Skip to main content
SpringerLink
Log in

Collision-Resistance from Multi-Collision-Resistance

  • Conference paper
  • First Online:
Advances in Cryptology – CRYPTO 2022 (CRYPTO 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13509))

Included in the following conference series:

Abstract

Collision-resistant hash functions (\(\textsf{CRH}\)) are a fundamental and ubiquitous cryptographic primitive. Several recent works have studied a relaxation of \(\textsf{CRH}\) called t-way multi-collision-resistant hash functions (\(t\text {-}\textsf{MCRH}\)). These are families of functions for which it is computationally hard to find a t-way collision, even though such collisions are abundant (and even \((t-1)\)-way collisions may be easy to find). The case of \(t=2\) corresponds to standard \(\textsf{CRH}\), but it is natural to study t-\(\textsf{MCRH}\) for larger values of t.

Multi-collision-resistance seems to be a qualitatively weaker property than standard collision-resistance. Nevertheless, in this work we show a non-blackbox transformation of any moderately shrinking t-\(\textsf{MCRH}\), for \(t \in \{3,4\}\), into an (infinitely often secure) \(\textsf{CRH}\). This transformation is non-constructive – we can prove the existence of a \(\textsf{CRH}\) but cannot explicitly point out a construction.

Our result partially extends to larger values of t. In particular, we show that for suitable values of \(t>t'\), we can transform a t-\(\textsf{MCRH}\) into a \(t'\)-\(\textsf{MCRH}\), at the cost of reducing the shrinkage of the resulting hash function family and settling for infinitely often security. This result utilizes the list-decodability properties of Reed-Solomon codes.

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 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
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.

    Their paper states this theorem for (t, n/2)-\(\textsf{MCRH}\), but their proof immediately extends to any \((t,\varOmega (n))\)-\(\textsf{MCRH}\). They define \(\textsf{MCRH}\) security as holding only against uniform adversaries and thus obtain a uniform i.o. \(\textsf{DCRH}\) secure against uniform adversaries. Under our definition of \(\textsf{MCRH}\) with security against non-uniform adversaries, their approach would also result in a non-uniform construction secure against non-uniform adversaries. They also construct \(\textsf{DCRH}\) from the average-case hardness of problems in \(\textsf{SZK}\), but this result is not relevant here.

  2. 2.

    Thanks to Iftach Haitner for pointing this out to us.

  3. 3.

    Actually, it may be the case that the shrinkage of the hash function we construct is larger than this \(\ell _f\). In such a case, we can simply pad the output of the hash function with 0’s to ensure that the shrinkage is exactly \(\ell '\) (without any effect on its collision-resistance properties).

  4. 4.

    We assume the field elements can be represented using \(\log _2(|\mathbb {F}|)\) bits (in the natural way) and that field operations (i.e., arithmetic operations as well as sampling of random field elements) can be performed in \(\textsf{polylog}(|\mathbb {F}|)\) time. See, e.g., [Sho88] for details.

  5. 5.

    Note that for \(\textsf{Gen}'\) to be non-trivial we must have \(\ell (n) > n/k\).

  6. 6.

    For sake of consistency we define the hash function w.r.t. “security parameter” n/k, since its domain is \(\{0,1\}^{n/k}\).

  7. 7.

    As in Footnote 5, this is only interesting if \(\ell (n) > (n-n/k)\).

  8. 8.

    This is the point where we use the adversary in a non-blackbox manner. Since the adversary is non-uniform, this also makes the construction non-uniform.

  9. 9.

    Sudan additionally established bounds on the algorithmic list-decoding properties of Reed-Solomon codes, whereas for our purposes a combinatorial bound (such as that established in [GRS00]) suffices.

