Substitution boxes (S-boxes) are important components in many modern-day symmetric key ciphers. Their study has attracted a great deal of attention over many years. The emergence of a variety of cryptosystem attacks has shown that substitutions must be designed with great care. Some general criteria such as high non-linearity and low autocorrelation have been proposed (providing some protection against attacks such as linear cryptanalysis and differential cryptanalysis). The design of appropriate S-boxes is a difficult task; several criteria must be traded off and the design space is huge. There has been little application of evolutionary search to the development of S-boxes. In this paper we show how a cost function that has found excellent single-out put Boolean functions can be generalised to provide improved results for small S-boxes.
This is a preview of subscription content,to check access.
Access this article
Biham, E. and Shamir, A., “Differential Cryptanalysis of DES-like Cryptosystems (Extended Abstract)”, inAdvances in Cryptology—Crypto’90 (Menezes, A.J. and Vanstone, S.A. eds.),LNCS, 537, pp. 2–21, Springer-Verlag, Berlin, 1990.
Clark, J. A. and Jacob, J. L., “Two Stage Optimisation in the Design of Boolean Functions”, in5th Australasian Conference on Information Security and Privacy (Dawson, E., Clark, A. and Boyd, C. eds.),ACIS 2000, LNCS,1841, pp. 242–254. Springe Verlag, July 2000.
Clark, J.A., Jacob, J.L., Maitra, S. and Stânicâ, P., “Almost Boolean functions: the Design of Boolean Functions by Spectral Inversion”,Computational Intelligence, 20, 3, pp. 450–462, August2004.
Clark, J.A., Jacob, J.L. and Stepney, S., “Searching for Cost Functions”, inConference on Evolutionary Computation-CEC-04, July 2004.
Clark, J.A., Jacob, J.L., Stepney, S., Maitra, S. and Millan, W., “Evolving Boolean Functions Satisfying Multiple Criteria”, inProgress in Cryptology—INDOCRYPT 2002, LNCS, 2551, pp. 246–259, Springer Verlag, 2002.
Heys, H. M., “A Tutorial on Linear and Differential Cryptanalysis”,Technical report, Electrical and Computer Engineering, University of Newfoundland, St. John’s, Newfoundland, Canada, 2004.
Kirkpatrick, S., Gelatt, Jr. C. D. and Vecchi, M. D., “Optimization by Simulated Annealing”,Science, 220, 4598, pp. 671–680,May 1983.
Matsui, M., “Linear Cryptanalysis Method for DES Cipher”, inAdvances in Cryptology—EuroCrypt ’93 (Helleseth, T. ed),LNCS, 765, pp. 386–397, Springer-Verlag, Berlin, 1993.
Millan, W., “How to Improve the Non-linearity of Bijective S-boxes”, in3rd Australian Conference on Information Security and Privacy (Boyd, C. and Dawson, E. eds.),LNCS, 1438, pp. 181–192, Springer-Verlag, April 1998.
Millan, W., Burnett, L., Carter, G., Clark, A. and Dawson, E., “Evolutionary Heuristics for Finding Cryptographically Strong S-Boxes”, inICICS 99, LNCS, 1726, pp. 263–274, Springer-Verlag, 1999.
Millan, W., Clark, A. and Dawson, E., “Smart Hill Climbing Finds Better Boolean Functions”, inWorkshop on Selected Areas on Cryptography, SAC 97, pp. 50–63, 1997.
Millan, W., Clark, A. and Dawson, E., “Boolean Function Design Using Hill Climbing Methods”, in4th Australian Conference on Information Security and Privacy (Schneier, B. ed),LNCS, 1587, pp. 1–11, Springer-Verlag, April 1999.
Millan, W., Clark, A. and Dawson, E., “Heuristic Design of Cryptographically Strong Balanced Boolean Functions”, inAdvances in Cryptology EUROCRYPT’98, LNCS, 1403, pp. 489–499, Springer-Verlag, 1998.
National Bureau of Standards, Data Encryption Standard,NBS FIPS PUB 46, 1976.
John A. Clark: He is Professor of Critical Systems at the University of York, where he leads the software testing, security and cryptography work. Much of this has been concerned with the application of meta-heuristic search.
Jeremy L. Jacob: He has a BSc. in Mathematics from the University of Hull, England, M.Sc. and D. Phil. in Computation from the University of Oxford, England and now works for the Univerity of York. His research interests include modelling secure systems and software engineering practices for secure systems.
Susan Stepney: She is Professor of computer Science at the University of York, and leads the Non-Standard Computation research group there. She is a member of the ACM, Fellow of the British Computer Society, and moderator of the UKCRC Grand Challenge in Non-Classical Computation. Her main research interests include novel applications of nature-inspired computation, modelling self-organising complex systems and designing and reasoning about emergent properties.
About this article
Cite this article
Clark, J.A., Jacob, J.L. & Stepney, S. The design of S-boxes by simulated annealing. New Gener Comput 23, 219–231 (2005). https://doi.org/10.1007/BF03037656