  This work presents a study of perturbations of symmetric Boolean functions.
In particular, it establishes a connection between exponential sums of these
perturbations and Diophantine equations of the form $$ \sum_{l=0}^n
\binom{n}{l} x_l=0,$$ where $x_j$ belongs to some fixed bounded subset $\Gamma$
of $\mathbb{Z}$. The concepts of trivially balanced symmetric Boolean function
and sporadic balanced Boolean function are extended to this type of
perturbations. An observation made by Canteaut and Videau for symmetric Boolean
functions of fixed degree is extended. To be specific, it is proved that,
excluding the trivial cases, balanced perturbations of fixed degree do not
exist when the number of variables grows. Some sporadic balanced perturbations
are presented. Finally, a beautiful but unexpected identity between
perturbations of two very different symmetric Boolean functions is also
included in this work.
