Abstract
Symmetric property is a special property of Boolean functions, which has attracted much study on it. This chapter presents fast Walsh transforms of symmetric Boolean functions, correlation immunity of symmetric functions, construction of symmetric resilient Boolean functions, and some cryptographic properties of majority functions being a special class of symmetric Boolean functions. The study on the correlation immunity of majority functions shows that majority functions have good asymptotical behavior of correlation immunity, i.e., although they are not correlation immune, they have, however, asymptotical correlation immunity.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Braeken, A., Preneel, B.: On the algebraic immunity of symmetric Boolean functions. In: Proceedings of Indocrypt 2005. LNCS 3797, pp. 35–48. Springer, Berlin/Heidelberg (2005)
Canteaut, A., Videau, M.: Symmetric Boolean functions. IEEE Trans. Inf. Theory IT-51(8), 2791–2811 (2005)
Chor, B., Goldreich, O., Hastad, J., Friedman, J., Rudich, S., Smolensky, R.: The bit extraction problem or t-resilient functions. In: Proceedings of 26th IEEE Symposium on Foundations of Computer Science, Portland, pp. 396–407 (1985)
Courtois, N., Meier, W.: Algebraic attacks on stream ciphers with linear feedback. In: Advances in Cryptology – Proceedings of Eurocrypt’03. LNCS 2656, pp. 345–359. Springer, Berlin/New York (2003)
Dalai, D.K., Maitra, S., Sarkar, S.: Basic theory in construction of Boolean functions with maximum possible annihilator immunity. Des. Codes Cryptogr. 40(1), 41–58 (2006)
Ding, C., Shan, W., Xiao, G.: The Stability Theory of Stream Ciphers. LNCSÂ 561, Springer, Berlin/New York (1991)
Friedman, J.: On the bit extraction problem. In: Proceedings 33rd IEEE Symposium on Foundations of Computer Science, Pittsburgh, pp. 314–319 (1992)
Gopalakrishnan, K., Hoffman, D.G., Stinson, D.R.: A note on a conjecture concerning symmetric resilient functions. Inf. Process. Lett. 47, 139–143 (1993)
Li, N., Qi, W.-F.: Symmetric Boolean functions depending on an odd number of variables with maximum algebraic immunity. IEEE Trans. Infor. Theory IT-52(5), 2271–2273 (2006)
Maitra, S., Sarkar, P.: Maximum nonlinearity of symmetric boolean functions on odd number of variables. IEEE Trans. Inf. Theory IT-48(9), 2626–2630 (2002)
Meier, W., Staffelbach, O.: Nonlinearity criteria for cryptographic functions. In: Advances in Cryptology – Proceedings of Eurocrypt’89. LNCS 434, pp. 549–562. Springer, Berlin/Heidelberg (1990)
Meier, W., Pasalic, E., Carlet, C.: Algebraic attacks and decomposition of Boolean functions. In: Advances in Cryptology – Proceedings of Eurocrypt’04. LNCS 3027, pp. 474–491. Springer, Berlin/Heidelberg (2004)
Nyberg, K.: Linear approximation of block ciphers, In: Advances in Cryptology – Proceedings of Eurocrypt’94. LNCS 950, pp. 439–444. Springer, Berlin/Heidelberg (1995)
Qu, C., Seberry, J., Pieprzyk, J.P.: On the symmetric property of homogeneous Boolean functions. In: Proceedings of Australian Conference on Information Security and Privacy (ACISP’99). LNCS 1587, pp. 26–35. Springer, Berlin/Heidelberg (1999)
Shannon, C.E.: Communication theory of secrecy systems. Bell Syst. Tech. J. 28, 59–88 (1949)
Siegenthaler, T.: Correlation-immunity of nonlinear combining functions for cryptographic applications. IEEE Trans. Inf. Theory IT-30(5), 776–780 (1984)
Siegenthaler, T.: Decrypting a class of stream ciphers using ciphertext only. IEEE Trans. Comput. C-34(1), 81–85 (1985)
Siegenthaler, T.: Cryptanalysts’ representation of nonlinearly filtered m-sequences. In: Advances in Cryptology – Proceedings of Eurocrypt’85. LNCS 219, pp. 103–110. Springer, Heidelberg (1986)
Siegenthaler, T.: Design of combiners to prevent divide and conquer attacks. In: Advances in Cryptology, Proceedings of Crypto’85. LNCS 218, pp. 237–279. Springer, Berlin/Heidelberg/New York (1986)
Stinson, D.R., Massey, J.L.: An infinite class of counterexamples to a conjecture concerning nonlinear resilient functions. J. Cryptol. 8, 167–173 (1995)
Stockmeyer, L.J.: On the combinational complexity of certain symmetric Boolean functions. Math. Syst. Theory 10, 323–336 (1977)
Wu, C.K., Dawson, E.: Correlation immunity and resiliency of symmetric Boolean functions. Theor. Comput. Sci. 312, 321–335 (2004)
Xiao, G.Z., Massey, J.L.: A spectral characterization of Correlation-immune combining functions. IEEE Trans. Inf. Theory IT-34(3), 569–571 (1988)
Zhao, Y., Li, H.: On bent functions with some symmetric properties. Discret. Appl. Math. 154, 2537–2543 (2006)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Wu, CK., Feng, D. (2016). The Symmetric Property of Boolean Functions. In: Boolean Functions and Their Applications in Cryptography. Advances in Computer Science and Technology. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48865-2_6
Download citation
DOI: https://doi.org/10.1007/978-3-662-48865-2_6
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-48863-8
Online ISBN: 978-3-662-48865-2
eBook Packages: Computer ScienceComputer Science (R0)