Notes on Reconstruction of Shadow Minimization Boolean Functions

A special class of monotone Boolean functions coming from shadow minimization theory of finite set-systems – KK-MBF functions – is considered. These functions are a descriptive model for systems of compatible groups of constraints, however, the class of all functions is unambiguously complex and it is sensible to study relatively simple subclasses of functions such as KK-MBF. Zeros of KK-MBF functions correspond to initial segments of lexicographic order on hypercube layers. This property is used to simplify the recognition. Lexicographic order applies priorities over constraints which is applicable property of practices. Query-based algorithms for KK-MBF functions are investigated in terms of their complexities.

We consider the problem of learning monotone Boolean functions over {0, 1}/sup n/ under the uniform distribution. Specifically, given a polynomial number of uniform random samples for an unknown monotone Boolean function f, and given polynomial completing time, we would like to approximate f as well as possible. We describe a simple algorithm that we prove achieves error at most 1/2-/spl Omega/(1//spl radic/n), improving on the previous best bound of 1/2-/spl Omega/((log/sup 2/ n)/n). We also prove that no algorithm, given a polynomial number of samples, can guarantee error 1/2-/spl omega/((log n)//spl radic/n), improving on the previous best hardness bound of O(1//spl radic/n). These lower bounds hold even if the learning algorithm is allowed membership queries. Thus this paper settles to an O(log n) factor the question of the best achievable error for learning the class of monotone Boolean functions with respect to the uniform distribution.

We study the function evaluation problem in the priced information framework introduced in [Charikar et al. 2002]. We characterize the best possible extremal competitive ratio for the class of game tree functions. Moreover, we extend the above result to the case when the cost of reading a variable depends on the value of the variable. In this new value dependent cost variant of the problem, we also exactly evaluate the extremal competitive ratio for the whole class of monotone Boolean functions.

The number and probability of canalizing functions

This paper investigates the optimum OBDD representation problem based on two classes of Boolean functions. The first class is defined by OBDDs, in which the number of non-terminal nodes is equal to the number of input variables. We refer to such OBDDs and their corresponding Boolean functions as thin OBDDs and thin Boolean functions. The second class is the thin factored Boolean functions, which is defined by factored forms with each variable appearing exactly once. Many interesting properties of these two classes of functions are presented. Based on which, a revised dynamic shortest cube first OBDD variable ordering algorithm is developed. This algorithm is shown to be optimum for thin factored Boolean functions.

International audience

Boolean functions play a central role in security applications because they constitute one of the basic primitives for modern cryptographic services. In the last decades, research on Boolean functions has been boosted due to the importance of security in many diverse public systems relying on such technology. A main focus is to find Boolean functions with specific properties. An open problem in this context is to find a balanced Boolean function with an 8-bit input and nonlinearity 118. Theoretically, such a function has been shown to exist, but it has not been found yet. In this work we focus on specific classes of Boolean functions, and analyze the landscape of results obtained by integrating algebraic and evolutionary computation (EC) based approaches. Results indicate that combinations of these approaches give better results although not reaching 118 nonlinearity.

Two Boolean functions are affine equivalent if one can be obtained from the other by applying an affine transformation to the input variables. For a long time, there have been efforts to investigate the affine equivalence of Boolean functions. Due to the complexity of the general problem, only affine equivalence under certain groups of permutations is usually considered. Boolean functions which are invariant under the action of cyclic rotation of the input variables are known as rotation symmetric (RS) Boolean functions. Due to their speed of computation and the prospect of being good cryptographic Boolean functions, this class of Boolean functions has received a lot of attention from cryptographic researchers. In this paper, we study affine equivalence for the simplest rotation symmetric Boolean functions, called MRS functions, which are generated by the cyclic permutations of a single monomial. Using Pólya’s enumeration theorem, we compute the number of equivalence classes, under certain large groups of permutations, for these MRS functions in any number

Several underlying structural and functional factors that determine the fault behavior of a network are not yet well understood. In this paper, we show that there exists a large class of Boolean functions, called root functions, which can never appear as faulty response in an irredundant two-level circuit even when any arbitrary multiple stuck-at faults are injected. Conversely, we show that any other Boolean function can appear as a faulty response in an irredundant realization of some root function under certain stuck-at faults. We characterize this new class of functions and show that for n variables, their number is exactly equal to the number of independent dominating sets (Harary and Livingston, Appl. Math. Lett., 1993) in a Boolean n-cube. Similar properties are observed for multiple-valued logic functions as well. Finally, we discuss its application to logic design and point out some open problems.

It is a difficult task to compute the r-th order nonlinearity of a given function with algebraic degree strictly greater than r > 1. Though lower bounds on the second order nonlinearity are known only for a few particular functions, the majority of which are cubic. We investigate lower bounds on the second order nonlinearity of cubic Boolean functions [Formula: see text], where [Formula: see text], dl = 2il + 2jl + 1, m, il and jl are positive integers, n > il > jl. Furthermore, for a class of Boolean functions [Formula: see text] we deduce a tighter lower bound on the second order nonlinearity of the functions, where [Formula: see text], dl = 2ilγ + 2jlγ + 1, il > jl and γ ≠ 1 is a positive integer such that gcd(n,γ) = 1. Lower bounds on the second order nonlinearity of cubic monomial Boolean functions, represented by fμ(x) = Tr(μx2i+2j+1), [Formula: see text], i and j are positive integers such that i > j, were obtained by Gode and Gangopadhvay in 2009. In this paper, we first extend the results of Gode and Gangopadhvay from monomial Boolean functions to Boolean functions with more trace terms. We further generalize and improve the results to a wider range of n. Our bounds are better than those of Gode and Gangopadhvay for monomial functions fμ(x). Especially, our lower bounds on the second order nonlinearity of some Boolean functions F(x) are better than the existing ones.