References

  1. Bitansky, N., Degwekar, A.: On the complexity of collision resistant hash functions: new and old black-box separations. In: Hofheinz, D., Rosen, A. (eds.) TCC 2019, Part I. LNCS, vol. 11891, pp. 422–450. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-36030-6_17

    Chapter  MATH  Google Scholar 

  2. Berman, I., Degwekar, A., Rothblum, R.D., Vasudevan, P.N.: Multi-collision resistant hash functions and their applications. In: Nielsen and Rijmen [NR18], pp. 133–161

    Google Scholar 

  3. Bitansky, N., Haitner, I., Komargodski, I., Yogev, E.: Distributional collision resistance beyond one-way functions. In: Ishai, Y., Rijmen, V. (eds.) EUROCRYPT 2019, Part III. LNCS, vol. 11478, pp. 667–695. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-17659-4_23

    Chapter  Google Scholar 

  4. Bitansky, N., Kalai, Y.T., Paneth, O.: Multi-collision resistance: a paradigm for keyless hash functions. In: Diakonikolas, I., Kempe, D., Henzinger, M. (eds.) Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2018, Los Angeles, CA, USA, 25–29 June 2018, pp. 671–684. ACM (2018)

    Google Scholar 

  5. Bitansky, N., Vaikuntanathan, V.: A note on perfect correctness by derandomization. In: Coron, J.-S., Nielsen, J.B. (eds.) EUROCRYPT 2017, Part II. LNCS, vol. 10211, pp. 592–606. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-56614-6_20

    Chapter  Google Scholar 

  6. Dvir, Z., Gutfreund, D., Rothblum, G.N., Vadhan, S.P.: On approximating the entropy of polynomial mappings. In: Chazelle, B. (ed.) Innovations in Computer Science - ICS 2011, Tsinghua University, Beijing, China, 7–9 January 2011, Proceedings, pp. 460–475. Tsinghua University Press (2011)

    Google Scholar 

  7. Dubrov, B., Ishai, Y.: On the randomness complexity of efficient sampling. In: Kleinberg, J.M. (ed.) Proceedings of the 38th Annual ACM Symposium on Theory of Computing, Seattle, WA, USA, 21–23 May 2006, pp. 711–720. ACM (2006)

    Google Scholar 

  8. Dwork, C., Naor, M.: Zaps and their applications. SIAM J. Comput. 36(6), 1513–1543 (2007)

    Article  MathSciNet  Google Scholar 

  9. Dwork, C., Naor, M., Reingold, O.: Immunizing encryption schemes from decryption errors. In: Cachin, C., Camenisch, J.L. (eds.) EUROCRYPT 2004. LNCS, vol. 3027, pp. 342–360. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-540-24676-3_21

    Chapter  Google Scholar 

  10. Goldreich, O.: The Foundations of Cryptography - Volume 2: Basic Applications. Cambridge University Press, Cambridge (2004)

    Google Scholar 

  11. Goldreich, O., Rubinfeld, R., Sudan, M.: Learning polynomials with queries: the highly noisy case. SIAM J. Discret. Math. 13(4), 535–570 (2000)

    Article  MathSciNet  Google Scholar 

  12. Haitner, I., Hoch, J.J., Reingold, O., Segev, G.: Finding collisions in interactive protocols - tight lower bounds on the round and communication complexities of statistically hiding commitments. SIAM J. Comput. 44(1), 193–242 (2015)

    Article  MathSciNet  Google Scholar 

  13. Holmgren, J., Lombardi, A.: Cryptographic hashing from strong one-way functions (or: one-way product functions and their applications). In: Thorup, M. (ed.) 59th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2018, Paris, France, 7–9 October 2018, pp. 850–858. IEEE Computer Society (2018)

    Google Scholar 

  14. Hsiao, C.-Y., Reyzin, L.: Finding collisions on a public road, or do secure hash functions need secret coins? In: Franklin, M. (ed.) CRYPTO 2004. LNCS, vol. 3152, pp. 92–105. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-540-28628-8_6

    Chapter  Google Scholar 

  15. Impagliazzo, R., Luby, M.: One-way functions are essential for complexity based cryptography (extended abstract). In: 30th Annual Symposium on Foundations of Computer Science, Research Triangle Park, North Carolina, USA, 30 October–1 November 1989, pp. 230–235. IEEE Computer Society (1989)

    Google Scholar 

  16. Joux, A.: Multicollisions in iterated hash functions. Application to cascaded constructions. In: Franklin, M. (ed.) CRYPTO 2004. LNCS, vol. 3152, pp. 306–316. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-540-28628-8_19

    Chapter  MATH  Google Scholar 

  17. Komargodski, I., Naor, M., Yogev, E.: White-box vs. black-box complexity of search problems: ramsey and graph property testing. In: Umans, C. (ed.) 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017, Berkeley, CA, USA, 15–17 October 2017, pp. 622–632. IEEE Computer Society (2017)

