Direct Construction of Maximally Dispersive and Near-Maximally Dispersive Matrices

Temel Kavramlar

This paper introduces direct construction methods for both recursive and nonrecursive MDS and NMDS matrices, which are crucial components in the design of secure and efficient cryptographic primitives.

Özet

The paper focuses on the construction of linear diffusion layers in symmetric-key cryptography, specifically exploring the direct construction of Maximum Distance Separable (MDS) and Near-MDS (NMDS) matrices.
Key highlights:

Presents direct construction methods for nonrecursive MDS and NMDS matrices using generalized Vandermonde matrices.
Introduces direct construction methods for recursive MDS and NMDS matrices, addressing the lack of such methods for recursive NMDS matrices in the literature.
Proposes a method for constructing involutory MDS and NMDS matrices.
Provides formal proofs for some commonly referenced folklore results in the literature of NMDS codes.
The paper is structured as follows:

Section 2 provides necessary notations and presents fundamental results, including useful results on NMDS codes.
Section 3 describes several direct construction methods for nonrecursive MDS and NMDS matrices.
Section 4 presents direct construction methods for recursive MDS and NMDS matrices.
Section 5 concludes the paper.

İstatistikler

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Alıntılar

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Önemli Bilgiler Şuradan Elde Edildi

On the Direct Construction of MDS and Near-MDS Matrices

by Kishan Chand... : arxiv.org 04-09-2024

https://arxiv.org/pdf/2306.12848.pdf

Daha Derin Sorular

How can the proposed direct construction methods for MDS and NMDS matrices be leveraged to design more secure and efficient cryptographic primitives

The proposed direct construction methods for MDS and NMDS matrices offer a more efficient and secure way to design cryptographic primitives. By directly constructing these matrices using generalized Vandermonde matrices, we can ensure that the resulting matrices are MDS or NMDS, which are crucial for achieving optimal diffusion in block ciphers and hash functions. These direct construction methods eliminate the need for exhaustive search methods, making the process more efficient, especially for larger order matrices. Additionally, the ability to construct NMDS matrices directly opens up new possibilities for achieving a better balance between security and efficiency in lightweight cryptography. Overall, leveraging these direct construction methods can lead to the development of cryptographic primitives that are both secure and resource-efficient.

What are the potential trade-offs between the security and efficiency of using MDS versus NMDS matrices as diffusion layers in lightweight cryptography

The potential trade-offs between the security and efficiency of using MDS versus NMDS matrices as diffusion layers in lightweight cryptography are significant. MDS matrices, with their optimal branch numbers, are preferred for their high level of security in diffusion layers. However, they may come with a higher computational cost due to their strict requirements. On the other hand, NMDS matrices, while having sub-optimal branch numbers, offer a better balance between security and efficiency. They provide a slower diffusion speed compared to MDS matrices but are more resource-efficient, making them suitable for lightweight implementations. The choice between MDS and NMDS matrices depends on the specific requirements of the cryptographic application, weighing the trade-offs between security and efficiency.

Can the insights from this work on the construction of involutory MDS and NMDS matrices lead to the development of new cryptographic primitives with improved properties

The insights from the construction of involutory MDS and NMDS matrices can indeed lead to the development of new cryptographic primitives with improved properties. Involutory matrices play a crucial role in cryptographic algorithms, especially in achieving properties like self-inverse transformations. By constructing involutory MDS and NMDS matrices directly using generalized Vandermonde matrices, we can enhance the security and efficiency of cryptographic primitives. These matrices can be utilized as diffusion layers in block ciphers and hash functions, providing a more robust defense against attacks while maintaining computational efficiency. The development of new cryptographic primitives based on these insights can lead to advancements in lightweight cryptography, ensuring a higher level of security with minimal resource requirements.