Adjacency Graphs, Irreducible Polynomials and Cyclotomy

We consider the adjacency graphs of linear feedback shift registers (LFSRs) with reducible characteristic polynomials. Let l(x) be a characteristic polynomial, and l(x) = l1(x)l2(x) · · · lr(x) be a decomposition of l(x) into co-prime factors. Firstly, we show a connection between the adjacency graph of FSR(l(x)) and the association graphs of FSR(li(x)), 1 ≤ i ≤ r. By this connection, the problem of determining the adjacency graph of FSR(l(x)) is decomposed to the problem of determining the association graphs of FSR(li(x)), 1 ≤ i ≤ r, which is much easier to handle. Then, we study the association graph of LFSRs with irreducible characteristic polynomials and give a relationship between these association graphs and the cyclotomic numbers over finite fields. At last, some applications are suggested.