Involutory Matrices Research Articles

Canonical form of Involutory Matrices over the Domain of Principal Ideals with Respect to Similarity Transformations

Canonical form of Involutory Matrices over the Domain of Principal Ideals with Respect to Similarity Transformations

Construction of lightweight involutory MDS matrices

In this paper, we propose an efficient method to find lightweight involutory MDS matrices. To obtain involutory matrices, we give a necessary and sufficient condition for judging the involutory MDS property and propose a search method. For the $$n\times n$$ involutory MDS matrices over $${\mathbb {F}}_{2^m}$$ , the amount of computation is reduced from $$2^{mn^2}$$ to $$2^{(mn^2)/2}$$ . Especially, we can exhaustively search for involutory MDS matrices when $$n=4$$ , and for larger n, we add additional restrictions to reduce the search range. As for finding lightweight ones, we use the permutation-equivalent class to extend the input such that the efficiency of the heuristic designed by Xiang et al. can be improved. Applying our method, we obtain a class of $$16\times 16$$ binary MDS matrices with branch number 5, which can be implemented with only 35 XOR gates. The results even reach the same implementation cost as the lightest non-involutory MDS matrix up to now. Concerning lightweight binary matrices with order 32, it is hard to obtain optimal results through search. Hence, we construct $$32\times 32$$ matrices with the lightweight $$16 \times 16$$ matrices that we found. In this way, we obtain two classes of $$ 4 \times 4 $$ involutory MDS matrices whose entries are $$ 8 \times 8 $$ binary matrices with 70 XOR gates while the previous lightest matrices with the same size cost 78 XOR gates. Moreover, we also generalize our search method to general cases and it is provable that the approach is feasible for MDS matrices of order 6 and 8 to achieve efficient search.

More about involutions in the group of almost-Riordan arrays

We investigate the involutory matrices in the recently introduced group of almost-Riordan arrays. We prove that each such involution is a conjugate of a diagonal involution.

Semi-involutory matrices and signed self-inverse

In this paper three new classes of matrices that have a computationally simple inverse are studied. Specifically, semi-involutory matrices are characterized for small orders and many interesting results are provided. Semi-involutory matrices can be thought of as a generalization of involutory matrices. In addition, signed semi-involutory matrices are studied and a natural generalization is introduced.

A construction of MDS involutory matrices using MDS self-dual codes: a preliminary result

The Maximum Distance Separable (MDS) matrices have an important application in cryptography, for example see [1]. In order to reduce the implementation complexities, it is important to consider MDS matrices which are involutory. Some constructions of involutory MDS matrices can be found in [2, 3]. In this paper, we give some properties related to a construction of involutory MDS matrices using orthogonal matrices derived from Euclidean self-dual MDS codes. Then, we do a computer search to find involutory MDS matrices with small sizes using classical simulated annealing algorithm. The proposed construction is different compared to the ones in [2, 3].

On the Construction of Involutory MDS Matrices over $$\mathbb{F}_{2^{m}}$$

This paper studies the problem of constructing lightweight involutory maximal distance separable (MDS) matrices. The authors find the exact lower bound of the XOR counts for 4 × 4 involutory MDS matrices over $$\mathbb{F}_{2^{4}}$$. Further, some new structures of 4 × 4 involutory MDS matrices over $$\mathbb{F}_{2^{m}}$$ are provided to construct involutory MDS matrices and the authors constructed the lightest 4 × 4 involutory MDS matrices over $$\mathbb{F}_{2^{8}}$$ so far by using these structures.

Orthogonal Bases of Involution in Hadamard Algebras

The notion of a Hadamard decomposition of a semisimple associative finite-dimensional complex algebra generalizes the notion of classical Hadamard matrices, which correspond to the case of commutative algebras. Algebras admitting a Hadamard decomposition are said to be Hadamard. Images of orthogonal bases of involution in Hadamard algebras under the canonical projections of these algebras onto their simple components are studied. Using a technique related to the study of central primitive idempotents of Hadamard algebras, we obtain a necessary condition for a family of involutory matrices of fixed order to be such an image. It is also shown that this necessary condition is not sufficient. We also present new proofs of results proved earlier.

