递归构造多个具有最优代数免疫度的平衡布尔函数

代数免疫度是针对代数攻击而提出来的一个新的密码学概念.要能够有效地抵抗代数攻击,密码系统中使用的布尔函数必须具有平衡性、较高的代数次数、较高的非线性度和较高的代数免疫度等.为了提高布尔函数的密码学性能,通过布尔函数仿射等价的方法,找出了所有具有最优代数免疫度的三变元布尔函数.由这些具有最优代数免疫度的三变元非线性布尔函数,递归构造了一类代数免疫度最优、代数次数较高的平衡布尔函数.给出了这类布尔函数非线性度的一个下界,偶数变元时,其下界严格大于Lobanov给出的下界.     

具有最优代数免疫度的2~m个变量的对称Boolean函数

本文给出了2m个变量的对称Boolean函数f具有最优代数免疫度AI2m(f)=2m-1的一个充分必要条件.由此得到一个递归公式,从而构造出全部具有最优代数免疫度的2m个变量的对称Boolean函数(m2).最后证明了这样的Boolean函数的个数为3·2m.     

布尔函数的代数免疫性与设计准则

本文讨论了布尔函数的重量与代数免疫性之间的关系,给出了判断布尔函数是否有低次零化子的一个充分条件,并对由几类传统的构造方法所获得的布尔函数的代数免疫性进行了分析.     

布尔函数的线性维数与非线性度

本文讨论了布尔函数的线性维数与非线性度的有关性质,证明了布尔函数变元个数,代数次数和线性维数之间的关系,给出了由线性维数计算二次布尔函数非线性度的公式。     

最优布尔函数的一个性质

Walsh谱只有3个值:0,±2m+2,且同时达到代数次数上界n-m-1和非线性度上界2n-1-2m+1的n元m阶弹性布尔函数(m>n/2-2)称为饱和最优函数(saturatedbest简写为SB).本文将给出关于SB函数非零谱值位置分布的一个性质,利用这一性质我们给出构造非线性度为56的4次7兀2阶弹性布尔函数的一种方法.     

R_0代数的根及其结构特征

在现有的基于中点与真布尔元对R0代数进行分类讨论的基础上,提出了R0代数的根的概念,并通过研究根的若干重要性质给出了含真布尔元的R0代数结构特征的一个精细刻画,从而完全解决了这类R0代数的结构问题.     

关于一类函数的代数逼近度的注记

设A（z)是函数列{fn(z)}的极限函数，已知^[4],当|fn(z)-A(z)|在z=0的阶满足一定的条件时，A（z)是超越函数。本文给出了极限函数的代数逼近度函数。     

具有最高代数次数的2n元n维Bent函数的构造

本文给出了代数次数达到最高的一类布尔置换的代数标准形 ;并用m序列的状态转移矩阵和所得置换 ,构造了一类代数次数达到最高的 2n元n维Bent函数 ,用这类函数所构造的S盒具有较高的安全强度 .     

单圈图的代数连通度的排序

n阶图G称为是一个单圈图,如果G是连通的,并且G的边数也是n.用U(n)表示所有n阶单圈图所成的集合.给出了当阶数n≥25时,代数连通度为前九大的n阶单圈图及它们的代数连通度.     

一类扩张仿射李代数上的非交换Poisson代数

讨论了以量子环面为坐标代数,零度v的A型扩张仿射李代数s1_N(CQ)上的非交换的Poisson代数,证明了它的导出李子代数上的结合乘积是平凡的.同时,给出了标度元素的结合积的形式.     

On the security of the Feng–Liao–Yang Boolean functions with optimal algebraic immunity against fast algebraic attacks

In the past few years, algebraic attacks against stream ciphers with linear feedback function have been significantly improved. As a response to the new attacks, the notion of algebraic immunity of a Boolean function f was introduced, defined as the minimum degree of the annihilators of f and f + 1. An annihilator of f is a nonzero Boolean function g, such that fg = 0. There is an increasing interest in construction of Boolean functions that possess optimal algebraic immunity, combined with other characteristics, like balancedness, high nonlinearity, and high algebraic degree. In this paper, we investigate a recently proposed infinite class of balanced Boolean functions with optimal algebraic immunity, optimum algebraic degree and much better nonlinearity than all the previously introduced classes of Boolean functions with maximal algebraic immunity. More precisely, we study the resistance of the functions against one of the new algebraic attacks, namely the fast algebraic attacks (FAAs). Using the special characteristics of the family members, we introduce an efficient method for the evaluation of their behavior against these attacks. The new algorithm is based on the well studied Berlekamp–Massey algorithm.     

