Abstract
In stream ciphers, we should use a t-resilient Boolean function f(X) with large nonlinearity to resist fast correlation attacks and linear attacks. Further, in order to be secure against an extension of linear attacks, we wish to find a t-resilient function f(X) which has a large distance even from low degree Boolean functions. From this point of view, we define a new covering radius p(t, r, n) as the maximum distance between a t-resilient function f(X) and the r-th order Reed-Muller code RM(r, n). We next derive its lower and upper bounds. Finally, we present a table of numerical bounds for p(t, r, n).
Chapter PDF
References
X.D. Hou. Some results on the covering radii of Reed-Muller codes. IEEE Transactions on Information Theory, IT-39:366–378, 1993.
X.D. Hou. Further results on the covering radii of the Reed-Muller codes. Designs, Codes and Cryptography, vol.3, pages 167–177, 1993.
X. Lai. Higher order derivatives and differential cryptanalysis. In Proceedings of Symposium on Communication, Coding and Cryptography, in honor of James L.Massey on the occasion of his 60’th birthday, February 10-13, 1994, Monte-Verita, Ascona Switzerland, 1994.
A.M. MacLoughlin. The covering radius of the (m-3)-rd order Reed-Muller codes and lower bounds on the (m-4)-th order Reed-Muller codes. SIAM Journal of Applied Mathematics, vol. 37, no. 2, October 1979.
P. Sarkar and S. Maitra. Nonlinearity bounds and constructions of resilient Boolean functions. Advances in Cryptology —CRYPTO 2000, LNCS 1880, pages 515–532, 2000.
J.R. Schatz. The second order Reed-Muller code of length 64 has covering radius 18. IEEE Transactions on Information Theory, IT-27(5):529–530 September 1981.
T. Siegentharler. Correlation-immunity of nonlinear combining functions for cryptographic applications. IEEE Transactions on Information Theory, IT-30(5):776–780 September 1984.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Iwata, T., Yoshiwara, T., Kurosawa, K. (2001). New Covering Radius of Reed-Muller Codes for t-Resilient Functions. In: Vaudenay, S., Youssef, A.M. (eds) Selected Areas in Cryptography. SAC 2001. Lecture Notes in Computer Science, vol 2259. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45537-X_6
Download citation
DOI: https://doi.org/10.1007/3-540-45537-X_6
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-43066-7
Online ISBN: 978-3-540-45537-0
eBook Packages: Springer Book Archive