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Further Results on Bent–Negabent Boolean Functions

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Security and Privacy

Part of the book series: Lecture Notes in Electrical Engineering ((LNEE,volume 744))

Abstract

Bent functions are optimal combinatorial objects having a lot of applications, in particular, in cryptography. Since their introduction, substantial efforts have been directed toward their study in the last three decades. In this paper, we investigate two families of functions possessing properties related to bentness: the so-called negabent and bent–negabent functions, and derive several results on their constructions and characterizations.

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References

  1. Calderbank R, Kantor WM (1986) The geometry of two-weight codes. Bull Lond Math Soc 18(2):97–122

    Article  MathSciNet  Google Scholar 

  2. Carlet C (2004) On the secondary constructions of resilient and bent functions. In: Coding, Cryptography and combinatorics (Progress in computer science and applied logic), 18 vol 23, Basel, Switzerland, Birkh\(\ddot{a}\)user, Verlag, , pp 3–28

    Google Scholar 

  3. Carlet C (2010) Boolean functions for cryptography and error correcting codes. chapter of the monography. In: Crama Y and Hammer P (eds) Boolean models and methods in mathematics, computer science, and engineering. Cambridge University Press, pp 257–397

    Google Scholar 

  4. Carlet C, Mesnager S (2016) Four decades of research on bent Functions. J Des Codes Cryptogr 78(1):5–50

    Article  MathSciNet  Google Scholar 

  5. Carlet C, Prouff E (2003) On plateaued functions and their constructions. In: Proceedings of fast software encryption FSE 2003, Lecture Notes in Computer Science 2887, pp 54–73

    Google Scholar 

  6. Cusick TW, St\(\breve{a}\)nic\(\breve{a}\) P (2009) Cryptographic Boolean functions and applications. Elsevier-Academic Press

    Google Scholar 

  7. Cohen G, Honkala I, Litsyn S, Lobstein A (1997) Covering codes, North Holland

    Google Scholar 

  8. Dillon JF (1972) A survey of bent functions. NSA Tech J Spec Issue 191–215

    Google Scholar 

  9. Dillon JF (1974) Elementary Hadamard difference sets. Ph.D. dissertation. University of Maryland

    Google Scholar 

  10. Mandal B, Singh B, Gangopadhyay S, Maitra S, Vetrivel V (2018) On non-existence of bent-negabent rotation symmetric Boolean functions. Discret Appl Math 236:1–6

    Article  MathSciNet  Google Scholar 

  11. Mesnager S (2016) Bent functions: fundamentals and results. Springer, Switzerland

    Book  Google Scholar 

  12. Mesnager S, Linear codes from functions. A concise encyclopedia of coding theory. To appear

    Google Scholar 

  13. Olsen JD, Scholtz RA, Welch LR (1982) Bent-function sequences. IEEE Transform Inf Theory 28(6):858–864

    Google Scholar 

  14. Parker MG (2000) The constant properties of Goley-Devis-Jedwab sequences. In: International symposium on information theory, Sorrento, Italy.http://www.ii.uib.no/matthew/mattweb.html

  15. Parker MG, Pott A (2007) On Boolean functions which are bent and negabent. SSC 2007, LNCS 4893. Springer, Heidelberg, pp 9–23

    Google Scholar 

  16. Pott A, Tan Y, Feng T (2010) Strongly regular graphs associated with ternary bent functions. J Comb Theory Ser A 117:668–682

    Google Scholar 

  17. Pott A, Tan Y, Feng T, Ling S (2011) Association schemes arising from bent functions. J. Des Codes Cryptogr 59:319–331

    Article  MathSciNet  Google Scholar 

  18. Riera C, Parker MG (2006) Generalized bent criteria for Boolean functions. IEEE Transform Inform Theory 52(9):4142–4159

    Article  MathSciNet  Google Scholar 

  19. Rothaus O (1976) On bent functions. J Comb Theory Ser A 20:300–305

    Google Scholar 

  20. Sarkar S (2012) Characterizing negabent Boolean functions over finite fields. SETA 2012, vol 7280, LNCS, Springer, Berlin, Heidelberg, 2, pp 77–88

