Abstract
In this paper we derive several important results towards a better understanding of propagation characteristics of resilient Boolean functions. We first introduce a new upper bound on nonlinearity of a given resilient function depending on the propagation criterion. We later show that a large class of resilient functions admit a linear structure; more generally, we exhibit some divisibility properties concerning the Walsh-spectrum of the derivatives of any resilient function. We prove that, fixing the order of resiliency and the degree of propagation criterion, a high algebraic degree is a necessary condition for construction of functions with good autocorrelation properties. We conclude by a study of the main constructions of resilient functions. We notably show how to avoid linear structures when a linear concatenation is used and when the recursive construction introduced in [11] is chosen.
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A. Canteaut and M. Trabbia, “Improved fast correlation attacks using parity-check equations of weight 4 and 5.” In Advances in Cryptology-EUROCRYPT 2000,number 1807 in Lecture Notes in Computer Science, pp. 573–588, Springer-Verlag,2000.
A. Canteaut, C. Carlet, P. Charpin, and C. Fontaine, “Propagation characteristics and correlation-immunity of highly nonlinear Boolean functions.” In Advancesin Cryptology-EUROCRYPT 2000, number 1807 in Lecture Notes in Computer Science, pp. 507–522, Springer-Verlag, 2000.
A. Canteaut, C. Carlet, P. Charpin, and C. Fontaine. “On cryptographic properties of the cosets of R(1,m).” IEEE Trans. Inform. Theory, 47(4):1494–1513, 2001.
C. Carlet, “On the coset weight divisibility and nonlinearity of resilient and correlation-immune functions.” In Sequences and their Applications-SETA’ 01,Discrete Mathematics and Theoretical Computer Science, pp. 131–144. Springer-Verlag, 2001.
C. Carlet, “On cryptographic propagation criteria for Boolean functions.” Information and Computation, number 151, pp. 32–56, 1999.
P. Charpin, and E. Pasalic, “On propagation characteristics of resilient functions.” In Research-report RR-4537, INRIA, September 2002.
S. Chee, S. Lee, D. Lee, and S.H. Sung, “On the correlation immune functions and their nonlinearity.” In Advances in Cryptology-ASIACRYPT’96, number 1163 in Lecture Notes in Computer Science, pp. 232–243, Springer-Verlag, 1996.
J. H. Evertse, “Linear structures in block ciphers.” In Advances in Cryptology-EUROCRYPT’ 87, number 304 in Lecture Notes in Computer Science, pp. 249–266, Springer Verlag, 1987.
M. Matsui, “Linear cryptanalysis method for DES cipher.” In Advances in Cryptology-EUROCRYPT’93, number 765 in Lecture Notes in Computer Science, pp. 386–397, Springer-Verlag, 1993.
W. Meier, and O. Staffelbach., “Nonlinearity criteria for cryptographic functions.” In Advances in Cryptology-EUROCRYPT’93, number 434 in Lecture Notes inComputer Science, pp. 549–562, Springer-Verlag, 1988.
E. Pasalic, T. Johansson, S. Maitra, and P. Sarkar., “New constructions of resilient and correlation immune Boolean functions achieving upper bounds on nonlinearity.” In Cryptology ePrint Archive, eprint.iacr.org, No. 2000/048, September, 2000.
B. Preneel, W.V. Leekwijck, L.V. Linden, R. Govaerts, and J. Vandewalle, “Propagation characteristics of Boolean functions.” In Advances in Cryptology-EUROCRYPT’ 90, number 437 in Lecture Notes in Computer Science, pp. 155–165, Springer-Verlag, 1990.
P. Sarkar and S. Maitra, “Nonlinearity bounds and constructions of resilient Boolean functions.” In Advances in Cryptology-EUROCRYPT 2000, number 1807 in Lecture Notes in Computer Science, pp. 515–532, Springer-Verlag, 2000.
T. Siegenthaler, “Correlation-immunity of nonlinear combining functions for cryptographic applications.” IEEE Trans. Inform. Theory, IT-30(5): 776–780, 1984.
Y. Tarannikov, “On resilient Boolean functions with maximal possible nonlinearity.” In Proceedings of Indocrypt 2000, number 1977 in Lecture Notes in Computer Science, pp. 19–30, Springer Verlag, 2000.
Y. V. Tarannikov, “New constructions of resilient Boolean functions with maximal nonlinearity.” In Fast Software Encryption-FSE 2001, to be published in Lecture Notes in Computer Science, pp. 70–81 (in preproceedings). Springer Verlag, 2001.
A.F. Webster and S.E. Tavares, “On the design of S-boxes.” In Advances in Cryptology-CRYPTO’85, number 219 in Lecture Notes in Computer Science, pp. 523–534, Springer-Verlag, 1985.
G. Xiao and J.L. Massey. “A spectral characterization of correlation-immune combining functions.” IEEE Trans. Inform. Theory, IT-34(3):569–571, 1988.
X.-M. Zhang and Y. Zheng, “GAC-the criterion for global avalanche characterics of cryptographic functions.” Journal of Universal Computer Science, vol. 1, no. 5, pp. 320–337, 1995.
X.-M. Zhang and Y. Zheng, “On relationship among avalanche, nonlinearity, and propagation criteria,” In Advances in Cryptology-Asiacrypt 2000, number 1976 in Lecture Notes in Computer Science, pp. 470–483, Springer-Verlag, 2000.
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Charpin, P., Pasalic, E. (2003). On Propagation Characteristics of Resilient Functions. In: Nyberg, K., Heys, H. (eds) Selected Areas in Cryptography. SAC 2002. Lecture Notes in Computer Science, vol 2595. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-36492-7_13
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DOI: https://doi.org/10.1007/3-540-36492-7_13
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