Construction Of Boolean Functions Research Articles

Constructions of Boolean Functions with Five-Valued Walsh Spectra and Their Applications

Constructions of Boolean Functions with Five-Valued Walsh Spectra and Their Applications

Toward more efficient heuristic construction of Boolean functions

Boolean functions have numerous applications in domains as diverse as coding theory, cryptography, and telecommunications. Heuristics play an important role in the construction of Boolean functions with the desired properties for a specific purpose. However, there are only sparse results trying to understand the problem’s difficulty. With this work, we aim to address this issue. We conduct a fitness landscape analysis based on Local Optima Networks (LONs) and investigate the influence of different optimization criteria and variation operators. We observe that the naive fitness formulation results in the largest networks of local optima with disconnected components. Also, the combination of variation operators can both increase or decrease the network size. Most importantly, we observe correlations of local optima’s fitness, their degrees of interconnection, and the sizes of the respective basins of attraction. This can be exploited to restart algorithms dynamically and influence the degree of perturbation of the current best solution when restarting.

An improved hybrid genetic algorithm to construct balanced Boolean function with optimal cryptographic properties

Boolean functions are used as nonlinear filter functions and combiner functions in several stream ciphers. The security of these stream ciphers largely depends upon cryptographic properties of Boolean functions. Finding a balanced Boolean function with optimal cryptographic properties is an open research problem in the cryptographic community. Since the number of n-variable Boolean functions is $$2^{2^n}$$ , it is not computationally feasible to search the entire space of such functions for cryptographically significant functions when $$n \ge 6$$ . In general, the construction of Boolean functions with optimal or near optimal cryptographically significant properties is formulated as combinatorial optimization problems. In this paper, we apply the Genetic algorithm with the integration of a local search procedure called hybrid GA (HGA) searching for the Boolean functions with high nonlinearity and low autocorrelation. The performance of the Genetic algorithm depends upon the tuning parameters such as crossover mechanism, mutation probability, selection criteria, and the choice of fitness/cost function. To achieve an optimal trade-off among two cryptographic properties, the choice of the cost function plays an important role for finding an optimal solution. So our main focus is to construct balanced Boolean functions using HGA with a new cost function and compare it with existing cost functions. Our experimental results shows that the HGA is more efficient than GA and the new cost function is more efficient than existing cost functions for reaching the optimal solutions.

Recent Results on Constructing Boolean Functions with (Potentially) Optimal Algebraic Immunity Based on Decompositions of Finite Fields

Boolean functions with optimal algebraic immunity (OAI functions) are important cryptographic primitives in the design of stream ciphers. During the past decade, a lot of work has been done on constructing such functions, among which mathematics, especially finite fields, play an important role. Notably, the approach based on decompositions of additive or multiplicative groups of finite fields turns out to be a very successful one in constructing OAI functions, where some original ideas are contributed by Tu and Deng (2012), Tang, et al. (2017), and Lou, et al. (2015). Motivated by their pioneering work, the authors and their collaborators have done a series of work, obtaining some more general constructions of OAI functions based on decompositions of finite fields. In this survey article, the authors review our work in this field in the past few years, illustrating the ideas for the step-by-step generalizations of previous constructions and recalling several new observations on a combinatorial conjecture on binary strings known as the Tu-Deng conjecture. In fact, the authors have obtained some variants or more general forms of Tu-Deng conjecture, and the optimal algebraic immunity of certain classes of functions we constructed is based on these conjectures.

Constructing Boolean Functions Using Blended Representations

In this paper, we study blended representations of Boolean functions, and construct the following two classes of Boolean functions. Two bounds on the r-order nonlinearity were given by Carlet in the IEEE TRANSACTIONS ON INFORMATION THEORY, vol. 54. In general, the second bound is better than the first bound. But it was unknown whether it is always better. Recently, Mesnager et al. constructed a class of Boolean functions where the second bound is strictly worse than the first bound, for r = 2. However, it is still an open problem for r ≥ 3. Using the blended representation, we construct a class of Boolean functions based on the trace function and show that the second bound can also be strictly worse than the first bound, for r = 3. The second class is based on the hidden weighted bit function, which seems to have the best cryptographic properties among all currently known functions.

