Designs, Codes and Cryptography

, Volume 47, Issue 1–3, pp 225–235 | Cite as

Two results on maximum nonlinear functions

  • Doreen Hertel
  • Alexander Pott


Maximum nonlinear functions \(F: \mathbb F_{2^m}\to \mathbb F_{2^m}\) are widely used in cryptography because the coordinate functions F β (x) := tr(β F(x)), \(\beta \in \mathbb F^{*}_{2^m}\) , have large distance to linear functions. Moreover, maximum nonlinear functions have good differential properties, i.e. the equations F(x + a) − F(x) = b, \(a,b \in \mathbb F_{2^m}, b\neq 0\) , have 0 or 2 solutions. Two classes of maximum nonlinear functions are the Gold power functions \(x^{2^{k}+1}\) , gcd(k, m) = 1, and the Kasami power functions \(x^{2^{2k}-2^{k}+1}\) , gcd(k, m) = 1. The main results in this paper are: (1) We characterize the Gold power functions in terms of the distance of their coordinate functions to characteristic functions of subspaces of codimension 2 in \(\mathbb F_{2^m}\) . (2) We determine the differential properties of the Kasami power functions if gcd(k,m) ≠ 1.


Maximum nonlinear Gold power function Walsh transform Difference set Finite field Kasami power function Almost perfect nonlinear 

AMS Classifications

05B10 05B25 


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© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Institute for Algebra and GeometryOtto-von-Guericke-University MagdeburgMagdeburgGermany

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