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Breaking permutation-based pseudorandom cryptographic schemes using distributed exact quantum algorithms

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Abstract

Quantum computing has made many important achievements in the field of cryptanalysis. However, due to the limitations of current physical system and technology, the realization of large-scale universal quantum computers is a long way off. Currently, small-scale quantum computers are easier to implement than large-scale universal quantum computers. Distributed quantum computing proposes an implementation architecture, which attempts to break down large-scale tasks into many sub-tasks distributed across multiple small-scale quantum computers. How to use small-scale quantum computers with fewer qubits and shallower quantum depths to complete large-scale tasks and improve the success rate of attacking symmetric cryptosystems is our concern. In this paper, we propose a distributed exact Simon’s algorithm, apply it to achieve quantum key recovery attacks on single-permutation-based pseudorandom cryptographic schemes with classical birthday bound security, and estimate the quantum resources of quantum circuits. Furthermore, we combine distributed exact Grover’s algorithm and distributed exact Simon’s algorithm to achieve quantum key recovery attacks on two-permutation-based pseudorandom cryptographic schemes with classical beyond birthday bound security and estimate the corresponding quantum resources. Our results show that (1) our algorithms are exact which means that the theoretical success probability of attacking pseudorandom cryptographic schemes is 100%; (2) the depth of circuit is in polynomial time which means that the theoretical depth of attacking symmetric cryptosystems is exponentially accelerated relative to the previous results; (3) the qubits of circuit don’t increase significantly. Our work is of great importance. It could lead to the rapid realization of effective quantum attacks against symmetric cryptosystems on small-scale quantum computers.

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References

  1. Shor, P.W.: Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J. Comput. 26(5), 1484–1509 (1997). https://doi.org/10.1137/S0097539795293172

    Article  MathSciNet  Google Scholar 

  2. Collins, D., Kim, K., Holton, W.: Deutsch–Jozsa algorithm as a test of quantum computation. Phys. Rev. A 58(3), 1633 (1998). https://doi.org/10.1103/PhysRevA.58.R1633

    Article  ADS  MathSciNet  Google Scholar 

  3. Xie, H., Yang, L.: Using Bernstein–Vazirani algorithm to attack block ciphers. Des. Codes Cryptogr. 87(5), 1161–1182 (2019). https://doi.org/10.1007/s10623-018-0510-5

    Article  MathSciNet  Google Scholar 

  4. Grover, L.K.: A fast quantum mechanical algorithm for database search. In: Miller, G.L. (ed.) Proceedings of the Twenty-Eighth Annual ACM symposium on the theory of computing, Philadelphia, Pennsylvania, USA, May 22-24, 1996, pp. 212–219. ACM, New York (1996). https://doi.org/10.1145/237814.237866

  5. Simon, D.R.: On the power of quantum computation. SIAM J. Comput. 26(5), 1474–1483 (1997). https://doi.org/10.1137/S0097539796298637

    Article  MathSciNet  Google Scholar 

  6. Harrow, A.W., Hassidim, A., Lloyd, S.: Quantum algorithm for linear systems of equations. Phys. Rev. Lett. 103(15), 150502 (2009). https://doi.org/10.1103/PhysRevLett.103.150502

    Article  ADS  MathSciNet  Google Scholar 

  7. Kuperberg, G.: A subexponential-time quantum algorithm for the dihedral hidden subgroup problem. SIAM J. Comput. 35(1), 170–188 (2005). https://doi.org/10.1137/S0097539703436345

    Article  MathSciNet  Google Scholar 

  8. Wang, Z., Hadfield, S., Jiang, Z., Rieffel, E.G.: Quantum approximate optimization algorithm for maxcut: a fermionic view. Phys. Rev. A 97(2), 022304 (2018). https://doi.org/10.1103/PhysRevA.97.022304

    Article  ADS  Google Scholar 

  9. Weinstein, Y.S., Pravia, M., Fortunato, E., Lloyd, S., Cory, D.G.: Implementation of the quantum Fourier transform. Phys. Rev. Lett. 86(9), 1889 (2001). https://doi.org/10.1103/PhysRevLett.86.1889

    Article  ADS  Google Scholar 

  10. Kadian, K., Garhwal, S., Kumar, A.: Quantum walk and its application domains: a systematic review. Comput. Sci. Rev. 41, 100419 (2021). https://doi.org/10.1016/j.cosrev.2021.100419

