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Time-Space Tradeoffs for Finding Multi-Collisions in Merkle-Damgård Hash Functions

Author Akshima

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Author Details

Akshima
  • NYU Shanghai, China

Funding

Most of this work was done while Akshima was a PhD student at University of Chicago and was supported in part by NSF Grant No. 1928767. Akshima is supported by Ministry of Education of P.R.C., through an Overseas Postdoc Special Program (2023), National Natural Science Foundation of China Grant No. 6210226 and NYTP Grant No. 2012120. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the funding institutions or the governments.

Acknowledgements

We thank the anonymous reviewers for their constructive comments on earlier drafts of the work. We thank David Cash and Siyao Guo for useful discussions.

Cite AsGet BibTex

Akshima. Time-Space Tradeoffs for Finding Multi-Collisions in Merkle-Damgård Hash Functions. In 5th Conference on Information-Theoretic Cryptography (ITC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 304, pp. 9:1-9:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ITC.2024.9

Abstract

We analyze the multi-collision resistance of Merkle-Damgård hash function construction in the auxiliary input random oracle model. Finding multi-collisions or m-way collisions, for some parameter m, in a hash function consists of m distinct input that have the same output under the hash function. This is a natural generalization of the collision finding problem in hash functions, which is basically finding 2-way collisions. Hardness of finding collisions, or collision resistance, is an important security assumption in cryptography. While the time-space trade-offs for collision resistance of hash functions has received considerable attention, this is the first work that studies time-space trade-offs for the multi-collision resistance property of hash functions based on the popular and widely used Merkle-Damgård (MD) constructions. In this work, we study how the advantage of finding m-way collisions depends on the parameter m. We believe understanding whether multi-collision resistance is a strictly easier property than collision resistance is a fundamental problem and our work facilitates this for adversaries with auxiliary information against MD based hash functions. Furthermore, in this work we study how the advantage varies with the bound on length of the m colliding inputs. Prior works [Akshima et al., 2020; Ashrujit Ghoshal and Ilan Komargodski, 2022; Akshima et al., 2022] have shown that finding "longer" collisions with auxiliary input in MD based hash functions becomes easier. More precisely, the advantage of finding collisions linearly depends on the bound on the length of colliding inputs. In this work, we show similar dependence for m-way collision finding, for any m ≥ 2. We show a simple attack for finding 1-block m-way collisions which achieves an advantage of Ω̃(S/mN). For 2 ≤ B < log m, we give the best known attack for finding B-blocks m-way collision which achieves an advantage of Ω̃(ST/m^{1/(B-1)}N) when m^{1/(B-1)}-way collisions exist on every salt. For B > log m, our attack achieves an advantage of Ω̃(STB/N) which is optimal when SB ≥ T and ST² ≤ N. The main results of this work is showing that our attacks are optimal for B = 1 and B = 2. This implies that in the auxiliary-input random oracle model, the advantage decreases by a multiplicative factor of m for finding 1-block and 2-block m-way collisions (compared to collision finding) in Merkle-Damgård based hash functions.

Subject Classification

ACM Subject Classification
  • Security and privacy → Cryptography
  • Security and privacy → Information-theoretic techniques
  • Mathematics of computing → Probability and statistics
Keywords
  • Collision
  • hash functions
  • multi-collisions
  • Merkle-Damgård
  • pre-computation
  • auxiliary input

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