Témata prací (Výběr práce)(verze: 368)
Detail práce
Přihlásit přes CAS
Constructions of APN permutations
Název práce v češtině: Kosntrukce APN permutací Constructions of APN permutations Nonlinear functions, almost perfect nonlinear functions, APN permutations, S-box 2014/2015 diplomová práce angličtina Katedra algebry (32-KA) Dr. rer. nat. Faruk Göloglu skrytý - zadáno a potvrzeno stud. odd. 10.05.2015 18.05.2015 02.06.2015 08.09.2016 00:00 26.07.2016 28.07.2016 08.09.2016 RNDr. Petr Lisoněk, Ph.D.
 Zásady pro vypracování In order to successfully complete the thesis, the student should understand the constructions of the recent papers [1,2], and explore possible generalization possibilities using theoretical or computational (or a combination of both) methods. The willing student can choose which method she/he wants. Since the problem of finding new APN permutations is a very difficult one, novel methods/constructions are not required to successfully complete the thesis. For instance applying the known technique of checking equivalence to permutations [3] to a subset of known classes would be regarded as a successful thesis.
 Seznam odborné literatury [1] C. Carlet, Relating three nonlinearity parameters of vectorial functions and building APN functions from bent functions, Des. Codes Cryptogr. 59 (1–3) (2011) 89–109. [2] Y. Zhou, A. Pott, A new family of semifields with 2 parameters, Adv. Math. 234 (2013) 43–60. [3] F. Göloğlu, Almost perfect nonlinear trinomials and hexanomials, Finite Fields and Their Applications, Volume 33, (2015), 258-282
 Předběžná náplň práce v anglickém jazyce Nonlinear functions, especially almost perfect nonlinear (APN) functions are important objects in cryptography. They are used in hash and block ciphers and provides confusion (a concept introduced by Claude Shannon) to the cryptosytem. To be able to use a nonlinear function as an S-Box in a block cipher it has to be a permutation. For instance AES uses the inverse function on GF(2,8) as its S-Box. APN functions provide best resistance against linear and differential cryptanalysis. The project is about APN permutations on the finite field GF(2,n). APN permutations exist for odd n, and n = 6. They do not exist for n = 2 and n = 4. The existence question for n = 2m > 6 is called the &quot;big problem&quot; by Dillon who found the APN permutation when n = 6. Inverse function on GF(2,8) is not APN, however it provides the best known nonlinearity. The question for n = 8 is extremely important, since the existence of such a permutation would imply the inverse function used in AES is not optimal. The project involves computer work as well as theoretical work. We will try to find theoretical and practical ways to construct APN permutations or try to show mathematically that the known functions are not equivalent to permutations for even n > 6.