  Rotation symmetric Boolean functions are invariant under circular translation
of indices. These functions have very rich cryptographic properties and have
been used in different cryptosystems. Recently, Thomas Cusick proved that
exponential sums of rotation symmetric Boolean functions satisfy homogeneous
linear recurrences with integer coefficients. In this work, a generalization of
this result is proved over any Galois field. That is, exponential sums over
Galois fields of rotation symmetric polynomials satisfy linear recurrences with
integer coefficients. In the particular case of $\mathbb{F}_2$, an elementary
method is used to obtain explicit recurrences for exponential sums of some of
these functions. The concept of trapezoid Boolean function is also introduced
and it is showed that the linear recurrences that exponential sums of trapezoid
Boolean functions satisfy are the same as the ones satisfied by exponential
sums of the corresponding rotations symmetric Boolean functions. Finally, it is
proved that exponential sums of trapezoid and symmetric polynomials also
satisfy linear recurrences with integer coefficients over any Galois field
$\mathbb{F}_q$. Moreover, the Discrete Fourier Transform matrix and some
Complex Hadamard matrices appear as examples in some of our explicit formulas
of these recurrences.
