On Almost Perfect Nonlinear mappings

Thierry Berger, Anne Canteaut, Pascale Charpin
and Yann Laigle-Chapuy
LACO (university of Limoges) and INRIA (projet CODES)

thierry.berger@unilim.fr
Anne.Canteaut@inria.fr
Pascale.Charpin@inria.fr
Yann.Laigle-Chapuy@inria.fr

Presented at ISIT 2005, Adelaide, Australie.
To appear in IEEE Trans. Inform. Theory.

Abstract

We investigate some open problems on Almost Perfect Nonlinear (APN) functions over a finite field of characteristic 2. We provide new characterizations of APN mappings and of APN permutations by means of their component functions. We also focus on the case of quadratic functions. Most notably, we prove that a class of quadratic functions cannot be APN. Our result strengthens the conjecture that all quadratic APN functions are power functions, up to equivalence.

Keywords : Almost bent mapping, almost perfect nonlinear mapping, power mapping, permutation polynomial.