Shifted plateaued functions and their differential properties
- PDF / 365,777 Bytes
- 15 Pages / 439.642 x 666.49 pts Page_size
- 83 Downloads / 190 Views
Shifted plateaued functions and their differential properties ˘ 2 ¨ Anbar1,2 · Canan Kas¸ıkc¸ı2 · Wilfried Meidl3 · Alev Topuzoglu Nurdagul Received: 28 November 2018 / Accepted: 27 February 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract A bent4 function is a Boolean function with a flat spectrum with respect to a certain unitary transform T . It was shown previously that a Boolean function f in an even number of variables is bent4 if and only if f + σ is bent, where σ is a certain quadratic function depending on T . Hence bent4 functions are also called shifted bent functions. Similarly, a Boolean function f in an odd number of variables is bent4 if and only if f + σ is a semibent function satisfying some additional properties. In this article, for the first time, we analyse in detail the effect of the shifts on plateaued functions, on partially bent functions and on the linear structures of Boolean functions. We also discuss constructions of bent and bent4 functions from partially bent functions and study the differential properties of partially bent4 functions, unifying the previous work on partially bent functions. Keywords Plateaued function · Bent4 function · Partially bent function · Relative difference set Mathematics Subject Classification (2010) 06E30 · 05B10
Canan Kas¸ıkc¸ı
[email protected] Nurdag¨ul Anbar [email protected] Wilfried Meidl [email protected] Alev Topuzo˘glu [email protected] 1
Johannes Kepler University, Altenbergerstrasse 69, 4040, Linz, Austria
2
Sabancı University, MDBF, Orhanlı, Tuzla, 34956 Istanbul, Turkey
3
Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Altenbergerstrasse 69, 4040, Linz, Austria
Cryptography and Communications
1 Introduction Let f : F2n → F2 be a Boolean function and c be an element in F2n . The unitary transform Vfc : F2n → C is defined in [2] as c Vfc (u) = (−1)f (x)+σ (x) i Trn (cx) (−1)Trn (ux) , (1.1) x∈F2n
where Trn (z) denotes the absolute trace of z ∈ F2n and σ c (x) is the Boolean function defined by i j (cx)2 (cx)2 . (1.2) σ c (x) = 0≤i k, 2k ≥ 2(t + 1), s = t − k, c = (c1 , c2 )∈ Fk2 ×Ft2 fixed (without loss of generality we assume c2 = 0, otherwise we permute the variables), and let S = {v0 , v1 , . . . , vt } ⊂ Ft2 satisfy the properties in Lemma 3.5. Our objective is to choose an injection π such that the Maiorana-McFarland s-plateaued function g(x, y) = π(x)·y from Fk2 ×Ft2 → F2 satisfies the following: g has trivial linear space (g) = {0}, which we can accomplish by including S in Im(π ), the shift is a c-s-plateaued function respectively a c-(s − 1)-plateaued function.
-
To obtain a shift which is c-s-plateaued, by Lemma 3.2 we require |Wg (β, γ )| = |Wg (β + c1 , γ + c2 )| which holds if and only if for all γ ∈ Ft2 , either γ , γ + c2 ∈ Im(π ) or γ , γ + c2 ∈ Im(π ). Such π exists since Ft2 can be expressed as the disjoint union of the sets {v, v + c2 }, v∈ Ft2 . Note that we have to include the elements vi + c2 , 0 ≤ i
Data Loading...