    Article  MathSciNet  Google Scholar 

  11. Leander, G., May, A.: Grover meets simon - quantumly attacking the fx-construction. In: Takagi, T., Peyrin, T. (eds.) Advances in vryptology—ASIACRYPT 2017—23rd international conference on the theory and applications of cryptology and information security, Hong Kong, China, December 3–7, 2017, proceedings, Part II. Lecture notes in computer science, vol. 10625, pp. 161–178. Springer, Heidelberg (2017). https://doi.org/10.1007/978-3-319-70697-9_6

  12. Zhou, B., Yuan, Z.: Quantum key-recovery attack on Feistel constructions: Bernstein–Vazirani meet grover algorithm. Quantum Inf. Process. 20(10), 330 (2021). https://doi.org/10.1007/s11128-021-03256-0

    Article  ADS  MathSciNet  Google Scholar 

  13. Li, H., Qiu, D., Luo, L.: Distributed Deutsch–Jozsa algorithm. Available at SSRN 4698889

  14. Qiu, D., Luo, L., Xiao, L.: Distributed Grover’s algorithm. Theoret. Comput. Sci. (2024). https://doi.org/10.1016/j.tcs.2024.114461

    Article  MathSciNet  Google Scholar 

  15. Tan, J., Xiao, L., Qiu, D., Luo, L., Mateus, P.: Distributed quantum algorithm for Simon’s problem. Phys. Rev. A 106(3), 032417 (2022). https://doi.org/10.1103/PhysRevA.106.032417

    Article  ADS  MathSciNet  Google Scholar 

  16. Zhou, X., Qiu, D., Luo, L.: Distributed Bernstein–Vazirani algorithm. Physica A 629, 129209 (2023). https://doi.org/10.1016/j.physa.2023.129209

    Article  MathSciNet  Google Scholar 

  17. Zhou, X., Qiu, D., Luo, L.: Distributed exact grover’s algorithm. Front. Phys. 18(5), 51305 (2023). https://doi.org/10.1007/s11467-023-1327-x

    Article  ADS  Google Scholar 

  18. Li, H., Qiu, D., Luo, L.: Distributed exact quantum algorithms for deutsch-jozsa problem. arXiv:2303.10663 (2023). https://doi.org/10.48550/arXiv.2303.10663

  19. Li, H., Qiu, D., Luo, L., Paulo, M.: Exact distributed quantum algorithm for generalized simon’s problem. arXiv:2307.14315 (2023). https://doi.org/10.48550/arXiv.2307.14315

  20. Long, G.-L.: Grover algorithm with zero theoretical failure rate. Phys. Rev. A 64(2), 022307 (2001). https://doi.org/10.1103/PhysRevA.64.022307

    Article  ADS  Google Scholar 

  21. Brassard, G., Hoyer, P., Mosca, M., Tapp, A.: Quantum amplitude amplification and estimation. Contemp. Math. 305, 53–74 (2002)

    Article  MathSciNet  Google Scholar 

  22. Guo, T., Wang, P., Hu, L., Ye, D.: Attacks on beyond-birthday-bound macs in the quantum setting. In: Cheon, J.H., Tillich, J. (eds.) Post-quantum cryptography—12th international workshop, PQCrypto 2021, Daejeon, South Korea, July 20-22, 2021, Proceedings. lecture notes in computer science, vol. 12841, pp. 421–441. Springer, Heidelberg (2021). https://doi.org/10.1007/978-3-030-81293-5_22

  23. Malviya, A.K., Tiwari, N., Chawla, M.: Quantum cryptanalytic attacks of symmetric ciphers: a review. Comput. Electr. Eng. 101, 108122 (2022). https://doi.org/10.1016/j.compeleceng.2022.108122

    Article  Google Scholar 

  24. Hosoyamada, A., Aoki, K.: On quantum related-key attacks on iterated even-mansour ciphers. IEICE Trans. Fundam. Electron. Commun. Comput. Sci. 102(1), 27–34 (2019). https://doi.org/10.1587/transfun.E102.A.27

    Article  ADS  Google Scholar 

  25. Kaplan, M., Leurent, G., Leverrier, A., Naya-Plasencia, M.: Breaking symmetric cryptosystems using quantum period finding. In: Robshaw, M., Katz, J. (eds.) Advances in Cryptology—CRYPTO 2016—36th annual international cryptology conference, Santa Barbara, CA, USA, August 14-18, 2016, proceedings, Part II. lecture notes in computer science, vol. 9815, pp. 207–237. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-53008-5_8

