**Almost Optimal Hash Sequence Traversal**

*Don Coppersmith and Markus Jakobsson*

**Abstract: **We introduce a novel technique for computation of
consecutive preimages of hash chains. Whereas traditional
techniques have a memory-times-computation complexity of
$O(n)$ per output generated, the complexity of our technique
is only $O(log^2 \, n)$, where $n$ is the length of the chain.

Our solution is based on the same principal amortization principle as \cite{J01}, and has the same asymptotic behavior as this solution. However, our solution decreases the real complexity by approximately a factor of two. Thus, the computational costs of our solution are approximately $ {1 \over 2} log_2 \, n$ hash function applications, using only a little more than $log_2 \, n$ storage cells.

A result of independent interest is the lower bounds we provide for the optimal (but to us unknown) solution to the problem we study. The bounds show that our proposed solution is very close to optimal. In particular, we show that there exists no improvement on our scheme that reduces the complexity by more than an approximate factor of two.

**Category / Keywords: **amortization, hash chain, pebbles, lower bound

**Publication Info: **FC '02

**Date: **received 25 Feb 2002

**Contact author: **mjakobsson at rsasecurity com

**Available format(s): **Postscript (PS) | Compressed Postscript (PS.GZ) | PDF | BibTeX Citation

**Version: **20020225:190306 (All versions of this report)

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