代数免疫度为1的布尔函数

布尔函数的代数免疫度是在流密码的代数攻击中所产生的重要概念.研究了代数免疫度为1的布尔函数,得到的主要结果有:对代数免疫度为1的布尔函数给出了一个谱刻画,给出了其个数的精确计数公式,最后给出了此类函数的非线性度的紧的上界.     

Correlation-Immune and Resilient Functions Over a Finite Alphabet and Their Applications in Cryptography

We extend the notions of correlation-immune functions and resilient functions to functions over any finite alphabet. A previous result due to Gopalakrishnan and Stinson is generalized as we give an orthogonal array characterization, a Fourier transform and a matrix characterization for correlation-immune and resilient functions over any finite alphabet endowed with the structure of an Abelian group. We then point out the existence of a tradeoff between the degree of the algebraic normal form and the correlation-immunity order of any function defined on a finite field and we construct some infinite families of t-resilient functions with optimal nonlinearity which are particularly well-suited for combining linear feedback shift registers. We also point out the link between correlation-immune functions and some cryptographic objects as perfect local randomizers and multipermutations.     

Characterization of Linear Structures

We study the notionof linear structure of a function defined from F ^mto F ⁿ, and in particular of a Boolean function.We characterize the existence of linear structures by means ofthe Fourier transform of the function. For Boolean functions,this characterization can be stated in a simpler way. Finally,we give some constructions of resilient Boolean functions whichhave no linear structure.     

On Binary Resilient Functions

Using a lifting formula for the coefficients of Boolean functions, we characterize binary resilient functions as binary matrices with certain row or column intersection properties. We give some new constructions of binary resilient functions based on this characterization. In particular, we show that the incidence matrix of a Steiner system can be used to construct binary resilient functions.     

Laced Boolean functions and subset sum problems in finite fields

In this paper, we investigate some algebraic and combinatorial properties of a special Boolean function on n variables, defined using weighted sums in the residue ring modulo the least prime p≥n. We also give further evidence relating to a question raised by Shparlinski regarding this function, by computing accurately the Boolean sensitivity, thus settling the question for prime number values p=n. Finally, we propose a generalization of these functions, which we call laced functions, and compute the weight of one such, for every value of n.     

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.     

Vectorial Resilient PC（l） of Order k Boolean Functions from AG-Codes

Propagation criteria and resiliency of vectorial Boolean functions are important for cryptographic purpose (see [1–4, 7, 8, 10, 11, 16]). Kurosawa, Stoh [8] and Carlet [1] gave a construction of Boolean functions satisfying PC(l) of order k from binary linear or nonlinear codes. In this paper, the algebraic-geometric codes over GF(2m) are used to modify the Carlet and Kurosawa-Satoh’s construction for giving vectorial resilient Boolean functions satisfying PC(l) of order k criterion. This new construction is compared with previously known results.     

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.     

Linearization of nonlinear filter generators and its application to cryptanalysis of stream ciphers

Nonlinear filter generators are commonly used as keystream generators in stream ciphers. A nonlinear filter generator utilizes a nonlinear filtering function to combine the outputs of a linear feedback shift register (LFSR) to improve the linear complexity of keystream sequences. However, the LFSR-based stream ciphers are still potentially vulnerable to algebraic attacks that recover the key from some keystream bits. Although the known algebraic attacks only require polynomial time complexity of computations, all have their own constraints. This paper uses the linearization of nonlinear filter generators to cryptanalyze LFSR-based stream ciphers. Such a method works for any nonlinear filter generators. Viewing a nonlinear filter generator as a Boolean network that evolves as an automaton through Boolean functions, we first give its linearization representation. Compared to the linearization representation in Limniotis et al. (2008), this representation requires lower spatial complexity of computations in most cases. Based on the representation, the key recoverability is analyzed via the observability of Boolean networks. An algorithm for key recovery is given as well. Compared to the exhaustive search to recover the key, using this linearization representation requires lower time complexity of computations, though it leads to exponential time complexity.     

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     

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.     

Boolean Functions: Degree and Support

In this paper we establish some properties about Boolean functions that allow us to relate their degree and their support. These properties allow us to compute the degree of a Boolean function without having to calculate its algebraic normal form. Furthermore, we introduce some linear algebra properties that allow us to obtain the degree of a Boolean function from the dimension of a linear or affine subspace. Finally we derive some algorithms and compute the average time to obtain the degree of some Boolean functions from its support.     

