Construction of orthomorphic $\mathrm{MDS}$ matrices with primitive characteristic polynomial

2021 ◽  
Vol 12 (4) ◽  
pp. 125-143
Author(s):  
O C Puente ◽  
R A De La Krus Khimenes
Keyword(s):  
Characteristic Polynomial

Проверочные матрицы линейных кодов с максимальным расстоянием ($\mathrm{MDS}$-матрицы) - важный элемент современных криптографических примитивов, обеспечивающий наилучшее рассеивание входных битов. В ряде работ изучались способы построения и описания $\mathrm{MDS}$-матриц для использования в низкоресурсной криптографии. Однако мало внимания уделялось влиянию приводимости предлагаемых $\mathrm{MDS}$-матриц, которая может позволить злоумышленнику использовать наличие нетривиальных инвариантных подпространств у соответствующих преобразований. В данной статье предлагаются некоторые методы построения $\mathrm{MDS}$-матриц с примитивными характеристическими многочленами, имеющие повышенную стойкость по отношению к атакам, основанным на инвариантных подпространствах.

scholarly journals Convergence of the spectral radius of a random matrix through its characteristic polynomial

2021 ◽  
Author(s):  
Charles Bordenave ◽  
Djalil Chafaï ◽  
David García-Zelada
Keyword(s):  
Spectral Radius ◽  
Characteristic Polynomial ◽  
Random Matrix

scholarly journals The normalized distance Laplacian

2021 ◽  
Vol 9 (1) ◽  
pp. 1-18
Author(s):  
Carolyn Reinhart
Keyword(s):  
Spectral Radius ◽  
Characteristic Polynomial ◽  
Adjacency Matrix ◽  
Connected Graph ◽  
Distance Matrix ◽  
Laplacian Matrix ◽  
Diagonal Matrix ◽  
The Matrix

Abstract The distance matrix 𝒟(G) of a connected graph G is the matrix containing the pairwise distances between vertices. The transmission of a vertex vi in G is the sum of the distances from vi to all other vertices and T(G) is the diagonal matrix of transmissions of the vertices of the graph. The normalized distance Laplacian, 𝒟𝒧(G) = I−T(G)−1/2 𝒟(G)T(G)−1/2, is introduced. This is analogous to the normalized Laplacian matrix, 𝒧(G) = I − D(G)−1/2 A(G)D(G)−1/2, where D(G) is the diagonal matrix of degrees of the vertices of the graph and A(G) is the adjacency matrix. Bounds on the spectral radius of 𝒟 𝒧 and connections with the normalized Laplacian matrix are presented. Twin vertices are used to determine eigenvalues of the normalized distance Laplacian. The distance generalized characteristic polynomial is defined and its properties established. Finally, 𝒟𝒧-cospectrality and lack thereof are determined for all graphs on 10 and fewer vertices, providing evidence that the normalized distance Laplacian has fewer cospectral pairs than other matrices.

scholarly journals Properties of characteristic polynomial and eigenvalues of antiadjacency matrix of directed unicyclic helm graph

2021 ◽  
Vol 1722 ◽  
pp. 012056
Author(s):  
Rizky P. Okfradifa ◽  
Siti Aminah ◽  
Kiki A. Sugeng
Keyword(s):  
Characteristic Polynomial

scholarly journals Multiple phases in a generalized Gross-Witten-Wadia matrix model

2020 ◽  
Vol 2020 (9) ◽  
Author(s):  
Jorge G. Russo ◽  
Miguel Tierz
Keyword(s):  
Partition Function ◽  
Matrix Models ◽  
Characteristic Polynomial ◽  
Matrix Model ◽  
Phase Structure ◽  
Unitary Matrix ◽  
Two Dimensional ◽  
Toeplitz Determinants ◽  
Dimensional Parameter ◽  
Planar Limit

Abstract We study a unitary matrix model of the Gross-Witten-Wadia type, extended with the addition of characteristic polynomial insertions. The model interpolates between solvable unitary matrix models and is the unitary counterpart of a deformed Cauchy ensemble. Exact formulas for the partition function and Wilson loops are given in terms of Toeplitz determinants and minors and large N results are obtained by using Szegö theorem with a Fisher-Hartwig singularity. In the large N (planar) limit with two scaled couplings, the theory exhibits a surprisingly intricate phase structure in the two-dimensional parameter space.

scholarly journals Deterministic computation of the characteristic polynomial in the time of matrix multiplication

2021 ◽  
pp. 101572
Author(s):  
Vincent Neiger ◽  
Clément Pernet
Keyword(s):  
Characteristic Polynomial ◽  
Matrix Multiplication

scholarly journals Properties of characteristic polynomial and eigenvalues of antiadjacency matrix of directed unicyclic tadpole graph

2021 ◽  
Vol 1836 (1) ◽  
pp. 012002
Author(s):  
Qomaruzzaman ◽  
S Aminah ◽  
K A Sugeng
Keyword(s):  
Characteristic Polynomial

scholarly journals Moments of the characteristic polynomial in the three ensembles of random matrices

2001 ◽  
Vol 34 (22) ◽  
pp. 4627-4639 ◽  
Author(s):  
Madan Lal Mehta ◽  
Jean-Marie Normand
Keyword(s):  
Random Matrices ◽  
Characteristic Polynomial

scholarly journals Characteristic polynomial and eigenvalues of anti-adjacency matrix of directed unicyclic corona graph

2021 ◽  
Vol 1836 (1) ◽  
pp. 012001
Author(s):  
N Hasyyati ◽  
K A Sugeng ◽  
S Aminah
Keyword(s):  
Characteristic Polynomial ◽  
Adjacency Matrix
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