Recursive constructions of NN-polynomials over GF(2s)GF(2s)

This paper presents procedures for constructing irreducible polynomials over GF(2s)GF(2s) with linearly independent roots (or normal polynomials or N-polynomials). For a suitably chosen initial N  -polynomial F0(x)∈GF(2s)F0(x)∈GF(2s) of degree n  , polynomials Fk(x)∈GF(2s)Fk(x)∈GF(2s) of degrees n2kn2k are constructed by iteratively applying the transformation x→x+x-1x→x+x-1, and their roots are shown to form a normal basis of GF(2sn2k)GF(2sn2k) over GF(2s)GF(2s). In addition, the sequences are shown to be trace compatible  , i.e., the trace map TGF(2sn2k+1)/GF(2sn2k)TGF(2sn2k+1)/GF(2sn2k) from GF(2sn2k+1)GF(2sn2k+1) onto GF(2sn2k)GF(2sn2k) maps the roots of Fk+1(x)Fk+1(x) onto those of Fk(x)Fk(x).