More Web Proxy on the site http://driver.im/

research-article

Complete Characterization of Generalized Bent and 2<sup>k</sup>-Bent Boolean Functions

Authors:

Keqin FengAuthors Info & Claims

IEEE Transactions on Information Theory, Volume 63, Issue 7

Pages 4668 - 4674

https://doi.org/10.1109/TIT.2017.2686987

Published: 01 July 2017 Publication History

Abstract

In this paper, we investigate properties of generalized bent Boolean functions and <inline-formula><tex-math notation="LaTeX">$2^{k}$</tex-math></inline-formula>-bent (i.e., negabent, octabent, hexadecabent, <italic>et al.</italic>) Boolean functions in a uniform framework. From the Hadamard matrices, Hodžić and Pasalic presented sufficient conditions for generalized bent functions. Using cyclotomic fields and the decomposition of generalized bent functions, we generalize their results, prove that Hodžić and Pasalic’s conditions of generalized bent functions are not only sufficient but also necessary, and completely characterize generalized bent functions in terms of their component functions. Furthermore, we present a secondary construction of bent functions or semi-bent functions from generalized bent functions. Finally, we give the relations of generalized bent functions and <inline-formula><tex-math notation="LaTeX">$2^{k}$</tex-math></inline-formula>-bent functions, demonstrate that <inline-formula><tex-math notation="LaTeX">$2^{k}$</tex-math></inline-formula>-bent functions are actually a special class of generalized bent functions, and completely characterize <inline-formula><tex-math notation="LaTeX">$2^{k}$</tex-math></inline-formula>-bent functions.

References

[1]

C. Carlet, “Boolean functions for cryptography and error correcting codes,” in Boolean Models and Methods in Mathematics, Computer Science, and Engineering. Cambridge, U.K.: Cambridge Univ. Press, Jun. 2010, pp. 257–397.

[2]

C. Carlet, “Vectorial boolean functions for cryptography,” in Boolean Methods and Models, Cambridge, U.K.: Cambridge Univ. Press, 2010, pp. 398–469.

[3]

C. Charnes, M. Rötteler, and T. Beth, “Homogeneous bent functions, invariants, and designs,” Designs, Codes Cryptograph., vol. 26, no. 1, pp. 139–154, Jun. 2002.

Digital Library

[4]

T. W. Cusick and P. Stanica, Cryptographic Boolean Functions and Applications. New York, NY, USA: Elsevier, 2009.

[5]

S. Hodžić, W. Meidl, and E. Pasalic. (May 2016). “Full characterization of generalized bent functions as (semi)-bent spaces, their dual, and the Gray image.” [Online]. Available: https://arxiv.org/abs/1605.05713

[6]

S. Hodžić and E. Pasalic, “Generalized bent functions—Some general construction methods and related necessary and sufficient conditions,” Cryptograph. Commun., vol. 7, no. 4, pp. 469–483, Dec. 2015.

Digital Library

[7]

S. Hodžić and E. Pasalic. (Jan. 2016). “Generalized bent functions— Sufficient conditions and related constructions.” [Online]. Available: https://arxiv.org/abs/1601.08084

[8]

S. Hodžić and E. Pasalic. (Apr. 2016). “Construction methods for generalized bent functions.” [Online]. Available: https://arxiv.org/abs/1601.08084

[9]

T. Martinsen, W. Meidl, and P. Stanica. (Nov. 2015). “Generalized bent functions and their gray images.” [Online]. Available: https://arxiv.org/abs/1511.01438

[10]

T. Martinsen, W. Meidl, S. Mesnager, and P. Stanica. (2016). “Decomposing generalized bent and hyperbent functions.” 491 [Online]. Available: https://arxiv.org/abs/1604.02830

[11]

S. Mesnager, Bent Functions, Fundamentals and Results. New York, NY, USA: Springer-Verlag, 2016.

[12]

J. Olsen, R. Scholtz, and L. Welch, “Bent-function sequences,” IEEE Trans. Inf. Theory, vol. 28, no. 6, pp. 858–864, Nov. 1982.

