MDS matrix

About: MDS matrix is a research topic. Over the lifetime, 102 publications have been published within this topic receiving 2000 citations.

The preservation of good cryptographyic properties of MDS matrix under direct exponent transformation

Abstract: Maximum Distance Separable (MDS) code has been studied for a long time in the coding theory and has been applied widely in cryptography. The methods for transforming an MDS into other ones have been proposed by many authors in the literature. These methods are called MDS matrix transformations in order to generate different MDS matrices (dynamic MDS matrices) from an existing one. In this paper, some new results on the preservation of many good cryptographic properties of MDS matrices under direct exponent transformation are presented. These good cryptographic properties include MDS, involutory, symmetric, recursive (exponent of a companion matrix), the number of 1's and distinct elements in a matrix, circulant and circulant-like . In addition, these results are shown to have important applications in constructing dynamic diffusion layers for block ciphers. The strength of the ciphers against developing cryptanalytic techniques can be enhanced by the dynamic MDS diffusion layers.

A method for linear transformation in substitution-permutation network symmetric-key block cipher

Abstract: One embodiment of the present invention is a method of linear transformation in Substitution-Permutation Network symmetric-key block cipher producing n x n key-dependent MDS matrices from given n x n MDS matrix by scalar multiplication and permutations of elements of given matrix where multiplicative scalar and permutations are derived from binary inputs of length l . The method comprising deriving multiplicative scalar from binary input; multiplying given matrix with multiplicative scalar, producing first intermediate matrix; deriving first permutation of n objects from binary input; permuting rows of first intermediate matrix according to first permutation, producing second intermediate matrix; deriving second permutation of n objects from binary input; and permuting columns of second intermediate matrix according to second permutation to produce final MDS matrix. Another embodiment of the present invention is a method of linear transformation in Substitution-Permutation Network symmetric-key block cipher producing n x n key-dependent MDS matrices from given n x n MDS matrix by scalar multiplication and permutations of elements of given matrix where multiplicative scalar and permutations are derived from binary inputs of length l . The method comprising deriving multiplicative scalar from the key (202); multiplying given matrix with multiplicative scalar to produce first intermediate matrix (204); deriving first permutation of n objects from the key (206); permuting rows of first intermediate matrix according to first permutation to produce second intermediate matrix (208); deriving second permutation of n objects from the key (304); and permuting columns of second intermediate matrix according to second permutation (212) to produce final MDS matrix (214).

On the Construction of the 4 x 4 Lightest Circulant MDS Matrices

Abstract: 4 x 4 MDS (Maximal Distance Separable) matrices with few XORs have a wide range of applications in many mainstream lightweight ciphers. For 4 x 4 circulant MDS matrices over GL(4,F2), they have at least 12 XOR operations. In this paper, by traversing their structure characteristics, the utter construction and the numeration of the lightest circulant MDS matrices are firstly investigated. Then the overall structure and the diagrams of these matrices are given. Finally the characteristics of permutation group in the lightest circulant MDS matrices above are found: they possess characteristics of symmetric group S4, and for a kind of particular MDS matrices, they can even form a Klein four-group in some ways.

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.

A New Statement about Direct Exponent of an MDS Matrix in Block Ciphers

Abstract: MDS code has been studied for a long time in the theory of error-correcting code and has been applied widely in cryptography. Some authors studied and proposed some methods for constructing MDS matrices which do not based on MDS code. Some MDS matrix transformations have been studied and direct exponent is such a transformation. In this paper we present some new results on direct exponent transformation to show the k* number (cycle) that direct p exponent of the MDS matrix fork times results in the original MDS matrix. In addition, the results are shown to have important applications in block ciphers.