PANG Shanqi, WANG Jing, WANG Xunan, WANG Xiaoli. Application of Orthogonal Array and Walsh Transform in Resilient Function[J]. Chinese Journal of Electronics, 2018, 27(2): 281-286. doi: 10.1049/cje.2017.09.011
 Citation: PANG Shanqi, WANG Jing, WANG Xunan, WANG Xiaoli. Application of Orthogonal Array and Walsh Transform in Resilient Function[J]. Chinese Journal of Electronics, 2018, 27(2): 281-286.

# Application of Orthogonal Array and Walsh Transform in Resilient Function

##### doi: 10.1049/cje.2017.09.011
Funds:  This work is supported by the National Natural Science Foundation of China (No.11571094, No.11171093).
• Rev Recd Date: 2017-06-07
• Publish Date: 2018-03-10
• The relationship between Walsh transform of a Boolean function and the orthogonality of some columns of its support table is investigated. This result improves the characterization of the orthogonality of Orthogonal array (OA). Stinson and Massey gave two construction methods of linear resilient functions, one is to use linear code, the other is to utilize the large set of orthogonal arrays and right cosets. And another major contribution is to show the equivalence of two methods.
•  W. Liang, X. Zeng and X. Yunge, "The periods of a class of nonlinear feedback shift register sequences", Chinese Journal of Electronics, Vol.25, No.2, pp.296-303, 2016. J. Du, Q. Wen, J. Zhang, et al., "Constraction and count of 1-resilient rotation symmetric Boolean functions on pr variables", Chinese Journal of Electronics, Vol.23, No.4, pp.816-820, 2014. Z. Zhuo, J. Chong and S. Wei, "Some properties of correlation function on generalized Boolean functions", Chinese Journal of Electronics, Vol.24, No.1, pp.166-169, 2015. Q. Wen, X. Niu and Y. Yang, The Boolean Functions in Modern Cryptology, Science Press, Beijing, China, pp.46-94, 2000. (in Chinese) S. Fu, C. Li, K. Matsuura, et al., "Construction of rotation symmetric Boolean functions with maximum algebraic immunity", International Conference on Cryptology and Network Security, Kanazawa, Japan, pp.402-412, 2009. S. Fu, L. Qu, C. Li, et al., "Balanced rotation symmetric Boolean functions with maximum algebraic immunity", IET Information Security, Vol.5, No.2, pp.93-99, 2011. K. Kurosawa, T. Satoh and K. Yamamoto, "Highly nonlinear t-resilient functions", Journal of Universal Computer Science, Vol.3, No.6, pp.721-729, 1997. P. Ke, L. Huang and S. Zhang, "Improved lower bound on the number of balanced symmetric functions over GF (p)", Information Sciences, Vol.179, No.5, pp.682-687, 2009. B. Chor, O. Goldreich, J. Hasted, et al., "The bit extraction problem or t-resilient functions", IEEE Symp. on Foundations of Computer Science, Portland, OR, USA, Vol.26, pp.396-407, 1985. C.H. Bennett, G. Brassard and J.M. Robert, "Privacy amplification by public discussion", SIAM Journal on Computing, Vol.17, No.2, pp.210-229, 1988. Y. Crama and P.L. Hammer, Boolean Models and Methods in Mathematics, Computer Science, and Engineering, Cambridge University Press, England, pp.257-397, 2010. Z. Saber, M.F. Uddin and A. Youssef, "On the existence of (9,3,5,240) resilient functions", IEEE Transactions on Information Theory, Vol.52, No.5, pp.2269-2270, 2006. K. Gopalakrishnan, D.G. Hoffman and D.R. Stinson, "A note on a conjecture concerning symmetric resilient functions", Information Processing Letters, Vol.47, No.3, pp.139-143, 1993. X. Zhang and Y. Zheng, "Cryptographically resilient functions", IEEE Transactions on Information Theory, Vol.43, No.5, pp.1740-1747, 