This paper studies the relations among several definitions of anonymity in the same attack environment. It is shown that one intuitive and two technical definitions we consider are asymptotically equivalent, and the indistinguishability-based technical definition is the strongest, i.e., the most secure when achieved, when the exact reduction cost is taken into account. We then extend our result to the threshold case where a subset of members cooperate to create a signature. The threshold setting makes the notion of anonymity more complex and yields a greater variety of definitions. We explore several notions and observe certain relation does not seem hold unlike the simple single-signer case. Nevertheless, we see that an indistinguishability-based definition is the most favorable in the threshold case. We also present a notion of anonymity with linkability and a simple generic construction.
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