Skip to main content
SpringerLink
Log in

Quantum Security Proofs Using Semi-classical Oracles

  • Conference paper
  • First Online:
Advances in Cryptology – CRYPTO 2019 (CRYPTO 2019)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 11693))

Included in the following conference series:

Abstract

We present an improved version of the one-way to hiding (O2H) Theorem by Unruh, J ACM 2015. Our new O2H Theorem gives higher flexibility (arbitrary joint distributions of oracles and inputs, multiple reprogrammed points) as well as tighter bounds (removing square-root factors, taking parallelism into account). The improved O2H Theorem makes use of a new variant of quantum oracles, semi-classical oracles, where queries are partially measured. The new O2H Theorem allows us to get better security bounds in several public-key encryption schemes.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 119.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 159.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Which allows to reprogram the random oracle at a location that is influenced by the adversary.

  2. 2.

    In Game 7 in [30], a secret \(\delta ^*\) is encrypted using a one-time secure encryption scheme, and the final step in the proof concludes that therefore \(\delta ^*\) cannot be guessed. However, Game 7 contains an oracle \( Dec ^{**}\) that in turn accesses \(\delta ^*\) directly, invalidating that argument.

  3. 3.

    Theorem 1 gives us different options how to define the right game. Conceptually simplest is variant (1) (it does not involve a semi-classical oracle in the right game), but it does not apply in all situations. The basic idea behind all variants is the same, namely that the adversary gets access to an oracle G that behaves differently on the set S of marked elements.

    In the present proof, we use specifically variant (4) because then Game 4 will be of a form that is particularly easy to analyze (the adversary has winning probability 0 there).

  4. 4.

    Choosing a different variant here would slightly change the formula below but lead to the same problems.

  5. 5.

    The reason for choosing this particular variant is that same as in footnote 3.

References

  1. Ambainis, A.: Quantum lower bounds by quantum arguments. J. Comput. Syst. Sci. 64(4), 750–767 (2002). https://doi.org/10.1006/jcss.2002.1826

    Article  MathSciNet  MATH  Google Scholar 

  2. Ambainis, A., Hamburg, M., Unruh, D.: Quantum security proofs using semi-classical oracles. IACR ePrint2018/904 (2019). Full version of this paper

  3. Ambainis, A., Rosmanis, A., Unruh, D.: Quantum attacks on classical proof systems: the hardness of quantum rewinding. In: 55th FOCS, pp. 474–483. IEEE Computer Society Press, October 2014

    Google Scholar 

  4. Balogh, M., Eaton, E., Song, F.: Quantum collision-finding in non-uniform random functions. In: Lange, T., Steinwandt, R. (eds.) PQCrypto 2018. LNCS, vol. 10786, pp. 467–486. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-79063-3_22

    Chapter  MATH  Google Scholar 

  5. Beals, R., Buhrman, H., Cleve, R., Mosca, M., de Wolf, R.: Quantum lower bounds by polynomials. J. ACM 48(4), 778–797 (2001). https://doi.org/10.1145/502090.502097

    Article  MathSciNet  MATH  Google Scholar 

  6. Bellare, M., Rogaway, P.: Random oracles are practical: a paradigm for designing efficient protocols. In: Ashby, V. (ed.) ACM CCS 1993, pp. 62–73. ACM Press, November 1993

    Google Scholar 

  7. Bellare, M., Rogaway, P.: Optimal asymmetric encryption. In: De Santis, A. (ed.) EUROCRYPT 1994. LNCS, vol. 950, pp. 92–111. Springer, Heidelberg (1995). https://doi.org/10.1007/BFb0053428

    Chapter  Google Scholar 

  8. Boneh, D., Dagdelen, Ö., Fischlin, M., Lehmann, A., Schaffner, C., Zhandry, M.: Random oracles in a quantum world. In: Lee, D.H., Wang, X. (eds.) ASIACRYPT 2011. LNCS, vol. 7073, pp. 41–69. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-25385-0_3

