Geodesic
Cryptographic properties of orthomorphic permutations
Prikladnaya Diskretnaya Matematika. Supplement, no. 13 (2020), pp. 29-31.

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

In this paper, we consider bijective mappings $F:\mathbb{Z}_2^n \rightarrow \mathbb{Z}_2^n$ called orthomorphisms such that the mappings $G(x) = F(x) \oplus x$ are also bijective. It is used in the Lai — Massey scheme as a mixing element between rounds and it also can be used to construct cryptographically strong $\mathrm{S}$-boxes. The main cryptographic properties are studied, namely nonlinearity and differential uniformity. It was revealed that, for $n=2,3,4$, the linear approximation tables of orthomorphisms consist of the values $0$ and $\pm 2^{n-1}$, and the difference distribution tables consist of the values $0$ and $2^n$. It turned out that orthomorphisms of a small number of variables are not resistant to linear and differential cryptanalysis.
Keywords: orthomorphic permutation, linear approximation table, difference distribution table. 
@article{PDMA_2020_13_a6,
     author = {J. P. Maksimluk},
     title = {Cryptographic properties of orthomorphic permutations},
     journal = {Prikladnaya Diskretnaya Matematika. Supplement},
     pages = {29--31},
     publisher = {mathdoc},
     number = {13},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/PDMA_2020_13_a6/}
}
TY  - JOUR
AU  - J. P. Maksimluk
TI  - Cryptographic properties of orthomorphic permutations
JO  - Prikladnaya Diskretnaya Matematika. Supplement
PY  - 2020
SP  - 29
EP  - 31
IS  - 13
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/PDMA_2020_13_a6/
LA  - ru
ID  - PDMA_2020_13_a6
ER  - 
%0 Journal Article
%A J. P. Maksimluk
%T Cryptographic properties of orthomorphic permutations
%J Prikladnaya Diskretnaya Matematika. Supplement
%D 2020
%P 29-31
%N 13
%I mathdoc
%U http://geodesic.mathdoc.fr/item/PDMA_2020_13_a6/
%G ru
%F PDMA_2020_13_a6
J. P. Maksimluk. Cryptographic properties of orthomorphic permutations. Prikladnaya Diskretnaya Matematika. Supplement, no. 13 (2020), pp. 29-31. http://geodesic.mathdoc.fr/item/PDMA_2020_13_a6/

[1] Nakahara J. Jr., Lai-Massey Cipher Designs. History, Design Criteria and Cryptanalysis, Springer, 2018, 726 pp. | MR | Zbl

[2] Mittenthal L., “Block substitutions using orthomorphic mappings”, Adv. Appl. Math., 16:1 (1995), 59–71 | DOI | MR | Zbl

[3] Beaulieu R., Shors D., Smith J., et al., The Simon and Speck Families Of Lightweight Block Ciphers, Cryptology Archive, Report 2013/404, 2013