1. Introduction

Finite fields are involved in all aspects of coding theory, but computations in these fields are not straightforward, unless special hardware is used [2]. Zech's logarithms were introduced (circa 1850) as an inevitable counterpart for the multiplication speedup obtained with discrete logarithms.

But now, the Zech function is useful for many other purposes than elementary additions. For example, it can be used for roots finding, and thus in error correcting codes [12]. Therefore, efficient time/space bargains are required for the management of that function, i.e., for building from scratch, and for subsequent storing/recovering of tables of that Zech function.

The present work, based upon a report written for the ``Logic and Algorithmic Seminar 1993/1994'' at Caen University, is organized as follows. Some general results about finite fields are recalled in part 2, with some broad outlines of their proofs. In part 3, the field is taken for example, and in part 4 an interpretation is given of the coset-orbit method developed in [6,5], method that reduces the requested storage space from to (roughly) for the fields .

In subsequent parts, a (new) probabilistic algorithm is developed for building from scratch a Zech table for a given field. It is shown that all the obtained information can be collected into a single natural integer, the "efficient exponent for ", whose knowledge enables a further rebuilding of the Zech table in a same order of time as sequential reading from a mass-storage.

Tables are given for the first fields .