Laurent Poinsot

Non Abelian bent functions

Perfect nonlinear functions from a finite group G to another one H are those functions f: G →H such that for all nonzero α ∈ G, the derivative

Bent functions on a finite nonabelian group

d_{\alpha}f: x \mapsto f(\alpha x) f(x)^{-1}

Non-Boolean Almost Perfect Nonlinear Functions on Non-Abelian Groups

is balanced. In the case where both G and H are Abelian groups, f: G →H is perfect nonlinear if, and only if, f is bent, i.e., for all nonprincipal character χ of H, the (discrete) Fourier transform of χ ∘ f has a constant magnitude equals to |G|. In this paper, using the theory of linear representations, we exhibit similar bentness-like characterizations in the cases where G and/or H are (finite) non Abelian groups. Thus we extend the concept of bent functions to the framework of non Abelian groups.

Generalized boolean bent functions

Abstract We introduce the notion of a bent function on a finite nonabelian group which is a natural generalization of the well-known notion of bentness on a finite abelian group due to Logachev, Salnikov and Yashchenko. Using the theory of linear representations and noncommutative harmonic analysis of finite groups we obtain several properties of such functions similar to the corresponding properties of traditional abelian bent functions.

A new characterization of group action-based perfect nonlinearity

The purpose of this paper is to present extended definitions and characterizations of the classical notions of APN and maximum nonlinear Boolean functions to deal with the case of mappings from a finite group K to another one N with the possibility that one or both groups are non-Abelian.

G-Perfect nonlinear functions

The notions of perfect nonlinearity and bent functions are closely dependent on the action of the group of translations over

Wronskian Envelope of a Lie Algebra

\mathbb{F}^{m}_{2}

The Dual Rings of an R-Coring Revisited

. Extending the idea to more generalized groups of involutions without fixed points gives a larger framework to the previous notions. In this paper we largely develop this concept to define G-perfect nonlinearity and G-bent functions, where G is an Abelian group of involutions, and to show their equivalence as in the classical case.

Differential (Monoid) Algebra and More

The left-regular multiplication is explicitly embedded in the notion of perfect nonlinearity. But there exist many other group actions. By replacing translations by another group action the new concept of group action-based perfect nonlinearity has been introduced. In this paper we show that this generalized concept of nonlinearity is actually equivalent to a new bentness notion that deals with functions defined on a finite Abelian group G that acts on a finite set X and with values in the finite-dimensional vector space of complex-valued functions defined on X.

Doubly perfect nonlinear boolean permutations

Perfect nonlinear functions are used to construct DES-like cryptosystems that are resistant to differential attacks. We present generalized DES-like cryptosystems where the XOR operation is replaced by a general group action. The new cryptosystems, when combined with G-perfect nonlinear functions (similar to classical perfect nonlinear functions with one XOR replaced by a general group action), allow us to construct systems resistant to modified differential attacks. The more general setting enables robust cryptosystems with parameters that would not be possible in the classical setting. We construct several examples of G-perfect nonlinear functions, both