Proof-of-work(PoW) is an algorithmic tool used to secure networks by imposing a computational cost on participating devices. Unfortunately, traditional PoW schemes require that correct devices perform significant computational work in perpetuity, even when the system is not under attack. We address this issue by designing general PoW protocols that ensure two properties. First, the fraction of identities in the system that are controlled by an attacker is a minority. Second, the computational cost of our protocol is comparable to the cost of an attacker. In particular, we present an efficient algorithm, GMCOM, which guarantees that the average computational cost to the good ID per unit time is O(J + sqrt(T(J+1))), where J is the average number of joins by the good IDs and T is the average computational spending of the adversary. Additionally, we discuss a precursor to this algorithm, CCOM, which guarantees an average computational cost to good IDs per unit time of O(J+T). We prove a lower bound showing that GMCOM's spending rate is asymptotically optimal among a large family of algorithms. Finally, we provide empirical evidence that our algorithms can be significantly more efficient than previous defenses under various attack scenarios.
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