Paper 2026/1835 Original: eprint.iacr.org/2026/1835 ↗

Wagner's $k$-Tree Revisited: Memory-Time Costs of Preimage Attacks on Incremental Hashes

Abstract

Wagner's $k$-tree algorithm solves the generalized birthday problem and underlies preimage attacks on randomize-then-combine incremental hashes (EUROCRYPT '97) such as iSHAKE, LtHash, and AdHash. In its full-index execution, the $2^{k-1}$ index entries dominate peak memory at large $k$, and index trimming narrows each entry but leaves their number intact. Tang et al. (TCHES '26) introduced post-retrieval for the single-chain algorithm and left the $2^k$ exponent of the $k$-tree setting as an open problem. We resolve it by extending post-retrieval to the $k$-tree algorithm. This cuts the peak working memory to $P_k\ell N = O(k^2 \ell N)$, removing the $2^{k-1}$ index entries from the forward pass at the cost of a $\Theta(k)$ time overhead. Free $\delta$-level caching shrinks this time factor at no memory cost. The trade-off is starkly asymmetric: it removes an exponential number of index entries for only a linear recovery-time overhead. As a practical application, we revisit the list-item-reduction landscape for the $k$-tree algorithm under the memory-time product metric, $\mathsf{MT} = M \cdot T$. The gain is regime-dependent: for small $k$ the index entries do not yet dominate, so the recovery overhead outweighs the saving. For large $k$ the saving dominates, lowering the optimized $\log_2 \mathsf{MT}$ from $4\sqrt{n}$ to $2\sqrt{2n}$ at leading order. For fixed-size iSHAKE preimage attacks, we save approximately $51$ and $122$ bits over state-of-the-art index trimming for iSHAKE-128 and iSHAKE-256, respectively, in the unlimited-block setting, narrowing to roughly $6$ and $8$ bits under block-count caps.