An application of newton formula on the computation of finite field trace

The paper described a novel method of computing the trace of an element from a given finite field GF(pⁿ). By applying the Newton Formula, this method shows a simplex and linear structure of traces of different elements, and the usual procedure of computing the products and sums of elements of GF(pⁿ) is not required. The paper also studied The structure of the minimal polynomial of an element from GF(pⁿ). A new fast way for computing the minimal polynomial of an element in GF(pⁿ) is derived through combining the structure with the trace computation.