Digital Library

[13]

M. G. Parker and A. Pott, “On Boolean functions which are bent and negabent,” in Proc. Int. Conf. Sequences, Subsequences, Consequences, Germany, Heidelberg, Jun. 2007, pp. 9–23.

[14]

A. Pott, “Nonlinear functions in abelian groups and relative difference sets,” Discrete Appl. Math., vol. 138, nos. 1–2, pp. 177–193, Mar. 2004.

Digital Library

[15]

O. S. Rothaus, “On ‘bent’ functions,” J. Combinat. Theory, Ser. A., vol. 20, no 3, pp. 300–305, May 1976.

[16]

C. Riera and M. G. Parker, “Generalized bent criteria for Boolean functions,” IEEE Trans. Inf. Theory vol. 52, no. 9, pp. 4142–4159, Jun. 2006.

Digital Library

[17]

K. U. Schmidt, “Quaternary constant-amplitude codes for multicode CDMA,” in Proc. IEEE Int. Symp. Inf. Theory (ISIT), Jun. 2007, pp. 2781–2785, [Online]. Available: http://arxiv.org/abs/cs.IT/0611162

[18]

K. U. Schmidt, “Complementary sets, generalized Reed–Muller codes, and power control for OFDM,” IEEE Trans. Inf. Theory, vol. 53, no. 2, pp. 808–814, Feb. 2007.

Digital Library

[19]

B. K. Singh, “Secondary constructions on generalized bent functions,” in Proc. IACR Cryptol. ePrint Arch., 2012, p. 17.

[20]

B. K. Singh, “On cross-correlation spectrum of generalized bent functions in generalized Maiorana-McFarland class,” Inf. Sci. Lett., vol. 2, no. 3, pp. 139–145, Sep. 2013.

[21]

P. Solé and N. N. Tokareva, “Connections between quaternary and binary bent functions,” Prikl. Diskr. Mat., vol. 1, pp. 16–18, 2009. [Online]. Available: http://eprint.iacr.org/ 2009/544

[22]

P. Stǎnicǎ, “On weak and strong $2^{k}$ -bent Boolean functions,” IEEE Trans. Inf. Theory, vol. 62, no. 5, pp. 2827–2835, May 2016.

Digital Library

[23]

P. Stǎnicǎ, T. Martinsen, S. Gangopadhyay, and B. K. Singh, “Bent and generalized bent Boolean functions,” Des. Codes Cryptograph., vol. 69, no. 1, pp. 77–94, Oct. 2013.

[24]

P. Stǎnicǎ, S. Gangopadhyay, A. Chaturvedi, A. K. Gangopadhyay, and S. Maitra, “Investigations on bent and negabent functions via the nega-Hadamard transform,” IEEE Trans. Inf. Theory, vol. 58, no. 6, pp. 4064–4072, Jun. 2012.

Digital Library

[25]

W. Su, A. Pott, and X. Tang, “Characterization of negabent functions and construction of bent-negabent functions with maximum algebraic degree,” IEEE Trans. Inf. Theory, vol. 59, no. 6, pp. 3387–3395, Jun. 2013.

Digital Library

[26]

N. Tokareva, Bent Functions-Results and Applications to Cryptography. New York, NY, USA: Academic, 2015.

[27]

L. C. Washington, “Introduction to cyclotomic fields,” in Graduate Texts in Math, vol. 83, 2nd ed. New York, NY, USA: Springer-Verlag, 1997.

Cited By

Meidl W(2022)A survey on p-ary and generalized bent functionsCryptography and Communications10.1007/s12095-022-00570-x14:4(737-782)Online publication date: 1-Jul-2022
https://dl.acm.org/doi/10.1007/s12095-022-00570-x
Meidl WPirsic I(2021)Bent and -Bent functions from spread-like partitionsDesigns, Codes and Cryptography10.1007/s10623-020-00805-z89:1(75-89)Online publication date: 1-Jan-2021
https://dl.acm.org/doi/10.1007/s10623-020-00805-z
Medina LParker MRiera CStănică P(2020)Root-Hadamard transforms and complementary sequencesCryptography and Communications10.1007/s12095-020-00440-412:5(1035-1049)Online publication date: 1-Sep-2020
https://dl.acm.org/doi/10.1007/s12095-020-00440-4
Show More Cited By

Index Terms

Complete Characterization of Generalized Bent and 2^k-Bent Boolean Functions

Index terms have been assigned to the content through auto-classification.