    Chapter  MATH  Google Scholar 

  9. Boneh, D., Eskandarian, S., Fisch, B.: Post-quantum EPID group signatures from symmetric primitives. Cryptology ePrint Archive, Report 2018/261 (2018). https://eprint.iacr.org/2018/261

  10. Boyer, M., Brassard, G., Høyer, P., Tapp, A.: Tight bounds on quantum searching. Fortschritte der Physik 46(4–5), 493–505 (1998)

    Article  Google Scholar 

  11. Chase, M., et al.: Post-quantum zero-knowledge and signatures from symmetric-key primitives. In: Thuraisingham, B.M., Evans, D., Malkin, T., Xu, D. (eds.) ACM CCS 2017, pp. 1825–1842. ACM Press, October/November 2017

    Google Scholar 

  12. Chen, M.S., Hülsing, A., Rijneveld, J., Samardjiska, S., Schwabe, P.: SOFIA: \(\cal{MQ}\)-based signatures in the QROM. In: Abdalla, M., Dahab, R. (eds.) PKC 2018, Part II. LNCS, vol. 10770, pp. 3–33. Springer, Heidelberg (Mar (2018). https://doi.org/10.1007/978-3-319-76581-5_1

    Chapter  Google Scholar 

  13. Derler, D., Ramacher, S., Slamanig, D.: Post-quantum zero-knowledge proofs for accumulators with applications to ring signatures from symmetric-key primitives. In: Lange, T., Steinwandt, R. (eds.) PQCrypto 2018. LNCS, vol. 10786, pp. 419–440. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-79063-3_20

    Chapter  MATH  Google Scholar 

  14. Eaton, E.: Leighton-Micali hash-based signatures in the quantum random-oracle model. In: Adams, C., Camenisch, J. (eds.) SAC 2017. LNCS, vol. 10719, pp. 263–280. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-72565-9_13

    Chapter  Google Scholar 

  15. Ebrahimi, E.E., Unruh, D.: Quantum collision-resistance of non-uniformly distributed functions: upper and lower bounds. Quantum Inf. Comput. 18(15&16), 1332–1349 (2018). http://www.rintonpress.com/xxqic18/qic-18-1516/1332-1349.pdf

  16. Fiat, A., Shamir, A.: How to prove yourself: practical solutions to identification and signature problems. In: Odlyzko, A.M. (ed.) CRYPTO 1986. LNCS, vol. 263, pp. 186–194. Springer, Heidelberg (1987). https://doi.org/10.1007/3-540-47721-7_12

    Chapter  Google Scholar 

  17. Fujisaki, E., Okamoto, T.: Secure integration of asymmetric and symmetric encryption schemes. J. Cryptol. 26(1), 80–101 (2013)

    Article  MathSciNet  Google Scholar 

  18. Galbraith, S.D., Petit, C., Silva, J.: Identification protocols and signature schemes based on supersingular isogeny problems. In: Takagi, T., Peyrin, T. (eds.) ASIACRYPT 2017. LNCS, vol. 10624, pp. 3–33. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-70694-8_1

    Chapter  Google Scholar 

  19. Hofheinz, D., Hövelmanns, K., Kiltz, E.: A modular analysis of the Fujisaki-Okamoto transformation. In: Kalai, Y., Reyzin, L. (eds.) TCC 2017. LNCS, vol. 10677, pp. 341–371. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-70500-2_12

    Chapter  MATH  Google Scholar 

  20. Hövelmanns, K., Kiltz, E., Schäge, S., Unruh, D.: Generic authenticated key exchange in the quantum random oracle model. Cryptology ePrint Archive, Report 2018/928 (2018). https://eprint.iacr.org/2018/928

