  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 $K=\mathbb{F}_p$ and $K=\mathbb{Q}$.
