Cryptology ePrint Archive: Report 2020/770

Time-Space Tradeoffs and Short Collisions in Merkle-Damgård Hash Functions

Akshima and David Cash and Andrew Drucker and Hoeteck Wee

Abstract: 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 $\Omega(ST^2/2^n)$, 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 $T$ blocks, which would limit their practical relevance. We prove several results related to improving these attacks to find short collisions. We first exhibit a simple attack for finding $B$-block-long collisions achieving advantage $\tilde{\Omega}(STB/2^n)$. We then study if this attack is optimal. We show that the prior technique based on the bit-fixing model (used for the $ST^2/2^n$ 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) order of $(S+T)/2^n$, $ST/2^n$ and $ST^2/2^n$ 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.

Category / Keywords: foundations / provable security / symmetric cryptography, time-memory tradeoffs, auxiliary input, hash functions, Merkle Damgård, random oracle, short collisions

Original Publication (with major differences): IACR-CRYPTO-2020

Date: received 22 Jun 2020

Contact author: akshima at uchicago edu,davidcash@uchicago edu,andy drucker@gmail com,wee@di ens fr

Available format(s): PDF | BibTeX Citation

Version: 20200624:075456 (All versions of this report)

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