It is common to call “virtual world” the set of abstractions given to humans by computers and networks. When the computers and the networks are emulated by programs, i.e. are virtual machines and virtual networks, there is one more layer of virtuality: a kind of virtual squared world. Virtual square (V ) is the project to study how to exploit the possibilities given by multi layered virtuality....

A square involution is a square permutation which is also an involution. In this paper we give the enumeration of square involutions, using purely combinatorial methods, by establishing a bijective correspondence with a class of lattice paths. As a corollary to our result, we enumerate various subclasses of square involutions, including the classes of triangular, decomposable, and fat involutions.

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...