A quantum algorithm for testing and learning resiliency of a Boolean function

  • Hongwei LiEmail author


A quantum algorithm to evaluate the resiliency of a Boolean function is explored. Recently, Chakraborty and Maitra (Cryptogr Commun 8(3):401–413, 2016) have provided quantum algorithms to check the non-resiliency of a Boolean function. However, the shortage of their algorithms is that they just output YES or NO. Refining one of the algorithms, a quantum algorithm is proposed here, which can describe the extent of the non-resiliency by \(\epsilon \)-almost resiliency under the condition NO.


Almost resilient function Resilient function Quantum algorithm Walsh spectrum 



This work was supported by the Science and Technology Project of Henan Province (China) under Grant No. 162102210103, Natural Science foundation of Henan Province (China) 162300410191.


  1. 1.
    Chakraborty, K., Maitra, S.: Application of Grover’s algorithm to check non-resiliency of a Boolean function. Cryptogr. Commun. 8(3), 401–413 (2016)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Li, H., Yang, L.: A quantum algorithm for approximating the influences of Boolean functions and its applications. Quantum Inf. Process. 14(6), 1787–1797 (2015)ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    Montanaro, A., de Wolf, R.: A Survey of Quantum Property Testing, arXiv:1310.2035v3 [quant-ph] 10 Dec (2014)
  4. 4.
    Chor, B., Goldreich, O., Hastad, J., Freidmann, J., Rudich, S., Smolensky, R.: The bit extraction problem or t-resilient functions. In: Proceedings of the 26th IEEE Symposium on Foundations of Computer Science, pp. 396–407 (1985)Google Scholar
  5. 5.
    Bennett, C.H., Brassard, G., Robert, J.-M.: Privacy amplification by public discussion. SIAM J. Comput. 17(2), 210–229 (1988)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Kurosawa, K., Johansson, T., Stinson, D.: Almost k-wise independent sample spaces and their applications. J. Cryptol. 14(4), 231–253 (2001)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Xiao, G.Z., Massey, J.L.: A spectral characterization of correlation-immune combining functions. IEEE Trans. Inf. Theory 34(3), 569–571 (1988)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Gopalakrishnan, K., Stinson, D.R.: Three characterizations of non-binary correlation-immune and resilient functions. Des. Codes Cryptogr. 5(3), 241–251 (1995)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Camion, P., Canteaut, A.: Correlation-immune and resilient functions over a finite alphabet and their applications in cryptography. Des. Codes Cryptogr. 16(2), 121–149 (1999)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Zhang, X.-M., Zheng, Y.: Cryptographically resilient functions. IEEE Trans. Inf. Theory 43(5), 1740–1747 (1997)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Sarkar, P., Maitra, S.: Construction of nonlinear Boolean functions with important cryptographic properties. In: Advances in Cryptology—EUROCRYPT 2000, Lecture Notes in Computer Science. Springer, pp. 485-506 (2000)Google Scholar
  12. 12.
    Zhang, W.G., Xiao, G.Z.: Constructions of almost optimal resilient Boolean functions on large even number of variables. IEEE Trans. Inf. Theory 55(12), 5822–5831 (2009)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Zhang, W.-G., Pasalic, E.: Generalized Maiorana–McFarland construction of resilient Boolean functions with high nonlinearity and good algebraic properties. IEEE Trans. Inf. Theory 60(10), 6681–6695 (2014)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Kurosawa, K., Matsumoto, R.: Almost security of cryptographic Boolean functions. IEEE Trans. Inf. Theory 50(11), 2752–2761 (2004)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Ke, P., Zhang, J., Wen, Q.: Results on almost resilient functions. ACNS 2006, LNCS, vol. 3989, pp. 421–432 (2006)Google Scholar
  16. 16.
    Canteaut, A., Carlet, C., Charpin, P., Fontaine, C.: On cryptographic properties of the cosets of R(1, m). IEEE Trans. Inf. Theory 47(4), 1494–1513 (2001)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Deutsch, D., Jozsa, R.: Rapid solution of problems by quantum computation. Proc. R. Soc. Lond. A439, 553–558 (1992)ADSMathSciNetCrossRefGoogle Scholar
  18. 18.
    Hoeffding, W.: Probability inequalities for sums of bounded random variables. Am. Stat. Assoc. J. 58(301), 13–30 (1963)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Bierbrauer, J., Schellwat, H.: Almost independent and weakly biased arrays: efficient constructions and cryptologic applications. CRYPTO 2000, LNCS, vol. 1880, pp. 533–543 (2000)Google Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsHenan Finance UniversityZhengzhouChina

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