Optimal annihilator immunity with suboptimal bound of other properties

In this paper we present constructions of Boolean functions based on the basic theory of annihilator immunity and bent functions. These constructions provide functions with the maximum possible annihilator immunity, and the nonlinearity and resiliency of the functions can also be calculated with specific cases. Here we have also described certain properties of Boolean functions that are necessary for resistance against algebraic and fast algebraic attacks. Annihilator immunity, nonlinearity and resiliency are important properties of a Boolean function to resist these attacks. There was no systematic attempt made to construct Boolean functions with maximum annihilator immunity that have good bound of nonlinearity and resiliency. We have constructed two different classes of desired Boolean functions through our newly developed method based on multiobjective optimization method(NSGA-II). In the first class, we have constructed bent functions(optimal nonlinearity) with sub-optimal bound of annihilator immunity while in other, Boolean functions having optimal bound of annihilator immunity with good bound of nonlinearity were developed. Our problem was multiobjective(nonlinearity, annihilator immunity and resiliency), hence a heuristic multiobjective optimization technique(NSGA-II) was used.

On Boolean functions with several flat spectra

In this paper, two constructions of Boolean functions which have at least two flat spectra with respect to {H, N}n are proposed. Some known results about bent-negabent functions can be seen as special cases of our results. Furthermore, some lower bounds on the numbers of flat spectra of Boolean functions with respect to {H, N}n or {I, N}n are given. In particular, we show that any Maiorana-McFarland bent function of n (n even) variables has at least $\frac {n}{2}+ 2^{\frac {n}{2}} $ flat spectra with respect to {H, N}n. Finally, following the work by Riera, Petrides, and Parker, we develop recursive formulae for the numbers of flat spectra of some structural quadratics, including star function, star-line function, and star-line-star function.

A New Method on Constructing Boolean Functions Satisfying the Strict Avalanche Criterion and Bounds on the Number of SAC Functions

Properties of Boolean functions satisfying the Strict Avalanche Criterion (SAC) are studied. A new method of constructing SAC functions is proposed, and it is proved that the Hamming weight of SAC functions with n variables belongs to the set W(n)={k|2n-2 3 2 2 − £ £ × n k , k is an even number}. Finally, the upper and lower bounds of the number of SAC functions are obtained.

Results on highly nonlinear Boolean functions with provably good immunity to fast algebraic attacks

In the last decade, algebraic and fast algebraic attacks are regarded as the most successful attacks on LFSR-based stream ciphers. Since the notion of algebraic immunity was introduced, the properties and constructions of Boolean functions with maximum algebraic immunity have been researched in a large number of papers. However, there are few results with respect to Boolean functions with provably good immunity against fast algebraic attacks. In previous literatures, only Carlet–Feng function was proven to have good immunity to fast algebraic attacks.In this paper, we first study a large family of highly nonlinear Boolean functions in terms of the immunity to fast algebraic attacks, which includes the functions of Tu–Deng, the functions of Tang et al. and the functions of Jin et al. Based on a sufficient and necessary condition for measuring the immunity of Boolean functions against fast algebraic attacks using bivariate polynomial representation, we propose an efficient method for estimating the immunity of the functions of such family. Then we prove that a family of 2k-variable Boolean functions, including the function recently constructed by Tang et al., are almost perfect algebraic immune for any integer k ≥ 3. More exactly, they achieve optimal algebraic immunity and almost perfect immunity to fast algebraic attacks. The functions of such family are balanced and have optimal algebraic degree. Besides, we prove a lower bound on their nonlinearity based on the work of Tang et al. which is better than that of Carlet–Feng function. It is also checked for 3 ≤ k ≤ 9 that the exact nonlinearity of such functions is very good, which is slightly smaller than that of Carlet–Feng function, and some functions of this family even have a slightly larger nonlinearity than Tang’s et al. function. To sum up, among the known functions with provably good immunity against fast algebraic attacks, the functions of this family make a trade-off between the exact value and the lower bound of nonlinearity.

Fast algebraic immunity of Boolean functions

Since 1970, Boolean functions have been the focus of a lot of attention in cryptography. An important topic in symmetric ciphers concerns the cryptographic properties of Boolean functions and constructions of Boolean functions with good cryptographic properties, that is, good resistance to known attacks. An important progress in cryptanalysis areas made in 2003 was the introduction by Courtois and Meier of algebraic attacks and fast algebraic attacks which are very powerful analysis concepts and can be applied to almost all cryptographic algorithms. To study the resistance against algebraic attacks, the notion of algebraic immunity has been introduced. In this paper, we use a parameter introduced by Liu and al., called fast algebraic immunity, as a tool to measure the resistance of a cryptosystem (involving Boolean functions) to fast algebraic attacks. We prove an upper bound on the fast algebraic immunity. Using our upper bound, we establish the weakness of trace inverse functions against fast algebraic attacks confirming a recent result of Feng and Gong.

