Geodesic
On affine classification of permutations on the space $GF(2)^3$
Diskretnaya Matematika, Tome 30 (2018) no. 3, pp. 77-87
Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We give an elementary proof that by multiplication on left and right by affine permutations $A,B\in AGL(3,2)$ each permutation $\pi:GF(2)^3\rightarrow GF(2)^3$ may be reduced to one of the 4 permutations for which the $3\times3$-matrices consisting of the coefficients of quadratic terms of coordinate functions have as an invariant the rank, which is either 3, or 2, or 1, or 0, respectively. For comparison, we evaluate the number of classes of affine equivalence by the Pólya enumerative theory.
Mots-clés : permutation, affine transformation
Keywords: Pólya theory, de Brouijn's theorem. 
@article{DM_2018_30_3_a6,
     author = {F. M. Malyshev},
     title = {On affine classification of permutations on the space $GF(2)^3$},
     journal = {Diskretnaya Matematika},
     pages = {77--87},
     year = {2018},
     volume = {30},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DM_2018_30_3_a6/}
}
TY  - JOUR
AU  - F. M. Malyshev
TI  - On affine classification of permutations on the space $GF(2)^3$
JO  - Diskretnaya Matematika
PY  - 2018
SP  - 77
EP  - 87
VL  - 30
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/DM_2018_30_3_a6/
LA  - ru
ID  - DM_2018_30_3_a6
ER  - 
%0 Journal Article
%A F. M. Malyshev
%T On affine classification of permutations on the space $GF(2)^3$
%J Diskretnaya Matematika
%D 2018
%P 77-87
%V 30
%N 3
%U http://geodesic.mathdoc.fr/item/DM_2018_30_3_a6/
%G ru
%F DM_2018_30_3_a6
F. M. Malyshev. On affine classification of permutations on the space $GF(2)^3$. Diskretnaya Matematika, Tome 30 (2018) no. 3, pp. 77-87. http://geodesic.mathdoc.fr/item/DM_2018_30_3_a6/

[1] Harrison M. A., “On the classification of Boolean functions by the general linear and affine groups”, J. SIAM, 12(2) (1964), 285–299 | MR | Zbl

[2] Lorens C. S., “Invertible Boolean functions”, IEEE Trans. Electr. Comput., EC-13(5) (1964), 529–541 | DOI | MR | Zbl

[3] Biryukov A., De Canniere C., Braeken A., Preneel B., “A Toolbox for cryptanalysis: linear and affine equivalence algorithms”, EUROCRYPT 2003, Lect. Notes Comput. Sci., 2656, 2003, 33–50 | DOI | MR | Zbl

[4] De Canniere C., Analysis and design of symmetric encryption algorithms, Doct. diss., K.U. Leuven, 2007

[5] Polya G., “Kombinatorische Anzahlbestimmungen für Gruppen und chemische Verbindungen”, Acta Math., 68 (1937), 145–254 | DOI | MR

[6] Brëin de N. G., “Obzor obobschenii perechislitelnoi teoremy Poia”, Perechislitelnye zadachi kombinatornogo analiza, Mir, M., 1979, 229–255; Bruijn de N.G., “A survey of generalizations of Polya's enumeration theorem”, Nieuw Archief voor Wiskunde, XIX (1971), 89–112 | MR | Zbl

[7] Burnside W., Theory of groups of finite order, 2nd ed., Cambridge, 1911 | MR

[8] Sachkov V.N., Kombinatornye metody diskretnoi matematiki, M.: “Nauka”, 1977