Design and Construction of Lightweight Semi Involutory Matrices Based on the Recursive Structures and Binary Sparse Matrices

matrices are one of the most important components in designing block ciphers. Based on the hardware terminologies, the acceptable speed of the implementation of  matrix and its inverse is one of the main features of  matrix for  structures. In this paper, a new type of binary block matrices called semi involutory is used such that the cost of implementation of these matrices and their inverses are equal. At first, by using binary linear functions over binary sparse matrices, a semi involutory matrix is proposed so that the cost of implementation of the proposed matrix is  bitwise  for  bit input. The structure of proposed matrix and its inverse are based on the structures, so they have suitable implementation from hardware point of view. Next, a semi involutory  matrix is proposed by applying structures such that the proposed  matrix is implemented with the bitwise  for  bit input. The proposed semi involutory  matrix is the major result, since the best known result in the implementation of a semi involutory  matrix for 8 bit input, based on the heuristic algorithm, is  bitwise  while  bitwise was obtained in this research. Moreover, the proposed  matrix is implemented with    for  bit input by applying suitable binary sparse matrices.