Raising αxto the yth power overGF(qn)can be performed by calculating αymodulo the minimum polynomial of αxand then multiplying the result by an n × n matrix overGF(q). The elements of the matrix are only a function of x and of the generating polynomial of the field. This principle offers a time-memory trade-off when exponentiating a fixed element ofGF(qn), where the multiplications (not the squarings) involved in the standard squareand- multiply process are traded for a reference to a stored n × n matrix. The operations which make use of the stored data consume time which is equivalent to a single multiplication operation over the field, and are performed continuously, where the timeconsuming part of the exponentiation process is performed independently of the stored data. It is then shown how the presented principle enables an efficient implementation overGF(qn)of some variations of Diffie-Hellman public-key distribution system.