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A recent work shows how we can optimize a tree based mode of operation for ahash 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 leavesat the same depth. In this work, we show that we can further optimize therunning time of such a mode by using a tree having leaves at all its levels. Wemake the assumption that the input message block has a size a multiple of thatof the digest and denote by $d$ the ratio block size over digest size. Therunning time is evaluated in terms of number of operations performed by thehash function, i.e. the number of calls to its underlying function. It turnsout that a digest can be computed in $\lceil \log_{d+1} (l/2) \rceil+2$evaluations of the underlying function using $\lceil l/2 \rceil$ processors,where $l$ is the number of blocks of the message. Other results of interest arediscussed, such as the optimization of the parallel running time for a tree ofrestricted height.

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