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 ALL Title Author Keyword Sponsors Type Publication date Submitted Time Subject Conference Name Source Categories KOS Subject Advisor ORCID Advanced ISCAS OpenIR  > 信息安全国家重点实验室  > 学位论文
 Subject: 计算机科学技术基础学科::数据安全与计算机安全 Title: 具有密码学意义的置换多项式的存在性及构造 Author: 李永强 Issued Date: 2011-11-21 Supervisor: 王明生 Major: 信息安全 Degree Grantor: 中国科学院研究生院 Place of Degree Grantor: 北京 Degree Level: 博士 Keyword: AB函数 ; APN函数 ; 置换多项式 ; EA等价 ; CCZ等价 Abstract: S盒为对称密码算法提供了混淆作用, 并且对于很多对称密码算法, S盒是其唯一的非线性部分, 因此S盒在对称密码中起着重要的作用. 考虑到实现的高效性, 在实际应用中, S盒通常设计为$\mathbb{F}_{2^{2n}}$上的置换. S盒的优劣通常由其抵抗线性分析和差分分析的能力来衡量. 因此, 构造$\mbf_{2^{2n}}$上的具有低差分均匀度, 高非线性度的置换是密码学中一个重要的问题.    差分均匀度, 非线性度是EA等价和CCZ等价的不变量, 但置换在这两个等价关系下并不保持. 因此, 在本文中, 我们详细讨论了从AB, APN函数出发, 利用EA等价以及CCZ等价构造$\mbf_{2^n}$上的置换多项式的可能性.    根据EA等价的定义, 如果存在置换与$F(x)\in\mbf_{2^n}[x]$ EA等价, 则存在线性多项式$L(x)\in\mbf_{2^n}[x]$, 使得$F(x)+L(x)$是$\mbf_{2^n}$上的置换. 在第三章中, 我们详细讨论了$\mbf_{2^n}$上形如$F(x)+L(x)$的置换多项式的存在性, 得到如下结果:    1. 当$F(x)$为幂函数时, 我们证明了对于某些特定$d$, 不存在线性多项式$L(x)\in\mbf_{2^n}[x]$, 使得$x^d+L(x)$是$\mbf_{2^n}$上的置换.    2. 当$F(x)$为Gold函数时, 我们证明了$x^{2^i+1}+L(x)$为置换当且仅当$n$为奇数且对于某个$\alpha\in\mbf_{2^n}^{*}$, 有$L(x)=\alpha x^{2^i}+\alpha^{2^i}x$.    3. 当$F(x)$为逆函数时, 我们证明了当$n\ge5$时, 不存在线性多项式$L(x)\in\mbf_{2^n}[x]$, 使得$x^{-1}+L(x)$是$\mbf_{2^n}$上的置换.    4. 当$F(x)$为AB函数$x^{2^i+1}+(x^{2^i}+x)\Tr(x^{2^i+1}+x)$时, 我们证明了当$m\ge2$时, $\mbf_{2^{2m+1}}$上不存在与$F(x)$ EA等价的置换多项式.该结果否证了Carlet在98年提出的任意一个AB函数都EA等价于一个置换''的猜想.    形如$L_1(F(x))+L_2(x)$的置换多项式在构造与$F(x)$ CCZ等价的函数中起着最本质的作用. 在第四章, 我们刻画了上述形式的置换多项式. 主要有如下结果:    1. 我们给出了一个存在线性多项式$L_1(x), L_2(x)$, 使得$L_1(F(x))+L_2(x)$为置换的充分条件.    2. 证明了对于任意的差分均匀度小于$2^n$的二次函数$F(x)\in\mbf_{2^n}[x]$, 都存在线性多项式$L_1(x), L_2(x)\in\mbf_{2^n}[x]$, 使得$L_1(F(x))+L_2(x)$是$\mbf_{2^n}$上的置换. 因此, 对于任意的二次AB, APN函数, 我们都可以构造与其CCZ等价的函数.    3. 对于$F(x)$为Gold函数, $n$为偶数, 我们找到了所有$\ker(L)\ge2^{n-2}$的线性多项式$L(x)$, 其使得$L(x^{2^i+1})+x$是$\mbf_{2^n}$上的置换多项式.    4. 对于逆函数, 我们也得到了部分结果.    在第五章, 我们讨论了利用三轮Feistel结构所构造的S盒的差分均匀度, 非线性度等密码学性质. 证明了其差分均匀度大于等于$2\delta(P_2)$.    在第六章, 我们给出了一个利用$\mbf_{2^{2m+1}}$上的二次APN置换构造$\mbf_{2^{2m}}$上具有已知最高非线性度的4差分均匀的置换的方法, 并利用Gold函数给出了几个具体的构造. English Abstract: S(ubstitution)-boxes play an important role in symmetriccryptography since they serve as the confusion part and in most cases are the only nonlinear part of round functions. For efficiency of implementations, S-boxes are often designed as permutations over $\mbf_{2^{2n}}$ in practice. The performance of these boxes is often measured by their resistance to linear and differential cryptanalysis. Therefore, constructing permutation polynomials with low differential uniformity and high nonlinearity over $\mbf_{2^{2n}}$ is of significant importance in cryptography.    Notice that the differential uniformity and nonlinearity offunctions are invariant under EA-equivalence and CCZ-equivalence, but permutation is not. Thus, we discuss the existence of permutation polynomials EA-equivalent or CCZ-equivalent to AB and APN polynomials in detail in the thesis.    