Optimal Tree Hash Modes: the Case of Trees Having their Leaves at All the Levels

A recent work shows how we can optimize a tree based mode of operation for a rate-1 hash function where the sizes of input message blocks and digest are the same, subject to the constraint that the involved tree structure has all its leaves at the same depth. In this note, we show that we can further optimize the running time of such a mode by using a tree having leaves at all its levels. It turns out that a digest can be computed in dlog2 le + 1 evaluations of the underlying function.