关于MDS变换的一个注记

A Note on MDS Mappings

MDS变换在分组密码设计中有着非常重要的作用。关于有限域上MDS矩阵的构造,已有丰富的理论研究成果。然而,关于模2n剩余类环上扩散变换的研究却非常少。因此,重点考虑模2n剩余类环上的扩散变换,研究模2n剩余类环上MDS变换的存在性及其构造问题,证明了模2~n剩余类环上不存在线性MDS变换。同时,基于正交拉丁方,给出了其上二元MDS变换的存在性定理及其记数,并给出了模2n剩余类环上二元MDS变换的构造方法。

MDS mappings are a fundamental building block in the construction of many block ciphers. Much theoretical study is done on the construction of MDS matrices over finite fields, while however, very little research on the diffusion transformation on the residual modules. This paper focuses on the diffusion transformation on the residual ring, and it is proved that there is no linear MDS transform on the residual ring. The existence and count for MDS mappings over the residue ring Z/（2^n） are explored, and based onthe orthogonal Latin squares, the way to donstruct the two variates MDS mappings also given.