Difference between revisions of "Plateaued Functions"

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(Created page with "= Background and Definition = A Boolean function <math>f : \mathbb{F}_{2^n} \rightarrow \mathbb{F}_2</math> is said to be ''plateaued'' if its Walsh transform takes at most t...")
 
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== Primary constructons ==
 
== Primary constructons ==
  
=== Maiorana-MacFarland Functions ===
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=== Generalization of the Maiorana-MacFarland Functions <ref name="camion1992">Camion P, Carlet C, Charpin P, Sendrier N. On Correlation-immune functions. InAdvances in Cryptology—CRYPTO’91 1992 (pp. 86-100). Springer Berlin/Heidelberg.</ref> ===
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The Maiorana-MacFarland class of bent functions can be generalized into the class of functions <math>f_{\phi,h}</math> of the form
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<div><math>f_{\phi,h}(x,y) = x \cdot \phi(y) + h(y)</math></div>
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for <math>x \in \mathbb{F}_2^r, y \in \mathbb{F}_2^s</math>, where <math>r</math> and <math>s</math> are any positive integers, <math>n = r + s</math>, <math>\phi : \mathbb{F}_2^s \rightarrow \mathbb{F}_2^r</math> is arbitrary and <math>h : \mathbb{F}_2^s \rightarrow \mathbb{F}_2</math> is any Boolean function.
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The Walsh transform of <math>f_{\phi,h}</math> takes the value
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<div><math>W_{f_{\phi,h}}(a,b) = 2^r \sum_{y \in \phi^{-1}(a)} (-1)^{b \cdot y + h(y)}</math></div>
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at <math>(a,b)</math>. If <math>\phi</math> is injective, resp. takes each value in its image set two times, then <math>f_{\phi,h}</math> is plateaued of amplitude <math>2^r</math>, resp. <math>2^{r+1}</math>.

Revision as of 13:57, 7 February 2019

Background and Definition

A Boolean function is said to be plateaued if its Walsh transform takes at most three distinct values, viz. and for some positive ineger called the amplitude of .

This notion can be naturally extended to vectorial Boolean functions by applying it to each component. More precisely, if is an -function, we say that is plateaued if all its component functions for are plateaued. If all of the component functions are plateaued and have the same amplitude, we say that is plateaued with single amplitude.

Equivalence relations

The class of functions that are plateaued with single amplitude is CCZ-invariant.

The class of plateaued functions is only EA-invariant.

Relations to other classes of functions

All bent and semi-bent Boolean functions are plateaued.

Any vectorial AB function is plateaued with single amplitude.

Constructions of Boolean plateaued functions

Primary constructons

Generalization of the Maiorana-MacFarland Functions [1]

The Maiorana-MacFarland class of bent functions can be generalized into the class of functions of the form

for , where and are any positive integers, , is arbitrary and is any Boolean function.

The Walsh transform of takes the value

at . If is injective, resp. takes each value in its image set two times, then is plateaued of amplitude , resp. .

  1. Camion P, Carlet C, Charpin P, Sendrier N. On Correlation-immune functions. InAdvances in Cryptology—CRYPTO’91 1992 (pp. 86-100). Springer Berlin/Heidelberg.