Related papers: The Landscape of Computing Symmetric $n$-Variable …

Let v(n) denote the number of compositions (ordered partitions) of a positive integer n into powers of 2. It appears that the function v(n) satisfies many congruences modulo 2^N. For example, for every integer B there exists (as k tends to…

We compute the nonlinearity of Boolean functions with Groebner basis techniques, providing two algorithms: one over the binary field and the other over the rationals. We also estimate their complexity. Then we show how to improve our…

To resist algebraic attack, a Boolean function should possess good algebraic immunity (AI). Several papers constructed symmetric functions with the maximum algebraic immunity $\lceil \frac{n}{2}\rceil $. In this correspondence we prove that…

Matrices are the most common representations of graphs. They are also used for the representation of algebras and cluster algebras. This paper shows some properties of matrices in order to facilitate the understanding and locating…

We associate to each Boolean function a polynomial whose evaluations represents the distances from all possible Boolean affine functions. Both determining the coefficients of this polynomial from the truth table of the Boolean function and…

An explicit algorithm is presented for testing whether two non-directed graphs are isomorphic or not. It is shown that for a graph of n vertices, the number of n independent operations needed for the test is polynomial in n. A proof that…

We show that the problem of counting the number of $n$-variable unate functions reduces to the problem of counting the number of $n$-variable monotone functions. Using recently obtained results on $n$-variable monotone functions, we obtain…

Using logic gates is the traditional way of designing logic circuits. However, most of the minimization algorithms concern a limited set of gates (complete sets), like sum of products, exclusive-or sum of products, NAND gates, NOR gates…

This paper presents a method to detect and recognize symmetries in Boolean functions. The idea is to use information theoretic measures of Boolean functions to detect sub-space of possible symmetric variables. Coupled with the new…

In this paper we generalize N-fold integer programs and two-stage integer programs with N scenarios to N-fold 4-block decomposable integer programs. We show that for fixed blocks but variable N, these integer programs are polynomial-time…

We investigate shift-invariant transformations, also known as rotation-symmetric vectorial Boolean functions, on $n$ bits that are induced from Boolean functions on $k$ bits, for $k\leq n$. We consider such transformations that are not…

Elementary symmetric polynomials $S_n^k$ are used as a benchmark for the bounded-depth arithmetic circuit model of computation. In this work we prove that $S_n^k$ modulo composite numbers $m=p_1p_2$ can be computed with much fewer…

In this report, we show that all n-variable Boolean function can be represented as polynomial threshold functions (PTF) with at most $0.75 \times 2^n$ non-zero integer coefficients and give an upper bound on the absolute value of these…

We define and study the complexity of robust polynomials for Boolean functions and the related fault-tolerant quantum decision trees, where input bits are perturbed by noise. We compare several different possible definitions. Our main…

We provide two sufficient and necessary conditions to characterize any $n$-bit partial Boolean function with exact quantum 1-query complexity. Using the first characterization, we present all $n$-bit partial Boolean functions that depend on…

We consider a problem, which we call secure grouping, of dividing a number of parties into some subsets (groups) in the following manner: Each party has to know the other members of his/her group, while he/she may not know anything about…

Exhibiting an explicit Boolean function with a large high-order nonlinearity is an important problem in cryptography, coding theory, and computational complexity. We prove lower bounds on the second-order, third-order, and higher-order…

In two-dimensional statistical physics, correlation functions of the O(N) and Potts models may be written as sums over configurations of non-intersecting loops. We define sums associated to a large class of combinatorial maps (also known as…

We give two approximation algorithms solving the Stochastic Boolean Function Evaluation (SBFE) problem for symmetric Boolean functions. The first is an $O(\log n)$-approximation algorithm, based on the submodular goal-value approach of…

An efficient pairwise Boolean matching algorithm to solve the problem of matching single-output specified Boolean functions under input negation and/or input permutation and/or output negation (NPN) is proposed in this paper. We present the…