Dynamical systems generated by iterations of multivariate polynomials with
slow degree growth have proved to admit good estimates of exponential sums
along their orbits which in turn lead to rather stronger bounds on the
discrepancy for pseudorandom vectors generated by these iterations. Here we add
new arguments to our original approach and also extend some of our recent
constructions and results to more general orbits of polynomial iterations which
may involve distinct polynomials as well. Using this construction we design a
new class of hash functions from iterations of polynomials and use our
estimates to motivate their "mixing" properties.