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Changing APN functions at two points


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.

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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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Correspondence to Nikolay S. Kaleyski.

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This article is part of the Topical Collection on Special Issue on Boolean Functions and Their Applications

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Kaleyski, N.S. Changing APN functions at two points. Cryptogr. Commun. 11, 1165–1184 (2019).

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  • Almost perfect nonlinear functions
  • Differential uniformity
  • Boolean functions

Mathematics Subject Classification (2010)

  • 94A60
  • 06E30