Exponential sums of symmetric Boolean functions are linear recurrent with integer coefficients. This was first established by Cai, Green and Thierauf in the mid nineties. Consequences of this result
has been used to study the asymptotic behavior of symmetric Boolean functions. Recently, Cusick extended it to rotation symmetric Boolean functions, which are functions with good cryptographic properties. In this article, we put all these results in the general context of Walsh transforms and some of its generalizations (nega-Hadamard transform, for example). Precisely, we show that Walsh transforms, for which exponential sums are just an instance, of symmetric and rotation symmetric Boolean functions satisfy linear recurrences with integer coefficients. We also provide a closed formula for the Walsh transform and nega-Hadamard transform of any symmetric Boolean functions. Moreover, using the techniques presented in this work, we show that some families of rotation symmetric Boolean functions are not bent when the number of variables is sufficiently large and provide asymptotic evidence to a conjecture of Stanica and Maitra.