Constructing Low-latency Involutory MDS Matrices with Lightweight Circuits

MDS matrices are important building blocks providing diffusion functionality for the design of many symmetric-key primitives. In recent years, continuous efforts are made on the construction of MDS matrices with small area footprints in the context of lightweight cryptography. Just recently, Duval and Leurent (ToSC 2018/FSE 2019) reported some 32 × 32 binary MDS matrices with branch number 5, which can be implemented with only 67 XOR gates, whereas the previously known lightest ones of the same size cost 72 XOR gates.In this article, we focus on the construction of lightweight involutory MDS matrices, which are even more desirable than ordinary MDS matrices, since the same circuit can be reused when the inverse is required. In particular, we identify some involutory MDS matrices which can be realized with only 78 XOR gates with depth 4, whereas the previously known lightest involutory MDS matrices cost 84 XOR gates with the same depth. Notably, the involutory MDS matrix we find is much smaller than the AES MixColumns operation, which requires 97 XOR gates with depth 8 when implemented as a block of combinatorial logic that can be computed in one clock cycle. However, with respect to latency, the AES MixColumns operation is superior to our 78-XOR involutory matrices, since the AES MixColumns can be implemented with depth 3 by using more XOR gates.We prove that the depth of a 32 × 32 MDS matrix with branch number 5 (e.g., the AES MixColumns operation) is at least 3. Then, we enhance Boyar’s SLP-heuristic algorithm with circuit depth awareness, such that the depth of its output circuit is limited. Along the way, we give a formula for computing the minimum achievable depth of a circuit implementing the summation of a set of signals with given depths, which is of independent interest. We apply the new SLP heuristic to a large set of lightweight involutory MDS matrices, and we identify a depth 3 involutory MDS matrix whose implementation costs 88 XOR gates, which is superior to the AES MixColumns operation with respect to both lightweightness and latency, and enjoys the extra involution property.

Constructing Low-latency Involutory MDS Matrices with Lightweight Circuits

MDS matrices are important building blocks providing diffusion functionality for the design of many symmetric-key primitives. In recent years, continuous efforts are made on the construction of MDS matrices with small area footprints in the context of lightweight cryptography. Just recently, Duval and Leurent (ToSC 2018/FSE 2019) reported some 32 × 32 binary MDS matrices with branch number 5, which can be implemented with only 67 XOR gates, whereas the previously known lightest ones of the same size cost 72 XOR gates.In this article, we focus on the construction of lightweight involutory MDS matrices, which are even more desirable than ordinary MDS matrices, since the same circuit can be reused when the inverse is required. In particular, we identify some involutory MDS matrices which can be realized with only 78 XOR gates with depth 4, whereas the previously known lightest involutory MDS matrices cost 84 XOR gates with the same depth. Notably, the involutory MDS matrix we find is much smaller than the AES MixColumns operation, which requires 97 XOR gates with depth 8 when implemented as a block of combinatorial logic that can be computed in one clock cycle. However, with respect to latency, the AES MixColumns operation is superior to our 78-XOR involutory matrices, since the AES MixColumns can be implemented with depth 3 by using more XOR gates.We prove that the depth of a 32 × 32 MDS matrix with branch number 5 (e.g., the AES MixColumns operation) is at least 3. Then, we enhance Boyar’s SLP-heuristic algorithm with circuit depth awareness, such that the depth of its output circuit is limited. Along the way, we give a formula for computing the minimum achievable depth of a circuit implementing the summation of a set of signals with given depths, which is of independent interest. We apply the new SLP heuristic to a large set of lightweight involutory MDS matrices, and we identify a depth 3 involutory MDS matrix whose implementation costs 88 XOR gates, which is superior to the AES MixColumns operation with respect to both lightweightness and latency, and enjoys the extra involution property.

Further results on generalized centro-invertible matrices

This paper deals with generalized centro-invertible matrices introduced by the authors in Lebtahi et al. (Appl. Math. Lett. 38, 106–109, 2014). As a first result, we state the coordinability between the classes of involutory matrices, generalized centro-invertible matrices, and {K}-centrosymmetric matrices. Then, some characterizations of generalized centro-invertible matrices are obtained. A spectral study of generalized centro-invertible matrices is given. In addition, we prove that the sign of a generalized centro-invertible matrix is {K}-centrosymmetric and that the class of generalized centro-invertible matrices is closed under the matrix sign function. Finally, some algorithms have been developed for the construction of generalized centro-invertible matrices.

New branches of non-supersymmetric attractors in N = 2 supergravity

In this paper we analyse non-supersymmetric single centred extremal black hole solutions in N=2 supergravity theory coupled to n vector multiplets with purely cubic pre-potential in four dimensions. We consider the algebraic attractor equations in their most general form at the black hole horizon. We explicitly construct a new class of solutions for these attractor equations. These solutions are characterised by a set of involutory matrices. These involutions are obtained from a constraint involving the parameters in the pre-potential and generate new attractor points in the moduli space.

Design of Lightweight Linear Diffusion Layers from Near-MDS Matrices

Near-MDS matrices provide better trade-offs between security and efficiency compared to constructions based on MDS matrices, which are favored for hardwareoriented designs. We present new designs of lightweight linear diffusion layers by constructing lightweight near-MDS matrices. Firstly generic n × n near-MDS circulant matrices are found for 5 ≤ n ≤9. Secondly, the implementation cost of instantiations of the generic near-MDS matrices is examined. Surprisingly, for n = 7 , 8, it turns out that some proposed near-MDS circulant matrices of order n have the lowest XOR count among all near-MDS matrices of the same order. Further, for n = 5 , 6, we present near-MDS matrices of order n having the lowest XOR count as well. The proposed matrices, together with previous construction of order less than five, lead to solutions of n × n near-MDS matrices with the lowest XOR count over finite fields F 2 m for 2 ≤ n ≤ 8 and 4 ≤ m ≤ 2048. Moreover, we present some involutory near-MDS matrices of order 8 constructed from Hadamard matrices. Lastly, the security of the proposed linear layers is studied by calculating lower bounds on the number of active S-boxes. It is shown that our linear layers with a well-chosen nonlinear layer can provide sufficient security against differential and linear cryptanalysis.