Recommendations

A Note on Semi-bent and Hyper-bent Boolean Functions
Information Security and Cryptology
Abstract
Semi-bent and hyper-bent funcitons as two classes of Boolean functions with low Walsh transform, are applied in cryptography and commnunications. This paper considers a new class of semi-bent quadratic Boolean function and a generalization of a ...
On generalized hyper-bent functions
Abstract
Hyper-bent Boolean functions were introduced in 2001 by Youssef and Gong (and initially proposed by Golomb and Gong in 1999 as a component of S-boxes) to ensure the security of symmetric cryptosystems but no cryptographic attack has been ...
Look into the Mirror: Evolving Self-dual Bent Boolean Functions
Genetic Programming
Abstract
Bent Boolean functions are important objects in cryptography and coding theory, and there are several general approaches for constructing such functions. Metaheuristics proved to be a strong choice as they can provide many bent functions, even ...

Comments

Please enable JavaScript to view thecomments powered by Disqus.

Information & Contributors

Information

Published In

cover image IEEE Transactions on Information Theory

IEEE Transactions on Information Theory Volume 63, Issue 7

July 2017

668 pages

ISSN:0018-9448

Issue’s Table of Contents

0018-9448 © 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

Publisher

IEEE Press

Publication History

Published: 01 July 2017

Qualifiers

Research-article

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

8
Total Citations
View Citations
0
Total Downloads

Downloads (Last 12 months)0
Downloads (Last 6 weeks)0

Reflects downloads up to 31 Dec 2024

Other Metrics

View Author Metrics

Citations

Cited By

Meidl W(2022)A survey on p-ary and generalized bent functionsCryptography and Communications10.1007/s12095-022-00570-x14:4(737-782)Online publication date: 1-Jul-2022
https://dl.acm.org/doi/10.1007/s12095-022-00570-x
Meidl WPirsic I(2021)Bent and -Bent functions from spread-like partitionsDesigns, Codes and Cryptography10.1007/s10623-020-00805-z89:1(75-89)Online publication date: 1-Jan-2021
https://dl.acm.org/doi/10.1007/s10623-020-00805-z
Medina LParker MRiera CStănică P(2020)Root-Hadamard transforms and complementary sequencesCryptography and Communications10.1007/s12095-020-00440-412:5(1035-1049)Online publication date: 1-Sep-2020
https://dl.acm.org/doi/10.1007/s12095-020-00440-4
Mesnager S(2020)On generalized hyper-bent functionsCryptography and Communications10.1007/s12095-019-00390-612:3(455-468)Online publication date: 1-May-2020
https://dl.acm.org/doi/10.1007/s12095-019-00390-6
Vahi AJafarali Jassbi S(2020)SEPAR: A New Lightweight Hybrid Encryption Algorithm with a Novel Design Approach for IoTWireless Personal Communications: An International Journal10.1007/s11277-020-07476-y114:3(2283-2314)Online publication date: 1-Oct-2020
https://dl.acm.org/doi/10.1007/s11277-020-07476-y
Mesnager SRiera CStănică P(2019)Multiple characters transforms and generalized Boolean functionsCryptography and Communications10.1007/s12095-019-00383-511:6(1247-1260)Online publication date: 1-Nov-2019
https://dl.acm.org/doi/10.1007/s12095-019-00383-5
Meidl WPott A(2019)Generalized bent functions into from the partial spread and the Maiorana-McFarland classCryptography and Communications10.1007/s12095-019-00370-w11:6(1233-1245)Online publication date: 1-Nov-2019
https://dl.acm.org/doi/10.1007/s12095-019-00370-w
Mesnager STang CQi Y(2017)Generalized Plateaued Functions and Admissible (Plateaued) FunctionsIEEE Transactions on Information Theory10.1109/TIT.2017.271580463:10(6139-6148)Online publication date: 12-Sep-2017
https://dl.acm.org/doi/10.1109/TIT.2017.2715804

View Options

View options

Media

Figures

Other

Tables

View Issue’s Table of Contents