Polynomial-Time Algorithms for Checking Some Properties of Boolean Functions Given by Polynomials

This special issue of the VLSI DESIGN: an International Journal of Custom-Chip Design, Simulation and Testing is devoted to spectral techniques, new types of decision trees and diagrams, their applications, efficient ways of their calculation and mutual relations between spectral techniques and classical decision diagrams. Manipulations and calculations with discrete functions are fundamental tasks in many areas of Computer Science and Engineering. Many problems in digital system design, simulation and testing can be expressed as a sequence of operations on discrete functions. The performance of Computer-Aided Design systems used in solving various problems in this area strongly depends on the efficiency of representations of discrete functions. Decision Diagrams (DDs) have proved very convenient data structure for discrete function representations, permitting manipulations and calculations with large functions efficiently in terms of time and space. In many applications, as for example those involving large matrices, conventional algorithms are significantly improved by using DDs. In logic design, such applications relate to basic problems of the design, verification, simulation and testing logical networks. Besides Boolean functions, some other types of discrete representations, such as multi-valued functions, cube sets, EXOR-based spectral and probabilistic formulas have to be dealt with. Hence the numbers of different decision trees and diagrams have been introduced for these various applications and representations. The spectral or polynomial representation of Boolean function is obtained by expressing it in terms of an orthogonal basis. The most popular basis is formed by discrete Walsh functions. Note that the spectral and the polynomial representation of a Boolean function are equivalent. The discrete orthogonal functions are referred to as the basis functions when the spectral or abstract harmonic terminology is used and as monomials when the polynomial terminology is used. Such polynomial representations have extensive applications in coding theory and cryptography as well as design of nonlinear filters. Spectral techniques based on Walsh, Haar, Arithmetic and Reed–Muller transforms have been used in digital logic design for more than 30 years. These techniques have been used for Boolean function classification, disjoint decomposition, identification of symmetries, parallel and serial linear decomposition, spectral translation synthesis (extraction of linear preand post-filters), multiplexer synthesis, threshold logic synthesis, state assignment and testing, and evaluation of logic complexity. The renewed interest in applications of spectral methods in design of VLSI digital circuits is caused by their excellent design for testability properties and in their efficiency in Boolean mapping problems. Furthermore, the practical applicability of these techniques was greatly improved with the development of efficient methods that allow calculating and operating on spectra of Boolean functions directly from reduced representations of such functions in the form of arrays of cubes or decision diagrams. Following from these recent developments, techniques have been presented for efficient spectral translation to decompose a circuit into a cascade of two sub circuits: linear block composed of EXOR gates fed by the primary input of the overall circuit. Such linear decomposition drastically simplifies the synthesis task. The choice of suitable linear transformation is based on a complexity measure assigned to each Boolean function that is heuristically related to the complexity of the final circuit implementation. To avoid the experimental costs of computing the Walsh transform, Binary Decision Diagrams (BDDs) based techniques are used. Recently the idea of linear transformations is applied to decision diagrams. By combining powerful spectral linear techniques with variable reordering techniques, it is possible to synthesize large Boolean functions with standard Computer-Aided Design tools that fail otherwise. The areas addressed by this special issue span spectral transforms, their applications and efficient calculation methods through decision diagrams. New spectral decision diagrams are also emerging that support larger data structures allowing more complex discrete representations to be implemented that was previously impossible.

Boolean function is one of the most fundamental computation models in theoretical computer science. The two most common representations of Boolean functions are Fourier transform and real polynomial form. Applying analytic tools under these representations to the study Boolean functions has led to fruitful research in many areas such as complexity theory, learning theory, inapproximability, pseudorandomness, metric embedding, property testing, threshold phenomena, social choice, etc. In this thesis, we focus on \emph{sparse representations} of Boolean function in both Fourier transform and polynomial form, and obtain the following new results. A classical result of Rothschild and van Lint asserts that if every non-zero Fourier coefficient of a Boolean function $f$ over $\mathbb{F}_2^{n}$ has the same absolute value, namely $|\hat{f}(\alpha)|=1/2^k$ for every $\alpha$ in the Fourier support of $f$, then $f$ must be the indicator function of some affine subspace of dimension $n-k$. Here we slightly generalize their result, and show that Boolean functions whose Fourier coefficients take values in the set $\{-2/2^k, -1/2^k, 0, 1/2^k, 2/2^k\}$ are indicator functions of two disjoint affine subspaces of dimension $n-k$ or four disjoint affine subspaces of dimension $n-k-1$. Our main technical tools are results from additive combinatorics which offer tight bounds on the affine span size of a subset of $\mathbb{F}_2^{n}$ when the doubling constant of the subset is small. For polynomial representation of Boolean function, we study the distribution of the number of non-zero coefficients of \emph{random} Boolean functions. For a random Boolean function $f:\{0,1\}^n\rightarrow \{0,1\}$, i.e., a function whose value at each point on the Boolean cube is chosen independently and uniformly at random from $\{0,1\}$, in real polynomial representation, we give several bounds and concentration results about the distribution of the sparsity of $f$.

