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 Diskr. Mat., 2016, Volume 28, Issue 4, Pages 100–121 (Mi dm1396)

On groups containing the additive group of the residue ring or the vector space

B. A. Pogorelova, M. A. Pudovkinab

a Academy of Cryptography of Russian Federation
b Bauman Moscow State Technical University

Abstract: Groups which are most frequently used as key addition groups in iterative block ciphers include the regular permutation representation $V_n^ +$ of the group of vector key addition, the regular permutation representation $\mathbb{Z}_{{2^n}}^ +$ of the additive group of the residue ring, and the regular permutation representation $\mathbb{Z}_{{2^n} + 1}^ \odot$ of the multiplicative group of a prime field (in the case where ${2^n} + 1$ is a prime number). In this work we consider the extension of the group ${G_n}$ generated by $V_n^ +$ and $\mathbb{Z}_{{2^n}}^ +$ by means of transformations and groups which naturally arise in cryptographic applications. Examples of such transformations and groups are the groups $\mathbb{Z}_{{2^d}}^ + \times V_{n - d}^ +$ and $V_{n - d}^ + \times \mathbb{Z}_{{2^d}}^ +$ and pseudoinversion over the field $GF({2^n})$ or over the Galois ring $GR({2^{md}}{,2^m})$.

Keywords: key addition group, additive regular group, wreath product of permutation groups, multiplicative group of the residue ring, Galois ring.

DOI: https://doi.org/10.4213/dm1396

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English version:
Discrete Mathematics and Applications, 2018, 28:4, 231–247

Bibliographic databases:

UDC: 512.541.4

Citation: B. A. Pogorelov, M. A. Pudovkina, “On groups containing the additive group of the residue ring or the vector space”, Diskr. Mat., 28:4 (2016), 100–121; Discrete Math. Appl., 28:4 (2018), 231–247

Citation in format AMSBIB
\Bibitem{PogPud16} \by B.~A.~Pogorelov, M.~A.~Pudovkina \paper On groups containing the additive group of the residue ring or the vector space \jour Diskr. Mat. \yr 2016 \vol 28 \issue 4 \pages 100--121 \mathnet{http://mi.mathnet.ru/dm1396} \crossref{https://doi.org/10.4213/dm1396} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3699325} \elib{http://elibrary.ru/item.asp?id=28119096} \transl \jour Discrete Math. Appl. \yr 2018 \vol 28 \issue 4 \pages 231--247 \crossref{https://doi.org/10.1515/dma-2018-0021} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000442245400003} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85053143943}