The Four Order MDS Matrices Over Finite Fields

In the diffusion transformation,if the number of MDS?Maximum Distance Separable,MDS?matrix branch were reached the maximum,the effect of diffusion would be achieved optimally.MDS matrix plays an extremely important role in some cryptographic algorithms based on significance of this characteristic in the study of diffusion layer.This paper focuses the four order MDS matrix over finite fields on existence,construction methods and count problems to research relatively.The main work is as followed:1.The approach of detecting randomly generated matrices to construct MDS matrices over finite fields.2.We utilize elementary transformation to change the four order matrix overF_2²and reduce some unavoidable discussions,which is based on the characteristic of elements in the four order matrix overF_2².In addition,it is proved by counter-proof that there is no four order MDS matrix overF_2².3.We take advantage of counter proof method and drawer principle to determine the maximum degree of the element in four order matrices overF_2³.Besides,for the sake of constructing four order matrices overF_2³,we must choose the way of the element of the undetermined matrix.Consequently,it's achieved to proof the existence of four order MDS matrices overF_2³.At the same time,the existence of four order MDS matrices can be extended to higher-order finite fields,too.4.Based on the different four order matrices are generated several times under restricted conditions overF_2⁴,the average number of the different four order MDS matrices is obtained.Then,for fitting and analyzing different cases under regression analysis,we attempt to research a comparative analysis of all cases.As a result,we finally find an optimal method to estimate the number of four order MDS matrices overF_2⁴ and efficiently deal with the issue of four order MDS matrices over F_2⁴.