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Windmill polynomials over fields of characteristic two

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Abstract

Chambers andSmeets [3] have designed a windmill arrangement of linear feedback shift registers (LFSRs) to generate pn-sequences overGF(2) with high speed. When the windmill hasv vanes, the associated minimal feedback polynomial (having degreen, relatively prime tov) can be taken to have the shapef 1(x v)+x n f 2(x −v), where the polynomialsf 1 andf 2 have degree ≤[n/v]. Their numerical evidence, whenv is divisible by 4, suggests that, surprisingly, there areno such windmill polynomials which are irreducible ifn≡±3 (mod 8), while about twice as many irreducible and primitive windmill polynomials as they expected occur ifn≡±1 (mod 8). A discussion of this behaviour is presented here with proofs. The brief explanation is that the Galois group of the underlying generic windmill polynomial overGF (4) is equal to the alternating groupA n .

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  1. Department of Mathematics, University of Glasgow, G12 8QW, Glasgow, Scotland

    Stephen D. Cohen

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  1. Stephen D. Cohen
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Cohen, S.D. Windmill polynomials over fields of characteristic two. Monatshefte für Mathematik 107, 291–301 (1989). https://doi.org/10.1007/BF01517357

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  • DOI: https://doi.org/10.1007/BF01517357

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