1-Resilient Boolean Functions on Even Variables with Almost Perfect Algebraic Immunity

Several factors (e.g., balancedness, good correlation immunity) are considered as important properties of Boolean functions for using in cryptographic primitives. A Boolean function is perfect algebraic immune if it is with perfect immunity against algebraic and fast algebraic attacks. There is an increasing interest in construction of Boolean function that is perfect algebraic immune combined with other characteristics, like resiliency. A resilient function is a balanced correlation-immune function. This paper uses bivariate representation of Boolean function and theory of finite field to construct a generalized and new class of Boolean functions on even variables by extending the Carlet-Feng functions. We show that the functions generated by this construction support cryptographic properties of 1-resiliency and (sub)optimal algebraic immunity and further propose the sufficient condition of achieving optimal algebraic immunity. Compared experimentally with Carlet-Feng functions and the functions constructed by the method of first-order concatenation existing in the literature on even (from 6 to 16) variables, these functions have better immunity against fast algebraic attacks. Implementation results also show that they are almost perfect algebraic immune functions.

A Proof of Constructions for Balanced Boolean Function with Optimum Algebraic Immunity

Algebraic immunity is a cryptographic criterion for Boolean functions used in cryptosystem to resist algebraic attacks. They usually should have high algebraic immunity. Chen proposed a first order recursive construction of Boolean functions and checked that they had optimum algebraic immunity for n 0.

Construction of Boolean functions with excellent cryptographic criteria using bivariate polynomial representation

A class of -variable Boolean functions with excellent cryptographic criteria is proposed in this paper, using bivariate polynomial representation (BPR). By comparing known Boolean functions created by the ‘BPR-method’, three conjectures on the relationship between cryptographic criteria and parameter settings are given as guidelines to the research. Then on the basis of certain combinatorial facts and computer experiments, we prove that our functions possess the optimal algebraic immunity k, and validate that, at least for , the functions preserve almost perfect immunity against fast algebraic attacks. In addition, we show the functions to be 1-resilient with the maximum algebraic degree of and give a proof of the lower bound for nonlinearity by means of Gauss sum. Our functions demonstrate great performance in meeting the desired cryptographic criteria for use in the filter model of pseudorandom generators.

CONSTRUCTING 2m-VARIABLE BOOLEAN FUNCTIONSWITH OPTIMAL ALGEBRAIC IMMUNITY BASED ON POLAR DECOMPOSITION OF $\mathbb{F}^\ast_{2^{2m}}$

Constructing 2m-variable Boolean functions with optimal algebraic immunity based on decomposition of additive group of the finite field [Formula: see text] seems to be a promising approach since Tu and Deng's work. In this paper, we consider the same problem in a new way. Based on polar decomposition of the multiplicative group of [Formula: see text], we propose a new construction of Boolean functions with optimal algebraic immunity. By a slight modification of it, we obtain a class of balanced Boolean functions achieving optimal algebraic immunity, which also have optimal algebraic degree and high nonlinearity. Computer investigations imply that this class of functions also behaves well against fast algebraic attacks.

A construction of Boolean functions with good cryptographic properties

The two important qualities of a cipher are security and speed. Frequently, to satisfy the security of a Boolean function primitive, speed may be traded-off. In this paper, we present a general construction that addresses both qualities. The idea of our construction is to manipulate a cryptographically strong base function and one of its affine equivalent functions, using concatenation and negation. We achieve security from the inherent qualities of the base function, which are preserved (or increased), and obtain speed by the simple Boolean operations. We present two applications of the construction to demonstrate the flexibility and efficiency of the construction.

