Balanced Boolean Functions Research Articles

Constructing Highly Nonlinear Cryptographic Balanced Boolean Functions on Learning Capabilities of Recurrent Neural Networks

Constructing Highly Nonlinear Cryptographic Balanced Boolean Functions on Learning Capabilities of Recurrent Neural Networks

A unified construction of weightwise perfectly balanced Boolean functions

At Eurocrypt 2016, Méaux et al. presented FLIP, a new family of stream ciphers that aimed to enhance the efficiency of homomorphic encryption frameworks. Motivated by FLIP, recent research has focused on the study of Boolean functions with good cryptographic properties when restricted to subsets of the space F2n. If an n-variable Boolean function has the property of balancedness when restricted to each set of vectors with fixed Hamming weight between 1 and n−1, it is a weightwise perfectly balanced (WPB) Boolean function. In the literature, a few algebraic constructions of WPB functions are known, in which there are some constructions that use iterative method based on functions with low degrees of 1, 2, or 4. In this paper, we generalize the iterative method and contribute a unified construction of WPB functions based on functions with algebraic degrees that can be any power of 2. For any given positive integer d not larger than m, we first provide a class of 2m-variable Boolean functions with a degree of 2d−1. Utilizing these functions, we then present a construction of 2m-variable WPB functions gm;d. In particular, gm;d includes four former classes of WPB functions as special cases when d=1,2,3,m. When d takes other integer values, gm;d has never appeared before. In addition, we prove the algebraic degree of the constructed WPB functions and compare the weightwise nonlinearity of WPB functions known so far in 8 and 16 variables.

Genetic Approach to Improve Cryptographic Properties of Balanced Boolean Functions Using Bent Functions

Recently, balanced Boolean functions with an even number n of variables achieving very good autocorrelation properties have been obtained for 12≤n≤26. These functions attain the maximum absolute value in the autocorrelation spectra (without considering the zero point) less than 2n2 and are found by using a heuristic search algorithm that is based on the design method of an infinite class of such functions for a higher number of variables. Here, we consider balanced Boolean functions that are closest to the bent functions in terms of the Hamming distance and perform a genetic algorithm efficiently aiming to optimize their cryptographic properties, which provides better absolute indicator values for all of those values of n for the first time. We also observe that among our results, the functions for 16≤n≤26 have nonlinearity greater than 2n−1−2n2. In the process, our search strategy produces balanced Boolean functions with the best-known nonlinearity for 8≤n≤16.

New Results on Boolean Functions with High Nonlinearity and Low Transparency Order

We improve the steepest descent algorithm and increase the double threshold parameter, which significantly improves the algorithm’s efficiency. And we design a new cost function so that in terms of search, various characteristics of Boolean functions can be taken into account simultaneously. Applying our algorithm, there are excellent results regarding the 9, 10, 11, and 12 variables. We find a Boolean function with a nonlinearity of 242 in 9 variables and the whole search space. Previously, this result only appeared in the rotational symmetry class. The best-achieved nonlinearity result for permutation (6, 5, 1, 4, 7, 2, 3, 0, 8) and (0, 7, 2, 5, 4, 1, 3, 6, 8) class is 238 and 239 introduced by Kavut in Information and Computation (2010). Still, applying our algorithm, we obtain a balanced Boolean function with a nonlinearity of 240 under the same permutation, indicating that our method is more general. Among the 11 variables, a Boolean function with a higher nonlinearity and a lower transparency level and the absolute value spectrum are maintained at a lower level. The algorithm performs well when considering all aspects of the property. There are similarly promising results in even-numbered variables.

A New Construction of Weightwise Perfectly Balanced Functions with High Weightwise Nonlinearity

The FLIP cipher was proposed at Eurocrypt 2016 for the purpose of meliorating the efficiency of fully homomorphic cryptosystems. Weightwise perfectly balanced Boolean functions meet the balancedness requirement of the filter function in FLIP ciphers, and the construction of them has attracted serious attention from researchers. Nevertheless, the literature is still thin. Modifying the supports of functions with a low degree is a general construction technique whose key problem is to find a class of available low-degree functions. We first seek out a class of quadratic functions and then, based on these functions, present the new construction of weightwise perfectly balanced Boolean functions by adopting an iterative approach. It is worth mentioning that the functions we construct have good performance in weightwise nonlinearity. In particular, some p-weight nonlinearities achieve the highest values in the literature for a small number of variables.

A new construction of weightwise perfectly balanced Boolean functions

In this paper, we first introduce a class of quartic Boolean functions. And then, the construction of weightwise perfectly balanced Boolean functions on $ 2^m $ variables are given by modifying the support of the quartic functions, where $ m $ is a positive integer. The algebraic degree, the weightwise nonlinearity, and the algebraic immunity of the newly constructed weightwise perfectly balanced functions are discussed at the end of this paper.