    Google Scholar 

  18. Komargodski, I., Naor, M., Yogev, E.: Collision resistant hashing for paranoids: dealing with multiple collisions. In: Nielsen and Rijmen [NR18], pp. 162–194

    Google Scholar 

  19. Komargodski, I., Yogev, E.: On distributional collision resistant hashing. In: Shacham, H., Boldyreva, A. (eds.) CRYPTO 2018, Part II. LNCS, vol. 10992, pp. 303–327. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-96881-0_11

    Chapter  Google Scholar 

  20. Lautemann, C.: BPP and the polynomial hierarchy. Inf. Process. Lett. 17(4), 215–217 (1983)

    Article  MathSciNet  Google Scholar 

  21. Naor, M.: Bit commitment using pseudo-randomness. In: Brassard, G. (ed.) CRYPTO 1989. LNCS, vol. 435, pp. 128–136. Springer, New York (1990). https://doi.org/10.1007/0-387-34805-0_13

    Chapter  Google Scholar 

  22. Nielsen, J.B., Rijmen, V. (eds.): Advances in Cryptology - EUROCRYPT 2018, Part II. LNCS, vol. 10821. Springer, Cham (2018)

    Google Scholar 

  23. Nandi, M., Stinson, D.R.: Multicollision attacks on some generalized sequential hash functions. IEEE Trans. Inf. Theory 53(2), 759–767 (2007)

    Article  MathSciNet  Google Scholar 

  24. Personal communication with the authors of [KNY18]

    Google Scholar 

  25. Shoup, V.: New algorithms for finding irreducible polynomials over finite fields. In: 29th Annual Symposium on Foundations of Computer Science, White Plains, New York, USA, 24–26 October 1988, pp. 283–290 (1988)

    Google Scholar 

  26. Simon, D.R.: Finding collisions on a one-way street: can secure hash functions be based on general assumptions? In: Nyberg, K. (ed.) EUROCRYPT 1998. LNCS, vol. 1403, pp. 334–345. Springer, Heidelberg (1998). https://doi.org/10.1007/BFb0054137

    Chapter  Google Scholar 

  27. Sudan, M.: Decoding of Reed Solomon codes beyond the error-correction bound. J. Complex. 13(1), 180–193 (1997)

    Article  MathSciNet  Google Scholar 

  28. Yu, H., Wang, X.: Multi-collision attack on the compression functions of MD4 and 3-Pass HAVAL. In: Nam, K.-H., Rhee, G. (eds.) ICISC 2007. LNCS, vol. 4817, pp. 206–226. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-76788-6_17

    Chapter  Google Scholar 

Download references

Acknowledgments

We thank Itay Berman and Akshay Degwekar for being part of much of our exploration of multicollision resistance, for helpful discussions, and for their collaboration in general. We also thank Iftach Haitner for helpful discussions.

Rothblum was supported in part by the Israeli Science Foundation (Grants No. 1262/18 and 2137/19), by grants from the Technion Hiroshi Fujiwara cyber security research center and Israel cyber directorate, and by the European Union. Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them.

Vasudevan was supported by funds from an NUS Presidential Young Professorship.

Author information

Authors and Affiliations

  1. Technion, Haifa, Israel

    Ron D. Rothblum

  2. National University of Singapore, Singapore, Singapore

    Prashant Nalini Vasudevan

Authors
  1. Ron D. Rothblum
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Prashant Nalini Vasudevan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Prashant Nalini Vasudevan .

Editor information

Editors and Affiliations

  1. New York University, New York, NY, USA

    Yevgeniy Dodis

  2. University of Florida, Gainesville, FL, USA

    Thomas Shrimpton

Rights and permissions

Reprints and permissions

Copyright information

© 2022 International Association for Cryptologic Research

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Rothblum, R.D., Vasudevan, P.N. (2022). Collision-Resistance from Multi-Collision-Resistance. In: Dodis, Y., Shrimpton, T. (eds) Advances in Cryptology – CRYPTO 2022. CRYPTO 2022. Lecture Notes in Computer Science, vol 13509. Springer, Cham. https://doi.org/10.1007/978-3-031-15982-4_17

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-15982-4_17

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-15981-7

  • Online ISBN: 978-3-031-15982-4

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Societies and partnerships

Search

Navigation