  26. Even, S., Mansour, Y.: A construction of a cipher from a single pseudorandom permutation. J. Cryptol. 10(3), 151–162 (1997). https://doi.org/10.1007/s001459900025

    Article  MathSciNet  Google Scholar 

  27. Cogliati, B., Lampe, R., Seurin, Y.: Tweaking even-mansour ciphers. In: Gennaro, R., Robshaw, M. (eds.) Advances in cryptology—CRYPTO 2015—35th annual cryptology conference, Santa Barbara, CA, USA, August 16-20, 2015, proceedings, Part I. Lecture notes in computer science, vol. 9215, pp. 189–208. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-47989-6_9

  28. Cogliati, B., Seurin, Y.: Analysis of the single-permutation encrypted Davies–Meyer construction. Des. Codes Cryptogr. 86(12), 2703–2723 (2018). https://doi.org/10.1007/S10623-018-0470-9

    Article  MathSciNet  Google Scholar 

  29. Zou, J., Wei, Z., Sun, S., Liu, X., Wu, W.: Quantum circuit implementations of AES with fewer qubits. In: Moriai, S., Wang, H. (eds.) Advances in Cryptology—ASIACRYPT 2020—26th international conference on the theory and application of cryptology and information security, Daejeon, South Korea, December 7-11, 2020, proceedings, Part II. Lecture notes in computer science, vol. 12492, pp. 697–726. Springer, Heidelberg (2020). https://doi.org/10.1007/978-3-030-64834-3_24

  30. Cai, B., Gao, F., Leander, G.: Quantum attacks on two-round even-mansour. Front. Phys. 10, 1028014 (2022). https://doi.org/10.3389/fphy.2022.1028014/full

    Article  Google Scholar 

  31. Cho, S., Kim, A., Choi, D., Choi, B., Seo, S.: Quantum modular multiplication. IEEE Access 8, 213244–213252 (2020). https://doi.org/10.1109/ACCESS.2020.3039167

    Article  Google Scholar 

  32. Chen, Y.L., Lambooij, E., Mennink, B.: How to build pseudorandom functions from public random permutations. In: Boldyreva, A., Micciancio, D. (eds.) Advances in Cryptology—CRYPTO 2019—39th annual international cryptology conference, Santa Barbara, CA, USA, August 18-22, 2019, Proceedings, Part I. Lecture Notes in Computer Science, vol. 11692, pp. 266–293. Springer, Heidelberg (2019). https://doi.org/10.1007/978-3-030-26948-7_10

  33. Chen, S., Lampe, R., Lee, J., Seurin, Y., Steinberger, J.P.: Minimizing the two-round even-mansour cipher. J. Cryptol. 31(4), 1064–1119 (2018). https://doi.org/10.1007/S00145-018-9295-Y

    Article  MathSciNet  Google Scholar 

  34. Dutta, A., Nandi, M., Talnikar, S.: Permutation based EDM: an inverse free BBB secure PRF. IACR Trans. Symmetric Cryptol. 2021(2), 31–70 (2021). https://doi.org/10.1016/j.ipl.2021.106172

    Article  Google Scholar 

  35. Shinagawa, K., Iwata, T.: Quantum attacks on sum of even-mansour pseudorandom functions. Inf. Process. Lett. 173, 106172 (2022). https://doi.org/10.1016/j.ipl.2021.106172

    Article  MathSciNet  Google Scholar 

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Acknowledgements

We would like to express our sincere thanks to editors and the anonymous reviewers for the valuable comments and suggestions.

Funding

This work was supported by the Open Fund of Advanced Cryptography and System Security Key Laboratory of Sichuan Province (Grant No.: SKLACSS-202315), National Natural Science Foundation of China (Grant Nos.: 62072207, U23B2002, 62272238, and 61902195), Guangdong Basic and Applied Basic Research Foundation (No. 2022A1515140090), and NUPTSF (Grant No.: NY219131).

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  1. School of Computer Science, Nanjing University of Posts and Telecommunications, Nanjing, 210023, China

    Ping Zhang

  2. Advanced Cryptography and System Security Key Laboratory of Sichuan Province, Chengdu, 610225, China

    Ping Zhang

  3. School of Computer Science and Engineering, Huizhou University, Huizhou, 516007, China

    Yiyuan Luo

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Zhang, P., Luo, Y. Breaking permutation-based pseudorandom cryptographic schemes using distributed exact quantum algorithms. Quantum Inf Process 23, 239 (2024). https://doi.org/10.1007/s11128-024-04424-8

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