Research On Diffusion Layer Of Block Cipher Based On Rotational-XOR

In the diffusion layer of block cipher,the construction of MDS linear transformation can achieve good diffusivity,and diffusion layer with good diffusion performance can resist differential cryptanalysis and linear cryptanalysisthere.There are several kinds of methods can achieve it.The MDS linear transformation based on Rotation-XOR only includes rotation and exclusive or operation,it is efficient to implement in hardware and software.In addition,it can enhance the ability of cryptographic algorithm to resist various cryptographic analysis.Hence,it has been adopted by many symmetric cryptographic algorithms,such as SMS4,ZUC and so on.Although there are many researches on the diffusion layer,most of them based on Rotation-XOR mainly focus on the case of block size of 4,and the researches on the diffusion layer on a larger scale often do not have a structure based on rotation and exclusive or operation.This paper investigates the necessary conditions of constructing MDS linear transformations and seeks MDS linear transformations which group size is 64 and the block size is 8 based on Rotation-XOR.The research content of this paper is as follows:(1)From the point of view of all linear transformations on the finite field,the construction method of MDS linear transformation is studied.Combined with the corresponding relationship between the linear transformation based on Rotation-XOR and cyclic matrix,we provide the MDS linear transformation screening method,which can be used in all finite fields.The method as the main basis for judging MDS linear transformation in this paper.(2)The number of MDS linear transformations with block size of 8 and group size of 64 is explored when the number of exclusive or terms is 9.By analyzing the properties of the first row matrix,it is proved that the linear transformation in this scale can be equivalent to one form,and the existence of MDS linear transformation in this form is studied.Then,we design algorithm to the verify theoretical results by the software of Magma.(3)we engage the MDS linear transformation with 11 exclusive or terms in the same scale,and determine the number of MDS linear transformation in this scale.Firstly,all the linear transformations in this scale are divided into three cases by equivalent partition,including at most one free term in one matrix,two free terms in the same matrix and three free terms in the same matrix.In these three cases,88 × 56× 55 ×54 linear transformations of this scale are equivalent divided into 15 forms.Then,the existence of MDS linear transformation in one form is proved by theory.Finally,we design algorithm to search for MDS linear transformations in the remaining 14 forms.The conclusion and the search method in this paper are instructive to the study of MDS diffusion layer with block size of 8.