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Application of Grover’s algorithm to check non-resiliency of a Boolean function

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Abstract

In this paper, we explore quantum algorithms to check the resiliency property of a Boolean function (in particular, when it is non-resilient). First we explain that Deutsch-Jozsa algorithm can be immediately used for this purpose. We further analyse how the quadratic improvement in query complexity can be obtained using Grover’s technique. While the worst case quantum query complexity to check the resiliency order is exponential in the number of input variables of the Boolean function, in our strategy one requires polynomially many measurements only. We also describe a subset of n-variable Boolean functions for which the algorithm works in polynomially many steps, i.e., we can achieve an exponential speed-up over best known classical algorithms.

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Notes

  1. Our analysis here does not require any specialized knowledge of the quantum paradigm. Instead the work is mostly related to combinatorial properties of Boolean functions. We show how such properties of Boolean functions can be exploited to achieve novel and improved results in the field of quantum algorithms.

  2. For quantum algorithms, we write “query complexity” instead of “time complexity” as we need to query some oracles, e.g., \(U_{f}, \mathcal {O}_{g}\) as described in Section 2.

  3. For more details on query complexity and measurements, refer to [11].

  4. We go for similar abuse of notation for the phase inversion oracle later.

References

  1. Bellare, M., Coppersmith, D., Hastad, J., Kiwi, M., Sudan, M.: Linearity testing over characteristic two. IEEE Trans. Inform. Theory 42, 1781 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  2. Boyer, M., Brassard, G., Hoeyer, P., Tapp, A.: Tight bounds on quantum searching. Fortsch. Phys. 46, 493–506 (1998) arXiv:quant-ph/9605034

    Article  Google Scholar 

  3. Braunstein, S.L., Choi, B.-S., Ghosh, S., Maitra, S.: Exact quantum algorithm to distinguish Boolean functions of different weights. J. Phys. A Math. Theor. 40, 8441–8454. doi:10.1088/1751-8113/40/29/017. published: 3 July 2007

  4. Buhrman, H., Fortnow, L., Newman, I., Röhrig, H.: Quantum property testing. SIAM J. Comput. 37(5), 1387–1400 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Demenkov, E., Kojevnikov, A., Kulikov, A., Yaroslavtsev, G.: New upper bounds on the Boolean circuit complexity of symmetric functions. Inf. Process. Lett. 110, 264–267 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Deutsch, D., Jozsa, R.: Rapid solution of problems by quantum computation. Proc. R. Soc. Lond. A439, 553–558 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  7. Grover, L.: A fast quantum mechanical algorithm for database search. In: Proceedings of 28th Annual Symposium on the Theory of Computing (STOC). Available at http://xxx.lanl.gov/abs/quant-ph/9605043, pp 212–219 (1996)

  8. Guo-Zhen, X., Massey, J.: A spectral characterization of correlation immune combining functions. IEEE Trans. Inf. Theory 34(3), 569–571 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  9. Maitra, S., Mukhopadhyay, P.: Deutsch-Jozsa algorithm revisited in the domain of cryptographically significant Boolean functions. In: International Journal on Quantum Information, vol. 3, No. 2, pp 359–370 (2005)

  10. Montanaro, A., de Wolf, R.: A survey of quantum property testing. Available at arXiv:1310.2035v2.pdf

  11. Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press (2010)

  12. Sarkar, P., Maitra, S.: Nonlinearity bounds and constructions of resilient Boolean functions. In: Advances in Cryptology - CRYPTO 2000, number 1880 in Lecture Notes in Computer Science, pp 515–532. Springer Verlag (2000)

  13. Tarannikov, Y.V.: On resilient Boolean functions with maximum possible nonlinearity. In: Progress in Cryptology - INDOCRYPT 2000, number 1977 in Lecture Notes in Computer Science, pp 19–30. Springer Verlag (2000)

  14. Zheng, Y., Zhang, X.M.: Improved upper bound on the nonlinearity of high order correlation immune functions. In: Selected Areas in Cryptography - SAC 2000, number 2012 in Lecture Notes in Computer Science, pp 264–274. Springer Verlag (2000)

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Acknowledgments

The authors like to thank the anonymous reviewers for their valuable comments. The authors also acknowledge the Centre of Excellence in Cryptology, Indian Statistical Institute, for relevant support towards this work.

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Correspondence to Subhamoy Maitra.

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This is a thoroughly revised version of the paper “Quantum algorithms to check Resiliency of a Boolean function” that has been presented in WCC 2013, April 15-19, 2013, Bergen, Norway.

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Chakraborty, K., Maitra, S. Application of Grover’s algorithm to check non-resiliency of a Boolean function. Cryptogr. Commun. 8, 401–413 (2016). https://doi.org/10.1007/s12095-015-0156-3

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Mathematics Subject Classfication (2010)

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