Geodesic
$p(x)$-Circulants over Finite Fields and Probability Methods of Their Construction
Matematičeskie zametki, Tome 96 (2014) no. 6, pp. 864-879.

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

In the paper, the algebra of $p(x)$-circulants over an arbitrary finite field is studied and algorithms of random equiprobable choice of elements in the subset of all invertible $p(x)$-circulants or in the subset of all $p(x)$-circulants with given value of the determinant are constructed. The specific feature of the algorithms under consideration is the minimization of time complexity and of the number of random elements used in the course of work of the algorithms.
Mots-clés : $p(x)$-circulant, algebra of $p(x)$-circulants.
Keywords: finite field, algorithm of random choice, random equiprobable choice, time complexity 
@article{MZM_2014_96_6_a5,
     author = {V. V. Gritsenko and A. Maevskiy},
     title = {$p(x)${-Circulants} over {Finite} {Fields} and {Probability} {Methods} of {Their} {Construction}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {864--879},
     publisher = {mathdoc},
     volume = {96},
     number = {6},
     year = {2014},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2014_96_6_a5/}
}
TY  - JOUR
AU  - V. V. Gritsenko
AU  - A. Maevskiy
TI  - $p(x)$-Circulants over Finite Fields and Probability Methods of Their Construction
JO  - Matematičeskie zametki
PY  - 2014
SP  - 864
EP  - 879
VL  - 96
IS  - 6
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2014_96_6_a5/
LA  - ru
ID  - MZM_2014_96_6_a5
ER  - 
%0 Journal Article
%A V. V. Gritsenko
%A A. Maevskiy
%T $p(x)$-Circulants over Finite Fields and Probability Methods of Their Construction
%J Matematičeskie zametki
%D 2014
%P 864-879
%V 96
%N 6
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2014_96_6_a5/
%G ru
%F MZM_2014_96_6_a5
V. V. Gritsenko; A. Maevskiy. $p(x)$-Circulants over Finite Fields and Probability Methods of Their Construction. Matematičeskie zametki, Tome 96 (2014) no. 6, pp. 864-879. http://geodesic.mathdoc.fr/item/MZM_2014_96_6_a5/

[1] V. V. Voevodin, E. E. Tyrtyshnikov, Vychislitelnye protsessy s teplitsevymi matritsami, Nauka, M., 1987 | MR | Zbl

[2] Zhao-Lin Jiang, Zong-Ben Xu, “Efficient algorithm for finding the inverse and the group inverse of FLS $r$-circulant matrix”, J. Appl. Math. Comput., 18:1-2 (2005), 45–57 | MR | Zbl

[3] F. Dzh. Mak-Vilyams, N. Dzh. A. Sloen, Teoriya kodov, ispravlyayuschikh oshibki, Svyaz, M., 1979

[4] D. Chillag, “Regular representations of semisimple algebras, separable field extensions, group characters,generalized circulants, and generalized cyclic codes”, Linear Algebra Appl., 218 (1995), 147–183 | DOI | MR | Zbl

[5] A. Mahalanobis, “The discrete logarithm problem in the group of non-singular circulant matrices”, Groups Complex. Cryptol., 2:1 (2010), 83–89 | DOI | MR | Zbl

[6] R. J. McEliece, A Public-Key Cryptosystem Based on Algebraic Coding Theory, Deep Space Network Progress Report, DSN PR 42-44, 1978, 114–116

[7] A. D. Wyner, “The Wire-Tap Channel”, Bell System Tech. J., 54:8 (1975), 1355–1387 | DOI | MR | Zbl

[8] L. H. Ozarow, A. D. Wyner, “Wire-tap channel. II”, Advances in Cryptology, Lecture Notes in Comput. Sci., 209, Springer, Berlin, 1985, 33–50 | MR | Zbl

[9] R. Lidl, G. Niderraiter, Konechnye polya, T. 1, Mir, M., 1988 | MR | Zbl

[10] F. R. Gantmakher, Teoriya matrits, Nauka, M., 1988 | MR | Zbl

[11] V. Shoup, A Computational Introduction to Number Theory and Algebra, Cambridge Univ. Press, Cambridge, 2009 | MR | Zbl

[12] D. Knut, Iskusstvo programmirovaniya. T. 2. Poluchislennye algoritmy, Mir, M., 2000 | MR | Zbl

[13] V. V. Gritsenko, Yu. V. Kosolapov, “Postroenie vsekh obratimykh tsirkulyantov nad polyami Galua”, Izv. Vuzov. Sev.-kavkazsk. region. Ser. Estestv. nauki, 2 (2012), 8–12