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Keywords:

  • S-boxes;
  • propagation characteristics;
  • bent functions;
  • semi-bent functions;
  • cross-correlation;
  • sum-of-squares indicator

Abstract

In this paper, several methods for constructing substitution boxes (S-boxes) with good cross-correlation properties are proposed. We firstly analyze the cross-correlation properties of bent functions and derive a sufficient condition that the absolute indicator Δf,g of two bent functions f and g achieve its lowest possible value 2n ∕ 2. More precisely, it is sufficient that f + g is also a bent function, which then implies that the absolute indicator of vectorial bent functions equals to 2n ∕ 2. This indicates an erroneous conclusion in by Zhou et al., claiming that if f is bent, then Δf,g = 2n ∕ 2 if and only if g is an affine function, which is not true. Furthermore, because of a strong relationship between the cross-correlation properties and disjoint spectra semi-bent functions, two classes of highly nonlinear vectorial semi-bent functions with very good cross-correlation properties are proposed. In particular, the first class of vectorial semi-bent functions introduced here compares favorably to other methods in terms of the cross-correlation properties of its component functions. In addition,

  1. A sufficient condition that the absolute indicator of two bent functions achieves its lowest value is derived.
  2. A construction of S-boxes with good auto-correlation properties from vectorial bent functions is given.
  3. Two classes of nonlinear vectorial semi-bent functions with good auto-correlation properties are proposed.

Copyright © 2014 John Wiley & Sons, Ltd.