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计算机工程

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一类具有最大分支数的16阶0-1矩阵构造

郭 磊1,郑浩然1,刘明伟2   

  1. (1. 解放军信息工程大学三院,郑州 450004;2. 空军西安飞行学院,西安 710300)
  • 收稿日期:2012-10-26 出版日期:2013-12-15 发布日期:2013-12-13
  • 作者简介:郭 磊(1986-),男,硕士研究生,主研方向:密码学;郑浩然,副教授;刘明伟,助理工程师
  • 基金资助:
    国家自然科学基金资助项目(61272041)

Construction of a 16×16 0-1 Matrice with Maximum Branch Number

GUO Lei 1, ZHENG Hao-ran 1, LIU Ming-wei 2   

  1. (1. The 3rd Institute, PLA Information Engineering University, Zhengzhou 450004, China; 2. The Air Force Xi’an Flight Academy, Xi’an 710300, China)
  • Received:2012-10-26 Online:2013-12-15 Published:2013-12-13

摘要: 具有最大分支数的0-1可逆矩阵被广泛应用于分组密码的扩散结构设计中。为构造16阶该类矩阵,将16阶0-1矩阵划分为以4阶0-1矩阵为单元的4阶块矩阵,根据特征和域上重量均为2的4维0-1向量相加后所得向量的重量分布特点,在行置换同构意义下构造满足某种特殊结构的4阶0-1矩阵单元组,以此为基础,根据Hadamard矩阵的结构特点,利用矩阵的分块构造思想,给出一类分支数达到最大值8的16阶0-1可逆矩阵和对合矩阵构造方法,并在行置换同构意义下给出对合矩阵的计数。

关键词: 分组密码, 扩散结构, 分支数, 0-1矩阵, Hadamard矩阵

Abstract: 0-1 invertible matrice which has the largest branch number is widely used in the design of diffusion structures in block ciphers. In view of how to construct such 16×16 matrix, this paper divides 16×16 matrix into 4×4 block matrix by 4×4 0-1 matrix as a unit. Using the weight distribution peculiarity of the sum of 4-dimensional 0-1vectors with weight 2 in field of characteristic 2, it constructs 4×4 0-1 matrix unit group with some special structures in the permutation of isomorphism. On the basis of the structure characteristic of Hadamard matrice, it presents the methods of constructing 16×16 invertible 0-1 matrice with maximum branch number 8 using the matrix block construction method. Further, it presents the methods of constructing 16×16 involutory 0-1 matrice with maximum branch number 8 and their number in the permutation of isomorphism.

Key words: block cipher, diffusion structure, branch number, 0-1 matrice, Hadamard matrice

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