94 research outputs found
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
with fixed XOR value of 1 cannot be an NMDS when raised to a power of
. 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 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 , respectively
Hankel Rhotrices and Constructions of Maximum Distance Separable Rhotrices over Finite Fields
Many block ciphers in cryptography use Maximum Distance Separable (MDS) matrices to strengthen the diffusion layer. Rhotrices are represented by coupled matrices. Therefore, use of rhotrices in the cryptographic ciphers doubled the security of the cryptosystem. We define Hankel rhotrix and further construct the maximum distance separable rhotrices over finite fields
Lifted MDS Codes over Finite Fields
MDS codes are elegant constructions in coding theory and have mode important
applications in cryptography, network coding, distributed data storage,
communication systems et. In this study, a method is given which MDS codes are
lifted to a higher finite field. The presented method satisfies the protection
of the distance and creating the MDS code over the by using MDS code over
$F_p.
Weighted Reed-Solomon convolutional codes
In this paper we present a concrete algebraic construction of a novel class
of convolutional codes. These codes are built upon generalized Vandermonde
matrices and therefore can be seen as a natural extension of Reed-Solomon block
codes to the context of convolutional codes. For this reason we call them
weighted Reed-Solomon (WRS) convolutional codes. We show that under some
constraints on the defining parameters these codes are Maximum Distance Profile
(MDP), which means that they have the maximal possible growth in their column
distance profile. We study the size of the field needed to obtain WRS
convolutional codes which are MDP and compare it with the existing general
constructions of MDP convolutional codes in the literature, showing that in
many cases WRS convolutional codes require significantly smaller fields.Comment: 30 page
- …