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We revisit the problem of finding B-block-long collisions in Merkle-Damg˚ard Hash Functions in the auxiliary-input random oracle model, in which an attacker gets a piece of S-bit advice about the random oracle and makes T oracle queries. Akshima, Cash, Drucker and Wee (CRYPTO 2020), based on the work of Coretti, Dodis, Guo and Steinberger (EUROCRYPT 2018), showed a simple attack for 2 ≤ B ≤ T (with respect to a random salt). The attack achieves advantage Ω( e ST B/2 n + T 2/2 n) where n is the output length of the random oracle. They conjectured that this attack is optimal. However, this so-called STB conjecture was only proved for B ≈ T and B = 2. Very recently, Ghoshal and Komargodski (CRYPTO 22) confirmed STB conjecture for all constant values of B, and provided an Oe(S 4T B2/2 n + T 2/2 n) bound for all choices of B. In this work, we prove an Oe((ST B/2 n)· max{1, ST2/2 n}+T 2/2 n) bound for every 2 < B < T. Our bound confirms the STB conjecture for ST2 ≤ 2 n, and is optimal up to a factor of S for ST2 > 2 n (note as T 2 is always at most 2n, otherwise finding a collision is trivial by the birthday attack). Our result subsumes all previous upper bounds for all ranges of parameters except for B = Oe(1) and ST2 > 2 n. We obtain our results by adopting and refining the technique of Chung, Guo, Liu, and Qian (FOCS 2020). Our approach yields more modular proofs and sheds light on how to bypass the limitations of prior techniques. Along the way, we obtain a considerably simpler and illuminating proof for B = 2, recovering the main result of Akshima, Cash, Drucker and Wee.
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Time-Space Tradeoffs and Short Collisions in Merkle-Damgård Hash Functions
We study collision-finding against Merkle-Damgård hashing in the random-oracle model by adversaries with an arbitrary S-bit auxiliary advice input about the random oracle and T queries. Recent work showed that such adversaries can find collisions (with respect to a random IV) with advantage 𝛺(𝑆𝑇2/2𝑛) , where n is the output length, beating the birthday bound by a factor of S. These attacks were shown to be optimal. We observe that the collisions produced are very long, on the order of T blocks, which would limit their practical relevance. We prove several results related to improving these attacks to find shorter collisions. We first exhibit a simple attack for finding B-block-long collisions achieving advantage 𝛺̃ (𝑆𝑇𝐵/2𝑛) . We then study if this attack is optimal. We show that the prior technique based on the bit-fixing model (used for the 𝑆𝑇2/2𝑛 bound) provably cannot reach this bound, and towards a general result we prove there are qualitative jumps in the optimal attacks for finding length 1, length 2, and unbounded-length collisions. Namely, the optimal attacks achieve (up to logarithmic factors) on the order of (𝑆+𝑇)/2𝑛 , 𝑆𝑇/2𝑛 and 𝑆𝑇2/2𝑛 advantage. We also give an upper bound on the advantage of a restricted class of short-collision finding attacks via a new analysis on the growth of trees in random functional graphs that may be of independent interest.
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- Award ID(s):
- 1453132
- PAR ID:
- 10252058
- Date Published:
- Journal Name:
- Annual International Cryptology Conference CRYPTO 2020: Advances in Cryptology – CRYPTO 2020
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1007/978-3-030-56784-2_6