Construction of rotation symmetric Boolean functions with optimal algebraic immunity and high nonlinearity

Recent research shows that the class of rotation symmetric Boolean functions is potentially rich in functions of cryptographic significance. In this paper, based on the knowledge of compositions of an integer, we present two new kinds of construction of rotation symmetric Boolean functions having optimal algebraic immunity on either odd variables or even variables. Our new functions are of much better nonlinearity than all the existing theoretical constructions of rotation symmetric Boolean functions with optimal algebraic immunity. Further, the algebraic degree of our rotation symmetric Boolean functions are also high enough.     

On extended algebraic immunity

Algebraic immunity (AI) measures the resistance of a Boolean function f against algebraic attack. Extended algebraic immunity (EAI) extends the concept of algebraic immunity, whose point is that a Boolean function f may be replaced by another Boolean function f ^c called the algebraic complement of f. In this paper, we study the relation between different properties (such as weight, nonlinearity, etc.) of Boolean function f and its algebraic complement f ^c . For example, the relation between annihilator sets of f and f ^c provides a faster way to find their annihilators than previous report. Next, we present a necessary condition for Boolean functions to be of the maximum possible extended algebraic immunity. We also analyze some Boolean functions with maximum possible algebraic immunity constructed by known existing construction methods for their extended algebraic immunity.     

A conjecture about binary strings and its applications on constructing Boolean functions with optimal algebraic immunity

In this paper, a combinatorial conjecture about binary strings is proposed. Under the assumption that the proposed conjecture is correct, two classes of Boolean functions with optimal algebraic immunity can be obtained. The functions in first class are bent, and then it can be concluded that the algebraic immunity of bent functions can take all possible values except one. The functions in the second class are balanced, and they have optimal algebraic degree and the best nonlinearity up to now.     

On the maximal component algebraic immunity of vectorial Boolean functions

Under study is the component algebraic immunity of vectorial Boolean functions. We prove a theorem on the correspondence between the maximal component algebraic immunity of a function and its balancedness. Some relationship is obtained between the maximal component algebraic immunity and matrices of a special form. We construct several functions with maximal component algebraic immunity in case of few variables.     

在仿射等价类中找具有好的密码学性质的布尔函数

The Boolean functions in an affine equivalence class are of the same algebraic degree and nonlinearity, but may satisfy different order of correlation immunity and propagation criterion. A method is presented in this paper to find Boolean functions with higher order correlation immunity or satisfying higher order propagation criterion in an affine equivalence class. 8 AES s-box functions are not better Boolean functions in their affine equivalence class.     

A systematic method of constructing Boolean functions with optimal algebraic immunity based on the generator matrix of the Reed–Muller code

Because of the recent algebraic attacks, optimal algebraic immunity is now an absolutely necessary (but not sufficient) property for Boolean functions used in stream ciphers. In this paper, we firstly determine the concrete coefficients in the linear expression of the column vectors with respect to a given basis of the generator matrix of Reed–Muller code, which is an important tool for constructing Boolean functions with optimal algebraic immunity. Secondly, as applications of the determined coefficients, we provide simpler and direct proofs for two known constructions. Further, we construct new Boolean functions on odd variables with optimal algebraic immunity based on the generator matrix of Reed–Muller code. Most notably, the new constructed functions possess the highest nonlinearity among all the constructions based on the generator matrix of Reed–Muller code, although which is not as good as the nonlinearity of Carlet–Feng function. Besides, the ability of the new constructed functions to resist fast algebraic attacks is also checked for the variable \(n=11,13\) and 15.     

Basic Theory in Construction of Boolean Functions with Maximum Possible Annihilator Immunity

So far there is no systematic attempt to construct Boolean functions with maximum annihilator immunity. In this paper we present a construction keeping in mind the basic theory of annihilator immunity. This construction provides functions with the maximum possible annihilator immunity and the weight, nonlinearity and algebraic degree of the functions can be properly calculated under certain cases. The basic construction is that of symmetric Boolean functions and applying linear transformation on the input variables of these functions, one can get a large class of non-symmetric functions too. Moreover, we also study several other modifications on the basic symmetric functions to identify interesting non-symmetric functions with maximum annihilator immunity. In the process we also present an algorithm to compute the Walsh spectra of a symmetric Boolean function with O(n²) time and O(n) space complexity. We use the term “Annihilator Immunity” instead of “Algebraic Immunity” referred in the recent papers [3–5, 9, 18, 19]. Please see Remark 1 for the details of this notational change     

Construction of balanced Boolean functions with high nonlinearity,good local and global avalanche characteristics

Boolean functions possessing multiple cryptographic criteria play an important role in the design of symmetric cryptosystems. The following criteria for cryptographic Boolean functions are often considered: high nonlinearity, balancedness, strict avalanche criterion, and global avalanche characteristics. The trade-off among these criteria is a difficult problem and has attracted many researchers. In this paper, two construction methods are provided to obtain balanced Boolean functions with high nonlinearity. Besides, the constructed functions satisfy strict avalanche criterion and have good global avalanche characteristics property. The algebraic immunity of the constructed functions is also considered.