Quantum Hardcore Functions by Complexity-Theoretical Quantum List Decoding

  • Akinori Kawachi
  • Tomoyuki Yamakami
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4052)


We present three new quantum hardcore functions for any quantum one-way function. We also give a “quantum” solution to Damgård’s question (CRYPTO’88) on his pseudorandom generator by proving the quantum hardcore property of his generator, which has been unknown to have the classical hardcore property. Our technical tool is quantum list-decoding of “classical” error-correcting codes (rather than “quantum” error-correcting codes), which is defined on the platform of computational complexity theory and cryptography (rather than information theory). In particular, we give a simple but powerful criterion that makes a polynomial-time computable code (seen as a function) a quantum hardcore for any quantum one-way function. On their own interest, we also give quantum list-decoding algorithms for codes whose associated quantum states (called codeword states) are “nearly” orthogonal using the technique of pretty good measurement.


Quantum Algorithm Pseudorandom Generator Quantum Fourier Transformation List Decode Hadamard Code 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Adcock, M., Cleve, R.: A quantum Goldreich-Levin theorem with cryptographic applications. In: Alt, H., Ferreira, A. (eds.) STACS 2002. LNCS, vol. 2285, pp. 323–334. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  2. 2.
    Akavia, A., Goldwasser, S., Safra, S.: Proving hard-core predicates using list decoding. In: Proc. FOCS 2003, pp. 146–157 (2003)Google Scholar
  3. 3.
    Ambainis, A., Iwama, K., Kawachi, A., Putra, R.H., Yamashita, S.: Robust quantum algorithms for oracle identification. In: Available at (2004), http://arxiv.org/abs/quant-ph/0411204
  4. 4.
    Atici, A., Servedio, R.: Improved bounds on quantum learning algorithms. In: To appear in Quantum Information Processing, Available also at http://arxiv.org/abs/quant-ph/0411140
  5. 5.
    Barg, A., Zhou, S.: A quantum decoding algorithm for the simplex code. In: Proc. Allerton Conference on Communication, Control and Computing (1998), http://citeseer.ist.psu.edu/barg98quantum.html
  6. 6.
    Bernstein, E., Vazirani, U.: Quantum complexity theory. SIAM J. Comput. 26(5), 1411–1473 (1997)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Buhrman, H., Newman, I., Rohrig, H., Wolf, R.d.: Robust quantum algorithms and polynomials. In: Diekert, V., Durand, B. (eds.) STACS 2005. LNCS, vol. 3404, pp. 593–604. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  8. 8.
    Damgård, I.B.: On the randomness of Legendre and Jacobi sequences. In: Goldwasser, S. (ed.) CRYPTO 1988. LNCS, vol. 403, pp. 163–172. Springer, Heidelberg (1990)Google Scholar
  9. 9.
    D. Deutsch and R. Jozsa. Rapid solution of problems by quantum computation. In Proc. Roy. Soc. London, A, Vol.439, pp.553–558, 1992.MATHMathSciNetGoogle Scholar
  10. 10.
    Eldar, Y.C., Forney Jr., G.D.: On quantum detection and the square-root measurement. IEEE Trans. Inform. Theory 47(3), 858–872 (2001)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Goldreich, O.: Foundations of Cryptography: Basic Tools. Cambridge University Press, Cambridge (2001)MATHCrossRefGoogle Scholar
  12. 12.
    Goldreich, O., Levin, L.A.: A hard-core predicate for all one-way functions. In: Proc. STOC 1989, pp. 25–32 (1989)Google Scholar
  13. 13.
    González Vasco, M.I., Näslund, M.: A survey of hard core functions. In: Proc. Workshop on Cryptography and Computational Number Theory, pp. 227–256. Birkhauser, Basel (2001)Google Scholar
  14. 14.
    Grover, L.K.: Quantum Mechanics helps in searching for a needle in a haystack. Phys. Rev. Lett. 79(2), 325–328 (1997)CrossRefGoogle Scholar
  15. 15.
    Guruswami, V., Sudan, M.: Extensions to the Johnson bound. In: Manuscript (2000), http://theory.csail.mit.edu/~madhu/
  16. 16.
    Hausladen, P., Wootters, W.K.: A ‘ pretty good’ measurement for distinguishing quantum states. J. Mod. Opt. 41, 2385–2390 (1994)MATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Holenstein, T., Maurer, U.M., Sjödin, J.: Complete classification of bilinear hard-core functions. In: Franklin, M. (ed.) CRYPTO 2004. LNCS, vol. 3152, pp. 73–91. Springer, Heidelberg (2004)Google Scholar
  18. 18.
    Høyer, P., Mosca, M., de Wolf, R.: Quantum search on bounded-error inputs. In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds.) ICALP 2003. LNCS, vol. 2719, pp. 291–299. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  19. 19.
    Sudan, M.: List decoding: Algorithms and applications. SIGACT News 31(1), 16–27 (2000)CrossRefGoogle Scholar
  20. 20.
    van Dam, W., Hallgren, S., Ip, L.: Quantum algorithms for some hidden shift problems. In: Proc. SODA 2003, pp. 489–498 (2003)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Akinori Kawachi
    • 1
  • Tomoyuki Yamakami
    • 2
  1. 1.Graduate School of Information Science and EngineeringTokyo Institute of TechnologyTokyoJapan
  2. 2.ERATO-SORST Quantum Computation and Information ProjectJapan Science and Technology AgencyTokyoJapan

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