Theory of transformation groups of polynomials over GF(2) with applications to linear shift register sequences

Abstract

The group of unimodular transformations on the roots of polynomials over GF(2) is considered, and those polynomials with symmetries in the unimodular group are identified. The cross-correlation function between two maximum-length linear shift register sequences of the same degree is shown to be computable as an explicit linear transformation, in matrix form, on either one of the sequences, regarded as a vector. The underlying vector space is the “cyclotomic algebra,” generated by the cyclotomic cosets, or by what Gauss termed the “periods” of the cyclotomic equation. A variety of numerical examples are worked in detail.