Which Irreducible Polynomials Divide Trinomials over GF(2)?

Abstract

The output sequence of a binary linear feedback shift register with k taps corresponds to a polynomial $f(x) = 1+x^{{a}_{1}}+x^{{a}_{2}}+...+x^{{a}_{k}}$, where the exponents a ₁, a ₂, ..., a _k = n are the positions of the taps, and n, the degree of f(x), is the length of the shift register. Different initial states of the shift register may give rise to different output sequences. The simplest shift registers to implement involve only two taps (k = 2). It is therefore of interest to know which irreducible polynomials f(x) divide trinomials, over GF(2), since the output sequences corresponding to f(x) can be obtained from a two-tap linear shift register (with a suitable initial state) if and only if f(x) divides some trinomial t(x) = x ^m + x ^a + 1 over GF(2). In this paper we develop the theory of which irreducible polynomials over GF(2) do, or do not, divide trinomials.