Geodesic
Piecewise-affine permutations of finite fields
Prikladnaâ diskretnaâ matematika, no. 4 (2015), pp. 5-23 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Piecewise-affine permutations (p.-a. p.) are defined on any field $\mathrm{GF}(q)$. They are a generalization of piecewise-linear permutations firstly introduced by A. B. Evans. Here some estimates for linear characteristics of p.-a. p. on $\mathrm{GF}(q)$ are given. In some cases, their exact values are pointed. Polynomials representing p.-a. p. are described. Under some conditions on $\sqrt{q-1}$, it is proved that piecewise-affine permutations form the full symmetric group of $\mathrm{GF}(q)$.
Keywords: finite field, piecewise-linear permutations, linear characteristic of permutations.
Mots-clés : piecewise-affine permutations 
@article{PDM_2015_4_a0,
     author = {A. D. Bugrov},
     title = {Piecewise-affine permutations of finite fields},
     journal = {Prikladna\^a diskretna\^a matematika},
     pages = {5--23},
     year = {2015},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/PDM_2015_4_a0/}
}
TY  - JOUR
AU  - A. D. Bugrov
TI  - Piecewise-affine permutations of finite fields
JO  - Prikladnaâ diskretnaâ matematika
PY  - 2015
SP  - 5
EP  - 23
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/PDM_2015_4_a0/
LA  - ru
ID  - PDM_2015_4_a0
ER  - 
%0 Journal Article
%A A. D. Bugrov
%T Piecewise-affine permutations of finite fields
%J Prikladnaâ diskretnaâ matematika
%D 2015
%P 5-23
%N 4
%U http://geodesic.mathdoc.fr/item/PDM_2015_4_a0/
%G ru
%F PDM_2015_4_a0
A. D. Bugrov. Piecewise-affine permutations of finite fields. Prikladnaâ diskretnaâ matematika, no. 4 (2015), pp. 5-23. http://geodesic.mathdoc.fr/item/PDM_2015_4_a0/

[1] Evans A. B., Orthomorphism Graphs of Groups, Lecture Notes in Mathematics, 1535, Springer Verlag, Berlin, 1992, 114 pp. | MR | Zbl

[2] Trishin A. E., “The nonlinearity index is a piecewise-linear substitution of the additive group of the field $F_{2^{n}}$”, Prikladnaya diskretnaya matematika, 2015, no. 4(30), 32–42 (in Russian)

[3] Lidl R., Niderrayter G., Finite Fields, v. 1, 2, Mir Publ., Moscow, 1988, 822 pp. (in Russian)

[4] Solodovnikov V. I., “On the coincidence of the class of bent-functions with the class of functions which are minimally close to linear functions”, Prikladnaya diskretnaya matematika, 2012, no. 3(17), 25–33 (in Russian)

[5] Kuz'min A. S., Markov V. T., Nechaev A. A., et al., “Bent and hyper-bent functions over a field of $2^l$ elements”, Problems of Information Transmission, 44:1 (2008), 12–33 | MR | Zbl

[6] Dixon J., Mortimer B., Permutation Groups, Springer Verlag, Berlin–N.Y., 1996, 346 pp. | MR | Zbl