Enhanced Boolean functions suitable for the filter model of pseudo-random generator

The filter model of pseudo-random generator (in stream ciphers) is currently the only one for which are known infinite classes of Boolean functions allowing to resist all the main known attacks. The combiner model, which is another possible way of using Boolean functions, requires the same properties as the filter model does, plus one extra criterion the Boolean function must fulfil: high order resiliency. No construction of functions is known which ensures all criteria for the combiner model, even if resiliency is taken in a weakened form, while such constructions are known for the filter model. But nonlinear functions used in this model must be in the particular form $$x_n+f(x_1,\dots ,x_{n-1})$$xn+f(x1,?,xn-1) to allow resistance to the distinguishing attacks for any choice of the tapping sequence. Much work has been done to construct and study Boolean functions allowing resistance to the main known attacks (the Berlekamp---Massey and ROnjom---Helleseth attacks, fast correlation attacks, algebraic attacks and fast algebraic attacks) on stream ciphers using the filter model. None of the found functions has the desired form above. Of course, we can take a function in $$n-1$$n-1 variables and add the extra variable $$x_n$$xn in order to obtain the desired form, but the algebraic immunity of the resulting function can be either equal to that of the original function $$f$$f (and it cannot then be optimal if $$n$$n is odd) or larger by 1. An increasement by 1 considerably impacts the complexity of algebraic attacks. Moreover, taking the best known constructions of functions and adapting them to the desired form result on functions which no longer ensure the best possible algebraic degree. This represents a gap in the research for Boolean functions usable in the filter model. In this paper we study the behavior of the cryptographic characteristics of a function when it is modified into the desired form and we study constructions of functions ensuring an optimal or almost-optimal tradeoff between all the necessary features in this form.

Two constructions of balanced Boolean functions with optimal algebraic immunity, high nonlinearity and good behavior against fast algebraic attacks

In this paper, two constructions of Boolean functions with optimal algebraic immunity are proposed. They generalize previous ones respectively given by Rizomiliotis (IEEE Trans Inf Theory 56:4014---4024, 2010) and Zeng et al. (IEEE Trans Inf Theory 57:6310---6320, 2011) and some new functions with desired properties are obtained. The functions constructed in this paper can be balanced and have optimal algebraic degree. Further, a new lower bound on the nonlinearity of the proposed functions is established, and as a special case, it gives a new lower bound on the nonlinearity of the Carlet-Feng functions, which is slightly better than the best previously known ones. For $$n\le 19$$n≤19, the numerical results reveal that among the constructed functions in this paper, there always exist some functions with nonlinearity higher than or equal to that of the Carlet-Feng functions. These functions are also checked to have good behavior against fast algebraic attacks at least for small numbers of input variables.

Hybrid classes of balanced Boolean functions with good cryptographic properties

Cryptographically strong Boolean functions play an imperative role in the design of almost every modern symmetric cipher. In this context, the cryptographic properties of Boolean functions, such as non-linearity, algebraic degree, correlation immunity and propagation criteria, are critically considered in the process of designing these ciphers. More recently, with the emergence of algebraic and fast algebraic attacks, algebraic immunity has also been included as an integral property to be considered. As a result, several constructions of Boolean functions with high non-linearity, maximal algebraic degree and optimal algebraic immunity have been devised since then. This paper focuses on some of these constructions and presents two hybrid classes of Boolean functions. The functions constructed within these classes possess maximal algebraic degree for balanced functions, optimal algebraic immunity, high non-linearity and good resistance to algebraic and fast algebraic attacks. A hybrid class of 1-resilient functions has also been proposed that also possesses high algebraic degree, optimal algebraic immunity, high non-linearity and good resistance to algebraic and fast algebraic attacks.

Balanced Boolean functions with optimum algebraic degree, optimum algebraic immunity and very high nonlinearity

It is a difficult challenge to construct Boolean functions with good cryptographic properties. In this paper, we construct an infinite class of even-variable balanced functions with optimum algebraic degree, optimum algebraic immunity and very high nonlinearity (higher than all other known balanced functions with optimum algebraic immunity). For any balanced Boolean function with optimum algebraic immunity, it is still unknown what is the highest nonlinearity possible. We achieve a higher nonlinearity than previous methods which gives a new lower bound on the maximum possible nonlinearity of balanced Boolean functions with optimum algebraic immunity.

A new method to construct Boolean functions with good cryptographic properties

To resist fast correlation attacks, Boolean functions used in stream ciphers should have high nonlinearity. n-variable bent functions have the maximum nonlinearity. However, they are not balanced and their algebraic degrees are at most n2. Therefore, they cannot be used directly as filter functions. In this paper, we give a new method to construct cryptographically significant Boolean functions. As an example, based on bent functions, we construct an infinite class of functions with good cryptographic properties: balancedness, optimum algebraic degree, almost optimum algebraic immunity and an almost optimum nonlinearity (higher than all other infinite classes of balanced functions with high algebraic immunity).