the square map is one of the functions used in cryptography. for instance, the square map is used in rabin encryption scheme, block cipher rc6 and stream cipher rabbit, in different forms. in this paper, we study statistical properties of the output of the square map as a vectorial boolean function. we obtain the joint probability distribution of arbitrary number of the upper and the lower bits...

Boolean Networks (BNs) are established models to qualitatively describe biological systems. The analysis of BNs might be infeasible for medium large due the state-space explosion problem. We propose a novel reduction technique called Backward Equivalence (BBE), which preserves some properties interest BNs. In particular, reduced provide compact representation by grouping variables that, if init...

We consider systems of equations of the form Xi=,IJ,a l &~LJ& i=l,...,n where A is the underlying alphabet, the Xi are variables, the Pi.0 are boolean functions in the variables J&B and each & is either the empty word or the empty set. The symbols and u denote concatenation and union of languages over A. We show that any such system has a unique solution which, moreover, is regular. These equat...

A charge (finitely additive measure) defined on a Boolean algebra of sets taking values in a group G is called a strictly nonzero (SNZ) charge if it takes the identity value in G only for the zero element of the Boolean algebra. A study of such charges was initiated by Rudiger Göbel and KPS Bhaskara Rao in 2002 [3]. Our main result is a solution to one of the questions posed in that paper: we s...

In this paper we address the problem of generating all elements obtained by the saturation of an initial set by some operations. More precisely, we prove that we can generate the closure of a boolean relation (a set of boolean vectors) by polymorphisms with a polynomial delay. Therefore we can compute with polynomial delay the closure of a family of sets by any set of “set operations”: union, i...

In order to study the relationship between random Boolean sets and some explanatory variables, this paper introduces a Propagation model. This model can be applied when corresponding Poisson process of the Boolean model is related to explanatory variables and the random grains are not affected by these variables. An approximation for the likelihood is used to find pseudo-maximum likelihood esti...