模2~n剩余类环上线性变换的异或线性分支数

On the XOR Linear Branch Numbers of Linear Transformations over Z/（2~n）

异或线性分支数是衡量分组密码扩散结构的扩散性能的一个重要指标,它对分组密码抵抗线性密码分析的能力有重要的影响.二元域上的非线性变换也常用作分组密码的扩散结构,本文给出了此类扩散结构的异或线性分支数的一个定义及其与分组密码抗线性逼近攻击能力的关系,证明了以模2n剩余类环上的线性变换为扩散结构的异或线性分支数等于将其奇系数换成1、偶系数换成0且将模2n加换成模2加所得的二元域上线性变换的异或线性分支数,从而将这类扩散结构的异或线性分支数归结为二元域上线性变换的异或线性分支数.

The xor linear branch numbers is used to evaluate the diffusion structure,and it indicates the security of the block cipher against linear cryptanalysis.The nonlinear transformations over the finite field GF（2） are also usually used for the diffusion structures of the block ciphers.This paper gives a definition of the xor linear branch numbers of this kind of diffusion structures and the relations between it as well as the strength of a cipher against liner cryptanalysis,and then we prove that the xor linear branch numbers of the diffusion structure over the residue class ring modulo Z/（2n） is equal to that of the diffusion structure over the finite field GF（2）,which we substitute 0 for the odd coefficient and 1 for the even coefficient and the linear transformations over Z/（2n） for the linear transformations over GF（2）.Consequently,we convert the problem of the xor linear branch numbers of the diffusion structure over Z/（2n） to that of the diffusion structure over the finite field GF（2）,which has been studied in many papers.