A systematic method of constructing weightwise almost perfectly balanced Boolean functions on an arbitrary number of variables

Boolean functions satisfying good cryptographic criteria when restricted to the set of vectors with constant Hamming weight play an important role in the well-known FLIP stream cipher. In this paper, we present a systematic method of constructing weightwise almost perfectly balanced Boolean functions on an arbitrary number of variables, which equal the direct sum of several known weightwise perfectly balanced Boolean functions. At the same time, we show two concrete constructions of weightwise almost perfectly balanced Boolean functions, whose k-weight nonlinearities and algebraic immunities are discussed at the end of this paper.

On Molecular Bonding Logic and Matrix Representation of Constant and Balanced Boolean Functions

Representing a bonding manifold of a molecule or molecular cluster by a graph given by a set of vertices associated with atoms and a set of edges imitating bonds, the bonding edge encoding formalism is defined on n-tuples qubits in terms of the NOT logic gate acting on the "non-bonded" string. This formalism is illustrated by the simplest diatomic and triatomic molecules whose adjacency matrices generate different quadratic Boolean functions, among which the balanced function appears. In this regard, we review the Deutsch–Jozsa quantum algorithm, well-known in quantum computing, that discriminates between the balanced and constant Boolean functions. A novel matrix representation of the constant-balancedquantum oracle within this algorithm is elaborated. The proposed approach is generalized to distinguish between constant and evenly balanced Boolean functions.

Constructions of balanced Boolean functions on even number of variables with maximum absolute value in autocorrelation spectra [formula omitted

The autocorrelation properties of Boolean functions are closely related to the Shannon’s concept of diffusion and can be accompanied with other cryptographic criteria (such as high nonlinearity and algebraic degree) for ensuring an overall robustness to various cryptanalytic methods. In a series of recent articles [14,9,15], the design methods of n-variable balanced Boolean functions n is strictly even) with small absolute indicator Δf < 2n/2 have been considered. Whereas the two first articles managed to solve this problem for relatively large n⩾46, a recent approach [15] has introduced a generic design framework achieving Δf < 2n/2 for even n⩾22. Based on a suitable modification of the method of Rothaus, used to construct new bent functions from known ones, we provide a generic iterative framework for designing balanced functions satisfying the condition Δf < 2n/2 and having overall good cryptographic properties for any even n⩾12. Even though the problem of specifying functions having Δf < 2n/2 for smaller n has been considered in [14,9,15] using various search algorithms, our method for the first time provides relatively simple iterative framework for variable spaces of more practical interest. Moreover, our approach can be efficiently applied to certain classes of initial functions (derived from partial spread bent functions) for deriving balanced functions with Δf < 2n/2 for relatively large n, namely for n⩾48 satisfying n≡0 mod 4 and n⩾54 with n≡2 mod 4. In the latter case, our nonlinearity bound is better than the one presented in [14].

On constructions of weightwise perfectly balanced Boolean functions

The recent FLIP cipher is an encryption scheme described by Meaux et al. at the conference EUROCRYPT 2016. It is based on a new stream cipher model called the filter permutator and tries to minimize some parameters (including the multiplicative depth). In the filter permutator, the input to the Boolean function has constant Hamming weight equal to the weight of the secret key. As a consequence, Boolean functions satisfying good cryptographic criteria when restricted to the set of vectors with constant Hamming weight play an important role in the FLIP stream cipher. Carlet et al. have shown that for Boolean functions with restricted input, balancedness and nonlinearity parameters continue to play an important role with respect to the corresponding attacks on the framework of FLIP ciphers. In particular, Boolean functions which are uniformly distributed over ${\mathbb {F}}_{2}$ on $E_{n,k}=\{x{\in \mathbb {F}_{2}^{n}}\mid \text {wt}(x)=k\}$ for every 0 < k < n are called weightwise perfectly balanced (WPB) functions, where wt(x) denotes the Hamming weight of x. In this paper, we firstly propose two methods of constructing weightwise perfectly balanced Boolean functions in 2k variables (where k is a positive integer) by modifying the support of linear and quadratic functions. Furthermore, we derive a construction of n-variable weightwise almost perfectly balanced Boolean functions for any positive integer n.

The lower bound of the weightwise nonlinearity profile of a class of weightwise perfectly balanced functions

Boolean functions satisfying good cryptographic criteria when restricted to the set of vectors with constant Hamming weight play an important role in the recent FLIP stream cipher. Although the nonlinearity of Boolean functions can be calculated rapidly by Walsh transform, the nonlinearity of the Boolean functions with restricted input seems to be very difficult to calculate. In this paper, a class of weightwise perfectly balanced Boolean functions with very simple algebraic normal form is proposed. Then, the cryptographic properties of the weightwise perfectly balanced Boolean functions are discussed. Especially, the lower bound of the weightwise nonlinearity of these 2m-variable functions is given for any positive integer m.

