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We present optimal binary pebbling algorithms for in-place reversal (backwardtraversal) of one-way hash chains. For a hash chain of length $2^k$, the numberof hashes performed in each output round does not exceed $\lceil k/2 \rceil$,whereas the number of hash values stored (pebbles) throughout is at most $k$.  We introduce a framework for rigorous comparison of explicit binary pebblingalgorithms, including simple speed-1 binary pebbling, Jakobsson's speed-2binary pebbling, and our optimal binary pebbling algorithm. Explicit schedulesdescribe for each pebble exactly how many hashes need to be performed in eachround. The optimal schedule turns out to be essentially unique and exhibits anice recursive structure, which allows for fully optimized implementations thatcan readily be deployed. In particular, we develop the first in-placeimplementations with minimal storage overhead (essentially, storing only hashvalues), and fast implementations with minimal computational overhead.

Binary Pebbling Algorithms for In-Place Reversal of One-Way Hash Chains