Since 2003, Algebraic Attacks have received a lot of attention in the cryptography literature. In this context, algebraic immunity quantifies the resistance of a Boolean function to the standard algebraic attack of the pseudo-random generators using it as a nonlinear Boolean function. A high value of algebraic immunity is now an absolutely necessary cryptographic criterion for a resistance to a...

Suppose X is a uniformly distributed N -dimensional binary vector and Y is obtained by passing X through a binary symmetric channel with crossover probability α. Recently, Courtade and Kumar postulates that I(f(X);Y) ≤ 1−Hb(α) for any Boolean function f [1]. In this paper, we provide a proof of the correctness of this conjecture. I. PROBLEM STATEMENT Let X be an N -dimensional binary random vec...

The Forward Problem (FB) of Boolean equations consists of finding solutions of a system of Boolean equations, or equivalently, a single Boolean equation of the form f(X) = 0 where f(X): B n → B and B is an arbitrary Boolean algebra. By contrast, the Inverse Problem (IB) of Boolean equations aims to reconstruct the equation f (X) = 0 given the set of solutions and hence to verify the correctness...

We identify a new restriction on Boolean circuits called bijectivity and prove that bijective Boolean circuits require exponential size to compute the Boolean permanent function. As consequences of this lower bound, we show exponential size lower bounds for: (a) computing the Boolean permanent using monotone multilinear circuits ; (b) computing the 0-1 permanent function using monotone arithmet...

We study the length of polynomials over nite simple non-Abelian groups needed to realize Boolean functions. We apply the results for bounding the length of 5-permutation branching programs recognizing a Boolean set. Moreover, for Boolean and general functions on these groups, we present upper bounds on the length of shortest polynomials computing an arbitrary nary Boolean or general function, o...

It is well known that there are two regimes in a standard one-dimensional Boolean percolation model: either the entire space covered a.s., or volume fraction strictly less than one. The aim of this work to demonstrate third possibility model with totally asymmetric grains on half-line: a.s. no unbounded component, but An explicit condition given characterizing existence an occupied component.