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**2005/096**

**Title:**

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

**Authors:**
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.

**ePrint:**
https://eprint.iacr.org/2005/096

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