Compactness of Hashing Modes and Efficiency beyond Merkle Tree
This work addresses an open problem in cryptography for designing secure hash functions with maximal efficiency, offering incremental improvements over existing tree-based methods.
The authors tackled the problem of designing optimally efficient and secure hash functions for large domains by introducing a new efficiency notion called compactness, and they presented two tree-based modes (ABR and ABR+) that achieve asymptotically optimal collision resistance with improved data processing compared to Merkle trees, where ABR+ also adds indifferentiability up to 2^{n/2-ε} queries.
We revisit the classical problem of designing optimally efficient cryptographically secure hash functions. Hash functions are traditionally designed via applying modes of operation on primitives with smaller domains. The results of Shrimpton and Stam (ICALP 2008), Rogaway and Steinberger (CRYPTO 2008), and Mennink and Preneel (CRYPTO 2012) show how to achieve optimally efficient designs of $2n$-to-$n$-bit compression functions from non-compressing primitives with asymptotically optimal $2^{n/2-ε}$-query collision resistance. Designing optimally efficient and secure hash functions for larger domains ($> 2n$ bits) is still an open problem. In this work we propose the new \textit{compactness} efficiency notion. It allows us to focus on asymptotically optimally collision resistant hash function and normalize their parameters based on Stam's bound from CRYPTO 2008 to obtain maximal efficiency. We then present two tree-based modes of operation -Our first construction is an \underline{A}ugmented \underline{B}inary T\underline{r}ee (ABR) mode. The design is a $(2^{\ell}+2^{\ell-1} -1)n$-to-$n$-bit hash function making a total of $(2^{\ell}-1)$ calls to $2n$-to-$n$-bit compression functions for any $\ell\geq 2$. Our construction is optimally compact with asymptotically (optimal) $2^{n/2-ε}$-query collision resistance in the ideal model. For a tree of height $\ell$, in comparison with Merkle tree, the $ABR$ mode processes additional $(2^{\ell-1}-1)$ data blocks making the same number of internal compression function calls. -While the $ABR$ mode achieves collision resistance, it fails to achieve indifferentiability from a random oracle within $2^{n/3}$ queries. $ABR^{+}$ compresses only $1$ less data block than $ABR$ with the same number of compression calls and achieves in addition indifferentiability up to $2^{n/2-ε}$ queries.