Constructing Small Generating Sets for the Multiplicative Groups of Algebras over Finite Fields.

We consider computational problems concerning algebras over finite fields. In particular, we propose an algorithm for finding a small generating set for the multiplicative group of GF(p)[x]/F, where p is a prime number and F in GF(p)[x] is an arbitrary polynomial. Based on this result, a new set of expander graphs can be explicitly constructed. In addition, we present algorithms for basis construction and decomposition of a given element with respect to the basis.