On the minimal fourier degree of symmetric Boolean functions

In this paper we give a new upper bound on the minimal degree of a nonzero Fourier coefficient in any non-linear symmetric Boolean function. Specifically, we prove that for every non-linear and symmetric f: {0, 1} ^k → {0, 1} there exists a set; \(\not 0 \ne S \subset [k]\) such that ¦S¦ = O(Γ(k)+√k, and \(\hat f(S) \ne 0\) where Γ(m)≤m ^0.525 is the largest gap between consecutive prime numbers in {1,..., m}. As an application we obtain a new analysis of the PAC learning algorithm for symmetric juntas, under the uniform distribution, of Mossel et al. [10]. Our bound on the degree is a significant improvement over the previous result of Kolountzakis et al. [8] who proved that ¦S¦=O(k=log k). We also show a connection between lower-bounding the degree of non-constant functions that take values in {0,1,2} and the question that we study here.     

On the Fourier spectrum of symmetric Boolean functions

We study the following questionWhat is the smallest t such that every symmetric boolean function on κ variables (which is not a constant or a parity function), has a non-zero Fourier coefficient of order at least 1 and at most t?We exclude the constant functions for which there is no such t and the parity functions for which t has to be κ. Let τ (κ) be the smallest such t. Our main result is that for large κ, τ (κ)≤4κ/logκ.The motivation for our work is to understand the complexity of learning symmetric juntas. A κ-junta is a boolean function of n variables that depends only on an unknown subset of κ variables. A symmetric κ-junta is a junta that is symmetric in the variables it depends on. Our result implies an algorithm to learn the class of symmetric κ-juntas, in the uniform PAC learning model, in time n ^o(κ) . This improves on a result of Mossel, O’Donnell and Servedio in [16], who show that symmetric κ-juntas can be learned in time n ^2κ/3.     

On generalized Boolean functions II

This cycle of papers is based on the concept of generalized Bolean functions introduced by the author in the first article of the series. Every generalized Boolean function f:Bⁿ→B can be written in a manner similar to the canonical disjunctive form using some function defined on A×B, where A is a finite subset of B containing 0 and 1. The set of those functions f is denoted by GBF_n[A]. In this paper the following questions are presented: (1) What is the relationship between GBF_n[A₁] and GBF_n[A₂] when A₁A₂. (2) What can be said about GBF_n[A₁∩A₂] and GBF_n[A₁A₂] in comparison with GBF_n[A₁]∩GBF_n[A₂] and GBF_n[A₁]GBF_n[A₂], respectively.     

On generalized Boolean functions i

In this paper we introduce the concept of generalized Boolean function. Such a function has its arguments and values in a Boolean algebra and can be written in a manner similar to the canonical disjunctive form, but instead of the product of simple or complemented variables, the product of values of certain functions is used. Every Boolean function is a generalized Boolean one but the converse is not true. The set of all generalized Boolean function “generated” by some fixed function is a Boolean algebra.     

On Boolean ranges of Banaschewski functions

We construct a countable lattice \({\varvec{\mathcal {S}}}\) isomorphic to a bounded sublattice of the subspace lattice of a vector space with two non-iso-morphic maximal Boolean sublattices. We represent one of them as the range of a Banaschewski function and we prove that this is not the case of the other. Hereby we solve a problem of F. Wehrung. We study coordinatizability of the lattice \({\varvec{\mathcal {S}}}\). We prove that although it does not contain a 3-frame, the lattice \({\varvec{\mathcal {S}}}\) is coordinatizable. We show that the two maximal Boolean sublattices correspond to maximal Abelian regular subalgebras of the coordinatizating ring.     

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 the Hamming distance between almost all Boolean functions

A precise estimate of the Hamming distance between almost all Boolean functions is presented.     

On the generation of false images of linear Boolean functions

The following game (test) problem is considered. Suppose that Alice has a function. Bob believes that her function is linear, which is not the case. For an arbitrary linear function, Alice needs to make Bob believe she has this particular function by showing him its actual values. Alice’s function can be constructed beforehand or can be random. The task is to estimate the required size of the domain of Alice’s function (its subfunction).     

On the fractional chromatic number of monotone self-dual Boolean functions

We compute the exact fractional chromatic number for several classes of monotone self-dual Boolean functions. We characterize monotone self-dual Boolean functions in terms of the optimal value of an LP relaxation of a suitable strengthening of the standard IP formulation for the chromatic number. We also show that determining the self-duality of a monotone Boolean function is equivalent to determining the feasibility of a certain point in a polytope defined implicitly.     

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.     

On the spectrum of cosine operator functions

In this paper we characterize the spectrum of strongly continuous cosine functions, defined in a Hilbert space, in terms of properties of the infinitesimal generator.     

On designated-weight Boolean functions with highest algebraic immunity

Algebraic immunity has been considered as one of cryptographically significant properties for Boolean functions. In this paper, we study ∑d-1 i=0 (ni)-weight Boolean functions with algebraic immunity achiev-ing the minimum of d and n - d + 1, which is highest for the functions. We present a simpler sufficient and necessary condition for these functions to achieve highest algebraic immunity. In addition, we prove that their algebraic degrees are not less than the maximum of d and n - d + 1, and for d = n1 +2 their nonlinearities equalthe minimum of ∑d-1 i=0 (ni) and ∑ d-1 i=0 (ni). Lastly, we identify two classes of such functions, one having algebraic degree of n or n-1.     

Normal Boolean functions

Dobbertin (Construction of bent functions and balanced Boolean functions with high nonlinearity, in: Fast Software Encryption, Lecture Notes in Computer Science, Vol. 1008, Springer, Berlin, 1994, pp. 61–74) introduced the normality of bent functions. His work strengthened the interest for the study of the restrictions of Boolean functions on k-dimensional flats providing the concept of k-normality. Using recent results on the decomposition of any Boolean functions with respect to some subspace, we present several formulations of k-normality. We later focus on some highly linear functions, bent functions and almost optimal functions. We point out that normality is a property for which these two classes are strongly connected. We propose several improvements for checking normality, again based on specific decompositions introduced in Canteaut et al. (IEEE Trans. Inform. Theory, 47(4) (2001) 1494), Canteaut and Charpin (IEEE Trans. Inform. Theory). As an illustration, we show that cubic bent functions of 8 variables are normal.