Geodesic
The differential uniformity of piecewise-linear substitutions over the field $\mathbb{F}_{2^{n}}$
Diskretnaya Matematika, Tome 35 (2023) no. 4, pp. 58-68.

Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper, we give lower and upper bounds on the differential uniformity of substitutions over the field $\mathbb{F}_{2^{n}}$ with restrictions to cosets of $H$ in $\mathbb{F}^{\times}_{2^{n}}$, $H\mathbb{F}^{\times}_{2^{n}}$, $|H|=l$, $l\cdot r=2^{n}-1$, being the maps $x\mapsto c_{i}x$, $c_{i}\in\mathbb{F}^{\times}_{2^{n}}$, $i=0,\dots,r-1$.
Keywords: block cipher nonlinear confusion components, permutation of a finite field, $s$-box, piecewise-linear function, adapted spectral-differential method. 
@article{DM_2023_35_4_a3,
     author = {A. V. Menyachikhin},
     title = {The differential uniformity of piecewise-linear substitutions over the field $\mathbb{F}_{2^{n}}$},
     journal = {Diskretnaya Matematika},
     pages = {58--68},
     publisher = {mathdoc},
     volume = {35},
     number = {4},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DM_2023_35_4_a3/}
}
TY  - JOUR
AU  - A. V. Menyachikhin
TI  - The differential uniformity of piecewise-linear substitutions over the field $\mathbb{F}_{2^{n}}$
JO  - Diskretnaya Matematika
PY  - 2023
SP  - 58
EP  - 68
VL  - 35
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/DM_2023_35_4_a3/
LA  - ru
ID  - DM_2023_35_4_a3
ER  - 
%0 Journal Article
%A A. V. Menyachikhin
%T The differential uniformity of piecewise-linear substitutions over the field $\mathbb{F}_{2^{n}}$
%J Diskretnaya Matematika
%D 2023
%P 58-68
%V 35
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/DM_2023_35_4_a3/
%G ru
%F DM_2023_35_4_a3
A. V. Menyachikhin. The differential uniformity of piecewise-linear substitutions over the field $\mathbb{F}_{2^{n}}$. Diskretnaya Matematika, Tome 35 (2023) no. 4, pp. 58-68. http://geodesic.mathdoc.fr/item/DM_2023_35_4_a3/

[1] Bugrov A. D., “Kusochno-affinnye podstanovki konechnykh polei”, Prikladnaya diskretnaya matematika, 4:30 (2015), 5–23

[2] Menyachikhin A. V., “Adaptirovannyi spektralno-raznostnyi metod postroeniya differentsialno 4-ravnomernykh kusochno-lineinykh podstanovok, ortomorfizmov, involyutsii polya $\mathbb{F}_{2^{n}}$”, Diskretnaya matematika, 35:2 (2023), 42–77 | DOI

[3] Menyachikhin A. V., “O lineinoi ekvivalentnosti kusochno-lineinykh podstanovok polya $\mathbb{F}_{2^{n}}$”, Diskretnaya matematika, 35:3 (2023), 37–44 | DOI

[4] Pogorelov B. A., Pudovkina M. A., “Klassy kusochno-kvaziaffinnykh preobrazovanii na obobschennoi 2-gruppe kvaternionov”, Diskretnaya matematika, 34:1 (2022), 103–125 | DOI

[5] Pogorelov B. A., Pudovkina M. A., “Klassy kusochno-kvaziaffinnykh podstanovok na diedralnoi, poludiedralnoi i modulyarnoi maksimalno-tsiklicheskoi 2-gruppakh”, Diskretnaya matematika, 34:2 (2022), 50–66 | DOI

[6] Pogorelov B. A., Pudovkina M. A., “O klasse stepennykh kusochno-affinnykh podstanovok na neabelevoi gruppe poryadka $2^{m}$, obladayuschei tsiklicheskoi podgruppoi indeksa dva”, PDM. Prilozhenie, 12 (2019), 27–29

[7] Sachkov V. N., “Kombinatornye svoistva differentsialno 2-ravnomernykh podstanovok”, Matem. voprosy kriptografii, 6:1 (2015), 159–179 | DOI | MR | Zbl

[8] Trishin A. E., “O pokazatele nelineinosti kusochno-lineinykh podstanovok additivnoi gruppy polya ${\rm {\mathbb F}}_{2^{n} } $”, Prikladnaya diskretnaya matematika, 4:30 (2015), 32–42

[9] Beierle C., Brinkmann M., Leander G., “Linearly self-equivalent APN permutations in small dimension”, IEEE Trans. Inf. Theory, 67:7 (2020), 4863–4875 | DOI | MR

[10] Brinkmann M., Leander G., “On the classification of APN functions up to dimension five”, Des. Codes Cryptogr., 49:1 (2008), 273–288 | DOI | MR | Zbl

[11] Browning K. A., Dillon J. F., McQuistan M. T., Wolfe A. J., “An APN permutation in dimension six”, Finite Fields: theory and applications, 518 (2010), 33–42 | DOI | MR | Zbl

[12] Chartrand G., Zhang P., A First Course in Graph Theory, Dover Publications, Mineola, New York, 2012, 464 pp.

[13] Dimitrov D., Ikica B., Škrekovski R., “Remarks on the Graovac–Ghorbani index of bipartite graphs”, Appl. Math. Comput., 293 (2017), 370–376 | MR | Zbl

[14] Evans A., Orthomorphisms Graphs of Groups, Springer-Verlag, Berlin, 1992, 114 pp. | MR

[15] Menyachikhin A. V., “Spectral-linear and spectral-differential methods for generating s-boxes having almost optimal cryptographic parameters”, Matem. voprosy kriptografii, 8:2 (2017), 97–116 | DOI | MR | Zbl

[16] Niederreiter H., Winterhof A., “Cyclotomic R-orthomorphisms of finite fields”, Discrete Mathematics, 295 (2005), 161–171 | DOI | MR | Zbl

[17] Nyberg K., “Differentially uniform mappings for cryptography”, EUROCRYPT 1993, Lect. Notes Comput. Sci., 765, 1994, 55–64 | DOI | MR | Zbl

[18] Nyberg K., Knudsen L.R., “Provable Security Against Differential Cryptanalysis”, CRYPTO 1992, Lect. Notes Comput. Sci., 740, 1992, 566–574 | DOI | MR

[19] Wan D., Lidl R., “Permutation polynomials of the form $x^{r}f(x^{q-1}/d)$ and their group structure”, Monatsh. Math., 112 (1991), 149–163 | DOI | MR | Zbl

[20] Wang A., Hou X., Liu B., Ma Y., “The Turán number for the edge blow-up of trees”, Discrete Mathematics, 344:12 (2021), 112627 | DOI | MR | Zbl

[21] Wells C., “Groups of permutation polynomials”, Monatsh. Math., 71:3 (1967), 248–262 | DOI | MR | Zbl