分组密码中置换理论的研究
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摘要
分组密码中置换理论的研究是近年来密码学研究上的热点之一，因为没有数据扩
展和数据压缩的分组密码本质上就是一个置换，一个分组密码体制的好坏与其使用的
置换有直接的关系。近年来随着差分分析和线性分析两种密码分析方法的引入和DES
的成功破译，人们愈来愈关心分组密码中置换理论的研究，掀起一个又一个寻找、构
造好的密码置换的高潮。本文正是在这样一个背景下研究了分组密码体制中的某些置
换理论，内容包括有限域上的置换多项式，有限域上的置换，正形置换及全距置换等，
作者所取得的主要研究成果如下：
（1）用线性代数方法和有限阿贝尔群上的特征标理论对有限域上的置换多项式组
的正交性进行了研究，展示了特征标理论在研究置换多项式方面的独到作用，为人们
研究置换多项式提供了一条思路；得到了正交多项式组的构造性定理；并指出，近来
人们研究的弹性函数可以看成正交多项式组的一种等价推广，于是，应用置换多项式
组的有关结果推出了弹性函数的一些性质；总结了人们在简化置换多项式组的正交性
的充要条件方面的结果，给出了正交多项式组的一个算法判别。
（2）利用关于置换多项式的研究结果，对有限域上的置换进行了研究，包括对布
尔置换，几乎完全非线性置换，高阶非线性置换的研究，这几类置换近来人们研究的
比较多，对前人在这些方面的结果进行了总结，给出了它们的一些性质及构造方法。
（3）正形置换是一类完全平衡映射，它具有良好的密码性质。本文对正形置换进
行了多方面的研究，推出了正形置换的许多性质，给出了正形置换的一般性构造方法；
阐述了正形置换与线性置换及拉丁方的关系，为正形置换的研究和应用提供了思路；
总结了人们在构造非线性正形置换方面的结果。
（4）全距置换也是近年来人们找到的一种密码置换源，从随机换位看，它具有良
好的密码性质。本文对全距置换进行了研究，给出了全距置换的一些必要条件，阐述
了全距置换与排序及排列的关系，为全距置换的研究提供了思路，并在引进全距拉丁
方概念的基础上给出了全距置换的一个较系统的构造方法，找到了全距置换的一下界，
为全距置换的应用打下了基础。
（5）在本文的研究中多次用到拉丁方，展示了拉丁方在密码学上的应用，并对拉
丁方的截集等问题做了研究。
The researches of permutation theoty of block ciphers is one of heat points in
cryptogrophy, because in fact block ciphers are permutation ,the security of a system of block
ciphers has much relation to permutation used in this system .The thesis is devoted to the
study of permutation theory of block ciphers, it consist of permutation polynomials over finite
field, permutation over finite field, orthomorphic permutation and quick trickle permutation.
Main contributions of this work are as follows:
? We investigate the regularity of permutation polynomials in n indeterminate over
finite field, show the use of characters of finite Abelian groups in study of the
permutation polynomials; obtain some theories on the construction of orthognomal
polynomials systems.
? We give a algorithm to determine orthognomal polynomials systems, and point out
resilient function is a extension of orthognomal polynomials systems in finite field
F.
? We study the Boolean permutation ,almost perfect nonlinear permutation and highly
nonlinear permutation .Because they play an important role in block ciphers ,so men
devote much attention to them in nearly years. We sunirnarize conclusion in these
aspect, give the methods of the construction of these permutation.
? A orthomorphic permutation is a perfectly balanced mapping, so orthomorphic
permutation has good property of cryptography .We give many properties about
orthomorphic permutation, this lay a reliable base to study orthomorphic
permutation.
? We research the construction of orthomorphic permutation through many methods,
give an example to use orthomorphic permutation to heighten security of block
ciphers.
? A n order quick trickle permutation is permutation of n objects (1,2, - ,n) ,such that
the spacings 1 梸 2, 2 ?3, etc. are all different ,so it has good cryptographic
properties. We give many necessary conditions of quick trickle permutation; and
research the graphic representation of quick trickle permutation.
