Abstract
Highly nonlinear Boolean functions occupy an important position in the design of secure block as well as stream ciphers. This paper proves two new lower bounds on the nonlinearity of Boolean functions. Based on the study of these new lower bounds, we introduce a class of highly nonlinear Boolean functions on odd dimensional spaces and show examples of such functions.
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C. M. Adams and S. E. Tavares. Generating and counting binary bent sequences. IEEE Transactions on Information Theory, IT-36 No. 5:1170–1173, 1990.
K. G. Beauchamp. Applications of Walsh and Related Functions with an Introduction to Sequency Functions. Microelectronics and Signal Processing. Academic Press, London, New York, Tokyo, 1984.
Claude Carlet. Partially-bent functions. Designs, Codes and Cryptography, 3:135–145, 1993.
G. D. Cohen, M. G. Karpovsky, Jr. H. F. Mattson, and J. R. Schatz. Covering radius — survey and recent results. IEEE Transactions on Information Theory, IT-31(3):328–343, 1985.
J. F. Dillon. A survey of bent functions. The NSA Technical Journal, pages 191–215, 1972. (unclassified).
C. Ding, G. Xiao, and W. Shan. The Stability Theory of Stream Ciphers, volume 561 of Lecture Notes in Computer Science. Springer-Verlag, Berlin, Heidelberg, New York, 1991.
F. J. MacWilliams and N. J. A. Sloane. The Theory of Error-Correcting Codes. North-Holland, Amsterdam, New York, Oxford, 1978.
M. Matsui. Linear cryptanalysis method for DES cipher. In Advances in Cryptology — EUROCRYPT'93, volume 765, Lecture Notes in Computer Science, pages 386–397. Springer-Verlag, Berlin, Heidelberg, New York, 1994.
B. Preneel, W. V. Leekwijck, L. V. Linden, R. Govaerts, and J. Vandewalle. Propagation characteristics of boolean functions. In Advances in Cryptology — EUROCRYPT'90, volume 437, Lecture Notes in Computer Science, pages 155–165. Springer-Verlag, Berlin, Heidelberg, New York, 1991.
O. S. Rothaus. On “bent” functions. Journal of Combinatorial Theory, Ser. A, 20:300–305, 1976.
J. Seberry, X. M. Zhang, and Y. Zheng. Nonlinearity and propagation characteristics of balanced boolean functions. Information and Computation, 119(1):1–13, 1995.
T. Siegenthaler. Correlation-immunity of nonlinear combining functions for cryptographic applications. IEEE Transactions on Information Theory, IT-30 No. 5:776–779, 1984.
H. Tanaka and T. Kaneko. A linear attack to the random generator by non linear combiner. In Proceedings of 1996 IEEE International Symposium on Information Theory and Its Applications, volume 1, pages 331–334, Victoria, B.C., Canada, September 1996.
R. Yarlagadda and J. E. Hershey. Analysis and synthesis of bent sequences. IEE Proceedings (Part E), 136:112–123, 1989.
X. M. Zhang and Y. Zheng. GAC — the criterion for global avalanche characteristics of cryptographic functions. Journal of Universal Computer Science, 1(5):316–333, 1995. (available at http://hgiicm.tu-graz.ac.at/).
X. M. Zhang and Y. Zheng. Auto-correlations and new bounds on the nonlinearity of boolean functions. In Advances in Cryptology — EUROCRYPT'96, volume 1070, Lecture Notes in Computer Science, pages 294–306. Springer-Verlag, Berlin, Heidelberg, New York, 1996.
X. M. Zhang and Y. Zheng. Characterizing the structures of cryptographic functions satisfying the propagation criterion for almost all vectors. Design, Codes and Cryptography, 7(1/2):111–134, 1996. special issue dedicated to Gus Simmons.
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© 1997 Springer-Verlag Berlin Heidelberg
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Zhang, XM., Zheng, Y. (1997). New lower bounds on nonlinearity and a class of highly nonlinear functions. In: Varadharajan, V., Pieprzyk, J., Mu, Y. (eds) Information Security and Privacy. ACISP 1997. Lecture Notes in Computer Science, vol 1270. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0027922
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DOI: https://doi.org/10.1007/BFb0027922
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