Science China Information Sciences

, Volume 58, Issue 11, pp 1–11 | Cite as

A public key cryptosystem based on data complexity under quantum environment

  • WanQing Wu
  • HuanGuo Zhang
  • HouZhen Wang
  • ShaoWu Mao
  • JianWei Jia
  • JinHui Liu
Research Paper Special Focus on Security of Cyberspace


Since the Shor algorithm showed that a quantum algorithm can efficiently calculate discrete logarithms and factorize integers, it has been used to break the RSA, EIGamal, and ECC classical public key cryptosystems. This is therefore a significant issue in the context of ensuring communication security over insecure channels. In this paper, we prove that there are no polynomial-size quantum circuits that can compute all Boolean functions (of which there are \({2^{{2^n}}}\) cases) in the standard quantum oracle model. Based on this, we propose the notion of data complexity under a quantum environment and suggest that it can be used as a condition for post-quantum computation. It is generally believed that NP-complete problems cannot be solved in polynomial time even with quantum computers. Therefore, a public key cryptosystem and signature scheme based on the difficulty of NP-complete problems and the notion of data complexity are presented here. Finally, we analyze the security of the proposed encryption and signature schemes.


public key cryptography information security NP-complete problem complexity theory quantum computation 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Deutsch D, Jozsa R. Rapid solution of problems by quantum computation. Math Phys Sci, 1992, 439: 553–558zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Bernstein E, Vazirani U. Quantum complexity theory. SIAM J Comput, 1997, 26: 1411–1473zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Simon D R. On the power of quantum computation. SIAM J Comput, 1997, 26: 1474–1483zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Grover L K. Quantum mechanics helps in searching for a needle in haystack. Phys Rev Lett, 1997, 79: 325–328CrossRefGoogle Scholar
  5. 5.
    Shor P W. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J Comput, 1997, 26: 1484–1509zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Mosca M, Ekert A. The hidden subgroup problem and eigenvalue estimation on a quantum computer. In: Proceedings of the 1st NASA International Conference on Quantum Computing and Quantum Communication. Berlin: Springer, 1999Google Scholar
  7. 7.
    Hallgren S, Russell A, Ta-Shma A. The hidden subgroup problem and quantum computation using group representations. SIAM J Comput, 2003, 32: 916–934zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Bennett C H, Brassard G. Quantum cryptography:public key distribution and coin tossing. In: Proceedings of IEEE International Conference on Computers, Systems and Signal Processing, Bangalor, 1984. 10–12Google Scholar
  9. 9.
    Bennett C H, Brassard G, Crépeau C, et al. Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. Phys Rev Lett, 1993, 70: 1895–1899zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Bennett C H, DiVincenzo D P, Smolin J A, et al. Mixed-state entanglement and quantum error correction. Phys Rev A, 1996, 54: 3824–3851MathSciNetCrossRefGoogle Scholar
  11. 11.
    Leung D W. Quantum vernam cipher. Quantum Inf Comput, 2002, 2: 14–34MathSciNetGoogle Scholar
  12. 12.
    Shi J J, Shi R H, Guo Y, et al. Batch proxy quantum blind signature scheme. Sci China Inf Sci, 2013, 56: 052115Google Scholar
  13. 13.
    Xiao F Y, Chen H W. Construction of minimal trellises for quantum stabilizer codes. Sci China Inf Sci, 2013, 56: 012306Google Scholar
  14. 14.
    Gehani A, Labean T H, Reif J H. DNA-based cryptography. In: Proceedings of the 5th Annual Meeting on DNA Based Computers, Cambridge, 2003. 233–249Google Scholar
  15. 15.
    Lu M X, Lai X J, Xiao G Z, et al. A symmetric key cryptography with DNA technology. Sci China Ser F-Inf Sci, 2007, 50: 324–333zbMATHCrossRefGoogle Scholar
  16. 16.
    Lai X J, Lu M X, Qin L, et al. Asymmetric encryption and signature method with DNA technology. Sci China Inf Sci, 2010, 53: 506–514MathSciNetCrossRefGoogle Scholar
  17. 17.
    Okamoto T, Tanaka K, Uchiyama S. Quantum public-key cryptosystems. In: Proceedings of 20th Annual International Cryptology Conference, Santa Barbara, 2000. 147–165Google Scholar
