# Fast Algebraic Attacks and Decomposition of Symmetric Boolean Functions

## TL;DR

This paper analyzes the vulnerability of symmetric Boolean functions to fast algebraic attacks, showing most are susceptible despite good algebraic immunity, and proves no symmetric functions are AAR, improving understanding of their security properties.

## Contribution

It provides a decomposition of symmetric Boolean functions and demonstrates their general weakness against fast algebraic attacks, also establishing that none are AAR functions.

## Key findings

- Most symmetric Boolean functions are vulnerable to fast algebraic attacks.
- No symmetric Boolean functions qualify as AAR functions.
- The paper improves bounds between algebraic degree and immunity.

## Abstract

Algebraic and fast algebraic attacks are power tools to analyze stream ciphers. A class of symmetric Boolean functions with maximum algebraic immunity were found vulnerable to fast algebraic attacks at EUROCRYPT'06. Recently, the notion of AAR (algebraic attack resistant) functions was introduced as a unified measure of protection against both classical algebraic and fast algebraic attacks. In this correspondence, we first give a decomposition of symmetric Boolean functions, then we show that almost all symmetric Boolean functions, including these functions with good algebraic immunity, behave badly against fast algebraic attacks, and we also prove that no symmetric Boolean functions are AAR functions. Besides, we improve the relations between algebraic degree and algebraic immunity of symmetric Boolean functions.