On the Construction of Near-MDS Matrices

The optimal branch number of MDS matrices makes them a preferred choice for designing diffusion layers in many block ciphers and hash functions. However, in lightweight cryptography, Near-MDS (NMDS) matrices with sub-optimal branch numbers offer a better balance between security and efficiency as a diffusion layer, compared to MDS matrices. In this paper, we study NMDS matrices, exploring their construction in both recursive and nonrecursive settings. We provide several theoretical results and explore the hardware efficiency of the construction of NMDS matrices. Additionally, we make comparisons between the results of NMDS and MDS matrices whenever possible. For the recursive approach, we study the DLS matrices and provide some theoretical results on their use. Some of the results are used to restrict the search space of the DLS matrices. We also show that over a field of characteristic 2, any sparse matrix of order $n\geq 4$ with fixed XOR value of 1 cannot be an NMDS when raised to a power of $k\leq n$. Following that, we use the generalized DLS (GDLS) matrices to provide some lightweight recursive NMDS matrices of several orders that perform better than the existing matrices in terms of hardware cost or the number of iterations. For the nonrecursive construction of NMDS matrices, we study various structures, such as circulant and left-circulant matrices, and their generalizations: Toeplitz and Hankel matrices. In addition, we prove that Toeplitz matrices of order $n>4$ cannot be simultaneously NMDS and involutory over a field of characteristic 2. Finally, we use GDLS matrices to provide some lightweight NMDS matrices that can be computed in one clock cycle. The proposed nonrecursive NMDS matrices of orders 4, 5, 6, 7, and 8 can be implemented with 24, 50, 65, 96, and 108 XORs over $\mathbb{F}_{2^4}$, respectively

On the Direct Construction of MDS and Near-MDS Matrices

The optimal branch number of MDS matrices makes them a preferred choice for designing diffusion layers in many block ciphers and hash functions. Consequently, various methods have been proposed for designing MDS matrices, including search and direct methods. While exhaustive search is suitable for small order MDS matrices, direct constructions are preferred for larger orders due to the vast search space involved. In the literature, there has been extensive research on the direct construction of MDS matrices using both recursive and nonrecursive methods. On the other hand, in lightweight cryptography, Near-MDS (NMDS) matrices with sub-optimal branch numbers offer a better balance between security and efficiency as a diffusion layer compared to MDS matrices. However, no direct construction method is available in the literature for constructing recursive NMDS matrices. This paper introduces some direct constructions of NMDS matrices in both nonrecursive and recursive settings. Additionally, it presents some direct constructions of nonrecursive MDS matrices from the generalized Vandermonde matrices. We propose a method for constructing involutory MDS and NMDS matrices using generalized Vandermonde matrices. Furthermore, we prove some folklore results that are used in the literature related to the NMDS code

Development of structures under the influence of heterogeneous flow field around rigid inclusions: insights from theoretical and numerical models

Rocks that are mechanically heterogeneous due to the presence of stiff or rigid inclusions floating in a ductile matrix, commonly show a variety of micro- to macro-scale structures developing under the influence of heterogeneous flow field in the neighbourhood of the inclusions. It is of fundamental importance to apprehend the nature of strain heterogeneity around inclusions to understand progressive development of structures associated with rigid inclusions such as strain shadow, foliation drag, porphyroclast mantle, porphyroblast inclusion trails, intragranular fractures, etc. The development of these diverse types of structures can be analyzed with the help of a suitable hydrodynamic theory. In this paper, we review different continuum models that have been proposed to characterize the heterogeneous flow field around rigid inclusions, focusing on recent developments. Recent studies reveal that Jeffery's [Proc. R. Soc. Lond. A 120 (1922) 161.] theory dealing with the motion of ellipsoidal rigid bodies in an infinitely extended viscous medium is more general in nature, and applicable for modeling the heterogeneous flow around both equant and inequant shapes of inclusions and ideal or non-ideal shear deformation of the matrix. The application of this theory, therefore, has advantages over other models, based on Lamb's [Lamb, H., 1932. Hydrodynamics. Cambridge University Press, Cambridge.] theory dealing with spherical inclusions. The review finally illustrates numerical simulations based on hydrodynamic theories, highlighting the controls of physical and kinematic factors on the progressive development of the structures mentioned above

Boudinage in multilayered rocks under layer-normal compression: a theoretical analysis

