Geodesic
On homogeneous matroids and block-schemes
Prikladnaya Diskretnaya Matematika. Supplement, no. 10 (2017), pp. 21-23.

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

This research is devoted to access control through ideal perfect secret sharing schemes and matroids. A matroid is homogeneous if all its circuits have equal cardinality, but possibly not all subsets of this cardinality are circuits. A linkage of such matroids with block-schemes including Steiner triple is revealed. It is proved that any matroid, in which co-hyperplanes are the Steiner triples, is homogeneous connected and separating if its cardinality is not less than seven. It is also proved that block-scheme, in which each pair of different elements appears in a single block, specifies the co-hyperplanes of a homogeneous connected separating matroid. Some hypotheses for further research are presented.
Keywords: secret sharing schemes, homogeneous matroids, block-schemes
Mots-clés : circuits. 
@article{PDMA_2017_10_a6,
     author = {N. V. Medvedev and S. S. Titov},
     title = {On homogeneous matroids and block-schemes},
     journal = {Prikladnaya Diskretnaya Matematika. Supplement},
     pages = {21--23},
     publisher = {mathdoc},
     number = {10},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/PDMA_2017_10_a6/}
}
TY  - JOUR
AU  - N. V. Medvedev
AU  - S. S. Titov
TI  - On homogeneous matroids and block-schemes
JO  - Prikladnaya Diskretnaya Matematika. Supplement
PY  - 2017
SP  - 21
EP  - 23
IS  - 10
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/PDMA_2017_10_a6/
LA  - ru
ID  - PDMA_2017_10_a6
ER  - 
%0 Journal Article
%A N. V. Medvedev
%A S. S. Titov
%T On homogeneous matroids and block-schemes
%J Prikladnaya Diskretnaya Matematika. Supplement
%D 2017
%P 21-23
%N 10
%I mathdoc
%U http://geodesic.mathdoc.fr/item/PDMA_2017_10_a6/
%G ru
%F PDMA_2017_10_a6
N. V. Medvedev; S. S. Titov. On homogeneous matroids and block-schemes. Prikladnaya Diskretnaya Matematika. Supplement, no. 10 (2017), pp. 21-23. http://geodesic.mathdoc.fr/item/PDMA_2017_10_a6/

[1] V. V. Yaschenko (obsch. red.), Vvedenie v kriptografiyu, Piter, SPb., 2001

[2] Parvatov N. G., “Sovershennye skhemy razdeleniya sekreta”, Prikladnaya diskretnaya matematika, 2008, no. 2(2), 50–57

[3] Bleikli G. R., Kabatyanskii G. A., “Obobschennye idealnye skhemy, razdelyayuschie sekret, i matroidy”, Problemy peredachi informatsii, 33:3 (1997), 102–110 | MR | Zbl

[4] Bolotova E. A., Konovalova S. S., Titov S. S., “Svoistva reshetok razgranicheniya dostupa, sovershennye shifry i skhemy razdeleniya sekreta”, Problemy bezopasnosti i protivod. terrorizmu, Materialy IV Mezhdunar. nauch. konf., v. 2, MTsNMO, M., 2009, 71–86

[5] Welsh D. J. A., Matroid Theory, Academic Press, 1976 | MR | Zbl

[6] Marti-Farre J., Padro C., “Secret sharing schemes on sparse homogeneous access structures with rank three”, Electronic J. Combinatorics, 1:1 (2004), Research Paper 72, 16 pp. | MR | Zbl

[7] Singer J., “A theorem in finite projective geometry and some applications to number theory”, Trans. Amer. Math., 17 (1938), 356–372 | MR

[8] Kholl M., Kombinatorika, Mir, M., 1970 | MR

[9] N. White (ed.), Theory of Matroids, Encyclopedia of Mathematics and its Applications, 26, Cambrige University Press, 1986 | MR | Zbl