Abstract
We develop an improved cost function to be used in simulated annealing followed by hill-climbing to find Boolean functions satisfying multiple desirable criteria such as high nonlinearity, low autocorrelation, balancedness, and high algebraic degree. Using this cost function that does not necessitate experimental search for parameter tuning, the annealing-based algorithm reaches the desired function profiles more rapidly. Some Boolean functions of eight and nine variables have been found, which are unattained in the computer search based literature, in terms of joint optimization of nonlinearity and autocorrelation. Global characteristics of eight-variable Boolean functions generated by algebraic construction or computer search are compared with respect to the sum-of-squared-errors in their squared spectra, which is also proportional to the sum-of-squared-errors in their autocorrelation function, the term ‘error’ denoting the deviation from bent function characteristics. Preliminary results consisting of cryptographically strong Boolean functions of nine, ten and eleven variables obtained using a three-stage optimization technique are also presented.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Cauteaut, A., Carlet, C., Charpin, P., Fontaine, C.: Propagation characteristics and correlation immunity of highly onlinear Boolean functions. In: Preneel, B. (ed.) EUROCRYPT 2000. LNCS, vol. 1807, pp. 507–522. Springer, Heidelberg (2000)
Clark, J.A., Jacob, J.L.: Two-Stage Optimisation in the Design of Boolean Functions. In: Clark, A., Boyd, C., Dawson, E.P. (eds.) ACISP 2000. LNCS, vol. 1841, pp. 242–254. Springer, Heidelberg (2000)
Clark, J.A., Jacob, J.L., Stepney, S., Maitra, S., Millan, W.: Evolving Boolean Functions Satisfying Multiple Criteria. In: Menezes, A., Sarkar, P. (eds.) INDOCRYPT 2002. LNCS, vol. 2551, pp. 246–259. Springer, Heidelberg (2002)
Clark, J.A.: Metaheuristic Search as a Cryptological Tool. D.Phil. Thesis. YCSY-2002-07. Department of Computer Science, University of York, York UK (December 2001), http://www.cs.york.ac.uk/ftpdir/reports
Dobbertin, H.: Construction of bent functions and balanced functions with high nonlinearity. In: Preneel, B. (ed.) FSE 1994. LNCS, vol. 1008, pp. 61–74. Springer, Heidelberg (1995)
Hou, X.D.: On the Norm and Covering Radius of First-Order Reed-Muller Codes. IEEE Transactions on Information Theory 43(3), 1025–1027 (1997)
Kirkpatrick, S., Gelatt Jr., C.D., Vecchi, M.P.: Optimization by simulated annealing. Science 220(4598), 671–680 (1983)
Maitra, S., Pasalic, E.: Further constructions of resilient Boolean functions with very high nonlinearity. IEEE Transactions on Information Theory 48(7), 1825–1834 (2002)
Maitra, S.: Highly nonlinear balanced Boolean functions with very good autocorrelation property. In: Workshop on Coding and Cryptograhy. WCC 2001, Paris, January 8-12. Electronic Notes in Discrete Mathematics, vol. 6, Elsevier Science, Amsterdam (2001)
Maitra, S.: Highly nonlinear balanced Boolean functions with good local and global avalanche characteristics. Information Processing Letters 83, 281–286 (2002)
Millan, W., Clark, A., Dawson, E.: An effective genetic algorithm for finding highly nonlinear Boolean functions. In: Han, Y., Quing, S. (eds.) ICICS 1997. LNCS, vol. 1334, pp. 149–158. Springer, Heidelberg (1997)
Millan, W., Clark, A., Dawson, E.: Heuristic Design of Cryptographically Strong Balanced Boolean Functions. In: Nyberg, K. (ed.) EUROCRYPT 1998. LNCS, vol. 1403, pp. 489–499. Springer, Heidelberg (1998)
Millan, W., Clark, A., Dawson, E.: Boolean function design using hill climbing methods. In: Pieprzyk, J.P., Safavi-Naini, R., Seberry, J. (eds.) ACISP 1999. LNCS, vol. 1587, pp. 1–11. Springer, Heidelberg (1999)
Patterson, N.J., Wiedemann, D.H.: The covering radius of the (215, 16) Reed-Muller code is at least 16276. IEEE Transactions on Information Theory IT-29(3), 354–356 (1983), see correction IT-36(2), 443 (1990)
Rothaus, O.S.: On Bent Functions. Journal of Combinatorial Theory, 300–305 (1976)
Stanica, P., Sung, S.H.: Improving the nonlinearity of certain balanced Boolean functions with good local and global avalanche characteristics. Information Processing Letters 79(4), 167–172 (2001)
Yücel, M.D.: Alternative Nonlinearity Criteria for Boolean Functions. Electrical and Electronics Engineering Departmental Memorandum, No.2001-1, Middle East Technical University (ODTÜ), Ankara, Türkiye, 20 pages (January 2001)
Zhang, X.M., Zheng, Y.: GAC - the criterion for global avalanche characteristics of cryptographic functions. Journal for Universal Computer Science 1(5), 316–333 (1995)
Zheng, Y., Zhang, X.M.: Improved upper bound on the nonlinearity of high order correlation immune functions. In: Stinson, D.R., Tavares, S. (eds.) SAC 2000. LNCS, vol. 2012, pp. 264–274. Springer, Heidelberg (2001)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Kavut, S., Yücel, M.D. (2003). Improved Cost Function in the Design of Boolean Functions Satisfying Multiple Criteria. In: Johansson, T., Maitra, S. (eds) Progress in Cryptology - INDOCRYPT 2003. INDOCRYPT 2003. Lecture Notes in Computer Science, vol 2904. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24582-7_9
Download citation
DOI: https://doi.org/10.1007/978-3-540-24582-7_9
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-20609-5
Online ISBN: 978-3-540-24582-7
eBook Packages: Springer Book Archive