give explicit classes of permutation polynomials over Fqn using some properties of Fq and some conditions of 2011, Akbary, Ghioca, and Wang. Additionally, recently Ellingsen et al. introduced a new concept, the c-Difference Distribution Table and the c-differential uniformity, by extending the usual differential notion. The motivation behind this new concept is based on having the ability to resist some known differential attacks, as shown by Borisov et. al. in 2002. In 2022, Hasan et al. gave an upper bound of the c-differential uniformity of the perturbed inverse function H via a trace function Tr(x2/(x+1) ). In their work, they also presented an open question on the exact c-differential uniformity of H . By using a new method based on algebraic curves over finite fields, we solve the open question in the case Tr(c)=1= Tr(1/c ) for c∈ F2n/{0,1} completely and we show that the exact c-differential uniformity of H is 8. In the remaining case, we almost completely solve the problem, and show that the c-differential uniformity of H is either 8 or 9.

Subject Keywords

Plateaued functions, Linear structure, Permutation polynomials, c-Differential Uniformity

URI

https://hdl.handle.net/11511/105394

Collections

Graduate School of Applied Mathematics, Thesis