The study of collision and avalanche effect in a family of pseudorandom binary sequences

In an earlier paper we studied collisions and avalanche effect in two of the most important constructions given for large families of binary sequences possessing strong pseudorandom properties. It turned out that one of the two constructions (which is based on the use of the Legendre symbol) is ideal from this point of view, while the other construction (which is based on the size of the modulo p residue of f(n) for some polynomial f(x) ∈ $ mathbb{F}_p $ mathbb{F}_p [x]) is not satisfactory since there are “many” collisions in it. Here it is shown that this weakness of the second construction can be corrected: one can take a subfamily of the given family which is just slightly smaller and collision free.