According to the definition of EA-equivalence, if there exists a permutation polynomial EA-equivalent to $F(x)\in\mbf_{2^n}[x]$, then there exists a linear polynomial $L(x)\in\mbf_{2^n}[x]$ such that $F(x)+L(x)$ is a permutation polynomial over $\mbf_{2^n}$. In Chapter 3, we discuss the existence of permutation polynomials of the type $F(x)+L(x)$ over $\mbf_{2^n}$ and obtain main results as follows:    1. As for power functions, we proved that for some special $d$, there are no permutations EA-equivalent to $x^d$ over $\mbf_{2^n}$.    2. As for Gold function, we proved that $x^{2^i+1}+L(x)$ is a permutation over $\mbf_{2^n}$ ifand only if $n$ is odd and $L(x)=\alphax^{2^i}+\alpha^{2^i}x$ for some $\alpha\in\mbf_{2^n}^{*}$.     3. As for inverse function, we proved that there do not exist linear polynomials $L(x)\in\mbf_{2^n}[x]$, such that $x^{-1}+L(x)$ is a permutation polynomial over $\mbf_{2^n}$ when $n\ge5$.    4. As for the AB function $x^{2^i+1}+(x^{2^i}+x)\Tr(x^{2^i+1}+x)$, we proved that when$m\ge2$, there are no permutations EA-equivalent to $F(x)$ over $\mbf_{2^{2m+1}}$. This result disproved the conjecture of Every AB function is EA-equivalent to a permutation'', which is raised by Carlet in 1998.    Permutation polynomials of the type $L_1(F(x))+L_2(x)$ are essential for constructing polynomials CCZ-equivalent to $F(x)$. We characterize the permutation polynomials of the above type in Chapter 4. We obtained main results as follows:    1. We give a sufficient condition for getting linear polynomials $L_1(x), L_2(x)\in\mbf_{2^n}[x]$ such that $L_1(F(x))+L_2(x)$ is a permutation over $\mbf_{2^n}$.    2. We proved that for any quadratic polynomials  $F(x)\in\mbf_{2^n}[x]$ with differential uniformity less than $2^n$, there always exist linear polynomials $L_1(x),L_2(x)\in\mbf_{2^n}[x]$, such that $L_1(F(x))+L_2(x)$ is a permutation over $\mbf_{2^n}$. Therefore, for any quadratic AB or APN functions, we always can construct AB or APN functions CCZ-equivalent to them.    3. As for Gold functions, we get all linear polynomials with $\ker(L)\ge2^{n-2}$, such that $L(x^{2^i+1})+x$ isa permutation over $\mbf_{2^n}$ when $n$ is even.    4. We also get some results for inverse function.    In chapter 5, we discuss the differential uniformity andnonlinearity of the S-boxes constructed by using three rounds Feistel structure. We proved that their differential uniformity always large than or equal to $2\delta(P_2)$.    In Chapter 6, we give a method for constructing differentially 4-uniform permutations with the best known nonlinearity over $\mbf_{2^{2m}}$ from quadratic APN permutations over $\mbf_{2^{2m+1}}$. Several constructions are given by using Gold functions. Content Type: 学位论文 URI: http://ir.iscas.ac.cn/handle/311060/14400 Appears in Collections: 信息安全国家重点实验室_学位论文

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 Recommended Citation: 李永强. 具有密码学意义的置换多项式的存在性及构造[D]. 北京. 中国科学院研究生院. 2011-11-21.
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