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Abstract. A new technique for combinational logic optimization is described. The technique is a two-step process. In the first step, the nonlinearity of a circuit -as measured by the number of non-linear gates it contains -is reduced. The second step reduces the number of gates in the linear components of the already reduced circuit. The technique can be applied to arbitrary combinational logic problems, and often yields improvements even after optimization by standard methods has been performed. In this paper we show the results of our technique when applied to the S-box of the Advanced Encryption Standard (AES [5]). This is an experimental proof of concept, as opposed to a full-fledged circuit optimization effort. Nevertheless the result is, as far as we know, the circuit with the smallest gate count yet constructed for this function. We have also used the technique to improve the performance (in software) of several candidates to the Cryptographic Hash Algorithm Competition. Finally, we have experimentally verified that the second step of our technique yields significant improvements over conventional methods when applied to randomly chosen linear transformations.

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The relative worst-order ratio, a relatively new measure for the quality of on-line algorithms, is extended and applied to the paging problem. We obtain results significantly different from those obtained with the competitive ratio. First, we devise a new deterministic paging algorithm, Retrospective-LRU, and show that, according to the relative worst-order ratio and in contrast with the competitive ratio, it performs better than LRU. Our experimental results, though not conclusive, are slightly positive and leave it possible that Retrospective-LRU or similar algorithms may be worth considering in practice. Furthermore, the relative worst-order ratio (and practice) indicates that LRU is better than the marking algorithm FWF, though all deterministic marking algorithms have the same competitive ratio. Look-ahead is also shown to be a significant advantage with this new measure, whereas the competitive ratio does not reflect that look-ahead can be helpful. Finally, with the relative worst-order ratio, as with the competitive ratio, no deterministic marking algorithm can be significantly better than LRU, but the randomized algorithm MARK is better than LRU.