Recursive Constructions of N-polynomials over GF (2s)

This paper presents procedures for constructing irreducible polynomials over GF(2s ) with linearly independent roots (or normal polynomials or N-polynomials). For a suitably chosen initial N-polynomial F0(x) ∈ GF(2s ) of degree n, polynomials Fk(x) ∈ GF(2s ) of degrees n2k are constructed by iteratively applying the transformation x → x + x−1, and their roots are shown to form a normal basis of GF(2sn2 k ) over GF(2s ). In addition, the sequences are shown to be trace compatible, i.e., the trace map TGF(2sn2k+1 )/GF(2sn2 ) from GF(2 sn2) onto GF(2sn2 k ) maps the roots of Fk+1(x) onto those of Fk(x). © 2007 Elsevier B.V. All rights reserved.