  Interesting properties of the partitions of a finite field $\mathbb F_q$
induced by the combination of involutions and trace maps are studied. The
special features of involutions of the form $\frac{u}{z}$, $u$ being a fixed
element of $\mathbb F_q$, are exploited to generate pseudorandom numbers, the
randomness resting on the uniform distribution of the images of zero-trace
elements among the sets of non-zero trace elements of $\mathbb F_q$.
