Abstract
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.
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Hertel, D., Pott, A. Two results on maximum nonlinear functions. Des. Codes Cryptogr. 47, 225–235 (2008). https://doi.org/10.1007/s10623-007-9124-z
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DOI: https://doi.org/10.1007/s10623-007-9124-z
Keywords
- Maximum nonlinear
- Gold power function
- Walsh transform
- Difference set
- Finite field
- Kasami power function
- Almost perfect nonlinear