scholarly journals The Group Involutory Matrix of the Combinations of Two Idempotent Matrices

2012 ◽  
Vol 2012 ◽  
pp. 1-17
Author(s):  
Lingling Wu ◽  
Xiaoji Liu ◽  
Yaoming Yu
Keyword(s):  
Involutory Matrix

We discuss the following problem: whenaP+bQ+cPQ+dQP+ePQP+fQPQ+gPQPQof idempotent matricesPandQ, wherea,b,c,d,e,f,g∈ℂanda≠0,b≠0, is group involutory.

scholarly journals DIVERSITY AES IN MIXCOLUMNS STEP WITH 8X8 CIRCULANT MATRIX

2021 ◽  
Vol 8 (9) ◽  
pp. 19-35
Author(s):  
Yan-Wen Chen ◽  
Jeng-Jung Wang ◽  
Yan-Haw Chen ◽  
Chong-Dao Lee
Keyword(s):  
Embedded Systems ◽  
Efficient Method ◽  
Matrix Multiplication ◽  
Circulant Matrix ◽  
Circulant Matrices ◽  
Matrix Operation ◽  
Branch Number ◽  
Involutory Matrix ◽  
Software Implementations ◽  
Encryption Decryption

In AES MixColumns operation, the branch number of circulant matrix is raised from 5 to 9 with 8´8 circulant matrices that can be enhancing the diffusion power. An efficient method to compute the circulant matrices in AES MixColumns transformation for speeding encryption is presented. Utilizing 8´8 involutory matrix multiplication is required 64 multiplications and 56 additions in in AES Mix-Columns transformation. We proposed the method with diversity 8´8 circulant matrices is only needed 19 multiplications and 57 additions. It is not only to encryption operations but also to decryption operations. Therefore, 8´8 circlant matrix operation with AES key sizes of 128bits, 192bits, and 256 bits are above 29.1%, 29.3%, and 29.8% faster than using 4´4 involutory matrix operation (16 multiplications, 12 additions), respectively. 8´8 circulant matrix encryption/decryption speed is above 78% faster than 8´8 involutory matrix operation. Ultimately, the proposed method for evaluating matrix multiplication can be made regular, simple and suitable for software implementations on embedded systems.

When is a linear combination of two idempotent matrices the group involutory matrix?

2006 ◽  
Vol 54 (6) ◽  
pp. 429-435 ◽  
Author(s):  
Jerzy K. Baksalary ◽  
Oskar Maria Baksalary
Keyword(s):  
Linear Combination ◽  
Involutory Matrix

scholarly journals On the singular value decomposition of (skew-)involutory and (skew-)coninvolutory matrices

2020 ◽  
Vol 8 (1) ◽  
pp. 1-13
Author(s):  
Heike Faßbender ◽  
Martin Halwaß
Keyword(s):  
Singular Value Decomposition ◽  
Singular Values ◽  
Singular Value ◽  
Singular Vectors ◽  
Involutory Matrix ◽  
Left And Right ◽  
Value Decomposition

AbstractThe singular values σ > 1 of an n × n involutory matrix A appear in pairs (σ, {1 \over \sigma }). Their left and right singular vectors are closely connected. The case of singular values σ = 1 is discussed in detail. These singular values may appear in pairs (1,1) with closely connected left and right singular vectors or by themselves. The link between the left and right singular vectors is used to reformulate the singular value decomposition (SVD) of an involutory matrix as an eigendecomposition. This displays an interesting relation between the singular values of an involutory matrix and its eigenvalues. Similar observations hold for the SVD, the singular values and the coneigenvalues of (skew-)coninvolutory matrices.

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