MDS (Maximum Distance Separable) matrices play a crucial role in modern block ciphers and hash functions due to their strong diffusion properties. Researchers continue to explore efficient implementations of MDS matrices to improve encryption and decryption performance. While circulant MDS matrices are not inherently involutory, they offer notable advantages in computational efficiency. The demand for secure and high-speed block cipher implementations remains critical. This paper presents new 8 × 8 circulant MDS matrices that satisfy the condition \(\:{A}^{m}=I\) . Our analysis reveals that these matrices outperform those used in the Whirlpool hash function, the Kalyna block cipher, and other existing proposals in terms of implementation efficiency. Beyond their practical performance, our investigation also uncovers a fundamental structural property of circulant MDS matrices over characteristic-2 fields. We formally prove that, for powers \(\:{A}^{{2}^{t}}\) , the resulting matrices exhibit a predictable and highly regular sparsity pattern determined solely by \(\:m\) and \(\:t\) . This structured sparsity provides a theoretical explanation for the efficiency of the proposed matrices and establishes a general design principle for generating other optimal circulant MDS matrices. We demonstrate that applying direct exponentiation and row permutation preserves the structural benefits of circulant matrices while maintaining the \(\:{A}^{m}=I\) property. Furthermore, we propose an algorithm to generate key-dependent circulant MDS matrices utilizing direct exponentiation, row permutation, and generator elements from finite fields. These matrices are integrated into AES to construct a dynamic block cipher. Security analysis and statistical evaluation confirm that the modified AES enhances resistance to advanced cryptanalysis while ensuring computational efficiency.