Boolean functions are fundamental building blocks in both discrete mathematics and computer science, with applications spanning from cryptography to coding theory. Bent functions, a subset of Boolean functions with maximal nonlinearity, are particularly valuable in cryptographic applications. This study introduces a novel equivalence relation among all Boolean functions and presents an algorithm to generate bent functions based on this relation. We systematically generated a collection of 10,000 bent functions over eight variables, all originating from the same equivalence class, and analyzed their structural complexity through rank determination. Our findings revealed the presence of at least five distinct affine classes of bent functions within this collection. By employing this construction, we devised an algorithm to generate a filter function capable of combining Boolean functions. This filter function can be dynamically adjusted based on a key, offering potential applications in symmetric cipher design, such as enhancing security or improving efficiency.

The class of polynomial-threshold functions is studied using harmonic analysis, and the results are used to derive lower bounds related to AC/sup 0/ functions. A Boolean function is polynomial threshold if it can be represented as a sign function of a sparse polynomial (one that consists of a polynomial number of terms). The main result is that polynomial-threshold functions can be characterized by means of their spectral representation. In particular, it is proved that a Boolean function whose L/sub 1/ spectral norm is bounded by a polynomial in n is a polynomial-threshold function, and that a Boolean function whose L/sub infinity //sup -1/ spectral norm is not bounded by a polynomial in n is not a polynomial-threshold function. Some results for AC/sup 0/ functions are derived. >

Hyper-bent functions as a subclass of bent functions attract much interest and it is elusive to completely characterize hyper-bent functions. Most of known hyper-bent functions are Boolean functions with Dillon exponents and they are often characterized by special values of Kloosterman sums. In this paper, we present a method for characterizing hyper-bent functions with Dillon exponents. A class of hyper-bent functions with Dillon exponents over \(\mathbb {F}_{2^{2m}}\) can be characterized by a Boolean function over \(\mathbb {F}_{2^m}\), whose Walsh spectrum takes the same value twice. Further, we show several classes of hyper-bent functions with Dillon exponents characterized by Kloosterman sum identities and the Walsh spectra of some common Boolean functions.

The total influence of a function is a central notion in analysis of Boolean functions, and characterizing functions that have small total influence is one of the most fundamental questions associated with it. The KKL theorem and the Friedgut junta theorem give a strong characterization of such functions whenever the bound on the total influence is \(o(\log n)\). However, both results become useless when the total influence of the function is \(\omega (\log n)\). The only case in which this logarithmic barrier has been broken for an interesting class of functions was proved by Bourgain and Kalai, who focused on functions that are symmetric under large enough subgroups of \(S_n\). In this paper, we build and improve on the techniques of the Bourgain–Kalai paper and establish new concentration results on the Fourier spectrum of Boolean functions with small total influence. Our results include: 1. A quantitative improvement of the Bourgain–Kalai result regarding the total influence of functions that are transitively symmetric. 2. A slightly weaker version of the Fourier-Entropy Conjecture of Friedgut and Kalai. Our result establishes new bounds on the Fourier entropy of a Boolean function f, as well as stronger bounds on the Fourier entropy of low-degree parts of f. In particular, it implies that the Fourier spectrum of a constant variance, Boolean function f is concentrated on \(2^{O(I[f]\log I[f])}\) characters, improving an earlier result of Friedgut. Removing the \(\log I[f]\) factor would essentially resolve the Fourier-Entropy Conjecture, as well as settle a conjecture of Mansour regarding the Fourier spectrum of polynomial size DNF formulas. Our concentration result for the Fourier spectrum of functions with small total influence also has new implications in learning theory. More specifically, we conclude that the class of functions whose total influence is at most K is agnostically learnable in time \(2^{O(K\log K)}\) using membership queries. Thus, the class of functions with total influence \(O(\log n/\log \log n)\) is agnostically learnable in \(\mathsf{poly}(n)\) time.

Given a Boolean function f, the f-isomorphism testing problem requires a randomized algorithm to distinguish functions that are identical to f up to relabeling of the input variables from functions that are far from being so. An important open question in property testing is to determine for which functions f we can test f-isomorphism with a constant number of queries. Despite much recent attention to this question, essentially only two classes of functions were known to be efficiently isomorphism testable: symmetric functions and juntas. We unify and extend these results by showing that all partially symmetric functions -- functions invariant to the reordering of all but a constant number of their variables -- are efficiently isomorphism-testable. This class of functions, first introduced by Shannon, includes symmetric functions, juntas, and many other functions as well. We conjecture that these functions are essentially the only functions efficiently isomorphism-testable. To prove our main result, we also show that partial symmetry is efficiently testable. In turn, to prove this result we had to revisit the junta testing problem. We provide a new proof of correctness of the nearly-optimal junta tester. Our new proof replaces the Fourier machinery of the original proof with a purely combinatorial argument that exploits the connection between sets of variables with low influence and intersecting families. Another important ingredient in our proofs is a new notion of symmetric influence. We use this measure of influence to prove that partial symmetry is efficiently testable and also to construct an efficient sample extractor for partially symmetric functions. We then combine the sample extractor with the testing-by-implicit-learning approach to complete the proof that partially symmetric functions are efficiently isomorphism-testable.