**Almost Perfect Nonlinear Monomials over GF($2^n$) for Infinitely Many $n$**

*David Jedlicka*

**Abstract: **I present some results towards a classification of power
functions with positive exponents that are Almost Perfect Nonlinear (APN),
or equivalently differentially 2-uniform, over ${\mathbb{F}}_{2^n}$ for
infinitely many $n$. APN functions are useful in constructing S-boxes in
AES-like cryptosystems. An application of Weil's theorem on absolutely
irreducible curves shows that a monomial $x^m$ is not APN over
${\mathbb{F}}_{2^n}$ for all sufficiently large $n$ if a related two
variable polynomial has an absolutely irreducible factor defined over
${\mathbb{F}}_{2}$. I will show that the latter polynomial's
singularities imply that except in three cases, all power functions have
such a factor. Two of these cases are already known to be APN for
infinitely many fields. A third case is still unproven. Some specific
cases of power functions have already been known to be APN over only
finitely many fields, but they will mostly follow from the main result
below.

**Category / Keywords: **Almost Perfect Nonlinear (APN), power function,

**Publication Info: **None

**Date: **received 23 Mar 2005, last revised 27 Sep 2005

**Contact author: **jedlicka at math utexas edu

**Available format(s): **Postscript (PS) | Compressed Postscript (PS.GZ) | PDF | BibTeX Citation

**Version: **20050927:145157 (All versions of this report)

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