Extended windmill polynomials

We present a generalization of a class of characteristic polynomials used for linear feedback shift registers (LFSRs). In previous works, several restrictions have been demonstrated for the windmill polynomials. Most notably, no irreducible windmill polynomial was found for a degree d = 3 mod 8. We show how to modify the original definition to overcome those restrictions. We also assess the security of our extended windmill generator considering the case of a filtered LFSR. This paper concerns LFSRs but it can be extended to any kind of shift registers including feedback with carry shift registers (FCSRs) and non-linear feedback shift registers (NLFSRs). We also establish the number of extended windmill polynomials for v = 4, 8, 16, 32 and 64 vanes up to the degree 160.