基于T-函数的密码组件设计与分析

T-functions, a mixture of basic algebraic operation and logical operations which are available for software and hardware implementation, can achieve quickly and occupy little resources in the same time. Moreover, with the complex nonlinear structure, the sequences generated by T-functions conform to the trend of platform based and nonlinear based stream cipher. Researches have shown that there exist T-functions which consist of a small amount of instruction and have good properties such as maximum length period. Faced with stream ciphers driven.primitive design requiring both better cryptography properties and less resource consumption, this provides a new choice. Our project is starting with the properties along with construction methods of single cycle T-function. To theoretically guarantee its application in the algorithm design, we will improve the theory of sequences generated by T-functions on pseudo random, consistency, relevance, sampling and differential properties, etc. Meanwhile, we will present the construction of single cycle T-functions with good properties and less resource, the methods that periods of all the coordinate sequences can reach the maximal value and the design of combination model driven primitive based on T-functions. This provides technical support for T- functions used in the design of modern driven primitives. On the one hand, the results of this study can solve the problems that few singe cycle T-functions can be used, truncation methods are with low efficiency and direct use may lead to much weakness. On the other hand, the results offer new ideas to the stream cipher design of nonlinear based driven primitives.

T-函数混合了基本代数运算和易于软硬件实现的逻辑运算，实现速度快，占用资源少，其生成序列天然具有了复杂的非线性结构，符合现代序列密码平台化和非线性化的发展趋势。特别是利用少量指令实现且具有最大周期等良好性质的单圈T-函数，为设计兼顾密码学性质和资源消耗的序列密码驱动组件提供了新的选择。本项目拟从单圈T-函数的密码学性质和构造方法研究入手，完善T-函数在序列伪随机性、一致性、相关性、采样性和差分性质等方面的理论，为其在算法设计中的应用提供深厚的理论保证；同时给出性质和效率兼顾的单圈T-函数设计方法、各分位序列都具有最大周期的组件构造方法和基于T-函数的组合模型驱动组件设计方法，为T-函数用于现代驱动组件设计提供技术支持。这些研究成果不仅可以解决单圈T-函数可用数量少、截尾方式效率低、直接使用弱点多等影响T-函数广泛应用的问题，也为序列密码驱动组件的非线性化设计提供崭新的思路。

T-函数是序列密码中典型的非线性组件。T-函数将基本代数运算和易于软硬件实现的逻辑运算相混合，不仅实现速度快、效率高，且其生成的序列天然具有复杂的非线性结构，它为兼顾密码学性质和资源消耗的序列密码驱动组件提供了一种新的选择。本文以T-函数的基本性质和构造方法进行了研究，主要成果包括：.1、针对字序列按“状态抽取”和按“分位抽取”两种使用方式，研究了单圈T-函数输出序列的平衡性、周期性、游程分布以及复杂度和自相关函数等密码学性质。证明了单圈T-函数抽取状态序列的线性复杂度为序列周期的一半，k-错复杂度具有良好的稳定性，自相关函数值大多数为0；证明了抽取分位级联序列具有良好的伪随机特性，给出了其线性复杂度和k-错线性复杂度，证明了当第一下降点相同时，抽取分位级联序列的稳定性好于抽取状态序列。.2、基于已有单圈T-函数，给出了新单圈T-函数的复合构造方法；构造了一类带有循环移位操作且适合软件快速实现的非线性函数，给出它成为单圈T-函数的充分条件；给出了一类多字单圈T-函数的s盒构造法，并给出了互不平移等价且不存在采样关系的此类单圈T-函数的个数。.3、通过对两个单圈T-函数各自的比特置换结果做“进位加”运算，构造了一类各分位序列周期都为2n的序列（n为T-函数字长）。该方法能够充分利用T-函数的各条分位序列，与截尾方法相比，具有更高的实现效率。. 上述成果中单圈T-函数的构造方法和最大分位周期序列的构造方法为序列密码非线性驱动组件的设计提供了更加丰富的资源；其密码学性质和设计原则为其在序列密码中的应用提供了重要的理论支撑。