    Google Scholar 

  21. Schmidt KU, Parker MG, Pott A (2008) Negabent functions in the Maiorana-McFarland Class. SETA 390–402

    Google Scholar 

  22. St\(\breve{a}\)nic\(\breve{a}\) P, Gangopadhyay S, Chaturvedi A, Gangopadhyay AK, Maitra S (2012) Investigations on bent and negabent functions via the Nega-Hadamard transform. IEEE Transform Inf Theory 58(6):4064–4072

    Google Scholar 

  23. St\(\breve{a}\)nic\(\breve{a}\) P, Mandal B, Maitra S (2019) The connection between quadratic bent-negabent functions and the Kerdock code. Appl Algebra Eng Commun Comput 30(5):387–401

    Google Scholar 

  24. Su W, Pott A, Tang X (2013) Investigations on bent and negabent functions via the nega-Hadamard transform. IEEE Transform Inf Theory 59(6):3387–3395

    Article  MathSciNet  Google Scholar 

  25. Tang C, Zhou Z, Qi Y, Zhang X, Fan C, Helleseth T (2017) Generic construction of bent functions and bent idempotents with any possible algebraic degrees. IEEE Trans Inf Theory 63(10):6149–6157

    Article  MathSciNet  Google Scholar 

  26. Zhang F, Wei Y, Pasalic E (2015) Constructions of bent-negabent functions and their relation to the completed Maiorana-McFarland class. IEEE Trans Inf Theory 61(3):1496–1506

    Article  MathSciNet  Google Scholar 

  27. Zhou Y, Qu L (2017) Constructions of negabent functions over finite fields. Cryptogr Commun 9:165–180

    Article  MathSciNet  Google Scholar 

  28. Zhuo Z, Chong J (2015) On negabent functions and nega-Hadamard transform. Math Problems Eng Article ID 959617. https://doi.org/10.1155/2015/959617

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Paris VIII, F-93526, Saint-Denis, France

    Sihem Mesnager & Bachir ben Moussat

  2. Laboratory Geometry, Analysis and Applications, LAGA, University Sorbonne Paris Nord, CNRS, UMR 7539, Villetaneuse, F-93430, 91120, Palaiseau, Telecom Paris, France

    Sihem Mesnager

  3. Laboratory Geometry, Analysis and Applications, LAGA, University Sorbonne Paris Nord, CNRS, UMR 7539, Villetaneuse, France

    Bachir ben Moussat

  4. School of Mathematical Science, Huaibei Normal University, Anhui, 235000, Huaibei, China

    Zepeng Zhuo

Authors
  1. Sihem Mesnager
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  2. Bachir ben Moussat
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  3. Zepeng Zhuo
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Corresponding author

Correspondence to Sihem Mesnager .

Editor information

Editors and Affiliations

  1. Department of Applied Mathematics, Naval Postgraduate School, Monterey, CA, USA

    Pantelimon Stănică

  2. Department of Computer Science and Engineering, Indian Institute of Technology Roorkee, Roorkee, India

    Sugata Gangopadhyay

  3. Department of Mathematics, National Institute of Technology Jamshedpur, Jamshedpur, Jharkhand, India

    Sumit Kumar Debnath

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Mesnager, S., ben Moussat, B., Zhuo, Z. (2021). Further Results on Bent–Negabent Boolean Functions. In: Stănică, P., Gangopadhyay, S., Debnath, S.K. (eds) Security and Privacy. Lecture Notes in Electrical Engineering, vol 744. Springer, Singapore. https://doi.org/10.1007/978-981-33-6781-4_5

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  • DOI: https://doi.org/10.1007/978-981-33-6781-4_5

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  • Publisher Name: Springer, Singapore

  • Print ISBN: 978-981-33-6780-7

  • Online ISBN: 978-981-33-6781-4

  • eBook Packages: Computer ScienceComputer Science (R0)

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