Susanta Samanta

Yogesh Kumar, Akshay Ankush Yadav, Susanta Samanta

In AES-like ciphers, diffusion layers are commonly instantiated using MDS matrices, since their optimal branch number yields strong diffusion guarantees and underpins classical resistance arguments against differential and linear cryptanalysis. However, Daemen and Rijmen (2009) showed that linear layers may still exhibit related-differential structure beyond what the MDS criterion captures, and Bardeh and Rijmen (2022) demonstrated that this phenomenon can be exploited in attacks on reduced-round AES. In this work, we systematically investigate the conditions under which linear layers avoid or exhibit these differentials, identifying matrix classes for which such structure is unavoidable. We first prove that every non-MDS matrix admits a nontrivial pair of related differentials, showing that the MDS property is necessary for avoiding them. We then establish that every odd-order symmetric MDS matrix admits related differentials, which rules out broad families of Cauchy-based constructions. We also substantially strengthen the circulant case by proving that related differentials are unavoidable for every circulant matrix of order $n$ with $n \not\equiv \pm 2 \pmod{12}$. Finally, we revisit the characterization of $3 \times 3$ MDS matrices over $\mathbb{F}_{2^m}$ for the absence of related differentials, and derive an explicit necessary and sufficient criterion in terms of $15$ polynomial constraints.

Kishan Chand Gupta, Sumit Kumar Pandey, Susanta Samanta

The optimal branch number of MDS matrices makes them a preferred choice for designing diffusion layers in many block ciphers and hash functions. Consequently, various methods have been proposed for designing MDS matrices, including search and direct methods. While exhaustive search is suitable for small order MDS matrices, direct constructions are preferred for larger orders due to the vast search space involved. In the literature, there has been extensive research on the direct construction of MDS matrices using both recursive and nonrecursive methods. On the other hand, in lightweight cryptography, Near-MDS (NMDS) matrices with sub-optimal branch numbers offer a better balance between security and efficiency as a diffusion layer compared to MDS matrices. However, no direct construction method is available in the literature for constructing recursive NMDS matrices. This paper introduces some direct constructions of NMDS matrices in both nonrecursive and recursive settings. Additionally, it presents some direct constructions of nonrecursive MDS matrices from the generalized Vandermonde matrices. We propose a method for constructing involutory MDS and NMDS matrices using generalized Vandermonde matrices. Furthermore, we prove some folklore results that are used in the literature related to the NMDS code.