1997. L. Chen and F. Fu, "On the constructions of new resilient functions from old ones", IEEE Transactions on Information Theory, Vol.45, No.6, pp.2077-2082, 1999. X. Li, Q. Zhou and H. Qian, "Balanced 2p-variable rotation symmetric Boolean functions with optimal algebraic immunity, good nonlinearity, and good algebraic degree", Journal of Mathematical Analysis and Applications, Vol.403, No.1, pp.63-71, 2013. J. Du, S. Fu, L. Qu, et al., "New constructions of q-variable 1-resilient rotation symmetric functions over Fp", Science China Information Sciences, Vol.59, No.7, pp.1-3, 2016. T. Beth, D. Jungnickel and H. Lenz, Design Theory, Cambridge University Press, England, pp.62-455, 1999. C.J. Colbourn and J.H. Dinitz, The CRC Handbook of Combinatorial Designs, Chapman and Hall, Boca Raton, USA, 2007. A.S. Hedayat, N.J.A. Sloane and J. Stufken, Orthogonal Arrays:Theory and Applications, Springer-Verlag, New York, USA, pp.302-305, 1999. C.J. Colbourn, S.S. Martirosyan, G.L. Mullen, et al., "Products of mixed covering arrays of strength two", Journal of Combinatorial Designs, Vol.14, No.2, pp.124-138, 2006. M. Rötteler and P. Wocjan, "Equivalence of decoupling schemes and orthogonal arrays", IEEE Transactions on Information Theory, Vol.52, No.9, pp.4171-4181, 2006. P. Wocjan, M. Rötteler, D. Janzing, et al., "Simulating Hamiltonians in quantum networks:Efficient schemes and complexity bounds", Physical Review A, Vol.65, No.4, pp.1-10, 2002. S. Pang, Y. Wang, J. Du, et al., "Iterative constructions of orthogonal arrays of strength t and orthogonal partitions", IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, Vol.E100-A, No.1, pp.308-311, 2017. Y. Zhang, S. Pang, Z. Jiao, et al., "Group partition and systems of orthogonal idempotents", Linear Algebra and Its Applications, Vol.278, No.1-3, pp.249-262, 1998. Y. Zhang, Y. Lu and S. Pang, "Orthogonal arrays obtained by orthogonal decomposition of projection matrices", Statistica Sinica, Vol.9, No.2, pp.595-604, 1999. S. Zhao, P. Li, R. Zhang, et al., "Construction of blocked two-level regular designs with general minimum lower order confounding", Journal of Statistical Planning and Inference, Vol.143, No.6, pp.1082-1090, 2013. Y. Zhang, S. Pang and Y. Wang, "Orthogonal arrays obtained by the generalized Hadamard product", Discrete Math, Vol.238, No.1-3, pp.151-170, 2001. S. Pang, "Construction of a new class of orthogonal arrays", Journal of Systems Science and Complexity, Vol.20, No.3, pp.429-436, 2007. S. Pang, Y. Zhu and Y. Wang, "A class of mixed orthogonal arrays obtained from projection matrix inequalities", Journal of Inequalities and Applications, Vol.2015, No.1, pp.1-9, 2015. D.R. Stinson and J.L. Massey, "An infinite class of counterexamples to a conjecture concerning nonlinear resilient functions", Journal of Cryptology, Vol.8, No.3, pp.167-173, 1995. D.R. Stinson, "Resilient functions and large sets of orthogonal arrays", Congressus Numerantium, Vol.92, pp.105-110, 1993. F.J. MacWilliams and N.J.A. Sloane, The Theory of ErrorCorrecting Codes, Elsevier, North Holland, pp.26-27, 1977. P. Sarkar and S. Maitra, "Nonlinearity bounds and constructions of resilient Boolean functions", Advances in CryptologyCRYPTO 2000, Springer, Berlin Heidelberg, Germany, pp.515-532, 2000. G.Z. Xiao and J.L. Massey, "A spectral characterization of correlation-immune combining functions", IEEE Transactions on Information Theory, Vol.34, No.3, pp.569-571, 1988.

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