Abstract
Sponge paradigm, used in the design of SHA-3, is an alternative hashing technique to the popular Merkle-Damgård paradigm. We revisit the problem of finding B-block-long collisions in sponge hash functions in the auxiliary-input random permutation model, in which an attacker gets a piece of S-bit advice about the random permutation and makes T (forward or inverse) oracle queries to the random permutation.
Recently, significant progress has been made in the Merkle-Damgård setting and optimal bounds are known for a large range of parameters, including all constant values of B. However, the sponge setting is widely open: there exist significant gaps between known attacks and security bounds even for \(B=1\).
Freitag, Ghoshal and Komargodski (CRYPTO 2022) showed a novel attack for \(B=1\) that takes advantage of the inverse queries and achieves advantage \(\widetilde{\varOmega }(\min (S^2T^2/2^{2c}\), \( (S^2T/2^{2c})^{2/3})+T^2/2^r)\), where r is bit-rate and c is the capacity of the random permutation. However, they only showed an \(\widetilde{O}(ST/2^c+T^2/2^r)\) security bound, leaving open an intriguing quadratic gap. For \(B=2\), they beat the general security bound by Coretti, Dodis, Guo (CRYPTO 2018) for arbitrary values of B. However, their highly non-trivial argument is quite laborious, and no better (than the general) bounds are known for \(B\ge 3\).
In this work, we study the possibility of proving better security bounds in the sponge setting. To this end,
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For \(B=1\), we prove an improved \(\widetilde{O}(S^2T^2/2^{2c}+S/2^c+T/2^c+T^2/2^r)\) bound. Our bound strictly improves the bound by Freitag et al., and is optimal for \(ST^2\le 2^c\).
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For \(B=2\), we give a considerably simpler and more modular proof, recovering the bound obtained by Freitag et al.
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We obtain our bounds by adapting the recent multi-instance technique of Akshima, Guo and Liu (CRYPTO 2022) which bypasses the limitations of prior techniques in the Merkle-Damgård setting. To complement our results, we provably show that the recent multi-instance technique cannot further improve our bounds for \(B=1,2\), and the general bound by Correti et al., for \(B\ge 3\).
Overall, our results yield state-of-the-art security bounds for finding short collisions and fully characterize the power of the multi-instance technique in the sponge setting.
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Notes
- 1.
In some practical sponge applications like SHA-3, this salt is usually set to 0. However, when we study the collision resistance of sponge hash functions in the auxiliary input model, such a fixed salt will make finding collisions trivial. [CDG18] identified this need for salting the hash functions for collision resistance in the auxiliary input model and so we are interested in the security bounds against a random initialization salt (just like what prior works [CDG18, ACDW20, AGL22, FGK22] did). See more details on the definition of the auxiliary input model below in Sect. 2.4.
- 2.
[CDG18] proved an \(\widetilde{O}(\frac{ST^2}{C}+\frac{T^2}{R})\) bound using presampling which implies an \((\widetilde{O}(\frac{ST^2}{C}+\frac{T^2}{R}))^S\) multi-instance security.
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Acknowledgements
We thank TCC reviewers for their constructive comments. Siyao Guo and Akshima are supported by National Natural Science Foundation of China Grant No.62102260, Shanghai Municipal Education Commission (SMEC) Grant No. 0920000169, NYTP Grant No. 20121201 and NYU Shanghai Boost Fund. The work was done while Xiaoqi Duan was a research assistant at Shanghai Qi Zhi Institute and supported by the Shanghai Qi Zhi Institute. Most of the work was done while Qipeng Liu was a Postdoctoral researcher in Simons Institute, supported in part by the Simons Institute for Theory of Computing, through a Quantum Postdoctoral Fellowship and by the DARPA SIEVE-VESPA grant No. HR00112020023. 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 supporting institutions.
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Akshima, Duan, X., Guo, S., Liu, Q. (2023). On Time-Space Lower Bounds for Finding Short Collisions in Sponge Hash Functions. In: Rothblum, G., Wee, H. (eds) Theory of Cryptography. TCC 2023. Lecture Notes in Computer Science, vol 14371. Springer, Cham. https://doi.org/10.1007/978-3-031-48621-0_9
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