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

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

Propagation of Rayleigh waves in non-homogeneous orthotropic elastic media under the effect of magnetic field

The influence of magnetic field on the propagation of Rayleigh waves in an inhomogeneous, orthotropic elastic solid medium has been discussed. The method of separation of variable is used to find the frequency equation of the surface waves. The obtained dispersion equations are in agreement with the classical results when magnetic field and non-homogeneity are neglected   Keywords: Inhomogeneity, Orthotropic elastic solid,   field, Magnetic field

A General Correlation Theorem

In 2001, Nyberg proved three important correlation theorems and applied them to several cryptanalytic contexts. We continue the work of Nyberg in a more theoretical direction. We consider a general functional form and obtain its Walsh transform. Two of Nyberg's correlation theorems are seen to be special cases of our general functional form. S-box look-up, addition modulo 2 and X-OR are three frequently occuring operations in the design of symmetric ciphers. We consider two methods of combining these operations and in each apply our main result to obtain the Walsh transform

Computing Partial Walsh Transform from the Algebraic Normal Form of a Boolean Function

We study the relationship between the Walsh transform and the algebraic normal form of a Boolean function. In the first part of the paper, we carry out a combinatorial analysis to obtain a formula for the Walsh transform at a certain point in terms of parameters derived from the algebraic normal form. The second part of the paper is devoted to simplify this formula and develop an algorithm to evaluate it. Our algorithm can be applied in situations where it is practically impossible to use the fast Walsh transform algorithm. Experimental results show that under certain conditions it is possible to execute our algorithm to evaluate the Walsh transform (at a small set of points) of functions on a few scores of variables having a few hundred terms in the algebraic normal form

A Metric on the Set of Elliptic Curves over Fp{\mathbf F}_p.

Elliptic Curves over finite field have found application in many areas including cryptography. In the current article we define a metric on the set of elliptic curves defined over a prime field ${\mathbf F}_p, p>3$

A 32-bit RC4-like Keystream Generator

In this paper we propose a new 32-bit RC4 like keystream generator. The proposed generator produces 32 bits in each iteration and can be implemented in software with reasonable memory requirements

Results on multiples of primitive polynomials and their products over GF(2)

AbstractLinear feedback shift registers (LFSR) are important building blocks in stream cipher cryptosystems. To be cryptographically secure, the connection polynomials of the LFSRs need to be primitive over GF(2). Moreover, the polynomials should have high weight and they should not have sparse multiples at low or moderate degree. Here we provide results on t-nomial multiples of primitive polynomials and their products. We present results for counting t-nomial multiples and also analyse the statistical distribution of their degrees. The results in this paper helps in deciding what kind of primitive polynomial should be chosen and which should be discarded in terms of cryptographic applications. Further the results involve important theoretical identities in terms of t-nomial multiples which were not known earlier