Boolean functions with a variety of secure cipher properties are the key factors to design cryptosystem with the ability to resist multiple cipher attacks and good safety performance. In this paper, using the derivative of the Boolean functions and the e-derivative defined by ourselves as the main research tools, we study algebraic immunity, correlation immunity and other cryptographic properties of the quadratic rotation symmetric Boolean functions. We determine the quadratic rotation symmetric Boolean functions which are H Boolean functions, and the range of weight distribution of the quadratic rotation symmetry H Boolean functions. Besides, we get the compatibility among propagation, balance, correlation immunity and algebraic immunity of the quadratic rotation symmetry H Boolean functions, and also focus on the relationship of balance, correlation immunity and dimension. Furthermore, we check the existence of the cubic rotation symmetry H Boolean functions, and obtain the relationship between existence and dimension of the cubic rotation symmetry H Boolean functions. Moreover, we obtain a more convenient method for solving annihilator. Such researches are important in cryptographic primitive designs, and have significance and role in the theory and application range of cryptosystems.

Algebraic immunity is an important cryptographic property for Boolean functions against algebraic attacks. Constructions of Boolean functions with the maximum algebraic immunity (MAI Boolean functions) by using univariate polynomial representation of Boolean functions over finite fields have received more and more attention. In this paper, how to obtain more MAI Boolean functions from a known MAI Boolean function under univariate polynomial representation is further investigated. The sufficient condition of Boolean functions having the maximum algebraic immunity obtained by changing a known MAI Boolean function under univariate polynomial representation is given. With this condition, more balanced MAI Boolean functions under univariate polynomial representation can be obtained. The algebraic degree and the nonlinearity of these Boolean functions are analyzed.

A circuit C compresses a function $${f : \{0,1\}^n\rightarrow \{0,1\}^m}$$f:{0,1}n?{0,1}m if given an input $${x\in \{0,1\}^n}$$x?{0,1}n, the circuit C can shrink x to a shorter l-bit string x? such that later, a computationally unbounded solver D will be able to compute f(x) based on x?. In this paper we study the existence of functions which are incompressible by circuits of some fixed polynomial size $${s=n^c}$$s=nc. Motivated by cryptographic applications, we focus on average-case $${(\ell,\epsilon)}$$(l,∈) incompressibility, which guarantees that on a random input $${x\in \{0,1\}^n}$$x?{0,1}n, for every size s circuit $${C:\{0,1\}^n\rightarrow \{0,1\}^{\ell}}$$C:{0,1}n?{0,1}l and any unbounded solver D, the success probability $${\Pr_x[D(C(x))=f(x)]}$$Prx[D(C(x))=f(x)] is upper-bounded by $${2^{-m}+\epsilon}$$2-m+∈. While this notion of incompressibility appeared in several works (e.g., Dubrov and Ishai, STOC 06), so far no explicit constructions of efficiently computable incompressible functions were known. In this work, we present the following results: (1)Assuming that E is hard for exponential size nondeterministic circuits, we construct a polynomial time computable boolean function $${f:\{0,1\}^n\rightarrow \{0,1\}}$$f:{0,1}n?{0,1} which is incompressible by size nc circuits with communication $${\ell=(1-o(1)) \cdot n}$$l=(1-o(1))·n and error $${\epsilon=n^{-c}}$$∈=n-c. Our technique generalizes to the case of PRGs against nonboolean circuits, improving and simplifying the previous construction of Shaltiel and Artemenko (STOC 14).(2)We show that it is possible to achieve negligible error parameter $${\epsilon=n^{-\omega(1)}}$$∈=n-?(1) for nonboolean functions. Specifically, assuming that E is hard for exponential size $${\Sigma_3}$$Σ3-circuits, we construct a nonboolean function $${f:\{0,1\}^n\rightarrow \{0,1\}^m}$$f:{0,1}n?{0,1}m which is incompressible by size nc circuits with $${\ell=\Omega(n)}$$l=Ω(n) and extremely small $${\epsilon=n^{-c} \cdot 2^{-m}}$$∈=n-c·2-m. Our construction combines the techniques of Trevisan and Vadhan (FOCS 00) with a new notion of relative error deterministic extractor which may be of independent interest.(3)We show that the task of constructing an incompressible boolean function $${f:\{0,1\}^n\rightarrow \{0,1\}}$$f:{0,1}n?{0,1} with negligible error parameter $${\epsilon}$$∈ cannot be achieved by "existing proof techniques". Namely, nondeterministic reductions (or even $${\Sigma_i}$$Σi reductions) cannot get $${\epsilon=n^{-\omega(1)}}$$∈=n-?(1) for boolean incompressible functions. Our results also apply to constructions of standard Nisan-Wigderson type PRGs and (standard) boolean functions that are hard on average, explaining, in retrospect, the limitations of existing constructions. Our impossibility result builds on an approach of Shaltiel and Viola (STOC 08).

