Abstract
Bent functions, that are useful in cryptographic applications, can be characterized in different ways. A recently formulated characterization is in terms of the Gibbs dyadic derivative. This characterization can be interpreted through permutation matrices associated with bent functions by this differential operator. We point out that these permutation matrices express some characteristic block structure and discuss a possible determination of it as a set of rules that should be satisfied by the corresponding submatrices. We believe that a further study of this structure can bring interesting results providing a deeper insight into features of bent functions.
Keywords
- Bent functions
- Walsh functions
- Dyadic derivatives
- Permutation matrices
This is a preview of subscription content, access via your institution.
Buying options
References
Cusic, T.W., Stănică, P.: Cryptographic Boolean Functions and Applications. Academic/Elsevier, Cambridge (2009)
Gibbs, J.E.: Walsh spectrometry, a form of spectral analysis well suited to binary digital computation, p. 24. National Physical Laboratory, Teddington (1967)
Gibbs, J.E.: Sine waves and Walsh waves in physics. In: Proc. Sympos. Applic. Walsh Functions, pp. 260–274, Washington, DC, 1970 March 31–April 3 (1970)
Gibbs, J.E., Gebbie, H.A.: Application of Walsh functions to transform spectroscopy. Nature 224(5223), 1012–1013 (1969).
Gibbs, J.E., Ireland, B.: Walsh Functions and differentiation. In: Schreiber, H., Sandy, G.F. (eds.) Applications of Walsh Functions and Sequency Theory, pp. 147–176. IEEE, New York (1974)
Karpovsky, M.G., Stanković, R.S., Astola, J.T.: Spectral Logic and Its Application in the Design of Digital Devices. Wiley, Hoboken (2008)
Pichler, F.: Walsh Functions and Linear System Theory. Tech. Res. Rept, Dept. of Electrical Engineering, University of Maryland, Report T-70-05 (1970)
Rothaus, O.S.: On ‘bent’ functions. J. Combin. Theory, Ser. A 20, 300–305 (1976)
Stanković, R.S., Moraga, C., Astola, J.T.: Fourier Analysis on Finite Non-Abelian Groups with Applications in Signal Processing and System Design. Wiley/IEEE Press, Hoboken (2005)
Stanković, R.S., Astola, J.T., Moraga, C., Stanković, M., Gajić, D.: Remarks on characterization of bent functions in terms of Gibbs dyadic derivatives, EUROCAST 2015. In: Moreno-Diaz, R., Pichler, F., Quesada-Arencibia, A. (eds.) Computer Aided Systems Theory – EUROCAST 2015, 15th International Conference, Las Palmas de Gran Canaria, February 8–13, 2015. Revised Selected Papers LNCS, vol. 9520, pp. 632–639. Springer, Berlin (2015)
Stanković, R.S., Stanković, M., Astola, J.T., Moraga, C.: Gibbs characterization of binary and ternary bent functions. In: Proceedings 46th International Symposium on Multiple-Valued Logic, pp. 205–210, Sapporo, May 18–20 (2016)
Stanković, R.S., Stanković, M., Astola, J.T., Moraga, C.: Towards the Gibbs characterization of a class of quaternary bent functions. In: Proceedings 47th International Symposium on Multiple-Valued Logic, pp. 73–78, Novi Sad, May 22–24 (2017)
Stanković, R.S., Stanković, M., Astola, J.T., Moraga, C.: Quaternary generalized Boolean bent functions obtained through permutation of binary Boolean bent functions. In: Proceedings 48th International Symposium on Multiple-Valued Logic, Linz, May 16–18 (2018)
Stanković, M., Moraga, C., Stanković, R. S., Some spectral invariant operations for functions with disjoint products in the polynomial form. In: Moreno-Diaz, R., Pichler, F., Quesada-Arencibia, A. (eds.) 16th International Conference Computer Aided Systems Theory – EUROCAST 2017, Las Palmas de Gran Canaria, February 19–24, 2017. Revised Selected Papers, LNCS, vol. 10672, Part 2, pp. 262–269 Springer, Berlin (2018)
Tokareva, N., Bent Functions Results and Applications to Cryptography. Elsevier, Amsterdam (2015). ISBN 978-0-12-802318-1
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Stanković, R.S., Stanković, M., Astola, J.T., Moraga, C. (2020). Towards the Structure of a Class of Permutation Matrices Associated with Bent Functions. In: Drechsler, R., Soeken, M. (eds) Advanced Boolean Techniques. Springer, Cham. https://doi.org/10.1007/978-3-030-20323-8_4
Download citation
DOI: https://doi.org/10.1007/978-3-030-20323-8_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-20322-1
Online ISBN: 978-3-030-20323-8
eBook Packages: EngineeringEngineering (R0)