Authors - Luong Tran Thi, Nguyen Van Long, Bac T. Nguyen, Hiep L Thi Abstract - Maximum Distance Separable (MDS) matrices play a crucial role in the design of diffusion layers in modern symmetric cryptographic primitives such as block ciphers, hash functions, and lightweight cryptographic schemes. Owing to their ability to achieve the optimal level of diffusion as measured by the branch number criterion, MDS matrices significantly enhance resistance against differential and linear cryptanalysis. However, the practical deployment of MDS matrices often faces challenges due to high computational cost, a large number of XOR operations, and substantial hardware resource requirements. Therefore, the construction of implementation-efficient MDS matrices has become an important research direction in modern cryptographic design. This paper presents a comprehensive survey of methods for constructing and optimizing MDS matrices with a focus on reducing implementation complexity in both software and hardware environments. Specifically, classical construction methods based on Cauchy, Vandermonde, and Reed–Solomon structures are reviewed, along with structured matrices such as circulant, recursive, and involutory forms. In addition, optimization techniques targeting XOR count, circuit depth, and memory usage are analyzed, and several open research directions in the design of efficient diffusion layers for modern cryptographic systems are discussed.
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