  21. Hülsing, A., Rijneveld, J., Song, F.: Mitigating multi-target attacks in hash-based signatures. In: Cheng, C.-M., Chung, K.-M., Persiano, G., Yang, B.-Y. (eds.) PKC 2016. LNCS, vol. 9614, pp. 387–416. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-49384-7_15

    Chapter  Google Scholar 

  22. Jiang, H., Zhang, Z., Chen, L., Wang, H., Ma, Z.: IND-CCA-secure key encapsulation mechanism in the quantum random oracle model, revisited. In: Shacham, H., Boldyreva, A. (eds.) CRYPTO 2018. LNCS, vol. 10993, pp. 96–125. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-96878-0_4

    Chapter  Google Scholar 

  23. Jiang, H., Zhang, Z., Ma, Z.: Key encapsulation mechanism with explicit rejection in the quantum random oracle model. In: Lin, D., Sako, K. (eds.) PKC 2019. LNCS, vol. 11443, pp. 618–645. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-17259-6_21

    Chapter  Google Scholar 

  24. Jiang, H., Zhang, Z., Ma, Z.: Key encapsulation mechanism with explicit rejection in the quantum random oracle model. Cryptology ePrint Archive, Report 2019/052 (2019). https://eprint.iacr.org/2019/052

  25. Leighton, F.T., Micali, S.: Large provably fast and secure digital signature schemes based on secure hash functions. US Patent 5,432,852 (1995)

    Google Scholar 

  26. Nielsen, M., Chuang, I.: Quantum Computation and Quantum Information, 1st edn. Cambridge University Press, Cambridge (2000)

    MATH  Google Scholar 

  27. Saito, T., Xagawa, K., Yamakawa, T.: Tightly-secure key-encapsulation mechanism in the quantum random oracle model. In: Nielsen, J.B., Rijmen, V. (eds.) EUROCRYPT 2018. LNCS, vol. 10822, pp. 520–551. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-78372-7_17

    Chapter  MATH  Google Scholar 

  28. Song, F., Yun, A.: Quantum security of NMAC and related constructions. In: Katz, J., Shacham, H. (eds.) CRYPTO 2017. LNCS, vol. 10402, pp. 283–309. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-63715-0_10

    Chapter  Google Scholar 

  29. Targhi, E.E., Tabia, G.N., Unruh, D.: Quantum collision-resistance of non-uniformly distributed functions. In: Takagi, T. (ed.) PQCrypto 2016. LNCS, vol. 9606, pp. 79–85. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-29360-8_6

    Chapter  MATH  Google Scholar 

  30. Targhi, E.E., Unruh, D.: Post-quantum security of the Fujisaki-Okamoto and OAEP transforms. In: Hirt, M., Smith, A. (eds.) TCC 2016. LNCS, vol. 9986, pp. 192–216. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-53644-5_8

    Chapter  MATH  Google Scholar 

  31. Unruh, D.: Quantum position verification in the random oracle model. In: Garay, J.A., Gennaro, R. (eds.) CRYPTO 2014. LNCS, vol. 8617, pp. 1–18. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-662-44381-1_1

    Chapter  Google Scholar 

  32. Unruh, D.: Non-interactive zero-knowledge proofs in the quantum random oracle model. In: Oswald, E., Fischlin, M. (eds.) EUROCRYPT 2015. LNCS, vol. 9057, pp. 755–784. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-46803-6_25

    Chapter  MATH  Google Scholar 

  33. Unruh, D.: Revocable quantum timed-release encryption. J. ACM 62(6), 49:1–49:76 (2015). Preprint on IACR ePrint 2013/606

    Google Scholar 

  34. Unruh, D.: Post-quantum security of Fiat-Shamir. In: Takagi, T., Peyrin, T. (eds.) ASIACRYPT 2017. LNCS, vol. 10624, pp. 65–95. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-70694-8_3

