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Backoff algorithms are used in many distributed systems where multiple devices contend for a shared resource. For the classic balls-into-bins problem, the number of singletons - those bins with a single ball - is important to the analysis of several backoff algorithms; however, existing analyses employ advanced probabilistic tools to obtain concentration bounds. Here, we show that standard Chernoff bounds can be used instead, and the simplicity of this approach is illustrated by re-analyzing some well-known backoff algorithms. 

A common tool to defend against Sybil attacks is proof-of-work, whereby computational puzzles are used to limit the number of Sybil participants. Unfortunately, current Sybil defenses require significant computational effort to offset an attack. In particular, good participants must spend computationally at a rate that is proportional to the spending rate of an attacker. In this paper, we present the first Sybil defense algorithm which is asymmetric in the sense that good participants spend at a rate that is asymptotically less than an attacker. In particular, if T is the rate of the attacker's spending, and J is the rate of joining good participants, then our algorithm spends at a rate of O( sqrt{TJ} + J ). We provide empirical evidence that our algorithm can be significantly more efficient than previous defenses under various attack scenarios. Additionally, we prove a lower bound showing that our algorithm's spending rate is asymptotically optimal among a large family of algorithms.