正形置换的一些新结论  

Some New Conclusions on Orthomorphisms

作　　者：童言[1] 张焕国[2,3] 池志强[1] 黄治华[1] 张剑[1] 

机构地区：[1]武汉数字工程研究所,湖北武汉430074 [2]武汉大学计算机学院,湖北武汉430072 [3]武汉大学空天信息安全与可信计算教育部重点实验室,湖北武汉430072

出　　处：《四川大学学报（工程科学版）》2013年第6期1-7,共7页Journal of Sichuan University (Engineering Science Edition)

基　　金：国家自然科学基金资助项目(60673071;60970115;61003267;91018008;61003268);国防预研项目(B0820132036;10113011010201)

摘　　要：正形置换在对称密码的设计中占有重要的地位。为了对正形置换的构造计数和性质进行进一步的分析探讨,首先指出戴宗铎等关于线性正形置换结构的结论中存在的问题,并根据修改后的结论,得到了最大线性正形置换的结构形式,进而实现了最大线性正形置换的完全无重复构造,而原先的构造方法会产生重复的结果;通过分析正形置换的补置换和仿射正形置换的关系,得到了正形置换的个数为2的(n+1)次方的倍数,比原来为2的n次方的倍数的结论更进了一步;给出了一种代数免疫度的定义,证明了这样定义的代数免疫度是Carlet-Charpin-Zinoviev等价不变量,并得到非仿射正形置换与它的补置换的差分均匀度、非线性度、代数次数和代数免疫度均相等。Orthomorphism plays an important role in the design of symmetric cryptography. To analyze its construction, counting and properties further, a problem in a conclusion about linear orthomorphism was pointed out and corrected. Then, with the corrected con- clusion, a non-redundant construction method to generate all maximal linear orthomorphisms was presented, while the previous method would produce repeatable results. The number of orthomorphism was proved to be a multiple of 2 to the power （ n ＋ 1 ） based on the re- lationship between affine orthomorphism and complementary permutation. At last, a definition of algebraic immunity was proposed and proved to be CCZ-eqnivalence-invariant. The algebraic immunity of a non-affine orthomorphism was also proved to be equal to that of complementary permutation of this orthomorphism. Same is the case with some other cryptographic properties, such as difference uni- formity, nonlinearity and algebraic degree.

关 键 词：对称密码 正形置换 最大线性正形置换 代数免疫度 CCZ等价 

分 类 号：TP309[自动化与计算机技术—计算机系统结构]