? On the definition of quick trickle Latin squares, we give a method of constructing
quick trickle permutation.
展和数据压缩的分组密码本质上就是一个置换，一个分组密码体制的好坏与其使用的
置换有直接的关系。近年来随着差分分析和线性分析两种密码分析方法的引入和DES
的成功破译，人们愈来愈关心分组密码中置换理论的研究，掀起一个又一个寻找、构
造好的密码置换的高潮。本文正是在这样一个背景下研究了分组密码体制中的某些置
换理论，内容包括有限域上的置换多项式，有限域上的置换，正形置换及全距置换等，
作者所取得的主要研究成果如下：
（1）用线性代数方法和有限阿贝尔群上的特征标理论对有限域上的置换多项式组
的正交性进行了研究，展示了特征标理论在研究置换多项式方面的独到作用，为人们
研究置换多项式提供了一条思路；得到了正交多项式组的构造性定理；并指出，近来
人们研究的弹性函数可以看成正交多项式组的一种等价推广，于是，应用置换多项式
组的有关结果推出了弹性函数的一些性质；总结了人们在简化置换多项式组的正交性
的充要条件方面的结果，给出了正交多项式组的一个算法判别。
（2）利用关于置换多项式的研究结果，对有限域上的置换进行了研究，包括对布
尔置换，几乎完全非线性置换，高阶非线性置换的研究，这几类置换近来人们研究的
比较多，对前人在这些方面的结果进行了总结，给出了它们的一些性质及构造方法。
（3）正形置换是一类完全平衡映射，它具有良好的密码性质。本文对正形置换进
行了多方面的研究，推出了正形置换的许多性质，给出了正形置换的一般性构造方法；
阐述了正形置换与线性置换及拉丁方的关系，为正形置换的研究和应用提供了思路；
总结了人们在构造非线性正形置换方面的结果。
（4）全距置换也是近年来人们找到的一种密码置换源，从随机换位看，它具有良
好的密码性质。本文对全距置换进行了研究，给出了全距置换的一些必要条件，阐述
了全距置换与排序及排列的关系，为全距置换的研究提供了思路，并在引进全距拉丁
方概念的基础上给出了全距置换的一个较系统的构造方法，找到了全距置换的一下界，
为全距置换的应用打下了基础。
（5）在本文的研究中多次用到拉丁方，展示了拉丁方在密码学上的应用，并对拉
丁方的截集等问题做了研究。
The researches of permutation theoty of block ciphers is one of heat points in
cryptogrophy, because in fact block ciphers are permutation ,the security of a system of block
ciphers has much relation to permutation used in this system .The thesis is devoted to the
study of permutation theory of block ciphers, it consist of permutation polynomials over finite
field, permutation over finite field, orthomorphic permutation and quick trickle permutation.
Main contributions of this work are as follows:
? We investigate the regularity of permutation polynomials in n indeterminate over
finite field, show the use of characters of finite Abelian groups in study of the
permutation polynomials; obtain some theories on the construction of orthognomal
polynomials systems.
? We give a algorithm to determine orthognomal polynomials systems, and point out
resilient function is a extension of orthognomal polynomials systems in finite field
F.
? We study the Boolean permutation ,almost perfect nonlinear permutation and highly
nonlinear permutation .Because they play an important role in block ciphers ,so men
devote much attention to them in nearly years. We sunirnarize conclusion in these
aspect, give the methods of the construction of these permutation.
? A orthomorphic permutation is a perfectly balanced mapping, so orthomorphic
permutation has good property of cryptography .We give many properties about
orthomorphic permutation, this lay a reliable base to study orthomorphic
permutation.
? We research the construction of orthomorphic permutation through many methods,
give an example to use orthomorphic permutation to heighten security of block
ciphers.
? A n order quick trickle permutation is permutation of n objects (1,2, - ,n) ,such that
the spacings 1 梸 2, 2 ?3, etc. are all different ,so it has good cryptographic
properties. We give many necessary conditions of quick trickle permutation; and
research the graphic representation of quick trickle permutation.
? On the definition of quick trickle Latin squares, we give a method of constructing
quick trickle permutation.
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