# Recursions associated to trapezoid, symmetric and rotation symmetric functions over Galois fields

## TL;DR

This paper generalizes the linear recurrence relations of exponential sums of rotation symmetric and trapezoid Boolean functions from binary fields to any Galois field, revealing deep algebraic structures and explicit formulas.

## Contribution

It extends known results about exponential sums satisfying linear recurrences from binary fields to arbitrary Galois fields, introducing trapezoid functions and explicit recurrence formulas.

## Key findings

- Exponential sums of rotation symmetric functions satisfy linear recurrences over any Galois field.
- Trapezoid Boolean functions share the same recurrence relations as rotation symmetric functions.
- Explicit formulas involve Discrete Fourier Transform and Hadamard matrices.

## Abstract

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.