In the Galois fieldsGF(2n), a polynomial basis with a small number of trace-one elements is desirable for its convenience in computing. To find new irreducible polynomialsg(x)overGF(2)with this property, we research into the auxiliary polynomialf(x)=(x+1)g(x)with roots{1,α1,α2,…,αn}, such that the symmetric polynomialssk=1+α1k+α2k+⋯+αnkare relative to the symmetric polynomials ofg(x). We introduce a new class of polynomials with the number “1” occupying most of the values in itssk. 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.
Efficient Bit-Parallel Systolic Polynomial Basis Multiplier over GF(28) based on Irreducible Polynomials