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Quantum Computational Cryptography

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Quantum Computation and Information

Part of the book series: Topics in Applied Physics ((TAP,volume 102))

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Abstract

As computational approaches to classical cryptography have succeeded in the establishment of the foundation of the network security, computational approaches even to quantum cryptography are promising, since quantum computational cryptography could offer richer applications than the quantum key distribution. Our project focused especially on the quantum one-wayness and quantum public-key cryptosystems. The one-wayness of functions (or permutations) is one of the most important notions in computational cryptography. First, we give an algorithmic characterization of quantum one-way permutations. In other words, we show a necessary and sufficient condition for quantum one-way permutations in terms of reflection operators. Second, we introduce a problem of distinguishing between two quantum states as a new underlying problem that is harder to solve than the graph automorphism problem. The new problem is a natural generalization of the distinguishability problem between two probability distributions, which are commonly used in computational cryptography. We show that the problem has several cryptographic properties and they enable us to construct a quantum publickey cryptosystem, which is likely to withstand any attack of a quantum adversary.

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References

  1. W. Diffie, M. E. Hellman: New directions in cryptography, IEEE Trans. Inf. Theory 22, 644–654 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  2. L. K. Grover: Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer, SIAM J. Comput. 26, 1484–1509 (1997)

    Article  MathSciNet  Google Scholar 

  3. D. Boneh, R. J. Lipton: Quantum cryptanalysis of hidden linear functions, in LNCS 963 (1995) pp. 424–437

    MATH  MathSciNet  Google Scholar 

  4. A. Kitaev: Quantum measurements and the abelian stabilizer problem, LANL Archive quant-ph/9511026 (1995)

    Google Scholar 

  5. S. Hallgren: Polynomial-time quantum algorithms for Pell’s equation and the principal ideal problem, Proc. 34th ACM Symp. Theory of Computing, pp. 653–658 (2002)

    Google Scholar 

  6. C. H. Bennett, G. Brassard: Quantum cryptography: Public key distribution and coin tossing, Proc. IEEE International Conf. Computers, Systems and Signal Processing, pp. 175–179 (1984)

    Google Scholar 

  7. D. Mayers: Unconditional security in quantum cryptography, J. Assoc. Comput. Mach. 48, 351–406 (2001)

    MathSciNet  Google Scholar 

  8. D. Mayers: Unconditionally secure quantum bit commitment is impossible, Phys. Rev. Lett. 78, 3414–3417 (1997)

    Article  ADS  Google Scholar 

  9. H. K. Lo, H. F. Chau: Is quantum bit commitment really possible?, Phys. Rev. Lett. 78, 3410–3413 (1997)

    Article  ADS  Google Scholar 

  10. M. Adcock, R. Cleve: A quantum Goldreich-Levin theorem with cryptographic applications, in LNCS 2285 (2002) pp. 323–334

    MATH  MathSciNet  Google Scholar 

  11. C. Crépeau, P. Dumais, D. Mayers, L. Salvail: Computational collapse of quantum state with application to oblivious transfer, in LNCS 2951 (2004) pp. 374–393

    Google Scholar 

  12. C. Crépeau, F. Légaré, L. Salvail: How to convert the flavor of a quantum bit commitment, in LNCS 2045 (2001) pp. 60–77

    MATH  Google Scholar 

  13. I. Damgård, S. Fehr, L. Salvail: Zero-knowledge proofs and string commitments withstanding quantum attacks, in LNCS 3152 (2004) pp. 254–272

    MATH  Google Scholar 

  14. P. Dumais, D. Mayers, L. Salvail: Perfectly concealing quantum bit commitment from any quantum one-way permutation, in LNCS 1807 (2000) pp. 300–315

    MATH  MathSciNet  Google Scholar 

  15. T. Okamoto, K. Tanaka, S. Uchiyama: Quantum public-key cryptosystems, in LNCS 1880 (2000) pp. 147–165

    MATH  MathSciNet  Google Scholar 

  16. A. Kawachi, H. Kobayashi, T. Koshiba, R. H. Putra: Universal test for quantum one-way permutations, in LNCS 3153 (2004) pp. 839–850

    MATH  MathSciNet  Google Scholar 

  17. A. Kawachi, H. Kobayashi, T. Koshiba, R. H. Putra: Universal test for quantum one-way permutations, Theor. Comput. Sci. 345, 370–385 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  18. A. Kawachi, T. Koshiba, H. Nishimura, T. Yamakami: Computational indistinguishability between quantum states and its cryptographic application, in LNCS 3494 (2005) pp. 268–284

    MathSciNet  Google Scholar 

  19. L. K. Grover: A fast quantum mechanical algorithm for database search, Proc. 28th ACM Symp. Theory of Computing, pp. 212–219 (1996)

    Google Scholar 

  20. G. Brassard, P. Høyer, M. Mosca, A. Tapp: Quantum amplitude amplification and estimation, in S. J. Lomonaco, Jr., H. E. Brandt (Eds.): Quantum Computation and Information, AMS Contemporary Mathematics 305 (2002) pp. 53–74

