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Detail práce
   Přihlásit přes CAS
Constructions of APN permutations
Název práce v češtině: Kosntrukce APN permutací
Název v anglickém jazyce: Constructions of APN permutations
Klíčová slova anglicky: Nonlinear functions, almost perfect nonlinear functions, APN permutations, S-box
Akademický rok vypsání: 2014/2015
Typ práce: diplomová práce
Jazyk práce: angličtina
Ústav: Katedra algebry (32-KA)
Vedoucí / školitel: Dr. rer. nat. Faruk Göloglu
Řešitel: skrytý - zadáno a potvrzeno stud. odd.
Datum přihlášení: 10.05.2015
Datum zadání: 18.05.2015
Datum potvrzení stud. oddělením: 02.06.2015
Datum a čas obhajoby: 08.09.2016 00:00
Datum odevzdání elektronické podoby:26.07.2016
Datum odevzdání tištěné podoby:28.07.2016
Datum proběhlé obhajoby: 08.09.2016
Oponenti: 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 "big
problem" 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.
 
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