Group actions based perfect nonlinearity

Laurent Poinsot (1) , Sami Harari (1)

1 PPF-STIC

In a recent paper, we generalized the notion of perfect nonlinearity of boolean functions by replacing the translations of a vector space on $\mathbb{F}_2$ by an Abelian group of fixed-point free involutions acting regularly on this vector space. We now show this generalization to be still valid when considering a finite nonempty set $X$ rather than a vector space on $\mathbb{F}_2$ and a faithful or regular action of a finite Abelian group $G$ on $X$. We also develop a dual characterization of this new concept through the Fourier transform as for the classical notion of perfect nonlinearity. By considering faithful actions we highlight a fundamental concept underlying to perfect nonlinearity that extends the classical notions. In short we integrate the traditional concepts within a more general and primitive framework.

Domaines: Théorie des groupes [math.GR]; Théorie de l'information et codage [math.IT]; Théorie de l'information [cs.IT]; Cryptographie et sécurité [cs.CR]

Laurent Poinsot, Sami Harari. Group actions based perfect nonlinearity. GESTS International Transactions on Computer Science and Engeneering, 2005, 12 (1), pp.1-14. ⟨hal-00460337⟩