A Boolean function f : {1,−1}n → {1,−1} is said to be sign-represented by a real polynomial p : Rn → R if sgn(p(x)) = f(x) for all x ∈ {1,−1}n. The PTF density of f is the minimum number of monomials in a polynomial that sign-represents f . It is well known that every n-variable Boolean function has PTF density at most 2n. However, in general, less monomials are enough. In this paper, we presen...

Algebraic attacks have received a lot of attention in studying security of symmetric ciphers. The function used in a symmetric cipher should have high algebraic immunity (AI) to resist algebraic attacks. In this paper we are interested in finding AI of Boolean power functions. We give an upper bound on the AI of any Boolean power function and a formula to find its corresponding low degree multi...

We consider the problem of deciding whether a given object has a given property or it is far from any object with the property, referred to as property testing. We focus on the case where the objects are Boolean functions, and we survey some of the previously known results about testing for properties such as the number of relevant variables and Fourier degree of a Boolean function. We present ...

where (~xn) ∈ B and B is a Boolean algebra. Only finite Boolean algebras are considered. The Boolean logic presented is based on the abstract Boolean algebra given by Huntington’s [14] postulates and on a special canonical form for a Boolean function given by Archie Blake [2]. The theory of reasoning is divided into functional and general parts, both of which seek antecedents or consequents of ...

Cancellations are known to be helpful in e cient algebraic computation of polynomials over elds. We de ne a notion of cancellation in Boolean circuits and de ne Boolean circuits that do not use cancellation to be non-cancellative. Non-cancellative Boolean circuits are a natural generalization of monotone Boolean circuits. We show that in the absence of cancellation, Boolean circuits require sup...

Generalisations of the bent property of a Boolean function are presented, by proposing spectral analysis of the Boolean function with respect to a well-chosen set of local unitary transforms. Quadratic Boolean functions are related to simple graphs and it is shown that the orbit generated by some graph transforms can be found within the spectra of certain unitary transform sets. The flat spectr...

We introduce the boolean convolution for probability measures on the unit circle. Roughly speaking, it describes the distribution of the product of two boolean independent unitary random variables. We find an analogue of the characteristic function and determine all infinitely divisible probability measures on the unit circle for the boolean convolution.

The largest Hamming distance between a Boolean function in $n$ variables and the set of all affine Boolean functions in $n$ variables is known as the covering radius $\rho_n$ of the $[2^n,n+1]$ Reed-Muller code. This number determines how well Boolean functions can be approximated by linear Boolean functions. We prove that \[ \lim_{n\to\infty}2^{n/2}-\rho_n/2^{n/2-1}=1, \] which resolves a conj...

Classification of Non-linear Boolean functions is a long-standing problem in the area of theoreticalcomputer science. In this paper, effort has been made to achieve a systematic classification of all n-variableBoolean functions, where only one affine Boolean function belongs to each class. Two different methods areproposed to achieve this classification. The first method is a recurs...

In this paper we introduce a nondeterministic counterpart to Reduced, Ordered Binary Decision Diagrams for the representation and manipulation of logic functions. ROBDDs are conceptually related to deterministic finite automata (DFA), accepting the language formed by the minterms of a function. This analogy suggests the use of nondeterministic devices as language recognizers. Unlike ROBDDs, the...