The Impact of Reversibility on Parallel Pebbling

The (parallel) classical black pebbling game is a helpful abstraction which allows us to analyze the resources (time, space, space-time, cumulative space) necessary to evaluate a function $f$ with a static data-dependency graph $G$ on a (parallel) computer. In particular, the parallel black pebbling game has been used as a tool to quantify the (in)security of Data-Independent Memory-Hard Functions (iMHFs). However, the classical black pebbling game is not suitable to analyze the cost of quantum preimage attack. Thus, Blocki et al. (TCC 2022) introduced the parallel reversible pebbling game as a tool to analyze resource requirements for a quantum computer. While there is an extensive line of work analyzing pebbling complexity in the (parallel) black pebbling game, comparatively little is known about the parallel reversible pebbling game. Our first result is a lower bound of $\Omega\left(N^{1+\sqrt{\frac{ 2-o(1)}{\log N}}} \right)$ on the reversible cumulative pebbling cost for a line graph on $N$ nodes. This yields a separation between classical and reversible pebbling costs demonstrating that the reversibility constraint can increase cumulative pebbling costs (and space-time costs) by a multiplicative factor of $N^{(\sqrt 2 + o(1))/\sqrt{\log N}}$ --- the classical pebbling cost (space-time or cumulative) for a line graph is just $\mathcal{O}(N)$. On the positive side, we prove that any classical parallel pebbling can be transformed into a reversible pebbling strategy whilst increasing space-time (resp. cumulative memory) costs by a multiplicative factor of at most $\mathcal{O}\left(N^{\sqrt{\frac{8}{\log N}}}\right)$ (resp. $\mathcal{O}\left(N^{\mathcal{O}(1)/\sqrt[4]{\log N}}\right)$). We also analyze the impact of the reversibility constraint on the cumulative pebbling cost of depth-robust and depth-reducible DAGs exploiting reversibility to improve constant factors in a prior lower bound of Alwen et al. (EUROCRYPT 2017). For depth-reducible DAGs we show that the state-of-the-art recursive pebbling techniques of Alwen et al. (EUROCRYPT 2017) can be converted into a recursive reversible pebbling attack without any asymptotic increases in pebbling costs. Finally, we extend a result of Blocki et al. (ITCS 2020) to show that it is Unique Games hard to approximate the reversible cumulative pebbling cost of a DAG $G$ to within any constant factor.