Abstract
Substitution boxes, aka S-boxes, are a key component of modern crypto-systems. Several studies and developments were carried out on the problem of building high-quality S-boxes in the last few years. Qualities of such boxes, such as nonlinearity and balance, steer the robustness of modern block ciphers. This work is concerned with the construction of highly nonlinear balanced Boolean functions. A deterministic optimization model which is the minimization of a polyhedral convex function on a convex polytope with 0–1 variables is introduced. A local deterministic optimization approach called DCA (Difference of Convex functions Algorithm) is investigated. For finding a good starting point of DCA we propose two versions of a combined DCA–GA (Genetic Algorithm) method. Numerical simulations prove that DCA is a promising approach for this problem. Moreover the combination of DCA–GA improves the efficiency of DCA and outperforms other standard approaches.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Canteaut, A., Carlet, C., Charpin, P., Fontaine, C.: Propagation characteristic and correlation-immunity of hight nonlinenar Boolean function. Advances in Cryptographie—EUROCRYPT 2000, LNCS, vol. 1807 Springer (2000)
Clark, J.A.: Optimisation heuristics for cryptology. PhD thesis, Faculty of Information Technology, Queensland University of Technology (1998)
Clark, J.A., Jeremy, J.L., Stepney, S.: The design of S-boxes by simulated annealing, CEC 2004: International Conference on Evolutionary Computation, Portland OR, June (2004), pp. 1533–1537. IEEE (2004)
Le Thi H.A., Pham D.T.: Solving a class of linearly constrained indefinite quadratic problems by dc algorithms. J. Glob. Optim. 11(3), 253–285 (1997)
Le Thi H.A, Pham D.T.: A continuous approach for globally solving linearly constrained quadratic zero-one programming problems. Optimization 50, 93–120 (2001)
Le Thi H.A., Pham D.T.: The DC (difference of convex functions) programming and DCA revisited with DC models of real world nonconvex optimization problems. Ann. Oper. Res. 133, 23–46 (2005)
Le Thi, H.A., Pham, D.T., Huynh, V.N.: Exact penalty techniques in DC programming. Technical Report LMI-INSA Rouen, pp. 1–28 (2005)
Menezes A., van Oorschot P., Vanstone S.: Handbook of Applied Cryptography. CRC Press, Boca Raton (1996)
Michalewicz Z.: Genetic Algorithms + Data Structures = Evolution Programs. Springer, New York (1994)
Pham, D.T., Le Thi, H.A.: Convex analysis approach to DC programming: theory, algorithms and applications. Acta Mathematica Vietnamica, dedicated to Professor Hoang Tuy on the occasion of his 70th birthday, 22, pp. 289–355 (1997)
Pham D.T., Le Thi H.A.: DC optimization algorithms for solving the trust region subproblem. SIAM J. Optim. 8, 476–505 (1998)
Reeves, C.R., Rowe, J.E.: Genetic Algorithms-Principles and Perspectives. ISBN: 0306480506, Kluwer (2002)
Sarkar, P., Maitra, S.: Construction of nonlinear Boolean functions with important cryptographic properties. In: Advances in Cryptographie–EUROCRYPT 2000, Lecture Note in Computer Science, vol. 1807 pp. 485–506. Springer (2000)
Seberry, J., Zhang, X.M., Zheng, Y.: Nonlinearly balanced Boolean functions and their propagation characteristics. Advances in Cryptographie—EUROCRYPT 2000 In: Advances in Cryptology—CRYPT0’93, Springer (1994)
Stallings W.: Cryptography and Network Security. 3rd edn. Prentice Hall, Englewood Cliffs (2003)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Le, H.M., Le Thi, H.A., Pham Dinh, T. et al. A combined DCA: GA for constructing highly nonlinear balanced boolean functions in cryptography. J Glob Optim 47, 597–613 (2010). https://doi.org/10.1007/s10898-009-9481-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10898-009-9481-4