There is no dearth of published literature on the design, implementation, analysis, or use of pseudo-random number generators or PRNGs. For example,    and the references therein, provide a broad overview and firm grounding for the subject. This report complements and elaborates upon the work of McKeever , who investigated PRNGs constructed in a non-commutative setting with the target application being so-called cryptographically secure PRNGs as discussed in  or . Novel "solutions" to the problem of designing cryptographically secure PRNGS continue to be proposed    , so despite the caution and skepticism required, the area remains active. The concept elaborated upon here is computation in a finite non-commutative object which is more than a matrix ring over a finite field. Specifically, we consider computation in a homomorphic image of a maximal order of an ordinary quaternion algebra. In Section Two we develop the necessary algebraic machinery. In Section Three we consider PRNG design in this computational setting. In Section Four we attempt some preliminary analysis of the PRNGs described. In Section Five we offer some final remarks and conclusions.
Gary Greenfield. On Quaternionic Pseudo-Random Number Generators. Technical paper (TR-96-01). Math and Computer Science Technical Report Series. Richmond, Virginia: Department of Mathematics and Computer Science, University of Richmond, May, 1996.