Showing papers on "MDS matrix published in 2021"

Construction of lightweight involutory MDS matrices

Abstract: 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.

Efficient Implementation of Lightweight MDS Matrix on FPGA

Abstract: Recently, studying of Maximum Distance Separable (MDS) matrix has become a topic of interest. The MDS matrix is the most important component of the diffusion layer in block ciphers. This paper introduces an optimized, low-cost hardware construction of Galois Field GF(2^8 ) 4×4 MDS matrix. The proposed design is implemented on Field programmable Gate Array (FPGA). The proposed design is synthesized targeting Virtex-7 FPGA using Xilinx ISE Design suite. Xilinx primitives LUT6 and LUT6_2 were used to control exactly the component placement in the design to maintain the minimum occupation area. The pipeline and parallel implementation techniques were used to improve the speed performance. The verification of the functionality of the proposed design has been proved using the ModelSim simulation tool. The synthesis result of the proposed design shows that, the new proposed architecture provides very competitive area and throughput trade-offs. In comparison with other related designs, the proposed design occupies the least area with the minimum time delay. The area of the developed MDS matrix design was significantly reduced, 68 LUT, with high throughput of 21.178 Gbps. The proposed design is a suitable candidate for lightweight cryptographic implementations.

Securing Digital Images using Stream Cipher and MDS Matrix

Abstract: This is an improved and extended version of the paper presented in CVIP 2020 conference. Stream ciphers are extensively used over a wide range of applications including security of digital data. In this paper, a method for securing different types of images (binary, gray scale, true color, and index) based on stream cipher (RC4A) and MDS (Maximum Distance Separable) matrix is proposed. The method adopts the framework of the Permutation-Substitution Network (PSN) of cryptography, and thus satisfies both confusion and diffusion properties required for a secure encryption algorithm. The proposed method encrypts a digital image into a random-like image from human visual as well as statistical point of view. Several encryption evaluation metrics, such as key sensitivity, chi-squared test, adjacent pixels correlation coefficient, irregular deviation, number of pixel change rate, unified averaged changed intensity, etc., are applied on test images taken from MATLAB IPT and USC-SIPI image database, to empirically assess the performance of the proposed method. The results of these statistical and security tests support the robustness of the proposed approach.

Construction of a set of circulant matrix submatrices for faster MDS matrix verification

Abstract: The present paper focuses on the construction of a set of submatrices of a circulant matrix such that it is a smaller set to verify that the circulant matrix is an MDS (maximum distance separable) one, comparing to the complete set of square submatrices needed in general case. The general MDS verification method requires to test for singular submatrices: if at least one square submatrix is singular the matrix is not MDS. However, the complexity of the general method dramatically increases for matrices of a greater dimension. We develop an algorithm that constructs a smaller subset of submatrices thanks to a simple structure of circulant matrices. The algorithm proposed in the paper reduces the size of the testing set by approximately two matrix orders.

Constructions of Lightweight MDS Diffusion Layers from Hankel Matrices

Abstract: Maximal distance separable (MDS) matrices are used as optimal diffusion layers in many block ciphers and hash functions Recently, the designers paid more attention to the lightweight MDS matrices because it can reduce the hardware resource In this paper, we give a new method to construct the lightweight MDS matrices We provide some theoretical results and two kinds of 4 × 4 lightweight Hankel MDS matrices We also prove that the 2s × 2s involution Hankel MDS matrix does not exist in finite field Furthermore, we searched the 4 × 4 Hankel MDS matrices over GL(4, F2) and GL(8, F2) that have the better s-XOR counts until now

On The Number of Different Entries in Involutory MDS Matrices over Finite Fields of Characteristic Two

Abstract: Two of many criteria of a good MDS matrix are being involutory and having few different elements. This paper investigates the number of different entries in an involutory MDS matrices of order 1, 2, 3, and 4 over finite fields of characteristic two. There are at least three and four different elements in an involutory MDS matrices with, respectively, order three and four, over finite fields of characteristic two.

On the number of different entries in involutory MDS matrices over finite fields of characteristic two

Abstract: Two of many criteria of a good MDS matrix are being involutory and having few different elements. This paper investigates the number of different entries in an involutory MDS matrices of order 1, 2, 3, and 4 over finite fields of characteristic two. There are at least three and four different elements in an involutory MDS matrices with, respectively, order three and four, over finite fields of characteristic two.