This paper presents a dynamic analysis of boudinage in multilayers of alternate brittle and ductile layers under layer-normal compression. Based on the mode of fracturing of individual brittle layers, boudinage is classified into three types: tensile fracture boudinage (Type 1), shear fracture boudinage (Type 2a) and extensional shear fracture boudinage (Type 2b). The layer-thickness ratio, Tr (=tb/td), and the strength ratio, F (=T/2ηε), between the brittle and the ductile units are the principal physical factors determining the type of boudinage. Type 1 boudinage develops rectangular boudins and occurs when Tr is low (<4.5) or F is high (>0.8). In contrast, Type 2a boudinage takes place when Tr is high (>8.5) or F is low (<0.5). The intermediate values of these factors delimit the field of extensional shear fracture boudinage. The square of fracture spacing or boudin width in Type 1 boudinage is linearly proportional to layer-thickness, whereas that in Type 2 boudinage shows a non-linear relationship with layer-thickness. The aspect ratio (Ar) of all the types of boudins is inversely proportional to layer-thickness ratio (Tr). However, Type 1 and Type 2 boudins, have contrasting aspect ratios, which are generally greater and less than 1, respectively

Rotation of single rigid inclusions embedded in an anisotropic matrix: a theoretical study

This paper presents a theoretical analysis of instantaneous rotation of elliptical rigid inclusions hosted in a foliated matrix under bulk tensile stress. The foliated matrix is modelled with orthotropic elastic rheology, considering two factors as measures of anisotropy: m = μ0/E01and n = E02/E01 , where μ0 is the shear modulus parallel to the foliation plane E01and E02 and are the Young moduli along and across the foliation, respectively. Normalized instantaneous inclusion rotation (θ) is plotted as a function of the bulk tension direction (α) with respect to the long axis of the inclusion, taking into account two parameters: (1) anisotropic factors m and n, and (2) the inclination of the foliation plane to the long axis of inclusion (θ). In the case of θ=0°, ω versus α variations are sinuous, showing maximum instantaneous rotation in positive and negative sense at α =45 and 135°, respectively, irrespective of m and n values. The magnitude of maximum ω increases with decrease in m, i.e. increasing degree of anisotropy in the matrix. On the other hand, decreasing the value of the anisotropic factor n results in decreasing instantaneous rotation. ω increases with the aspect ratio R of inclusion, assuming an asymptotic value when R is large. This asymptotic value is larger for lower values of m. In case of θ ≠0°, ω versus α variations are asymmetrical, showing maximum instantaneous rotation at varying inclusion orientation for different m. For given m and n, with increase in θ the sense of instantaneous rotation reverses at a critical value of θ

Role of weak flaws in nucleation of shear zones: an experimental and theoretical study

This paper investigates the role of inherent weak flaws in the formation of plastic zones in deforming solids to understand the development of geological shear zones. Physical experiments were carried out on polymethylmethaacrylate (PMMA) models containing single and multiple circular cylindrical flaws under compression, maintaining plane strain condition. Models with single flaws show development of shear zones against a flaw in the form of conjugate sets with an average dihedral angle of 84° and oriented at an angle of 42° to the bulk compression direction. The shear zones are generally tapered, with increasing width away from the flaw. In models with multiple flaws, shear zones nucleated against individual flaws, which propagated and coalesced with one another, forming through-going, band-like shear zones with inclination varying from 35° to 53° with the bulk compression direction. With an increase in flaw concentration, the through-going shear zones defined persistent conjugate sets. We applied the plane theory of elasticity for numerical simulations of ductile shear zones under the influence of a single circular weak flaw. The pattern of shear zones yielded from numerical runs grossly matches with those observed in physical model experiments. Theoretical analysis demonstrates that the presence of a flaw promotes nucleation of shear zones at a bulk stress below the yield strength of matrix. This critical stress is a non-linear function of the flaw-matrix competence contrast, and decreases asymptotically with increasing competence contrast

