A note on cyclotomic polynomials and linear feedback shift registers

Abstract

Linear Feedback Shift Registers (LFSR) are tools commonly used in cryptography in many contexts, for example as pseudo-random numbers generators. In this paper we characterize LFSR with certain symmetry properties. Related to this question we also classify polynomials f satisfying the property that if  is a root of f then f (a^{deg f}) = 0. The classification heavily depends on the choice of the fields of coefficients of the polynomial; we consider the cases K = F_p and K = Q.