Abstract
New criteria of extended resiliency and extended immunity of vectorial Boolean functions, such as S-boxes for stream or block ciphers, were recently introduced. They are related to a divide-and-conquer approach to algebraic attacks by conditional or unconditional equations. Classical resiliency turns out to be a special case of extended resiliency and as such requires more conditions to be satisfied. In particular, the algebraic degrees of classically resilient S-boxes are restricted to lower values. In this paper, extended immunity and extended resiliency of S-boxes are studied and many characterisations and properties of such S-boxes are established. The new criteria are shown to be necessary and sufficient for resistance against the divide-and-conquer algebraic attacks by conditional or unconditional equations.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Bennett, C.H., Brassard, G., Robert, J.M.: Privacy amplification by public discussion. SIAM J. Computing 17, 210–229 (1988)
Bierbrauer, J., Gopalakrishnan, K., Stinson, D.R.: Bounds on resilient functions and orthogonal arrays. In: Desmedt, Y.G. (ed.) CRYPTO 1994. LNCS, vol. 839, pp. 247–256. Springer, Heidelberg (1994)
Camion, P., Carlet, C., Charpin, P., Sendrier, N.: On correlation-immune functions. In: Feigenbaum, J. (ed.) CRYPTO 1991. LNCS, vol. 576, pp. 87–100. Springer, Heidelberg (1992)
Carlet, C.: Improving the algebraic immunity of resilient and nonlinear functions and constructing bent functions (2004), http://eprint.iacr.org/2004/276/
Carlet, C., Prouff, E.: Vectorial functions and covering sequences. In: Mullen, G.L., Poli, A., Stichtenoth, H. (eds.) Fq7 2003. LNCS, vol. 2948, pp. 215–248. Springer, Heidelberg (2004)
Cheon, J.H.: Nonlinear vector resilient functions. In: Kilian, J. (ed.) CRYPTO 2001. LNCS, vol. 2139, pp. 458–469. Springer, Heidelberg (2001)
Cheon, J.H., Lee, D.H.: Resistance of S-boxes against algebraic attacks. In: Roy, B., Meier, W. (eds.) FSE 2004. LNCS, vol. 3017, pp. 83–94. Springer, Heidelberg (2004)
Chor, B., Goldreich, O., Håstad, J., Friedman, J., Rudich, S., Smolensky, R.: The bit extraction problem or t-resilient functions. In: Proc. 26th IEEE Symp. on Foundations of Computer Science, pp. 396–407 (1985)
Courtois, N.: Fast algebraic attacks on stream ciphers with linear feedback. In: Boneh, D. (ed.) CRYPTO 2003. LNCS, vol. 2729, pp. 176–194. Springer, Heidelberg (2003)
Courtois, N., Meier, W.: Algebraic attacks on stream ciphers with linear feedback. In: Biham, E. (ed.) EUROCRYPT 2003. LNCS, vol. 2656, pp. 345–359. Springer, Heidelberg (2003)
Courtois, N., Pieprzyk, J.: Cryptanalysis of block ciphers with overdefined systems of equations. In: Zheng, Y. (ed.) ASIACRYPT 2002. LNCS, vol. 2501, pp. 267–287. Springer, Heidelberg (2002)
Dalai, D., Gupta, K., Maitra, S.: Results on algebraic immunity for cryptographically significant Boolean functions. In: Canteaut, A., Viswanathan, K. (eds.) INDOCRYPT 2004. LNCS, vol. 3348, pp. 92–106. Springer, Heidelberg (2004)
Dalai, D., Gupta, K., Maitra, S.: Results on algebraic immunity for cryptographically significant Boolean functions: Construction and analysis in term of algebraic immunity. In: Gilbert, H., Handschuh, H. (eds.) FSE 2005. LNCS, vol. 3557, pp. 98–111. Springer, Heidelberg (2005)
Friedman, J.: On the bit extraction problem. In: Proc. 33rd IEEE Symp. on Foundations of Computer Science, pp. 314–319 (1992)
Golić, J.D.: Vectorial Boolean functions and induced algebraic equations (2004), http://eprint.iacr.org/2004/225/
Gupta, K.C., Sarkar, P.: Improved construction of nonlinear resilient s-boxes. In: Zheng, Y. (ed.) ASIACRYPT 2002. LNCS, vol. 2501, pp. 466–483. Springer, Heidelberg (2002)
Hall Jr., M.: Combinatorial Theory. Ginn-Blaisdell, Waltham (1967)
Meier, W., Pasalic, E., Carlet, C.: Algebraic attacks and decomposition of Boolean functions. In: Cachin, C., Camenisch, J.L. (eds.) EUROCRYPT 2004. LNCS, vol. 3027, pp. 474–491. Springer, Heidelberg (2004)
O’Nan, M.: Linear Algebra. Harcourt Brace Jovanovich, New York (1976)
Pasalic, E., Maitra, S.: Further constructions of resilient Boolean functions with very high nonlinearity. IEEE Transactions on Information Theory 48(7), 1825–1834 (2002)
Sarkar, P., Maitra, S.: Nonlinearity bounds and constructions of resilient Boolean functions. In: Bellare, M. (ed.) CRYPTO 2000. LNCS, vol. 1880, pp. 515–532. Springer, Heidelberg (2000)
Siegenthaler, T.: Correlation-immunity of nonlinear combining functions for cryptographic applications. IEEE Transactions on Information Theory 30(5), 776–779 (1984)
Stinson, D.R.: Resilient functions and large sets of orthogonal arrays. Congressus Numerantium 92, 105–110 (1993)
Zhang, X.M., Zheng, Y.: Cryptographically resilient functions. IEEE Transactions on Information Theory 43(5), 1740–1747 (1997)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Pieprzyk, J., Zhang, XM., Golić, J.D. (2006). Characterisations of Extended Resiliency and Extended Immunity of S-Boxes. In: Won, D.H., Kim, S. (eds) Information Security and Cryptology - ICISC 2005. ICISC 2005. Lecture Notes in Computer Science, vol 3935. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11734727_19
Download citation
DOI: https://doi.org/10.1007/11734727_19
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-33354-8
Online ISBN: 978-3-540-33355-5
eBook Packages: Computer ScienceComputer Science (R0)