Skip to main content
SpringerLink
Log in

Cryptography and Algorithmic Randomness

  • Published:
Theory of Computing Systems Aims and scope Submit manuscript

Abstract

The secure instantiation of the random oracle is one of the major open problems in modern cryptography. We investigate this problem using concepts and methods of algorithmic randomness. In modern cryptography, the random oracle model is widely used as an imaginary framework in which the security of a cryptographic scheme is discussed. In the random oracle model, the cryptographic hash function used in a cryptographic scheme is formulated as a random variable uniformly distributed over all possibility of the function, called the random oracle. The main result of this paper is to show that, for any secure signature scheme in the random oracle model, there exists a specific computable function which can instantiate the random oracle while keeping the security originally proved in the random oracle model. In modern cryptography the generic group model is used also for a similar purpose to the random oracle model. We show that the same results hold for the generic group model. In the process of proving the results, we introduce the notion of effective security, demonstrating the importance of this notion in modern cryptography.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. Normally, Martin-Löf random is defined with fixing the total recursive function \(f\colon \mathbb {N}^{+}\to \mathbb {Q}\cap (0,\infty )\) to the form f(n)=2 −n . However, the relaxation of the function f as in Definition 1 does not alter the class of Martin-Löf random infinite binary sequences.

  2. Normally, a probabilistic (uniform) polynomial-time Turing machine is called a probabilistic polynomial-time adversary when it is used as an adversary against a cryptographic scheme.

  3. In fact, \(\mathsf {ML\text {-}TEST}_{\mathrm {\Pi }}^{\mathsf {EUF\text {-}ACMA}}\) contains only Schnorr tests, where a Schnorr test is defined as a Martin-Löf test \(\mathcal {C}\subset \mathbb {N}^{+}\times \{0,1\}^{*}\) such that \(\mathcal {L}\left ({[{\mathcal {C}_{n}}]^{\prec }}\right )\) is computable uniformly in n. For the detail of Schnorr tests, see e.g. Section 3.5 of Nies [23].

  4. Lemma 3 can be used to prove the non-existence of universal Schnorr test for the notion of Schnorr randomness for an infinite binary sequence. See Fact 3.5.9 of [23] for the detail.

  5. See [15, Chapter 4] for the detail of collision resistant hash function.

  6. As such S ′, the set T∪{s∈S∣t is a prefix of s} suffices, where T={t} if there is a prefix s∈S of t and T=∅ otherwise.

  7. In order to prove Theorem 13, it is suffice to use the inequality 2n≥n d which holds for all n≥((d+1)/ ln2)d+1 and not for all n≥d 2 as in Lemma 4. The former follows immediately from the inequality e x≥x which holds for all \(x\in \mathbb {R}\). However, we prefer a more “insightful” polynomial lower bound n≥d 2 than the super-exponential lower bound n≥((d+1)/ ln2)d+1. See Section 11 for further remarks.

References

  1. Bellare, M., Rogaway, P.: Random oracles are practical: a paradigm for designing efficient protocols. In: Proceedings of 1st ACM Conference Computing, Communications and Security, ACM, pp. 62–73 (1993)

  2. Bellare, M., Boldyreva, A., Palacio, A.: An uninstantiable random-oracle-model scheme for a hybrid-encryption problem. In: Proceedings of EUROCRYPT 2004, Lecture Notes in Computer Science, vol. 3027, pp. 171–188. Springer (2004)

  3. Bienvenu, L., Merkle, W., Nies, A.: Solovay functions and K-triviality. In: Proceedings of the 28th Symposium on Theoretical Aspects of Computer Science (STACS2011), pp. 452–463 (2011)

  4. Brattka, V., Miller, J., Nies, A.: Randomness and differentiability (2012). preprint

  5. Canetti, R., Goldreich, O., Halevi, S: The random oracle methodology, revisited. J. ACM 51, 557–594 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  6. Chaitin, G. J.: On the length of programs for computing finite binary sequences. J. Assoc. Comput. Mach. 13, 547–569 (1966)

    Article  MATH  MathSciNet  Google Scholar 

  7. Chaitin, G. J.: A theory of program size formally identical to information theory. J. Assoc. Comput. Mach. 22, 329–340 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  8. Chaitin, G. J.: Algorithmic Information Theory. Cambridge University Press, Cambridge (1987)

    Book  Google Scholar 

  9. Dent, A. W.: Adapting the weaknesses of the random oracle model to the generic group model. In: Proceedings of ASIACRYPT 2002, Lecture Notes in Computer Science, vol. 2501, pp. 100–109. Springer (2002)

