Abstract
In this paper we consider functions \(F : {\mathbb F}^{m}_{2} \rightarrow {\{\pm\}}\) which satisfy certain linear and differential properties. The investigation of these properties is motivated by applications in cryptography.
The linear property that we are interested in is “correlation immunity”, the differential properties are known under the name of “avalanche criteria”. It is not our purpose to construct new correlation immune functions or new functions with good differential properties, but we will describe known constructions (Maiorana-McFarland construction) and its variations in terms of group rings. This is (notationally) a quite useful description since it yields immediate further generalizations and it gives easy ways to obtain bounds on the maximum nonlinearity of the functions.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Beth, T., Jungnickel, D., Lenz, H.: Design Theory, 2nd edn. Cambridge University Press, Cambridge (1999)
Canteaut, A., Carlet, C., Charpin, P., Fontaine, C.: On cryptographic properties of the cosets of R(1,m). IEEE Trans. Inform. Theory 47, 1494–1513 (2001)
Carlet, C.: A larger class of cryptographic boolean functions via a study of the maiorana-mcFarland construction. In: Yung, M. (ed.) CRYPTO 2002. LNCS, vol. 2442, pp. 549–564. Springer, Heidelberg (2002)
Carlet, C.: On the confusion and diffusion properties of Maiorana-McFarland’s and extended Maiorana-McFarland’s functions. J. Complexity 20, 182–204 (2004)
Choi, S., Yang, K.: Autocorrelation properties of resilient functions and three-valued almost-optimal functions satisfying pC(p). In: Helleseth, T., Sarwate, D., Song, H.-Y., Yang, K. (eds.) SETA 2004. LNCS, vol. 3486, pp. 167–171. Springer, Heidelberg (2005)
Cusick, T.W.: On constructing balanced correlation immune functions, in Sequences and their applications (Singapore, 1998), Springer Ser. Discrete Math. Theor. Comput. Sci., pp. 184–190. Springer, London (1999)
Dubuc, S.: Characterization of linear structures. Des. Codes Cryptogr. 22, 33–45 (2001)
Jungnickel, D.: On automorphism groups of divisible designs. Canad. J. Math. 34, 257–297 (1982)
Pott, A.: Finite Geometry and Character Theory. Lecture Notes in Mathematics, vol. 1601. Springer, Berlin (1995)
Wolfmann, J.: Bent functions and coding theory. In: Difference sets, sequences and their correlation properties (Bad Windsheim, 1998) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 542, pp. 393–418. Kluwer Acad. Publ., Dordrecht (1999)
Zheng, Y., Zhang, X.-M.: On relationships among avalanche, nonlinearity, and correlation immunity. In: Okamoto, T. (ed.) ASIACRYPT 2000. LNCS, vol. 1976, pp. 470–482. Springer, Heidelberg (2000)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Pott, A. (2005). Group Algebras and Correlation Immune Functions. In: Helleseth, T., Sarwate, D., Song, HY., Yang, K. (eds) Sequences and Their Applications - SETA 2004. SETA 2004. Lecture Notes in Computer Science, vol 3486. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11423461_34
Download citation
DOI: https://doi.org/10.1007/11423461_34
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-26084-4
Online ISBN: 978-3-540-32048-7
eBook Packages: Computer ScienceComputer Science (R0)