Algorithms for solving the inverse problem associated with [formula omitted

In previous papers, the authors introduced and characterized a class of matrices called {K,s+1}-potent. Also, they established a method to construct these matrices. The purpose of this paper is to solve the associated inverse problem. Several algorithms are developed in order to find all involutory matrices K satisfying KAs+1K=A for a given matrix A∈Cn×n and a given natural number s. The cases s=0 and s≥1 are separately studied since they produce different situations. In addition, some examples are presented showing the numerical performance of the methods.

Several Anzahl theorems of matrices over Galois rings and their applications

Let R denote the Galois ring of characteristic ps and cardinality psh. In this paper, we determine the Smith normal forms of matrices over R, compute the number of the orbits of m×n matrices under the group GLm(R)×GLn(R) and the length of each orbit. Moreover, we discuss their applications to association schemes, systems of linear equations, generalized invertible matrices, idempotent matrices and involutory matrices.

Generalized centro-invertible matrices with applications

Centro-invertible matrices are introduced by R.S. Wikramaratna in 2008. For an involutory matrix R , we define the generalized centro-invertible matrices with respect to R to be those matrices A such that R A R = A − 1 . We apply these matrices to a problem in modular arithmetic. Specifically, algorithms for image blurring/deblurring are designed by means of generalized centro-invertible matrices. In addition, if R 1 and R 2 are n × n involutory matrices, then there is a simple bijection between the set of all centro-invertible matrices with respect to R 1 and the set with respect to R 2 .

Optimum Dynamic Diffusion of Block Cipher Based on Maximum Distance Separable Matrices

Maximum Distance Separable matrices became the state of the art as a diffusion component in block cipher design for example those MDS matrices used in algorithms such as AES and Twofish. This paper firstly reviews the relation between coding theory and cryptography in the context of providing optimal diffusion. Secondly, The Vandermonde and Cauchy based methodologies introduced by Mahdi Sajadieh et al. and J. Nakahara respectively for generating Involutory MDS matrices that are proposed to provide full block diffusion in order to decrease number of rounds of a block cipher were assessed. Finally Punctured MDS matrices are proposed to provide dynamicity of a block cipher, which guaranteed to provide optimum diffusion that should be considered in security proof against Linear and Differential Cryptanalysis of a block cipher.

Linear maps on matrix algebra Jordan derivable at involutory matrices

Let be a positive integer and be a unital commutative ring with 2 invertible. By , we mean the algebra consisting of all matrices over , and by , we denote the identity matrix in . We call an involutory matrix if . The Jordan product ‘’ is defined by . A linear transformation on is called derivable (resp., Jordan derivable) at involutory matrices if (resp., ) for any involutory matrix . In this paper, we show that a linear transformation on is derivable, or Jordan derivable, at involutory matrices if it is an inner derivation.

Chaos control of a new 3D autonomous system by stability transformation method

This paper sheds new insights into the stability transformation method (STM) for chaos control of dynamical system. The STM is applied to stabilize both multiple equilibrium points and unstable periodic orbits (UPOs) embedded in the chaotic attractor of a new 3D autonomous system. Firstly, the different equilibrium points of chaotic system are stabilized with STM by choosing specific initial points and involutory matrices. Then, the stability matrix is derived based on a priori information of equilibrium points, which indicates that the non-involutory matrix can also be used to control the unstable equilibrium points of chaotic systems. Finally, it is found that the STM can be regarded as a special form of speed feedback control method (SFCM), which facilitates the practical implementation of STM scheme. The chaotic system can be controlled by adopting the corresponding feedback control strategy once the stability matrices are determined. In this way, the blindness of choice for control strategy in the SFCM is avoided, and the difficulty for determining the state variables to be controlled is overcome.

Tripotency of a linear combination of two involutory matrices and a tripotent matrix that mutually commute

This paper concerns with the tripotency of a linear combination of three matrices, which has a background in statistical theory. We demonstrate all the possible cases that lead to the tripotency of a linear combination of two involutory matrices and a tripotent matrix that mutually commute. By utilizing block technique and returning the partitioned matrices to their original forms, we derive the sufficient and necessary conditions such that a linear combination of three mutually commuting involutory matrices is tripotent.

Algorithms for [formula omitted]-potent matrix constructions

In this paper, we deal with {K,s+1}-potent matrices. These matrices generalize all the following classes of matrices: k-potent matrices, periodic matrices, idempotent matrices, involutory matrices, centrosymmetric matrices, mirrorsymmetric matrices, circulant matrices, among others. Several applications of these classes of matrices can be found in the literature. We develop algorithms in order to compute {K,s+1}-potent matrices and {K,s+1}-potent linear combinations of {K,s+1}-potent matrices. In addition, some examples are presented in order to show the numerical performance of the method.