In this study, we investigated the relationship between similarity measures and entropy for fuzzy sets. First, we developed fuzzy entropy by using the distance measure for fuzzy sets. We pointed out that the distance between the fuzzy set and the corresponding crisp set equals fuzzy entropy. We also found that the sum of the similarity measure and the entropy between the fuzzy set and the corresponding crisp set constitutes the total information in the fuzzy set. Finally, we derived a similarity measure from entropy and showed by a simple example that the maximum similarity measure can be obtained using a minimum entropy formulation.

This paper presents a simplifying method of the two previous fault attacks to pairing and the Miller algorithms based on a practical fault assumption. Our experimental result shows that the assumption is feasible and easy to implement.