  18. 18.
    Bernstein D J, Buchmann J, Dahmen E. Post-quantum Cryptography. Berlin: Springer, 2000Google Scholar
  19. 19.
    Wang H Z, Zhang H G, Wang Z Y, et al. Extended multivariate public key cryptosystems with secure encryption function. Sci China Inf Sci, 2011, 54: 1161–1171zbMATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Mu L W, Liu X C, Liang C L. Improved construction of LDPC convolutional codes with semi-random parity-check matrices. Sci China Inf Sci, 2014, 57: 022304CrossRefGoogle Scholar
  21. 21.
    Biham E, Shamir A. Differential cryptanalysis of DES-like cryptosystems. J Cryptol, 1991, 4: 3–72zbMATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Biham E, Shamir A. Differential cryptanalysis of the full 16-round DES. In: Proceedings of the 12th Annual International Cryptology Conference, Santa Barbara, 1993. 487–496Google Scholar
  23. 23.
    Feng D G. Cryptanalysis (in Chinese). Beijing: Tsinghua University Press, 2000Google Scholar
  24. 24.
    Bennett C H, Brassard G, Vazirani U, et al. Strengths and weaknesses of quantum computing. SIAM J Comput, 1997, 26: 1510–1523zbMATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    Sleator T, Weinfurter H. Realizable universal quantum logic gates. Phys Rev Lett, 1995, 74: 4087–4090zbMATHMathSciNetCrossRefGoogle Scholar
  26. 26.
    Barenco A, Deutsch D, Ekert A, et al. Conditional quantum dynamics and logic gates. Phys Rev Lett, 1995, 74: 4083–4086CrossRefGoogle Scholar
  27. 27.
    Monroe C, Meekhof D M, King B E, et al. Demonstration of a fundamental quantum logic gate. Phys Rev Lett, 1995, 75: 4714–4717zbMATHMathSciNetCrossRefGoogle Scholar
  28. 28.
    Vedral V V, Barenco A, Ekert A. Quantum networks for elementary arithmetic operations. Phys Rev A, 1996, 54: 147–153MathSciNetCrossRefGoogle Scholar
  29. 29.
    Beckman D, Chari A N, Devabhaktuni S, et al. Efficient networks for quantum factoring. Phys Rev A, 1996, 54: 1034–1063MathSciNetCrossRefGoogle Scholar
  30. 30.
    Christof Z. Fast versions of Shor’s quantum factoring algorithm. arXiv: quant-ph/9806084Google Scholar
  31. 31.
    Parker S, Plenio M B. Efficient factorization with a single pure qubit and logN mixed qubits. Phys Rev Lett, 2000, 85: 3049–3052CrossRefGoogle Scholar
  32. 32.
    Susan L. Protecting Information: from Classical Error Correction to Quantum Cryptograph. Cambridge: Cambridge University Press, 2006Google Scholar
  33. 33.
    de Riedmatten H, Afzelius M, Staudt M U, et al. A solid-state light-matter interface at the single-photon level. Nature, 2008, 456: 773–777CrossRefGoogle Scholar
  34. 34.
    Mariantoni M, Wang H, Yamamoto T, et al. Implementing the quantum von Neumann architecture with superconducting circuits. Science, 2011, 334: 61–65CrossRefGoogle Scholar
  35. 35.
    Kashefi E, Kent A, Vedral V, et al. Comparison of quantum oracles. Phys Rev A, 2002, 65: 050304CrossRefGoogle Scholar
  36. 36.
    Nielsen M A, Chuang I L. Quantum Computation and Quantum Information. Cambridge: Cambridge University Press, 2010zbMATHCrossRefGoogle Scholar
  37. 37.
    Hastad J. Tensor rank is NP-complete. J Algorithms, 1990, 11: 644–654zbMATHMathSciNetCrossRefGoogle Scholar
  38. 38.
    Hillar C J, Lim L-H. Most tensor problems are NP hard. J ACM, 2013, 60: 45MathSciNetCrossRefGoogle Scholar
  39. 39.
    Mao S, Zhang H G, Wu W Q, et al. A resistant quantum key exchange protocol and its corresponding encryption scheme. China Commun, 2014, 11: 124–134CrossRefGoogle Scholar
  40. 40.
    Schneier B. Applied Cryptography: Protocols, Algorithms, and Source Code in C. New York: Wiley, 1996zbMATHGoogle Scholar
  41. 41.
    Wu W Q, Zhang H G, Mao S W, et al. Quantum algorithm to find invariant linear structure of MD hash functions. Quantum Inf Process, 2015, 14: 813–829zbMATHMathSciNetCrossRefGoogle Scholar
  42. 42.
    Wu W Q, Zhang H G, Wang H Z, et al. Polynomial-time quantum algorithms for finding the linear structures of Boolean function. Quantum Inf Process, 2015, 14: 1215–1226MathSciNetCrossRefGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • WanQing Wu
    • 1
  • HuanGuo Zhang
    • 1
  • HouZhen Wang
    • 1
    • 2
  • ShaoWu Mao
    • 1
  • JianWei Jia
    • 1
  • JinHui Liu
    • 1
  1. 1.Computer SchoolWuhan UniversityWuhanChina
  2. 2.State Key Laboratory of CryptologyBeijingChina

Personalised recommendations