Paper 2005/096

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

David Jedlicka


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.

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Published elsewhere. None
Almost Perfect Nonlinear (APN)power function
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jedlicka @ math utexas edu
2005-09-27: revised
2005-04-05: received
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      author = {David Jedlicka},
      title = {Almost Perfect Nonlinear Monomials over GF($2^n$) for Infinitely Many $n$},
      howpublished = {Cryptology ePrint Archive, Paper 2005/096},
      year = {2005},
      note = {\url{}},
      url = {}
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