  While exploring desirable properties of hash functions in cryptography, the
author was led to investigate three notions of functions with scattering or
"diffusive" properties, where the functions map between binary strings of fixed
finite length. These notions of diffusion ask for some property to be fulfilled
by the Hamming distances between outputs corresponding to pairs of inputs that
lie on the endpoints of edges of an $n$-dimensional hypercube. Given the
dimension of the input space, we explicitly construct such functions for every
dimension of the output space that allows for the functions to exist.
