【作者】 张卫国；

【导师】 肖国镇；

【作者基本信息】 西安电子科技大学 ， 密码学， 2006， 博士

【摘要】 密码函数在流密码、分组密码的设计中扮演着重要角色.本文研究了密码函数中的若干重要问题,取得以下主要结果:1)利用Maiorana-McFarland构造法构造出一类Plateaued函数,这种密码函数可以满足多个密码学准则:平衡性、高非线性度、适当阶数的相关免疫性、严格雪崩准则、不存在非零线性结构、好的GAC性质等.2)引入多输出Plateaued函数的概念,讨论了其密码学性质和构造方法.给出构造[ n , k ]不相交码集合的有效方法.用这种方法在n≥2k时,可以找到一个基数是2n ? k+ ?? ( n ?k )/k??的不相交码集合.并指出在n < 2k时,不存在基数大于1的不相交码集合.给出构造[ n, k ,≥?? d/2 ??]不相交码集合的方法.利用不相交码集合构造出具有高非线性度的多输出弹性Plateaued函数.3)给出可分布尔函数和可分Plateaued函数的一些性质;提出两个度量密码函数不可分性的指标:不可分度和λ-不可分度.4)给出k -正规布尔函数代数免疫阶的上界;给出判定Bent函数正规性的一个算法.5)利用毗连非线性函数的方法构造出一大类弹性函数,可以限定条件使构造的函数达到Siegenthalor界,同时也考虑了这类函数的非线性度等密码学性质;通过毗连2d个满足某些条件的Plateaued函数构造出具有高非线性度的弹性函数.6)给出计算乘积多项式周期的方法和公式,并将其用于计算卷积序列的周期.更多还原

【Abstract】 Cryptographic Boolean functions play an important role in both stream ciphers and block ciphers. In this dissertation, some important problems on cryptographic Boolean functions are investigated. The main results are as follows:1) A class of plateaued functions has been get by way of using the Maiorana- McFarland construction. A variety of desirable criteria for functions with cryptographic application could be satisfied: balancedness, high nonlinearity, correlation immunity of reasonably high order, strict avalanche criterion, non- existence of non-zero linear structures and good global avalanche characteristics etc.2) The notion of multi-output plateaued functions is introduced and some methods to construct this kind of cryptographic functions are provided. An effective method for finding a set of [ n , k ] disjoint linear codes is presented. When n≥2k, we could find a set of [ n , k ] disjoint linear codes with cardinality 2 n ? k + ?? ( n ?k )/k??; When n < 2k, there does not exist a set a disjoint linear codes with cardinality at least 2. The method on constructing a set of [ n , k ,≥? d/2 ?] disjoint linear codes is also be considered. We show how, thanks to our method, a (9,2,1) multi- output plateaued functions with highly nonlinearity could be constructed.3) We derive several results towards a better understanding the characterization of separable Boolean functions. Some properties of separable plateaued functions are also given. Two indicators related to the inseparability of cryptographic functions are introduced.4) We derive an upper bound on the algebraic immunity of a k-normal Boolean function. An effective algorithm to check whether a given bent function is normal or not is present.5) A large class of resilient functions is constructed via concatenating nonlinear resilient functions. We show that how this technique can be used to construct resilient functions which could achieve Siegenthalor bound. The nonlinearity of such functions is also considered. By way of concatenating 2d plateaued functions on F2 n ?d, which possess some properties, a resilient function of n variables with highly nonlinearity could be constructed.6) We show how to calculate periods of product polynomials via combinatorial techniques on factorization. A general formula for the period of a product polynomial is present. Then, the result is applied to compute the period of a convolution sequence.更多还原