    Google Scholar 

  21. M. Blum, S. Micali: How to generate cryptographically strong sequences of pseudo-random bits, SIAM J. Comput. 13, 850–864 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  22. A. C.-C. Yao: Theory and applications of trapdoor functions, Proc. 23rd IEEE Symp. Foundations of Computer Science, pp. 80–91 (1982)

    Google Scholar 

  23. O. Goldreich, L. A. Levin: A hard-core predicate for all one-way functions, Proc. 21st ACM Symp. Theory of Computing, pp. 25–32 (1989)

    Google Scholar 

  24. J. Håstad, R. Impagliazzo, L. A. Levin, M. Luby: A pseudorandom generator from any one-way function, SIAM J. Comput. 28, 1364–1396 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  25. A. W. Schrift, A. Shamir: Universal tests for nonuniform distributions, J. Cryptol. 6, 119–133 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  26. E. Kashefi, H. Nishimura, V. Vedral: On quantum one-way permutations, Quantum Inf. Comput. 2, 379–398 (2002)

    MathSciNet  MATH  Google Scholar 

  27. R. Impagliazzo, M. Naor: Efficient cryptographic schemes provably as secure as subset sum, J. Cryptol. 9, 199–216 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  28. M. Ajtai, C. Dwork: A public-key cryptosystem with worst-case/average-case equivalence, Proc. 29th ACM Symp. Theory of Computing, pp. 284–293 (1997)

    Google Scholar 

  29. O. Regev: New lattice-based cryptographic constructions, J. Assoc. Comput. Mach. 51, 899–942 (2004)

    MathSciNet  MATH  Google Scholar 

  30. O. Regev: On lattices, learning with errors, random linear codes and cryptography, Proc. 37th ACM Symp. Theory of Computing, pp. 84–93 (2005)

    Google Scholar 

  31. S. Goldwasser, S. Micali: Probabilistic encryption, J. Comput. Syst. Sci. 28, 270–299 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  32. O. Regev: Quantum computation and lattice problems, SIAM J. Comput. 33, 738–760 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  33. S. Hallgren, A. Russell, A. Ta-Shma: The hidden subgroup problem and quantum computation using group representations, SIAM J. Comput. 32, 916–934 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  34. M. Grigni, L. J. Schulman, M. Vazirani, U. Vazirani: Quantum mechanical algorithms for the nonabelian hidden subgroup problem, Combinatorica 24, 137–154 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  35. J. Kempe, A. Shalev: The hidden subgroup problem and permutation group theory, Proc. 16th ACM-SIAM Symp. Discrete Algorithms, pp. 1118–1125 (2005)

    Google Scholar 

  36. C. Moore, A. Russell: Tight results on multiregister fourier sampling: Quantum measurements for graph isomorphism require entanglement, LANL Archive quant-ph/0511149 (2005)

    Google Scholar 

  37. S. Hallgren, M. Rötteler, P. Sen: Limitations of quantum coset states for graph isomorphism, LANL Archive quant-ph/0511148 (2005)

    Google Scholar 

  38. M. Bellare, A. Desai, D. Pointcheval, P. Rogaway: Relations among notions of security for public-key encryption schemes, in LNCS 1462 (1998) pp. 26–45

    MATH  Google Scholar 

  39. E. M. Luks: Permutation groups and polynomial-time computation, in L. Finkelstein, W. M. Kantor (Eds.): Groups and Computation, DIMACS Series in Discrete Mathematics and Theoretical Computer Science 11 (1993) pp. 139–175

    Google Scholar 

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Author information

Authors and Affiliations

  1. Graduate School of Information Science and Engineering, Tokyo Institute of Technology, Ookayama 2-12-1, Meguro-ku, Tokyo, 152-8552, Japan

    Akinori Kawachi

  2. Department of Information and Computer Sciences, Saitama University, 255 Shimo-Ohkubo, Sakura-ku, Saitama, 338-8570, Japan

    Takeshi Koshiba

Authors
  1. Akinori Kawachi
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  2. Takeshi Koshiba
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Editor information

Editors and Affiliations

  1. Graduate School of Information, Science and Technology, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-8656, Japan

    Hiroshi Imai

  2. ERATO Quantum Computation and Information Project, Japan Science and Technology Agency, 201 Daini Hongo White Bldg 5-28-3, Hongo, Bunkyo-ku, Tokyo, 113-0033, Japan

    Hiroshi Imai  & Masahito Hayashi  & 

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© 2006 Springer-Verlag Berlin Heidelberg

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Kawachi, A., Koshiba, T. (2006). Quantum Computational Cryptography. In: Imai, H., Hayashi, M. (eds) Quantum Computation and Information. Topics in Applied Physics, vol 102. Springer, Berlin, Heidelberg . https://doi.org/10.1007/3-540-33133-6_7

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