Single-Layer Fractal Hash Chain Traversal with Almost Optimal Complexity

Abstract

We study the problem of traversing a hash chain with dynamic helper points (called pebbles). Basically, two kinds of algorithms for this problem are known to date. Jakobsson algorithm is a single-layer fractal algorithm with the computational cost of ⌈logn ⌉ (hash evaluations per chain link) and ⌈logn ⌉ pebbles. Coppersmith-Jakobsson algorithm is a complicated double-layer fractal algorithm that improves efficiency at the expense of simplicity; with a complex movement pattern and some extra pebbles, it reduces the computational cost by half. Specifically, Coppersmith-Jakobsson algorithm requires $\lfloor \frac{1}{2}\log n \rfloor$ hash evaluations per chain link and ⌈logn ⌉ + ⌈log(logn + 1) ⌉ pebbles, which attains an almost optimal complexity. We introduce a new hash chain traversal algorithm that achieves both simplicity and efficiency. While our algorithm is based on the simple single-layer fractal structure of the Jakobsson algorithm, it reduces the computational cost by half without using extra pebbles; specifically, $\lceil \frac{1}{2}\log n \rceil$ hash evaluations per chain link and ⌈logn ⌉ pebbles are needed.