Topic

# MDS matrix

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

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13 Jul 2000

TL;DR: In this article, a method for determining an MDS matrix given the optimal complexity in the combination with the S-box, and a ciphering device adopting the MDS matrices obtained by the method are provided.

Abstract: PROBLEM TO BE SOLVED: To solve the problem that an S-box and an MDS matrices comprised in a ciphering device as the system components offset their effects against each other in spite of the X-box and the MDS intended to realize optimal complexity according to the design policies independent of each other, therefore, the ciphering device has had a probability of being rather unsafe. SOLUTION: By evaluating complexity of a result of multiplication of each candidate of MDS matrix elements by the matrix elements of a given S-box; evaluating, based on a evaluation result of this complexity, the complexity of the combinations of the matrix element candidates composing the MDS matrix; further, evaluating similar complexity also of an inverse matrix to this MDS matrix; and based on the evaluation results of the complexity of these matrices, a method for determining an MDS matrix giving the optimal complexity in the combination with the S-box, and a ciphering device adopting the MDS matrix obtained by the method are provided.

3 citations

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01 Jan 2018

TL;DR: This paper proves that the MDS condition, which guarantees the existence of MDS matrices with a prescribed set of zeros over large fields, is in fact sufficient for existence of such matrices over small fields.

Abstract: An MDS matrix is a matrix whose minors all have full rank. A question arising in coding theory is what zero patterns can MDS matrices have. There is a natural combinatorial characterization (called the MDS condition) which is necessary over any field, as well as sufficient over very large fields by a probabilistic argument. Dau et al. (ISIT 2014) conjectured that the MDS condition is sufficient over small fields as well, where the construction of the matrix is algebraic instead of probabilistic. This is known as the GM-MDS conjecture. Concretely, if a $k \times n$ zero pattern satisfies the MDS condition, then they conjecture that there exists an MDS matrix with this zero pattern over any field of size $|\mathbb{F}| \ge n+k-1$. In recent years, this conjecture was proven in several special cases. In this work, we resolve the conjecture.

3 citations

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TL;DR: It is proved that the proposed MDS matrices with a small number of both different elements and XOR gates are efficient in terms of implementation performance, and it is shown that the multi-MDS matrix generator inherits the dynamical properties of the high-dimensional Cat map, improving the resistance of diffusion layers against the powerful techniques of cryptanalysis.

3 citations

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11 Mar 2013

TL;DR: WIDEA is a family of block ciphers designed by Junod and Macchetti in 2009 as an extension of IDEA to larger block sizes and larger key sizes.

Abstract: WIDEA is a family of block ciphers designed by Junod and Macchetti in 2009 as an extension of IDEA to larger block sizes (256 and 512 bits for the main instances WIDEA-\(4\) and WIDEA-\(8\)) and larger key sizes (512 and 1024 bits, respectively). WIDEA-\(w\) is composed of \(w\) parallel copies of the IDEA block cipher, with an MDS matrix to provide diffusion between them. An important motivation was to use WIDEA to design a hash function.

3 citations

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TL;DR: The fixed XOR operation in AES rounds is replaced with a dual dynamic XOR table by using a 3D chaotic map and results show that the proposed method is better than the original AES.

Abstract: An efficient approach to secure information is critically needed at present. Cryptography remains the best approach to achieve security. On this basis, the national institute of standards and technology (NIST) selected Rijndael, which is a symmetric block cipher, as the advanced encryption standard (AES). The MixColumns transformation of this cipher is the most important function within the linear unit and the major source of diffusion. Dynamic MixColumns transformation can be used to enhance the AES security. In this study, a method to enhance the AES security is developed on the basis of two methods. The first method is an extension of a previous study entitled “A novel Approach for Enhancing Security of Advance Encryption Standard using Private XOR Table and 3D chaotic regarding to Software quality Factor.” In the current study, the fixed XOR operation in AES rounds is replaced with a dual dynamic XOR table by using a 3D chaotic map. The dual dynamic XOR table is based on 4 bits; one is used for even rounds, and the other is used for odd rounds. The second method is dynamic MixColumns transformation, where the maximum distance separable (MDS) matrix of the MixColumns transformation, which is fixed and public in every round, is changed with a dynamic MDS matrix, which is private, by using a 3D chaotic map. A 3D chaotic map is used to generate secret keys. These replacements enhance the AES security, particularly the resistance against attacks. Diehard and NIST tests, entropy, correlation coefficient, and histogram are used for security analysis of the proposed method. C++ is used to implement the proposed and original algorithms. MATLAB and LINX are used for the security analysis. Results show that the proposed method is better than the original AES.

3 citations