Designs, Codes and Cryptography

, Volume 59, Issue 1–3, pp 207–222

Bounds on the degree of APN polynomials: the case of x−1 + g(x)



In this paper we consider APN functions \({f:\mathcal{F}_{2^m}\to \mathcal{F}_{2^m}}\) of the form f(x) = x−1 + g(x) where g is any non \({\mathcal{F}_{2}}\)-affine polynomial. We prove a lower bound on the degree of the polynomial g. This bound in particular implies that such a function f is APN on at most a finite number of fields \({\mathcal{F}_{2^m}}\). Furthermore we prove that when the degree of g is less than 7 such functions are APN only if m ≤ 3 where these functions are equivalent to x3.


Symmetric cryptography Sbox Differential cryptanalysis Almost perfect nonlinear 

Mathematics Subject Classification (2000)

94A60 11T71 14G50 


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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of MathematicsTechnical University of DenmarkLyngbyDenmark
  2. 2.Institut of Mathematiques of LuminyC.N.R.S.Marseille Cedex 9France

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