Constructing Boolean Functions With Potential Optimal Algebraic Immunity Based on Additive Decompositions of Finite Fields

TL;DR

This paper introduces a method to construct Boolean functions with high algebraic degree, nonlinearity, and potential optimal algebraic immunity using additive decompositions of finite fields, with implications for cryptographic security.

Contribution

The paper presents a novel construction approach for Boolean functions based on finite field decompositions, achieving functions with optimal algebraic immunity under a conjecture, and demonstrates their cryptographic robustness.

Findings

Constructed classes of Boolean functions with high algebraic degree and nonlinearity.

Achieved functions with potential optimal algebraic immunity under a conjecture.

Functions show good resistance to fast algebraic attacks in small cases.

Abstract

We propose a general approach to construct cryptographic significant Boolean functions of $(r+1)m$ variables based on the additive decomposition $\mathbb{F}_{2^{rm}}\times\mathbb{F}_{2^m}$ of the finite field $\mathbb{F}_{2^{(r+1)m}}$, where $r$ is odd and $m\geq3$. A class of unbalanced functions are constructed first via this approach, which coincides with a variant of the unbalanced class of generalized Tu-Deng functions in the case $r=1$. This class of functions have high algebraic degree, but their algebraic immunity does not exceeds $m$, which is impossible to be optimal when $r>1$. By modifying these unbalanced functions, we obtain a class of balanced functions which have optimal algebraic degree and high nonlinearity (shown by a lower bound we prove). These functions have optimal algebraic immunity provided a combinatorial conjecture on binary strings which generalizes the Tu-Deng conjecture is true. Computer investigations show that, at least for small values of number of variables, functions from this class also behave well against fast algebraic attacks.