A note on cyclotomic polynomials and Linear Feedback Shift Registers

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