On the transparency order relationships between one Boolean function and its decomposition functions

The concept of transparency order (denoted by TO) is an important criterion of (n,m)-functions to resist against differential power analysis (DPA). In this paper, we give several transparency order relationships of some Boolean functions. We give the lower bound on the transparency order for Boolean function, and obtain the transparency order relationship between one Boolean function and its decomposition Boolean functions. Furthermore, we deduce one relationship among TO(f⊕g), TO(f), TO(g) and TO(fg) for any n-variable Boolean functions f,g. Finally, we study the transparency order for the sum function and for the product function between two variable-disjoint Boolean functions, and calculate the transparency order distributions of 4-variable and 5-variable balanced Boolean functions, respectively.

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.

Non-classical computing problems: Toward novel type of quantum computing problems

Quantum teleportation draws our attention to propose a new type of problems which can not be solved using classical computers. In this paper, we propose one of these problems. Concretely, this paper extends the definition of Deutsch’s problem to decide whether a black box Uf applied on a given unknown qubit α|0〉+β|1〉, such that |α|>0,|β|>0,and|α|≠|β|, is constant or balanced Boolean function, besides, estimation of |α| and |β|. Although, this problem is very simple but it can not be solved using classical computers, because qubit can not be implemented physically using classical computers. A novel quantum algorithm based on principle of entanglement measure is proposed to solve this problem. IBMs 5-qubit quantum computer (ibmqx4) is used to realize the proposed algorithm experimentally.

Оценка нелинейности сбалансированных булевых функций, порождённых обобщённой конструкцией Доббертина

Оценка нелинейности сбалансированных булевых функций, порождённых обобщённой конструкцией Доббертина

Intrinsic Resiliency of S-Boxes Against Side-Channel Attacks–Best and Worst Scenarios

Constructing S-boxes that are inherently resistant against side-channel attacks is an important problem in cryptography. By using an optimal distinguisher under an additive Gaussian noise assumption, we clarify how a defender (resp., an attacker) can make side-channel attacks as difficult (resp., easy) as possible, in relation with the auto-correlation spectrum of Boolean functions. We then construct balanced Boolean functions that are optimal for each of these two scenarios. Generalizing the objectives for an S-box, we analyze the auto-correlation spectra of some well-known S-box constructions in dimensions at most 8 and compare their intrinsic resiliency against side-channel attacks. Finally, we perform several simulations of side-channel attacks against the aforementioned constructions, which confirm our theoretical approach.

New Constructions of Balanced Boolean Functions with Maximum Algebraic Immunity, High Nonlinearity and Optimal Algebraic Degree

This paper consists of proposal of two new constructions of balanced Boolean function achieving a new lower bound of nonlinearity along with high algebraic degree and optimal or highest algebraic immunity. This construction has been made by using representation of Boolean function with primitive elements. Galois Field, used in this representation has been constructed by using powers of primitive element such that greatest common divisor of power and is 1. The constructed balanced variable Boolean functions achieve higher nonlinearity, algebraic degree of , and algebraic immunity of for odd , for even . The nonlinearity of Boolean function obtained in the proposed constructions is better as compared to existing Boolean functions available in the literature without adversely affecting other properties such as balancedness, algebraic degree and algebraic immunity.

A construction of highly nonlinear Boolean functions with optimal algebraic immunity and low hardware implementation cost

In this paper we put forward a method to construct balanced Boolean functions by modifying the Tang–Carlet–Tang functions with the help of functions in Generalized Maiorana–McFarland class. Essential cryptographic properties, for instance, nonlinearity, algebraic immunity and immunity to fast algebraic attacks are considered. A new lower bound on the nonlinearity of the functions that are contained in our construction is obtained. It is slightly better than the known result. Owing to a combinatorial fact, it can be demonstrated that the proposed functions have optimal algebraic immunity. Meanwhile, they also possess high nonlinearity and a moderate behavior against fast algebraic attacks. Apart from theoretical contributions, hardware implementation efficiency on the proposed functions is also analyzed. Since the participation of GMM class component, experimental results imply that functions constructed in this paper have a small implementation size and employ fewer logic gates compared with the construction containing an iterative structure.

On the linear structures of balanced functions and quadratic APN functions

The set of linear structures of most known balanced Boolean functions is non-trivial. In this paper, some balanced Boolean functions whose set of linear structures is trivial are constructed. We show that any APN function in even dimension must have a component whose set of linear structures is trivial. We determine a general form for the number of bent components in quadratic APN functions in even dimension and some bounds on the number are produced. We also count bent components in any quadratic power functions.

Construction of weightwise perfectly balanced Boolean functions with high weightwise nonlinearity

In this paper, we firstly introduce a kind of weightwise almost perfectly balanced Boolean functions. And then, a construction of weightwise perfectly balanced Boolean functions on 2q+2 variables is given by modifying the support of the weightwise almost perfectly balanced functions, where q is a non-negative integer. The algebraic degree, the weightwise nonlinearity, and the algebraic immunity of the newly constructed weightwise perfectly balanced functions are discussed at the end of this paper.