Construction Of Lightweight Hadamard MDS Matrix Of Order 8

In 1949,the American mathematician Shannon proposed confusion and diffusion as the two basic principles of cryptographic design.Generally speaking,the linear layer is used as the diffusion layer component and the S-box is used as the confusion layer component in block cipher.In order to improve the security of the cryptographic algorithm,the designer use the MDS matrix as the diffusion layer of the block cipher to resist attacking.The most typical example is the column confusion matrix in AES,so the MDS matrix is widely studied.There is a method that is using Hadamard matrix.Because of the special nature of the Hadamard matrix,the Hadamard MDS matrix has become one of the research hotspots.Our purpose is to find Hadamard MDS matrix of order 8 with low XOR number.There may be a large numbers of submatrix determinants that are repeatedly cal-culated in the general judgment method,so we first optimized the judgment algorithm of the MDS matrix.We use the submatrix determinant of the low order to calculate the high order.Compared with the general method,the efficiency of the optimized judgment algorithm is increased by 7 times for the MDS matrix of order 8.The main purpose of this article is to find the low XOR number Hadamard MDS matrix of order 8.We first decompose Hadamard matrix of order 8 into the product of the 43 elementary transformation matrices,but the amount of calculation is too large.Therefore,we transform Hadamard matrix of order 8 into an upper triangular.This process reduces the 43 elementary transformation matrices that need to be considered to 19,which greatly reduces the computational complexity.We focused on the decom-position of the upper triangular matrix,and then optimized the decomposition method.And we filter the elements to reconstruct the involution Hadamard MDS matrix of order8.Finally,we obtained a involution Hadamard MDS matrix of order 8,and its XOR number is 397 in the F₂8.However,the best result before is 425.