Cryptography and Communications

, Volume 11, Issue 6, pp 1165–1184 | Cite as

Changing APN functions at two points

  • Nikolay S. KaleyskiEmail author
Part of the following topical collections:
  1. Special Issue on Boolean Functions and Their Applications


We investigate a construction in which a vectorial Boolean function G is obtained from a given function F over \(\mathbb {F}_{2^{n}}\) by changing the values of F at two points of the underlying field. In particular, we examine the possibility of obtaining one APN function from another in this way. We characterize the APN-ness of G in terms of the derivatives and in terms of the Walsh coefficients of F. We establish that changing two points of a function F over \(\mathbb {F}_{2^{n}}\) which is plateaued (and, in particular, AB) or of algebraic degree deg(F) < n − 1 can never give a plateaued (and AB, in particular) function for any n ≥ 5. We also examine a particular case in which we swap the values of F at two points of \(\mathbb {F}_{2^{n}}\). This is motivated by the fact that such a construction allows us to obtain one permutation from another. We obtain a necessary and sufficient condition for the APN-ness of G which we then use to show that swapping two points of any power function over a field \(\mathbb {F}_{2^{n}}\) with n ≥ 5 can never produce an APN function. We also list some experimental results indicating that the same is true for the switching classes from Edel and Pott (Adv. Math. Commun. 3(1):59–81, 2009), and conjecture that the Hamming distance between two APN functions cannot be equal to two for n ≥ 5.


Almost perfect nonlinear functions Differential uniformity Boolean functions 

Mathematics Subject Classification (2010)

94A60 06E30 



This research was co-funded by the Trond Mohn Foundation (formerly the Bergen Research Foundation). I would like to thank my supervisors Lilya Budaghyan and Claude Carlet, as well as Tor Helleseth and Nian Li for their ongoing interest and support.


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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of InformaticsUniversity of BergenBergenNorway

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