GF(2^n)-bent functions

Laurent Poinsot (1)

1 LIPN - Laboratoire d'Informatique de Paris-Nord

A function from a finite Abelian group G and with values in the unit circle T of the complex field is called bent if its Fourier transform (i.e., the decomposition of f in the basis of characters of G) has a constant magnitude equals to the number of elements of G. In this contribution we define a modulo 2 notion of characters by allowing the characters of an elementary finite Abelian p-group G to take their values in the multiplicative group $GF(2^n)$ (with $p=2^{n}-1$) of the roots of the unity in the finite field $GF(2^n)$ with $2^n$ elements rather than in the complex roots of the unity T. We show that this kind of characters forms an orthogonal basis of the $GF(2^n)$-vector space of functions from G to $GF(2^n)$ that permits us to define a modulo 2 version of the Fourier transform (as a decomposition of a vector in this basis of characters). We show that many classical properties of the Fourier transform still hold for this characteristic 2 version. In particular, we can define an appropriate notion of bent functions, called $GF(2^n)$-bent functions, with respect to this Fourier transform. Finally we construct a class of $GF(2^n)$-bent functions and we also study their relations with classical and group action versions of perfect nonlinearity.

Mots clés: Bent functions; Perfect nonlinearity; Finite Abelian groups; Theory of characters; Fourier transform; Group actions

Domaines: Algèbre commutative [math.AC]; Cryptographie et sécurité [cs.CR]

Laurent Poinsot. GF(2^n)-bent functions. Advances and Applications in Discrete Mathematics, 2009, 3 (1), pp.1-46. ⟨hal-00460347⟩