Nonlinear Dyn (2013) 73:1795–1801 DOI 10.1007/s11071-013-0904-x ORIGINAL PAPER An efficient technique for the construction of substitution box with chaotic partial differential equation Majid Khan · Tariq Shah · Muhammad Asif Gondal Received: 8 October 2012 / Accepted: 6 April 2013 / Published online: 26 April 2013 © Springer Science+Business Media Dordrecht 2013 Abstract In this manuscript, we proposed a novel for- mation of the nonlinear component of block cipher. The projected method is chaos based. We are merg- ing two different structures namely the Kuramoto– Sivashinsky equation as a chaotic system and use the Galois field (GF) as an algebraic structure. We design an innovative block cipher with the help of the planned chaotic scheme. We investigated some standard prop- erties of our proposed nonlinear component with al- ready existing standard results for block ciphers. The results of the analysis authenticate that the designed cryptosystem is reliable for secure communication. Keywords Chaos · S-boxes · Kuramoto–Sivashinsky equation 1 Introduction Chaos theory is an area of applied sciences which is applicable in numerous subjects comprise physics, engineering, coding and cryptography, fuzzy theory, M. Khan ( ) · T. Shah Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan e-mail: mk.cfd1@gmail.com M.A. Gondal Department of Sciences and Humanities, National University of Computer and Emerging Sciences, Islamabad, Pakistan economics, biology, stochastic theory, and philosophy. Chaos theory investigates the performance of those dy- namical systems which are extremely subtle to initial conditions. The slight alterations in initial situations yield extensively deviating results for chaotic systems. The deterministic nature of these systems does not make them predictable. Chaotic behavior can be de- tected in numerous natural systems, such as weather. Justification of such conduct may be pursued through examination of a chaotic mathematical model, itera- tive maps, and systems of differential equations. The fundamental effort that relates chaos and cryp- tography was given by Shannon [1]. The argotic in- volvement, unsystematic performance, and elemen- tary features of chaos that is subtlety to initial condi- tions, intersections with simple structures of cryptol- ogy. This relation is employed in the set-up of novel cryptosystems. The construction of substitution boxes (S-Box) is essential part in block ciphers found in chaotic systems [2–10]. In [11], Adam et al. suggested a novel design of S- box of size n × n. In [12], Detombe et al. designed S- box of size 5 × 5, which is resistant against differential cryptanalysis [13]. Even though the designed S-Box is resistant against differential cryptanalysis, it yields acceptable results only for those S-Boxes having odd-numbered dimen- sions [14, 15]. Jakimoski and Kocarev proposed a four-step method employing chaotic maps in order to generate S- Boxes [16]. The proposed method had good crypto-