Generalized nonlinearity of $ S$-boxes

Sugata Gangopadhyay; Goutam Paul; Nishant Sinha; Pantelimon Stǎnicǎ

doi:10.3934/amc.2018007

Article Contents

2018, Volume 12, Issue 1: 115-122. Doi: 10.3934/amc.2018007

This issue Previous Article Singleton bounds for R-additive codes Next Article Trace description and Hamming weights of irreducible constacyclic codes

Generalized nonlinearity of $ S$-boxes

1.
Department of Computer Science and Engineering, Indian Institute of Technology Roorkee, Roorkee 247667, India
2.
Cryptology and Security Research Unit, R. C. Bose Centre for Cryptology and Security, Indian Statistical Institute, Kolkata 700108, India
3.
Department of Applied Mathematics, Naval Postgraduate School, Monterey, CA 93943-5216, USA

* Corresponding author: Goutam Paul
* Corresponding author: Goutam Paul

Received: August 31, 2016

Published: February 2018

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Abstract

While analyzing $ S$-boxes, or vectorial Boolean functions, it is of interest to approximate its component functions by affine functions. In the usual attack models, it is assumed that all input vectors to an $ S$-box are equiprobable. The nonlinearity of an $ S$-box is defined, subject to this assumption. In this paper, we explore the possibility of linear cryptanalysis of an $ S$-box by introducing biased inputs and thus propose a generalized notion of nonlinearity along with a generalization of the Walsh-Hadamard spectrum of an $ S$-box.

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Mathematics Subject Classification: Primary: 06E30, 11T71; Secondary: 94A60.

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Table 1. Maximum bias without and with biased inputs for all DES S-boxes.

$F$	$S_1$	$S_2$	$S_3$	$S_4$	$S_5$	$S_6$	$S_7$	$S_8$
$\displaystyle \max_{{\bf{u}} \in {\mathbb{F}}_2^n}\epsilon({\bf{u}}, {\bf{v}} \cdot F)$	0.219	0.219	0.219	0.156	0.219	0.188	0.281	0.188
$\displaystyle \max_{{\bf{u}} \in {\mathbb{F}}_2^n} \epsilon^{(p)}_{{\mathcal{S}}}({\bf{u}}, {\bf{v}}\cdot F)$	0.494	0.494	0.497	0.489	0.494	0.491	0.494	0.494

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