A novel technique is described for computing the trace over GF(2) of an element from a given finite field GF(2m). This technique requires a primitive polynomial of degree m and a division circuit only, i.e. the usual knowledge of a table of powers of a primitive element of GF(2m) is not required. The computation of the minimal polynomial of an element of GF(2m) is derived as a function of the trace and of a sub-trace function