Hledání APN permutací ve známých APN funkcích

• Thesis title in Czech: Hledání APN permutací ve známých APN funkcích
• Thesis title in English: Search for APN permutations among known APN functions
• English key words: vectorial Boolean functions, APN permutations, CCZ–equivalence, computational proof
• Academic year of topic announcement: 2017/2018
• Thesis type: diploma thesis
• Thesis language: angličtina
• Department: Department of Algebra (32-KA)
• Supervisor: doc. Faruk Göloglu, Dr. rer. nat.
• Author: hidden - assigned and confirmed by the Study Dept.
• Date of registration: 19.06.2018
• Date of assignment: 20.06.2018
• Confirmed by Study dept. on: 21.06.2018
• Date and time of defence: 18.09.2018 09:00
• Date of electronic submission: 20.07.2018
• Date of submission of printed version: 20.07.2018
• Date of proceeded defence: 18.09.2018
• Opponents: prof. RNDr. Aleš Drápal, CSc., DSc.

Guidelines

In order to successfully complete the thesis,
* the student should understand the algorithms in [1] and improve it for special cases,
* understand the notion of equivalence of APN functions and implement as an algorithm,
* prove some theoretical results which would lead to theoretical (in-)equivalence proofs for (some) infinite families and/or to practical algorithms for extension degrees n >= 12, as far as the computing time is reasonable.

References

[1] K. A. Browning, J. F. Dillon, M. T. McQuistan, and A. J. Wolfe, "An APN permutation in dimension six," in Finite Fields: Theory and Applications, Contemp. Math., 518, Amer. Math. Soc., Providence, RI, pp. 33–42, (2010).

[2] F. Gologlu and P. Langevin, "APN families which are not equivalent to permutations", preprint (2017).

Preliminary scope of work in English

Nonlinear permutations are used frequently in cryptography. APN (Almost Perfect Nonlinear) permutations provide best known security against differential cryptanalysis. Unfortunately whether they exist in any even extension of the binary field is not known. For extension degrees n=2 and n=4 they do not exist. In [1] authors gave the first known example in extension degree n=6. They also showed for n < 12, no known APN function is equivalent to permutations (except for the case in n=6).

The project is about giving equivalence results (negative or positive) for larger values of the extension degree (n >= 12) and possibly for some infinite families independent of the extension degree. Note that, in [2] authors have given some inequivalence results for the families Gold and Kasami. The project involves research on similar results, i.e., (in-)equivalence results for different families.

Project should involve both theoretical (proving conditions related to equivalence to permutations) and practical research (developing algorithms and writing computer programs).