Geodesic
Nonabelian key addition groups and $\otimes _{\mathbf{W}}$-markovian property of block ciphers
Matematičeskie voprosy kriptografii, Tome 11 (2020), pp. 107-131.

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

For an Abelian key addition group $\left( {X, \otimes } \right)$ and a partition ${\bf{W}} = \{ {W_0},\ldots ,{W_{r-1}}\}$ of a set $X$ we had introduced ${ \otimes _{\bf{W}}}$-markovian transformations and ${ \otimes _{\bf{W}}}$-markovian ciphers. The ${ \otimes _{\bf{W}}}$-markovian condition is required to validate different generalizations of differential technique. In this paper, we study ${ \otimes _{\bf{W}}}$-markovian ciphers and transformations on an nonabelian group $\left( {X, \otimes } \right)$. We get restrictions on the structure of groups $(X, \otimes )$, $\left\langle {{g_k}|k \in X} \right\rangle $ and blocks of a nontrivial partition ${\bf{W}}$ as a consequence of the condition of partial preservation of $\bf{W}$ by the round function ${g_k}\colon X \to X$ for all $k \in X$. For all nonabelian groups of the order ${2^m}$ with a cyclic subgroup having index $2$ we describe classes of ${ \otimes _{\bf{W}}}$-markovian permutations. 
@article{MVK_2020_11_a6,
     author = {B. A. Pogorelov and M. A. Pudovkina},
     title = {Nonabelian key addition groups and $\otimes _{\mathbf{W}}$-markovian property of block ciphers},
     journal = {Matemati\v{c}eskie voprosy kriptografii},
     pages = {107--131},
     publisher = {mathdoc},
     volume = {11},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MVK_2020_11_a6/}
}
TY  - JOUR
AU  - B. A. Pogorelov
AU  - M. A. Pudovkina
TI  - Nonabelian key addition groups and $\otimes _{\mathbf{W}}$-markovian property of block ciphers
JO  - Matematičeskie voprosy kriptografii
PY  - 2020
SP  - 107
EP  - 131
VL  - 11
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MVK_2020_11_a6/
LA  - ru
ID  - MVK_2020_11_a6
ER  - 
%0 Journal Article
%A B. A. Pogorelov
%A M. A. Pudovkina
%T Nonabelian key addition groups and $\otimes _{\mathbf{W}}$-markovian property of block ciphers
%J Matematičeskie voprosy kriptografii
%D 2020
%P 107-131
%V 11
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MVK_2020_11_a6/
%G ru
%F MVK_2020_11_a6
B. A. Pogorelov; M. A. Pudovkina. Nonabelian key addition groups and $\otimes _{\mathbf{W}}$-markovian property of block ciphers. Matematičeskie voprosy kriptografii, Tome 11 (2020), pp. 107-131. http://geodesic.mathdoc.fr/item/MVK_2020_11_a6/

[1] Lai X., Massey J. L., Murphy S., “Markov ciphers and differential cryptanalysis”, EUROCRYPT 1991, Lect. Notes Comput. Sci., 547, 1991, 17–38 | DOI | MR | Zbl

[2] Pogorelov B. A., Pudovkina M. A., “Razbieniya na bigrammakh i markovost algoritmov blochnogo shifrovaniya”, Matematicheskie voprosy kriptografii, 8:1 (2017), 5–40 | MR

[3] Pogorelov B. A., Pudovkina M. A., “Podstanovochnye gomomorfizmy algoritmov blochnogo shifrovaniya i ${ \otimes _{\mathbf{W}}}$-markovost”, Matematicheskie voprosy kriptografii, 9:3 (2018), 109–126 | MR

[4] Kemeni D., Snell D., Konechnye tsepi Markova, Nauka, M., 1970, 272 pp. | MR

[5] Kholl M., Teoriya grupp, IL, M., 1962, 468 pp.

[6] Humphreys J. F., A course in group theory, Oxford Univ. Press, 1996, 279 pp. | MR | Zbl

[7] Pogorelov B. A., Pudovkina M. A., “${ \otimes _{\bf{W}}}$-markovost XSL-algoritmov blochnogo shifrovaniya, svyazannaya so svoistvami sloev raundovoi funktsii”, Matematicheskie voprosy kriptografii, 10:1 (2019), 115–143 | MR

[8] Pogorelov B. A., “Primitivnye gruppy podstanovok, soderzhaschie $2^m$-tsikl”, Algebra i logika, 19:2 (1980), 236–247 | MR

[9] Sachkov V. N., Vvedenie v kombinatornye metody diskretnoi matematiki, 2-e izd., MTsNMO, M., 2004, 424 pp.

[10] Vinberg E. B., Kurs algebry, 2-e izd., ispr. i dop., Faktorial Press, M., 2001, 544 pp.

[11] Glukhov M. M., Elizarov V. P., Nechaev A. A., Algebra, 2-e izd., ispr. i dop., Lan, SPb., 2015, 608 pp.