The preservation of the coefficient of fixed points of an MDS matrix under direct exponent transformation

MDS (Maximum Distance Separable) code has been studied for a long time in the theory of error-correcting code and has been applied widely in cryptography. Some authors studied and proposed some methods for constructing MDS matrices which do not base on MDS codes. Some MDS matrix transformations have been studied and direct exponent is such a transformation. In this paper, we present some new results on the preservation of the number of fixed points of an MDS matrix under direct exponent transformation. In addition, the important applications of these results will be shown in block ciphers.