Abstract
We establish, for the first time, an explicit and simple lower bound on the nonlinearity N f of a Boolean function f of n variables satisfying the avalanche criterion of degree p, namely, Nf≥ 2n-1 . 2n-1-1/2p. We also show that the lower bound is tight, and identify all the functions whose nonlinearity attains the lower bound. As a further contribution of this paper, we prove that except for very few cases, the sum of the degree of avalanche and the order of correlation immunity of a Boolean function of n variables is atmost n-2. These new results further highlight the significance of the fact that while avalanche property is in harmony with nonlinearity, it goes against correlation immunity.
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E. Biham and A. Shamir. Differential cryptanalysis of DES-like cryptosystems. Journal of Cryptology, Vol. 4, No. 1:3–72, 1991.
P. Camion, C. Carlet, P. Charpin, and N. Sendrier. On correlation-immune functions. In Advances in Cryptology-CRYPTO’91, volume 576 of Lecture Notes in Computer Science, pages 87–100. Springer-Verlag, Berlin, Heidelberg, New York, 1991.
C. Carlet and P. Codes. On the propagation criterion of degree l and order k. InAdvances in Cryptology-EUROCRYPT’98, volume 1403 of Lecture Notes in Computer Science, pages 462–474. Springer-Verlag, Berlin, Heidelberg, New York, 1998.
Claude Carlet. Partially-bent functions. Designs, Codes and Cryptography, 3:135–145, 1993.
D. Coppersmith. The development of DES, 2000. (Invited talk at CRYPTO2000).
H. Feistel. Cryptography and computer privacy. Scientific American, 228(5):15–23, 1973.
**ao Guo-Zhen and J. L. Massey. A spectral characterization of correlationimmune combining functions. IEEE Transactions on Information Theory, 34(3):569–571, 1988.
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 DESc cipher. In Advances in Cryptology-EUROCRYPT’93, volume 765of Lecture Notes in Computer Science, pages 386–397. Springer-Verlag, Berlin, Heidelberg, New York, 1994.
W. Meier and O. Staffelbach. Nonlinearity criteria for cryptographic functions. In Advances in Cryptology-EUROCRYPT’89, volume 434 of Lecture Notes in Computer Science, pages 549–562. Springer-Verlag, Berlin, Heidelberg, New York, 1990.
K. Nyberg. On the construction of highly nonlinear permutations. InAdvances in Cryptology-EUROCRYPT’92, volume 658 of Lecture Notes in Computer Science, pages 92–98. Springer-Verlag, Berlin, Heidelberg, New York, 1993.
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 437of 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, S er. A, 20:300–305, 1976.
J. Seberry, X. M. Zhang, and Y. Zheng. On constructions and nonlinearity of correlation immune functions. In Advances in Cryptology-EUROCRYPT’93, volume 765 of Lecture Notes in Computer Science, pages 181–199. Springer-Verlag, Berlin, Heidelberg, New York, 1994.
J. Seberry, X. M. Zhang, and Y. Zheng. Nonlinearity and propagation characteristics of balanced boolean functions. Information and Computation, 119(1):1–13, 1995.
C. E. Shannon. Communications theory of secrecy system. Bell Sys. Tech. Journal, Vol. 28:656–751, 1949.
T. Siegenthaler. Correlation-immunity of nonlinear combining functions for cryptographic applications. IEEE Transactions on Information Theory, IT-30 No. 5:776–779, 1984.
A. F. Webster and S. E. Tavares. On the design of S-boxes. In Advances in Cryptology-CRYPTO’85, volume 219 of Lecture Notes in Computer Science, pages 523–534. Springer-Verlag, Berlin, Heidelberg, New York, 1986.
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. Auto-correlations and new bounds on the nonlinearity of boolean functions. In Advances in Cryptology-EUROCRYPT’96, volume 1070 of 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.
X. M. Zhang and Y. Zheng. Cryptographically resilient functions. IEEE Transactions on Information Theory, 43(5):1740–1747, 1997.
X. M. Zhang and Y. Zheng. On plateaued functions. IEEE Transactions on Information Theory, 2000. (accepted).
Y. Zheng and X. M. Zhang. Plateaued functions. In Advances in Cryptology-ICICS’99, volume 1726of Lecture Notes in Computer Science, pages 284–300. Springer-Verlag, Berlin, Heidelberg, New York, 1999.
Y. Zheng and X. M. Zhang. Improved upper bound on the nonlinearity of high order correlation immune functions. In Selected Areas in Cryptography, 7th Annual International Workshop, SAC2000, volume xxxx of Lecture Notes in Computer Science, pages xxx–xxx. Springer-Verlag, Berlin, Heidelberg, New York, 2000. now in Preceedings pages 258–269.
Y. Zheng and X. M. Zhang. Strong linear dependence and unbiased distribution of non-propagative vectors. In Selected Areas in Cryptography, 6th Annual International Workshop, SAC’99, volume 1758 of Lecture Notes in Computer Science, pages 92–105. Springer-Verlag, Berlin, Heidelberg, New York, 2000.
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Zheng, Y., Zhang, XM. (2000). On Relationships among Avalanche, Nonlinearity, and Correlation Immunity. In: Okamoto, T. (eds) Advances in Cryptology — ASIACRYPT 2000. ASIACRYPT 2000. Lecture Notes in Computer Science, vol 1976. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44448-3_36
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