    Chapter  Google Scholar 

  35. Yoo, Y., Azarderakhsh, R., Jalali, A., Jao, D., Soukharev, V.: A post-quantum digital signature scheme based on supersingular isogenies. In: Kiayias, A. (ed.) FC 2017. LNCS, vol. 10322, pp. 163–181. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-70972-7_9

    Chapter  Google Scholar 

  36. Zalka, C.: Grover’s quantum searching algorithm is optimal. Phys. Rev. A 60, 2746–2751 (1999). https://arxiv.org/abs/quant-ph/9711070

    Article  Google Scholar 

  37. Zhandry, M.: How to construct quantum random functions. In: 53rd FOCS, pp. 679–687. IEEE Computer Society Press, October 2012

    Google Scholar 

  38. Zhandry, M.: Secure identity-based encryption in the quantum random oracle model. In: Safavi-Naini, R., Canetti, R. (eds.) CRYPTO 2012. LNCS, vol. 7417, pp. 758–775. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-32009-5_44

    Chapter  MATH  Google Scholar 

  39. Zhandry, M.: A note on the quantum collision and set equality problems. Quantum Inf. Comput. 15(7&8) (2015)

    Google Scholar 

Download references

Acknowledgements

Thanks to Daniel Kane, Eike Kiltz, and Kathrin Hövelmanns for valuable discussions. Ambainis was supported by the ERDF project 1.1.1.5/18/A/020. Unruh was supported by institutional research funding IUT2-1 of the Estonian Ministry of Education and Research, the United States Air Force Office of Scientific Research (AFOSR) via AOARD Grant “Verification of Quantum Cryptography” (FA2386-17-1-4022), the Mobilitas Plus grant MOBERC12 of the Estonian Research Council, and the Estonian Centre of Exellence in IT (EXCITE) funded by ERDF.

Author information

Authors and Affiliations

  1. University of Latvia, Riga, Latvia

    Andris Ambainis

  2. Rambus Security Division, San Francisco, USA

    Mike Hamburg

  3. University of Tartu, Tartu, Estonia

    Dominique Unruh

Authors
  1. Andris Ambainis
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mike Hamburg
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Dominique Unruh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Andris Ambainis or Mike Hamburg .

Editor information

Editors and Affiliations

  1. Georgia Institute of Technology, Atlanta, GA, USA

    Alexandra Boldyreva

  2. University of California at San Diego, La Jolla, CA, USA

    Daniele Micciancio

A Optimality of Corollary 1

A Optimality of Corollary 1

Lemma 8

If \(S=\{x\}\) where , then there is a q-query algorithm such that

Proof

The algorithm is as follows:

  • Make the first query with amplitude \(1/\sqrt{N}\) in all positions.

  • Between queries, transform the state by the unitary \(U:=2E/N-I\) where E is the matrix containing 1 everywhere. That U is unitary follows since \(U^\dagger U=4E^2/N^2-4E/N+I=I\) using \(E^2=NE\).

One may calculate by induction that the final non-normalized state has amplitude

$$ \left( 1-\frac{2}{N}\right) ^{q-1} \cdot \frac{1}{\sqrt{N}} $$

in all positions except for the xth one (where the amplitude is 0), so its squared norm is

As a function of 1 / N, this expression’s derivatives alternate on [0, 1 / 2], so it is below its second-order Taylor expansion:

This completes the proof.    \(\square \)

Rights and permissions

Reprints and permissions

Copyright information

© 2019 International Association for Cryptologic Research

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Ambainis, A., Hamburg, M., Unruh, D. (2019). Quantum Security Proofs Using Semi-classical Oracles. In: Boldyreva, A., Micciancio, D. (eds) Advances in Cryptology – CRYPTO 2019. CRYPTO 2019. Lecture Notes in Computer Science(), vol 11693. Springer, Cham. https://doi.org/10.1007/978-3-030-26951-7_10

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-26951-7_10

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-26950-0

  • Online ISBN: 978-3-030-26951-7

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Societies and partnerships

Search

Navigation