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Quantum algorithms for learning the algebraic normal form of quadratic Boolean functions

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Abstract

Quantum algorithms for the analysis of Boolean functions have received a lot of attention over the last few years. The algebraic normal form (ANF) of a linear Boolean function can be recovered by using the Bernstein–Vazirani (BV) algorithm. No research has been carried out on quantum algorithms for learning the ANF of general Boolean functions. In this paper, quantum algorithms for learning the ANF of quadratic Boolean functions are studied. We draw a conclusion about the influences of variables on quadratic functions, so that the BV algorithm can be run on them. We study the functions obtained by inversion and zero-setting of some variables in the quadratic function and show the construction of their quantum oracle. We introduce the concept of “club” to group variables that appear in quadratic terms and study the properties of clubs. Furthermore, we propose a bunch of algorithms for learning the full ANF of quadratic Boolean functions. The most efficient algorithm, among those we propose, provides an O(n) speedup over the classical one, and the number of queries is independent of the degenerate variables.

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References

  1. Arute, F., Arya, K., Babbush, R., Bacon, D., Bardin, J.C., Barends, R., et al.: Quantum supremacy using a programmable superconducting processor. Nature 574(7779), 505–510 (2019)

    Article  ADS  Google Scholar 

  2. Shor, P.W.: Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM Rev. 41(2), 303–332 (1999)

    Article  MathSciNet  ADS  Google Scholar 

  3. Grover, L.K.: A fast quantum mechanical algorithm for database search. In: Proceedings of the 28th Annual ACM Symposium on Theory of Computing, pp. 212–219 (1996)

  4. Simon, D.R.: On the power of quantum computation. SIAM J. Comput. 26(5), 1474–1483 (1997)

    Article  MathSciNet  Google Scholar 

  5. Bernstein, E., Vazirani, U.: Quantum complexity theory. SIAM J. Comput. 26(5), 1411–1473 (1993)

    Article  MathSciNet  Google Scholar 

  6. Matsui, M.: Linear cryptanalysis method for DES Cipher. EUROCRYPT 1993, 386–397 (1994)

    MATH  Google Scholar 

  7. Millan, W.: Low order approximation of cipher functions. CPA 1995, 144–155 (1995)

    MATH  Google Scholar 

  8. Wang, Q., Tan, C.H.: On the second-order nonlinearity of the hidden weighted bit function. Discrete Appli. Math. 215, 197–202 (2016)

    Article  MathSciNet  Google Scholar 

  9. Shi, D., Sun, S., Sasaki, Y., Li, C., Hu, L.: Correlation of quadratic Boolean functions: cryptanalysis of all versions of full \(\sf MORUS\). CRYPTO 2019, 180–209 (2019)

  10. Dillon, J.F.: Elementary Hadamard difference sets. (Doctoral dissertation) (1974)

  11. McFarland, R.L.: A family of difference sets in non-cyclic groups. J. Comb. Theory Ser. A. 15(1), 1–10 (1973)

    Article  MathSciNet  Google Scholar 

  12. Camion, P., Carlet, C., Charpin, P., Sendrier, N.: On correlation-immune functions. In: CRYPTO 1991, 86–100 (1991)

  13. Floess, D., Andersson, E., Hillery, M.: Quantum algorithms for testing and learning Boolean functions. Math. Struct. Comput. Sci. 23(2), 386–398 (2013)

    Article  MathSciNet  Google Scholar 

  14. 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)

    Article  MathSciNet  ADS  Google Scholar 

  15. Li, H., Yang, L.: A quantum algorithm to approximate the linear structures of Boolean functions. Math. Struct. Comput. Sci. 28(1), 1–13 (2018)

    Article  MathSciNet  Google Scholar 

  16. Xie, H., Yang, L.: Using Bernstein–Vazirani algorithm to attack block ciphers. Des. Codes Crypt. 87(5), 1161–1182 (2019)

    Article  MathSciNet  Google Scholar 

  17. Zhang, F., Hao, X., Wei, Y., Zhou, Y.: Quantum period finding based on the Bernstein–Vazirani algorithm. Quantum Inf. Comput. 20(1&2), 65–84 (2020)

    MathSciNet  Google Scholar 

  18. Li, H.: A quantum algorithm for testing and learning resiliency of a Boolean function. Quantum Inf. Process. 18, 51 (2019)

    Article  MathSciNet  ADS  Google Scholar 

  19. Chakraborty, K., Chattopadhyay, A., Maitra, S.: Quantum algorithms to check resiliency, symmetry and linearity of a Boolean function. https://eprint.iacr.org/2013/232.pdf (2013). Accessed 26 Nov (2019)

  20. Chakraborty, K., Maitra, S.: Improved quantum test for linearity of a Boolean function. arXiv:1306.6195 (2013). Accessed 26 Nov (2019)

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Acknowledgements

X. Hao and F. Zhang were supported in part by the National Natural Science Foundation of China (No. 61972400), the Natural Science Foundation of Jiangsu Province (No. BK20181352) and the Guangxi Key Laboratory of Cryptography and Information Security (No. GCIS201712).

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Authors and Affiliations

  1. School of Computer Science and Technology, China University of Mining and Technology, Xuzhou, 221116, People’s Republic of China

    Xuexuan Hao, Fengrong Zhang, Shixiong Xia & Yong Zhou

  2. Guangxi Key Laboratory of Cryptography and Information Security, Guilin University of Electronic Technology, Guilin, 541004, People’s Republic of China

    Xuexuan Hao & Fengrong Zhang

  3. Mine Digitization Engineering Research Center of Ministry of Education, China University of Mining and Technology, Xuzhou, 221116, People’s Republic of China

    Xuexuan Hao, Fengrong Zhang, Shixiong Xia & Yong Zhou

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  1. Xuexuan Hao
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  2. Fengrong Zhang
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  3. Shixiong Xia
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  4. Yong Zhou
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Correspondence to Fengrong Zhang.

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Hao, X., Zhang, F., Xia, S. et al. Quantum algorithms for learning the algebraic normal form of quadratic Boolean functions. Quantum Inf Process 19, 273 (2020). https://doi.org/10.1007/s11128-020-02778-3

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