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Cryptography and Communications

, Volume 9, Issue 2, pp 301–314 | Cite as

Cryptographic Boolean functions with biased inputs

  • Sugata GangopadhyayEmail author
  • Aditi Kar Gangopadhyay
  • Spyridon Pollatos
  • Pantelimon Stănică
Article

Abstract

While performing cryptanalysis, it is of interest to approximate a Boolean function in n variables \(f: {\mathbb {F}_{2}^{n}} \rightarrow \mathbb {F}_{2}\) by affine functions. Usually, it is assumed that all the input vectors to a Boolean function are equiprobable while mounting affine approximation attack or fast correlation attacks. In this paper we consider a more general case when each component of the input vector to f is independent and identically distributed Bernoulli variates with the parameter p. Since our scope is within the area of cryptography, we initiate an analysis of cryptographic Boolean functions under the previous considerations and derive expression of the analogue of Walsh–Hadamard transform and nonlinearity in the case under consideration. We observe that if we allow p to take up complex values then a framework involving quantum Boolean functions can be introduced, which provides a connection between Walsh-Hadamard transform, nega-Hadamard transform and Boolean functions with biased inputs.

Keywords

Boolean functions Quantum Boolean functions Bias Walsh–Hadamard transform Nega-Hadamard transform 

Mathematics Subject Classification (2010)

94A60 94C10 81P94 06E30 

Notes

Acknowledgments

The authors thank Dr. Aalok Misra of the Department of Physics, Indian Institute of Technology Roorkee for extremely helpful discussions on quantum mechanics.

References

  1. 1.
    Cusick, T.W., Stănică, P.: Cryptographic Boolean functions and applications. Elsevier–Academic Press (2009)Google Scholar
  2. 2.
    Erdős, P., Rényi, A.: On the evolution of random graphs. Publ. Math. Inst. Hungar. Acad. Sci. 5, 17–61 (1960)MathSciNetGoogle Scholar
  3. 3.
    Friedgut, E., Kalai, G.: Every monotone graph property has a sharp threshold. Proc. AMS. 124(10), 2293–3002 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Keller, N., Mossel, E., Schlank, T.: A note on the entropy/influence conjecture. Discrete Math. 312(22), 3364–3372 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Lu, Y., Desmedt, Y.: Bias analysis of a certain problem with applications to E0 and Shannon ciper. ICISC LNCS 6829(2011), 16–28 (2010)zbMATHGoogle Scholar
  6. 6.
    Montanaro, A., Osborne, T.J.: Quantum Boolean functions, Chicago J. Theor. Comput. Sci. Article 1, pages 1–45,. http://cjtcs.cs.uchicago.edu/ , arXiv:0810.2435v5 (2010)
  7. 7.
    Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press (2000)Google Scholar
  8. 8.
    O’Donnell, R.: Analysis of Boolean functions. Cambridge University Press (2014)Google Scholar
  9. 9.
    O’Donnell, R., Tan, L-Y.: A composition theorem for the Fourier entropy-influence conjecture, ICALP (1) 2013 780–791. arXiv:1304.1347v1 (2013)
  10. 10.
    Parker, M.G.: Generalised S-box nonlinearity, NESSIE Public Document, 11.02.03: NES/DOC/UIB/WP5/020/AGoogle Scholar
  11. 11.
    Parker, M.G., Pott, A.: On Boolean functions which are bent and negabent. In: Proc. Int. Conf. Sequences, Subsequences, Consequences, vol. 4893, pp 9–23 (2007)Google Scholar
  12. 12.
    Riera, C.: Spectral Properties of Boolean functions, Graphs and Graph States. PhD. thesis University of Bergen (2005)Google Scholar
  13. 13.
    Riera, C., Parker, M.G.: Generalized bent criteria for Boolean functions. IEEE Trans. Inf. Theory 52(9), 4142–4159 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Rothaus, O.S.: On bent functions. J. Combin. Theory, Ser. A 20, 300–305 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Schmidt, K.U., Parker, M.G., Pott, A.: Negabent functions in the Maiorana–McFarland class. In: Golomb, S.W., Parker, M.G., Pott, A., Winterhof, A. (eds.) . SETA 2008, LNCS 5203, pp 390–402. Springer, Heidelberg (2008)Google Scholar
  16. 16.
    Stănică, P., Gangopadhyay, S., Chaturvedi, A., Gangopadhyay, A.K., Maitra, S.: Investigations on bent and negabent functions via the nega–Hadamard transform. IEEE Trans. Inf. Theory 58(6), 4064–4072 (2012)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Sugata Gangopadhyay
    • 1
    Email author
  • Aditi Kar Gangopadhyay
    • 2
  • Spyridon Pollatos
    • 3
  • Pantelimon Stănică
    • 3
  1. 1.Department of Computer Science and EngineeringIndian Institute of Technology RoorkeeRoorkeeIndia
  2. 2.Department of MathematicsIndian Institute of Technology RoorkeeRoorkeeIndia
  3. 3.Department of Applied MathematicsNaval Postgraduate SchoolMontereyUSA

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