On the distribution of auto-correlation value of balanced Boolean functions

Yu Zhou

doi:10.3934/amc.2013.7.335

Article Contents

2013, Volume 7, Issue 3: 335-347. Doi: 10.3934/amc.2013.7.335

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On the distribution of auto-correlation value of balanced Boolean functions

Yu Zhou^1,

1.
Science and Technology on Communication Security Laboratory, Chengdu, Sichuan 610041

Received: November 2012

Revised: February 2013

Published: August 2013

Abstract / Introduction Related Papers Cited by

Abstract

In this paper, we study the lower bound on the sum-of-square indicator of balanced Boolean functions obtained by Son, et al. in 1998, and give a sufficient and necessary condition under which balanced Boolean functions achieve this lower bound. We introduce a new general class of balanced Boolean functions in $n$ variables $(n\geq 4)$ with optimal auto-correlation distribution, and we study two sub-classes more explicitely. Finally, we study the sets of Boolean functions having a same auto-correlation distribution, and derive a lower bound on the number of elements in such set.

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Mathematics Subject Classification: 06E30, 94A60.

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References

[1]	C. M. Adams and S. E. Tavares, Generating and counting binary bent sequences, IEEE Trans. Inform. Theory, 36 (1990), 1170-1173.doi: 10.1109/18.57222.
[2]	A. Canteaut, C. Carlet, P. Charpin and C. Fontaine, Propagation characteristics and correlation immunity of hightly nonlinear Boolean functions, in "EUROCRYPT 2000,'' (2000), 507-522.doi: 10.1007/3-540-45539-6_36.
[3]	A. Canteaut, C. Carlet, P. Charpin and C. Fontaine, On cryptographic properties of the coset of $R(1,m)$, IEEE Trans. Inform. Theory, 47 (2001), 1494-1513.doi: 10.1109/18.923730.
[4]	C. Carlet, Partially bent functions, in "Advances in Cryptology-Crypto'92,'' Springer-Verlag, (1993), 280-291.doi: 10.1007/3-540-48071-4_19.
[5]	C. Carlet, Boolean functions for cryptography and error correcting codes, in "Boolean Models and Methods in Mathematics, Computer Science, and Engineering'' (eds. Y. Crama and P. Hammer), Cambridge University Press, Cambridge, (2010), 257-397.doi: 10.1017/CBO9780511780448.011.
[6]	P. Charpin and E. Pasalic, On propagation characteristics of resilient functions, in "Selected Areas in Cryptography,'' Springer, Berlin, (2003), 175-195.doi: 10.1007/3-540-36492-7_13.
[7]	C. Ding and P. Sarkar, Personal communications, 2000.
[8]	G. Gong and K. Khoo, Additive autocorrelation of resilient Boolean functions, in "Selected Areas in Cryptography,'' Springer, Berlin, (2004), 275-290.doi: 10.1007/978-3-540-24654-1_20.
[9]	S. Hirose and K. Ikeda, Nonlinearity criteria of Boolean functions, Kyoto University, 1994, available online at http://fuee.u-fukui.ac.jp/~hirose/publication/kuis-tech-rep-94-0002.pdf
[10]	S. Maitra, Highly nonlinear balanced Boolean functions with very good autocorrelation property, in "Proc. Workshop on Coding and Cryptography 2001,'' Elsevier, (2001), 355-364.
[11]	B. Preneel, "Analysis and Design of Cryptographic Hash Functions,'' Ph.D thesis, Katholieke Universiteit te Leuven, 1993.
[12]	B. Preneel, W. Leekwijck, L. V. Linden, et al., Propagation characteristics of Boolean functions, in "Advances in Cryptology-Eurocrypt'90,'' Springer-Verlag, Berlin, (1991), 155-165.doi: 10.1007/3-540-46877-3_14.
[13]	J. J. Son, J. I. Lim, S. Chee and S. H. Sung, Global avalanche characteristics and nonlinearity of balanced Boolean functions, Inform. Proc. Letters, 65 (1998), 139-144.doi: 10.1016/S0020-0190(98)00009-X.
[14]	S. H. Sung, S. Chee and C. Park, Global avalanche characteristics and propagation criterion of balanced Boolean functions, Inform. Proc. Letters, 69 (1999), 21-24.doi: 10.1016/S0020-0190(98)00184-7.
[15]	A. F. Webster, "Plaintext/Ciphertext Bit Dependencies in Cryptographic System,'' Master's thesis, Dep. Electrical Engineering, Queen's University, Ontario, Cannada, 1985.
[16]	X. M. Zhang and Y. L. Zheng, GAC- the criterion for global avalanche characteristics of cryptographic functions, J. Universal Comp. Sci., 1 (1995), 316-333.
[17]	X. M. Zhang and Y. L. Zheng, Characterizing the structures of cryptographic functions satisfying the propagation criterion for almost all vectors, Des. Codes Crypt., 7 (1996), 111-134.doi: 10.1007/BF00125079.
[18]	Y. L. Zheng and X. M. Zhang, On the plateaued functoins, IEEE Trans. Inform. Theory, 47 (2001), 1215-1223.doi: 10.1109/18.915690.
[19]	Y. Zhou, M. Xie and G. Z. Xiao, On the global avalanche characteristics of two Boolean functions and the higher order nonlinearity, Inform. Sci., 180 (2010), 256-265.doi: 10.1016/j.ins.2009.09.012.
[20]	Y. Zhou, W. G. Zhang, J. Li, X. F. Dong and G. Z. Xiao, The autocorrelation distribution of balanced Boolean function, Frontier Comp. Sci., 7 (2013), 272-278.doi: 10.1007/s11704-013-2013-x.
[21]	Y. Zhou, W. Z. Zhang, S. X. Zhu and G. Z. Xiao, The global avalanche characteristics of two Boolean functions and algebraic immunity, Int. J. Comp. Math., 89 (2012), 2165-2179.doi: 10.1080/00207160.2012.712689.