Abstract
We apply autocorrelation and Walsh coefficients for the investigation of correlation immune and resilient Boolean functions. We prove new lower bound for the absolute indicator of resilient functions that improves significantly (for m > (n - 3)/2) the bound of Zheng and Zhang [18] on this value. We prove new upper bound for the number of nonlinear variables in high resilient Boolean function. This result supersedes the previous record. We characterize all possible values of resiliency orders for quadratic functions and give a complete description of quadratic Boolean functions that achieve the upper bound on resiliency. We establish new necessary condition that connects the number of variables, the resiliency and the weight of an unbalanced nonconstant correlation immune function and prove that such functions do not exist for m > 0.75n - 1.25. For high orders of m this surprising fact supersedes the well-known Bierbrauer-Friedman bound [8], [1] and was not formulated before even as a conjecture. We improve the upper bound of Zheng and Zhang [18] for the nonlinearity ofhigh order correlation immune unbalanced Boolean functions and establish that for high orders of resiliency the maximum possible nonlinearity for unbalanced correlation immune functions is smaller than for balanced.
Chapter PDF
Similar content being viewed by others
Keywords
References
J. Bierbrauer, Bounds on orthogonal arrays and resilient functions, Journal of Combinatorial Designs, V. 3, 1995, pp. 179–183.
J. Bierbrauer, K. Gopalakrishnan, D. R. Stinson, Orthogonal arrays, resilient functions, error correcting codes and linear programming bounds, SIAM Journal of Discrete Mathematics, V. 9, 1996, pp. 424–452.
R. Canetti, Y. Dodis, S. Halevi, E. Kushilevitz, A. Sahai, Exposure-resilient functions and all-or-nothing transforms, In Advanced in Cryptology: Eurocrypt 2000, Proceedings, Lecture Notes in Computer Science, V. 1807, 2000, pp. 453–469.
A. Canteaut, C. Carlet, P. Charpin, C. Fontaine, Propagation characteristics and correlation-immunity of highly nonlinear Boolean functions, In Advanced in Cryptology: Eurocrypt 2000, Proceedings, Lecture Notes in Computer Science, V. 1807, 2000, pp. 507–522.
C. Carlet, Partially-bent functions, In Advanced in Cryptology: Crypto 1992, Proceedings, Lecture Notes in Computer Science, V. 740, 1992, pp. 280–291.
B. Chor, O. Goldreich, J. Hastad, J. Friedman, S. Rudich, R. Smolensky, The bit extraction problem or t-resilient functions, IEEE Symposium on Foundations of Computer Science, V. 26, 1985, pp. 396–407.
W. Feller, An introduction to probability theory and its applications, John Wiley & Sons, New York, 3rd edition, 1968.
J. Friedman, On the bit extraction problem, Proc. 33rd IEEE Symposium on Foundations ofComputer Science, 1992, pp. 314–319.
Xiao Guo-Zhen, J. Massey, A spectral characterization ofcorrelation-imm une combining functions, IEEE Transactions on Information Theory, V. 34, No 3, May 1988, pp. 569–571.
V. Levenshtein, Split orthogonal arrays and maximum independent resilient systems off unctions, Designs, Codes and Cryptography, V. 12, 1997, pp. 131–160.
F. J. Mac Williams, N. J. A. Sloane, The theory oferror correcting codes, North-Holland, Amsterdam, 1977.
P. Sarkar, S. Maitra, Nonlinearity bounds and constructions of resilient Boolean functions, In Advanced in Cryptology: Crypto 2000, Proceedings, Lecture Notes in Computer Science, V. 1880, 2000, pp. 515–532.
T. Siegenthaler, Correlation-immunity of nonlinear combining functions for cryptographic applications, IEEE Transactions on Information theory, V. IT-30, No 5, 1984, p. 776–780.
Yu. Tarannikov, On resilient Boolean functions with maximal possible nonlinearity, Proceedings ofIndo crypt 2000, Lecture Notes in Computer Science, V. 1977, pp. 19–30, Springer-Verlag, 2000.
Yu. Tarannikov, New constructions of resilient Boolean functions with maximal nonlinearity, Preproceedings of8th Fast Software Encryption Workshop, Yokohama, Japan, April 2–4, 2001, pp.70–81.
Yu. Tarannikov, D. Kirienko, Spectral analysis ofhigh order correlation immune functions, Proceedings of 2001 IEEE International Symposium on Information Theory ISIT2001, Washington, DC, USA, June 2001, p. 69, full version is available at Cryptology ePrint archive (http://eprint.iacr.org/), Report 2000/050, October 2000, 8 pp.
X. M. Zhang, Y. Zheng, GAC — the criterion for global avalanche characteristics and nonlinearity ofcryptographic functions, Journal ofUniv ersal Computer Science, V. 1, 1995, pp. 136–150.
Y. Zheng, X. M. Zhang, Improved upper bound on the nonlinearity ofhigh order correlation immune functions, Selected Areas in Cryptography, 7th Annual International Workshop, SAC2000, Lecture Notes in Computer Science, V. 2012, pp. 264–274, Springer-Verlag, 2001.
Y. Zheng, X. M. Zhang, New results on correlation immune functions, The 3nd International Conference on Information Security and Cryptology (ICISC 2000), Seoul, Korea, Lecture Notes in Computer Science, V. 2015, pp. 49–63, Springer-Verlag, 2001.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Tarannikov, Y., Korolev, P., Botev, A. (2001). Autocorrelation Coefficients and Correlation Immunity of Boolean Functions. In: Boyd, C. (eds) Advances in Cryptology — ASIACRYPT 2001. ASIACRYPT 2001. Lecture Notes in Computer Science, vol 2248. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45682-1_27
Download citation
DOI: https://doi.org/10.1007/3-540-45682-1_27
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-42987-6
Online ISBN: 978-3-540-45682-7
eBook Packages: Springer Book Archive