Hledání APN permutací ve známých APN funkcích
| Název práce v češtině: | Hledání APN permutací ve známých APN funkcích |
|---|---|
| Název v anglickém jazyce: | Search for APN permutations among known APN functions |
| Klíčová slova anglicky: | vectorial Boolean functions, APN permutations, CCZ–equivalence, computational proof |
| Akademický rok vypsání: | 2017/2018 |
| 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: | 19.06.2018 |
| Datum zadání: | 20.06.2018 |
| Datum potvrzení stud. oddělením: | 21.06.2018 |
| Datum a čas obhajoby: | 18.09.2018 09:00 |
| Datum odevzdání elektronické podoby: | 20.07.2018 |
| Datum odevzdání tištěné podoby: | 20.07.2018 |
| Datum proběhlé obhajoby: | 18.09.2018 |
| Oponenti: | prof. RNDr. Aleš Drápal, CSc., DSc. |
| Zásady pro vypracování |
| 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. |
| Seznam odborné literatury |
| [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). |
| Předběžná náplň práce v anglickém jazyce |
| 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). |