# Showing papers on "MDS matrix published in 2016"

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20 Mar 2016TL;DR: With this method, it is shown that circulant involutory MDS matrices, which have been proved do not exist over the finite field $$\mathbb {F}_{2^m}$$, can be constructed by using non-commutative entries.

Abstract: In the present paper, we investigate the problem of constructing MDS matrices with as few bit XOR operations as possible. The key contribution of the present paper is constructing MDS matrices with entries in the set of $$m\times m$$ non-singular matrices over $$\mathbb {F}_2$$ directly, and the linear transformations we used to construct MDS matrices are not assumed pairwise commutative. With this method, it is shown that circulant involutory MDS matrices, which have been proved do not exist over the finite field $$\mathbb {F}_{2^m}$$, can be constructed by using non-commutative entries. Some constructions of $$4\times 4$$ and $$5\times 5$$ circulant involutory MDS matrices are given when $$m=4,8$$. To the best of our knowledge, it is the first time that circulant involutory MDS matrices have been constructed. Furthermore, some lower bounds on XORs that required to evaluate one row of circulant and Hadamard MDS matrices of order 4 are given when $$m=4,8$$. Some constructions achieving the bound are also given, which have fewer XORs than previous constructions.

57 citations

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14 Aug 2016TL;DR: This paper investigates which field representation, that is which choice of basis, allows for an optimal implementation of finite field multiplications with one fixed element, and constructs new MDS matrices which outperform or are on par with all previous results when focusing on a round-based hardware implementation.

Abstract: In this paper we consider the fundamental question of optimizing finite field multiplications with one fixed element. Surprisingly, this question did not receive much attention previously. We investigate which field representation, that is which choice of basis, allows for an optimal implementation. Here, the efficiency of the multiplication is measured in terms of the number of XOR operations needed to implement the multiplication. While our results are potentially of larger interest, we focus on a particular application in the second part of our paper. Here we construct new MDS matrices which outperform or are on par with all previous results when focusing on a round-based hardware implementation.

55 citations

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20 Mar 2016TL;DR: In this paper, the authors proposed cyclic matrices, which preserve the benefits of circulant matrices and have the potential of being self-invertible, and obtained the MDS matrices with the least XOR gates requirement for dimensions from 3 × 3 to 8 × 8.

Abstract: In this article, we analyze the circulant structure of generalized circulant matrices to reduce the search space for finding lightweight MDS matrices. We first show that the implementation of circulant matrices can be serialized and can achieve similar area requirement and clock cycle performance as a serial-based implementation. By proving many new properties and equivalence classes for circulant matrices, we greatly reduce the search space for finding lightweight maximum distance separable MDS circulant matrices. We also generalize the circulant structure and propose a new class of matrices, called cyclic matrices, which preserve the benefits of circulant matrices and, in addition, have the potential of being self-invertible. In this new class of matrices, we obtain not only the MDS matrices with the least XOR gates requirement for dimensions from $$3 \times 3$$ to $$8 \times 8$$ in $${\text {GF}}2^4$$ and $${\text {GF}}2^8$$, but also involutory MDS matrices which was proven to be non-existence in the class of circulant matrices. To the best of our knowledge, the latter matrices are the first of its kind, which have a similar matrix structure as circulant matrices and are involutory and MDS simultaneously. Compared to the existing best known lightweight matrices, our new candidates either outperform or match them in terms of XOR gates required for a hardware implementation. Notably, our work is generic and independent of the metric for lightweight. Hence, our work is applicable for improving the search for efficient circulant matrices under other metrics besides XORi¾?gates.

40 citations

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TL;DR: The minimum value of XOR counts of 4 × 4 MDS matrices over F 2 4 and F 2 8 are reported, and it is proved that Toeplitz matrices cannot be both MDS and involutory.

Abstract: MDS matrices are used as building blocks of diffusion layers in block ciphers, and XOR count is a metric that estimates the hardware implementation cost. In this paper we report the minimum value of XOR counts of 4 × 4 MDS matrices over F 2 4 and F 2 8 , respectively. We give theoretical constructions of Toeplitz MDS matrices and show that they achieve the minimum XOR count. We also prove that Toeplitz matrices cannot be both MDS and involutory. Further we give theoretical constructions of 4 × 4 involutory MDS matrices over F 2 4 and F 2 8 that have the best known XOR counts so far: for F 2 4 our construction gives an involutory MDS matrix that actually improves the existing lower bound of XOR count, whereas for F 2 8 , it meets the known lower bound.

26 citations

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04 Nov 2016TL;DR: Format-Preserving Encryption is a symmetric key cryptographic primitive that is used to achieve this functionality and almost all the existing FPE schemes are based on Feistel construction and have efficiency issues.

Abstract: Commonly used encryption methods treat the plaintext merely as a stream of bits, disregarding any specific format that the data might have. In many situations, it is desirable and essential to have the ciphertext follow the same format as the plaintext. Moreover, ciphertext length expansion is also not allowed in these situations. Encryption of credit card numbers and social security numbers are the two most common examples of this requirement. Format-Preserving Encryption (FPE) is a symmetric key cryptographic primitive that is used to achieve this functionality. Initiated by the work of Black and Rogaway (CT-RSA 2002), many academic solutions have been proposed in literature that have focused on designing efficient FPE schemes. However, almost all the existing FPE schemes are based on Feistel construction and have efficiency issues.

8 citations

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TL;DR: Some new results on the preservation of many good cryptographic properties of MDS matrices under direct exponent transformation are presented and are shown to have important applications in constructing dynamic diffusion layers for block ciphers.

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.

4 citations

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TL;DR: This paperroduces a special generalization of circulant matrices called block circulants withcirculant blocks, which can be used to construct MDS matrices and proves that their inverse matrices can be implemented efficie ntly.

Abstract: Maximum distance separable (MDS) codes introduce MDS matrices which not only have applications in coding theory but also are of great importance in the design of block ciphers. It has received a great amount of attention. In this paper, we first i ntroduce a special generalization of circulant matrices called block circulants with circulant blocks, which can be used to construct MDS matrices. Then we investigate some interesting and useful properties of this class of matrices and prove that their inverse matrices can be implemented efficie ntly. Furthermore, we present some 4×4 and8×8 efficient MDS matrices of this class which are suitable for MD S diffusion layer. Compared with previous results, our construction provides better ef ficiency for the implementation of both the matrix and the its inverse matrix.

2 citations