  10. Downey, R. G., Hirschfeldt, D. R.: Algorithmic Randomness and Complexity. Springer-Verlag, New York (2010)

    Book  MATH  Google Scholar 

  11. Fischlin, M., Lehmann, A, Ristenpart, T., Shrimpton, T., Stam, M., Tessaro, S.: Random oracles with(out) programmability. Proceedings of ASIACRYPT 2010, Lecture Notes in Computer Science, Springer-Verlag, vol. 6477, pp. 303–320. (2010)

  12. Goldreich, O.: Foundations of Cryptography: Volume 1 – Basic Tools. Cambridge University Press, New York (2001)

    Book  Google Scholar 

  13. Goldreich, O.: Foundations of Cryptography: Volume 2 – Basic Applications. Cambridge University Press, New York (2004)

    Book  Google Scholar 

  14. Impagliazzo, R., Rudich, S.: Limits on the provable consequences of one-way permutations. In: Proceedings of CRYPTO’88, Lecture Notes in Computer Science, Springer-Verlag, vol. 403, pp. 8–26. (1990)

  15. Katz, J., Lindell, Y.: Introduction to Modern Cryptography. Chapman & Hall/CRC Press (2007)

  16. Kolmogorov, A. N.: Three approaches to the quantitative definition of information. Probl. Inform. Transm. 1(1), 1–7 (1965)

    MathSciNet  Google Scholar 

  17. Leurent, G., Nguyen, P. Q.: How risky is the random-oracle model? Proceedings of CRYPTO 2009, Lecture Notes in Computer Science, Springer-Verlag, vol. 5677, pp. 445–464. (2009)

  18. Martin-Löf, P.: The definition of random sequences. Inf. Control. 9, 602–619 (1966)

    Article  MATH  Google Scholar 

  19. Maurer, U., Wolf, S.: Lower bounds on generic algorithms in groups. In: Proceedings of EUROCRYPT’98, Lecture Notes in Computer Science, Springer-Verlag, vol. 1403, pp. 72–84. (1998)

  20. Maurer, U.: Abstract models of computation in cryptography. Proceedings of Cryptography and Coding 2005, Lecture Notes in Computer Science, Springer-Verlag, vol. 3796, pp. 1–12. (2005)

  21. Miller, J., Yu, L.: On initial segment complexity and degrees of randomness. Trans. Amer. Math. Soc. 360, 3193–3210 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  22. Moriyama, D., Nishimaki, R., Okamoto, T.: Theory of Public-Key Cryptography Industrial and Applied Mathematics Series vol. 2. JSIAM, Kyoritsu Shuppan Co., Ltd., Tokyo, 2011. In Japanese

  23. Nies, A.: Computability and Randomness. Oxford University Press, New York (2009)

    Book  MATH  Google Scholar 

  24. Pour-El, M. B., Richards, J. I.: Computability in Analysis and Physics. Perspectives in Mathematical Logic. Springer-Verlag, Berlin (1989)

    Book  Google Scholar 

  25. Schnorr, C.-P.: A unified approach to the definition of a random sequence. Math. Syst. Theory 5, 246–258 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  26. Schnorr, C.-P.: Process complexity and effective random tests. J. Comput. Syst. Sci. 7, 376–388 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  27. Shoup, V.: Lower bounds for discrete logarithms and related problems. Proceedings of EUROCRYPT’97, Lecture Notes in Computer Science, Springer-Verlag, vol. 1233, pp. 256–266. (1997)

  28. Solomonoff, R. J.: A formal theory of inductive inference. Part I and Part II. Inform. Control vol. 7, pp. 1–22, 1964; vol. 7, pp. 224–254 (1964)

  29. Solovay, R. M.: Draft of a paper (or series of papers) on Chaitin’s work... done for the most part during the period of Sept.–Dec. 1974,” unpublished manuscript, IBM Thomas J. Watson Research Center, Yorktown Heights, New York, May 1975, pp. 215

  30. Weihrauch, K.: Computable Analysis. Springer-Verlag, Berlin (2000)

    Book  MATH  Google Scholar 

Download references

Acknowledgments

This work was supported by JSPS KAKENHI Grant Number 23340020 and by the Ministry of Economy, Trade and Industry of Japan.

Author information

Authors and Affiliations

  1. Research and Development Initiative, Chuo University 1–13–27 Kasuga, Bunkyo-ku, Tokyo, 112-8551, Japan

    Kohtaro Tadaki & Norihisa Doi

Authors
  1. Kohtaro Tadaki
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Norihisa Doi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kohtaro Tadaki.

Additional information

A part of this work was presented at the Seventh International Conference on Computability, Complexity and Randomness (CCR 2012), July 2-6, 2012, Cambridge, Great Britain.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tadaki, K., Doi, N. Cryptography and Algorithmic Randomness. Theory Comput Syst 56, 544–580 (2015). https://doi.org/10.1007/s00224-014-9545-9

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00224-014-9545-9

Keywords

Search

Navigation