This special issue of Theory of Computing Systems contains articles which relate to talks given at the 8th International Conference on Computability, Complexity and Randomness (CCR) in Moscow from the 23rd to the 27th of September 2015. The CCR conference series is devoted to algorithmic randomness, Kolmogorov complexity, and their relationship with computability theory, complexity theory, logic and reverse mathematics. The contributions to this special issue reflect both the interests of reserachers in Russia, in particular Kolmogorov and computational complexity, and new directions in the field of algorithmic randomness such as the focus on normal numbers and connections to ergodic theory. No conference proceedings have been published; this special issue is the only publication relating to the confernece. All contributions are a significant extension of the conference presentations, and underwent the usual rigorous journal reviewing process. The article Polynomial-time algorithms for checking some properties of Boolean functions given by polynomials by Svetlana Selezneva and Anton Bukhman presents a polynomial time algorithm, which tests whether a Boolean function represented by a multivariate polynomial over GF(2) is even, that is, has zero derivative in the direction of (1, . . . , 1). The approach is based on the notion of the characteristic of monomials in the polynomial representation of Boolean functions. In Normality and finite-state dimension of Liouville numbers Satyadev Nandakumar gives a new construction of a Liouville number which is normal for a given base. Vladimir V’yugin in On Stability of Probability Laws with Respect to Small Violations of Algorithmic Randomness examines the robustness of ergodic theorems when the assumption of randomness is relaxed only a little. For example he shows that

On the modulo degree complexity of Boolean functions

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.

Algebraic thickness is an important property of Boolean functions. In this paper we give the relationship between algebraic thickness of Boolean functions with disjoint variables and their algebraic thickness, and derive some algebraic thicknesses of special Boolean functions. Furthermore, we prove the restrictive relationship of algebraic thickness between Boolean function and complement Boolean function, by this result the upper bounds on algebraic thickness of Boolean functions with small hamming weights and their complement Boolean functions are given. Finally, we summarize some lower or upper bounds on algebraic thickness of common Boolean functions.

Revealing the canalizing structure of Boolean functions: Algorithms and applications

Given a Boolean function f on n variables, a Disjoint Sum-of-Products (DSOP) of f is a set of products (ANDs) of subsets of literals whose sum (OR) equals f, such that no two products cover the same minterm of f. DSOP forms are a special instance of partial DSOPs, i.e. the general case where a subset of minterms must be covered exactly once and the other minterms (typically corresponding to don’t care conditions of f) can be covered any number of times. We discuss finding DSOPs and partial DSOPs with a minimal number of products, a problem theoretically connected with various properties of Boolean functions and practically relevant in the synthesis of digital circuits. Finding an absolute minimum is hard, in fact we prove that the problem of absolute minimization of partial DSOPs is NP-hard. Therefore it is crucial to devise a polynomial time heuristic that compares favorably with the known minimization tools. To this end we develop a further piece of theory starting from the definition of the weight of a cube c as a functions of the number of fragments induced on other cubes by the selection of c, and show how cube weights can be exploited for building a class of minimization heuristics for DSOP and partial DSOP synthesis. A set of experiments conducted on major benchmark functions show that our method, with a family of variants, always generates better results than the ones of previous heuristics, including the method based on a BDD representation of f.

A family of subsets of an n-element set is k-intersecting if the intersection of every k subsets in the family is nonempty. A family is maximalk-intersecting if no subset can be added to the family without violating the k-intersection property. There is a one-to-one correspondence between the families of subsets and Boolean functions defined as follows: To each family of subsets, assign the Boolean function whose unit tuples are the characteristic vectors of the subsets.We show that a family of subsets is maximal 2-intersecting if and only if the corresponding Boolean function is monotone and selfdual. Asymptotics for the number of such families is obtained. Some properties of Boolean functions corresponding to k-intersecting families are established fork > 2.

Safe recursion revisited I: Categorical semantics for lower complexity

Branching problems are considered as a nonuniform model of computation in complexity theory as well as a data structure for boolean functions in several applications. In many applications (e.g., verification), exact representations are required. For learning boolean functions f on the basis of classified examples, it is sufficient to produce the representation of a function g approximating f. This motivates the investigation of the size of the smallest branching program approximating f. Although several non-approximability results are contained in the papers on randomized branching programs, these results often do not work for the uniform distribution (which is the most important one in applications). Here, the following non-approximability results are presented. (1) It is proven that a simple function from the branching program literature requires exponential size to be approximated with respect to the uniform distribution by OBDDs, which are the most important type of branching programs in applications. (2) The first truly exponential lower bound on the size of approximating syntactic read-k-times branching programs with respect to the uniform distribution and error probability 1/2-2/sup -/spl Omega/(n)/, n the input size, is shown. In order to improve upon the so far best results for error probabilities smaller than 1/3, a strong combinatorial lemma from a recent paper of Ajtai on linear-length branching programs is exploited.

On the Nonapproximability of Boolean Functions by OBDDs and Read-k-Times Branching Programs