Exact relations between nonlinearity and algebraic immunity

We sharpen some lower bounds on the higher order nonlinearity of a Boolean function in terms of the value of its algebraic immunity and obtain new tight bounds. We prove a universal tight lower bound, which enables us to reduce the problem of estimating higher order nonlinearity to finding the dimension of certain linear subspaces in the space of Boolean functions. As a simple corollary of this result, we obtain all previously known estimates in this area. For polynomials with disjoint terms, finding the dimension of those linear subspaces reduces to a simple combinatorial inspection. We prove a tight lower bound on the second order nonlinearity of a Boolean function in terms of the value of its algebraic immunity.     

Algebraic Functions,Calculus Style

We investigate the structure and properties of the explicit algebraic functions described in calculus texts and their relation to algebraic functions. A structure theorem enables us to construct a large number of examples. An example of an algebraic function that is not explicitly algebraic is studied, and an abstract algebraic context is provided for the theory.     

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.     

On Some Special Classes of Sequential Dynamical Systems

Sequential Dynamical Systems (SDSs) are mathematical models for analyzing simulation systems. We investigate phase space properties of some special classes of SDSs obtained by restricting the local transition functions used at the nodes. We show that any SDS over the Boolean domain with symmetric Boolean local transition functions can be efficiently simulated by another SDS which uses only simple threshold and simple inverted threshold functions, where the same threshold value is used at each node and the underlying graph is d-regular for some integer d. We establish tight or nearly tight upper and lower bounds on the number of steps needed for SDSs over the Boolean domain with 1-, 2- or 3-threshold functions at each of the nodes to reach a fixed point. When the domain is a unitary semiring and each node computes a linear combination of its inputs, we present a polynomial time algorithm to determine whether such an SDS reaches a fixed point. We also show (through an explicit construction) that there are Boolean SDSs with the NOR function at each node such that their phase spaces contain directed cycles whose length is exponential in the number of nodes of the underlying graph of the SDS.AMS Subject Classification: 68Q10, 68Q17, 68Q80.     

Games played by Boole and Galois

We define an infinite class of 2-pile subtraction games, where the amount that can be subtracted from both piles simultaneously is an extended Boolean function f of the size of the piles, or a function over GF(2). Wythoff's game is a special case. For each game, the second player winning positions are a pair of complementary sequences. Sample games are presented, strategy complexity questions are discussed, and possible further studies are indicated. The motivation stems from the major contributions of Professor Peter Hammer to the theory and applications of Boolean functions.     

On the arithmetic Walsh coefficients of Boolean functions

We generalize to the arithmetic Walsh transform (AWT) some results which were previously known for the Walsh–Hadamard transform of Boolean functions. We first generalize the classical Poisson summation formula to the AWT. We then define a generalized notion of resilience with respect to an arbitrary statistical measure of Boolean functions. We apply the Poisson summation formula to obtain a condition equivalent to resilience for one such statistical measure. Last, we show that the AWT of a large class of Boolean functions can be expressed in terms of the AWT of a Boolean function of algebraic degree at most three in a larger number of variables.     

On second-order nonlinearity and maximum algebraic immunity of some bent functions in \mathcal{PS}^{+}

The rth-order nonlinearity and algebraic immunity of Boolean function play a central role against several known attacks on stream and block ciphers. Since its maximum equals the covering radius of the rth-order Reed-Muller code, it also plays an important role in coding theory. The computation of exact value or high lower bound on the rth-order nonlinearity of a Boolean function is very complected/challenging problem, especially when r>1. In this article, we identify a subclass of \({\mathcal{D}}_{0}\) type bent functions constructed by modifying well known Dillon functions having sharper bound on their second-order nonlinearity. We further, identify a subclass of bent functions in \({\mathcal {PS}}^{+}\) class with maximum possible algebraic immunity. The result is proved by using the well known conjecture proposed by Tu and Deng (Des. Codes Cryptogr. 60(1):1–14, 2011). To obtain rth-order nonlinearity (r>2), that is, whole nonlinearity profile of the constructed bent functions is still an open problem.     

A novel algorithm enumerating bent functions

Based on the relationship between the Walsh spectra of a Boolean function at partial points and the Walsh spectra of its subfunctions, and on the binary Möbius transform, a novel algorithm is developed, which can theoretically construct all bent functions. Practically we enumerate all bent functions in 6 variables. With the restriction on the algebraic normal form, the algorithm is also efficient in more variables case. For example, enumeration of all homogeneous bent functions of degree 3 in 8 variables can be done in one minute with a P4 1.7 GHz computer; the nonexistence of homogeneous bent functions in 10 variables of degree 4 is computationally proved.