On the Polynomial Basis of GF(2n) Having a Small Number of Trace-One Elements

In the Galois fields GF(2n), a polynomial basis with a small number of trace-one elements is desirable for its convenience in computing. To find new irreducible polynomials g(x) over GF(2) with this property, we research into the auxiliary polynomial f(x)=(x+1)g(x) with roots {1,α1,α2,…,αn}, such that the symmetric polynomials sk=1+α1k+α2k+⋯+αnk are relative to the symmetric polynomials of g(x). We introduce a new class of polynomials with the number “1” occupying most of the values in its sk. This indicates that the number “0” occupies most of the values of the traces of the elements {α1,α2,…,αn}. This new class of polynomial gives us an indirect way to find irreducible polynomials having a small number of trace-one elements in their polynomial bases