Constructing $2m$-variable Boolean functions with optimal algebraic immunity based on polar decomposition of $\mathbb{F}_{2^{2m}}^*$
This work addresses the need for secure Boolean functions in cryptography, offering an incremental improvement over existing methods for constructing such functions.
The paper tackles the problem of constructing Boolean functions with optimal algebraic immunity for cryptographic applications, proposing a new method based on polar decomposition of the multiplicative group of a finite field, resulting in balanced functions that achieve optimal algebraic immunity, degree, and high nonlinearity, with computer tests showing good resistance to fast algebraic attacks.
Constructing $2m$-variable Boolean functions with optimal algebraic immunity based on decomposition of additive group of the finite field $\mathbb{F}_{2^{2m}}$ seems to be a promising approach since Tu and Deng's work. In this paper, we consider the same problem in a new way. Based on polar decomposition of the multiplicative group of $\mathbb{F}_{2^{2m}}$, we propose a new construction of Boolean functions with optimal algebraic immunity. By a slight modification of it, we obtain a class of balanced Boolean functions achieving optimal algebraic immunity, which also have optimal algebraic degree and high nonlinearity. Computer investigations imply that this class of functions also behave well against fast algebraic attacks.