An analysis of anisotropy of rocks containing shape fabrics of rigid inclusions

This paper presents a theoretical basis for estimation of mechanical anisotropy in homogeneous rocks containing shape fabrics of rigid inclusions. The analysis is based on two types of viscous models: one containing linear fabrics of prolate (a > b = c) inclusions (cf. L-tectonite) and the other containing planar fabrics of oblate (a < b = c) inclusions (cf. S-tectonite). Models show contrasting bulk viscosities in stretching (normal viscosity) and shearing (shear viscosity) parallel to the fabric. The axial ratio R (= a/b) and the volume concentration (ρv) of rigid inclusions appear to be the principal parameters in determining the viscosity contrast. In anisotropic models with linear fabrics, normal viscosity (ηp) increases monotonically with increase in R, whereas shear viscosity (ηs) increases to a maximum, and then drops down to a near-stationary value. In anisotropic models with planar fabrics, the normal viscosity increases little with increasing flatness of inclusions, but the variation assumes a steep gradient when the latter is large. Shear viscosity, on the other hand, is relatively less sensitive to the shape of inclusions. The ratio of normal and shear viscosities, conventionally described as anisotropy factor δ, in both the models is always greater than 1, indicating that normal viscosity will be essentially greater than shear viscosity, irrespective of the axial ratio of inclusions forming the fabric. Models with a linear fabric show contrasting normal viscosities in pure shear flow along and across the linear fabric. The anisotropy is expressed by the ratio of longitudinal and transverse normal viscosities (anisotropic factor σ). It is revealed that the transverse viscosity is essentially less than the longitudinal viscosity, as observed in test models

Numerical models of flow patterns around a rigid inclusion in a viscous matrix undergoing simple shear: implications of model parameters and boundary conditions

The hydrodynamic models that have recently been developed to investigate the nature of flow around coherent, rigid inclusions in simple shear reveal two contrasting patterns with eye-shaped and bow-tie shaped separatrix, even though all the models are based on Navier-Stokes law. In order to find the cause of this variance, this paper reviews the existing models in the light of different boundary conditions imposed on individual models. Scrutiny of the models reveals that inclusion-matrix systems, when considered infinitely extended in space, develop eye-shaped flows. However, those with finite dimensions essentially display bow-tie shaped flows. Using a finite element method (FEM), we advance the study to show the additional effects of model/inclusion dimension ratio (DR) and model aspect ratio (AR) under different boundary conditions. In the flow with bow-tie shaped separatrix, the regions of back flow define a nearly semi-circular geometry when DR is low (<2). These regions assume a semi-elliptical shape with increasing DR. The distance of stagnation points from the inclusion is found to increase non-linearly with DR. Model results suggest that transformation of a flow with eye-shaped separatrix to that with bow-tie shaped separatrix can occur due to increasing AR under a specific boundary condition. Applying FEM results in geological situations thus requires the appropriate choice of dimensional parameters of the model as well as the kinematic conditions imposed at the model boundaries

Flow and strain patterns at the terminations of tapered shear zones

With the help of corner flow theory, this paper numerically analyzes the deformation pattern at the terminations of tapered shear zones, the walls of which are rigid and move parallel to each other in opposite directions. The overall flow pattern is characterized by curvilinear particle paths that show convexity towards and opposite to the tapering direction respectively for low (< 5°) and high (> 10°) inclinations of the wall verging opposite to the sense of wall movement. In tapered shear zones there are two distinct fields of instantaneous shortening and extension parallel to the direction of wall movement. Numerical models reveal that the finite strain distributions are generally asymmetrical with larger strain concentration occurring near the wall verging opposite to sense of wall movement. The S-foliation trajectories show a curvilinear pattern, convexing against the tapering direction. The analysis of rotationality (vorticity) indicates that the sense of vorticity near the synthetically verging wall is reverse to the sense of wall movement; however Wk is one everywhere within the shear zone

Problem of folding in ductile shear zones: a theoretical and experimental investigation

The paper describes micro-scale folds within a narrow ductile shear zone of the Peninsular Gneissic Complex, South India. The characteristics of the folds indicate that they have formed by buckling on the mylonitic foliation parallel to the C fabric. This raises the question of how a buckling instability could develop on the mylonitic foliation, as there can be no overall shortening along the shear direction. On the basis of natural observations we show that mylonitization involves sericitization locally along the C fabrics, forming discrete mechanically weak zones, which perturb the homogeneous shear stress field in their neighborhood, and induce shortening along the bulk shear direction, leading to buckling instabilities on the mylonitic foliation. The weak zone model is supported with results obtained from physical experiments. Theoretical analysis shows that the weak bodies with large length-to-thickness ratios perturb the shear stress field in the shear zone significantly, developing compressive stresses along the bulk shear direction, required for buckle instabilities to occur on the mylonitic foliation