Abstract
The notion of local primitivity for a quadratic 0, 1-matrix of size n × n is extended to any part of the matrix which need not be a rectangular submatrix. A similar generalization is carried out for any set B of pairs of initial and final vertices of the paths in an n-vertex digraph, B ⊆ {(i, j) : 1 ≤ i, j ≤ n}. We establish the relationship between the local B-exponent of a matrix (digraph) and its characteristics such as the cyclic depth and period, the number of nonprimitive matrices, and the number of nonidempotentmatrices in the multiplicative semigroup of all quadratic 0, 1-matrices of order n, etc. We obtain a criterion of B-primitivity and an upper bound for the B-exponent. We also introduce some new metric characteristics for a locally primitive digraph Γ: the k, r-exporadius, the k, r-expocenter, where 1 ≤ k, r ≤ n, and the matex which is defined as the matrix of order n of all local exponents in the digraph Γ. An example of computation of the matex is given for the n-vertex Wielandt digraph. Using the introduced characteristics, we propose an idea for algorithmically constructing realizable s-boxes (elements of round functions of block ciphers) with a relatively wide range of sizes.
Similar content being viewed by others
References
S. N. Kyazhin and V. M. Fomichev, “Local Primitiveness of Graphs and Nonnegative Matrices,” Prikl. Diskretn. Mat. No. 3, 68–80 (2014).
V. M. Fomichev, “On Characteristics of Local Primitive Matrices and Digraphs,” Prikl. Diskretn. Mat. Prilozh. No. 10, 96–99 (2017).
V. M. Fomichev, D. I. Zadorozhnyi, A. M. Koreneva, D. M. Lolich, and A. V. Yuzbashev, “On Algorithmic Implementation of s-Boxes,” in Proceedings of XIX Scientific-Practical Conference “Rus-Cripto,” Moscow, Russia, March 21–24, 2017 (Available at http://www.ruscrypto.ru/resource/summary/rc2017/02_fomitchev_zadorozhny_koreneva_lolich_yuzbashev.pdf (accessed Dec. 29, 2017).
V. M. Fomichev and S. N. Kyazhin, “Local Primitivity ofMatrices and Graphs,” Diskretn. Anal. Issled.Oper. 24 (1), 97–119 (2017) [J. Appl. Indust.Math. 11 (1), 26–39 (2017)].
V. M. Fomichev and D. A. Melnikov, Cryptographic Methods of Information Security, Part 1: Mathematical Aspects (YURAIT, Moscow, 2016) [in Russian].
T. P. Berger, J. Francq, M. Minier, and G. Thomas, “Extended Generalized Feistel Networks Using Matrix Representation to Propose a New Lightweight Block Cipher: Lilliput,” IEEE Trans. Comput. 65 (7), 2074–2089 (2016).
T. P. Berger, M. Minier, and G. Thomas, “Extended Generalized Feistel Networks UsingMatrix Representation,” in Selected Areas in Cryptography (Revised Selected Papers. 20th International Conference on SAC, Burnaby, Canada, August 14–16, 2013) (Springer, Heidelberg, 2014).
R. A. Brualdi and B. Liu, “Generalized Exponents of Primitive Directed Graphs,” J. Graph Theory 14, 483–499 (1990).
Y. Huang and B. Liu, “Generalized r-Exponents of Primitive Digraphs,” Taiwan. J.Math. 15 (5), 1999–2012 (2011).
B. Liu, “Generalized Exponents of BooleanMatrices,” Linear Algebra Appl. 373, 169–182 (2003).
Z. Miao and K. Zhang, “The Local Exponent Sets of Primitive Digraphs,” Linear Algebra Appl. 307, 15–33 (2000).
J. Shen and S. Neufeld, “Local Exponents of Primitive Digraphs,” Linear Algebra Appl. 268, 117–129 (1998).
T. Suzaki and K. Minematsu, “Improving the Generalized Feistel,” in Fast Software Encryption (Revised Selected Papers from 17th International Workshop FSE, Seoul, Korea, February 7–10, 2010) (Springer, Heidelberg, 2010).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © V.M. Fomichev, 2018, published in Diskretnyi Analiz i Issledovanie Operatsii, 2018, Vol. 25, No. 2, pp. 124–143.
Rights and permissions
About this article
Cite this article
Fomichev, V.M. Semigroup and Metric Characteristics of Locally Primitive Matrices and Digraphs. J. Appl. Ind. Math. 12, 243–254 (2018). https://doi.org/10.1134/S1990478918020059
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1990478918020059