Simple method to find the trace of an arbitrary element of a finite field V.C. da Rocha Jr. * & G. Markarian † Institute of Integrated Information Systems, School of Electronic & Electrical Engineering, Leeds LS2 9JT, UK Abstract A novel technique is described for computing the trace over GF (2) of an element from a given finite field GF (2 m ). 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 (2 m ) is not required. The computation of the minimal polynomial of an element of GF (2 m ) is derived as a function of the trace and of a sub-trace function. 1 Introduction The practical use of finite fields [1]-[2] and, in particular, the use of the struc- ture of the Galois field GF (2 m ) [3]-[4] for digital signal processing, is nowadays commonplace in applications of both error-correcting codes and cryptography [5]-[6]. In such applications it is very often necessary to compute products and sums of elements of a finite field, e.g., when computing the minimal polynomial [2, pp.54-60] of an element of a finite field. The trace [3, p.90] is a very useful analytic tool in finite fields. In this Letter we show how to compute the trace over GF (2) of an element from a given finite field GF (2 m ), using just a binary primitive polynomial of degree m and a division circuit, i.e., the usual knowledge of a table of powers of a primitive element of GF (2 m ) is not required. The complexity for computing the trace is thus drastically reduced since division circuits are simpler and readily implementable [8, p.210]. We then show how to use the trace and a trace derived function (sub-trace) to compute the minimal polynomial over the binary field, denoted as GF (2), of an element β, where β ∈ GF (2 m ). We remark that it is just as easy to extend these results for a finite field GF (p) and its extensions, where p is a prime number. An alternative technique for computing minimal polynomials over finite fields was given in [7], which however requires knowledge of a table of powers of a primitive element of GF (2 m ). * v.c.rocha@leeds.ac.uk, on leave from Comms. Res. Group, Dept. of Electronics & Sys- tems - UFPE, CP7800, CEP 50711-970 Recife PE, BRAZIL. The research of this author received partial support from the Brazilian National Council for Scientific and Technological Development - CNPq, Grant 201102/2004-8. † g.markarian@leeds.ac.uk 1