The majority of fault-tolerant distributed algorithms are designed assuming a nominal corruption model, in which at most a fraction $f_n$ of parties can be corrupted by the adversary. However, due to the infamous Sybil attack, nominal models are not sufficient to express the trust assumptions in open (i.e., permissionless) settings. Instead, permissionless systems typically operate in a weighted model, where each participant is associated with a weight and the adversary can corrupt a set of parties holding at most a fraction $f_w$ of the total weight. In this paper, we suggest a simple way to transform a large class of protocols designed for the nominal model into the weighted model. To this end, we formalize and solve three novel optimization problems, which we collectively call the weight reduction problems, that allow us to map large real weights into small integer weights while preserving the properties necessary for the correctness of the protocols. In all cases, we manage to keep the sum of the integer weights to be at most linear in the number of parties, resulting in extremely efficient protocols for the weighted model. Moreover, we demonstrate that, on weight distributions that emerge in practice, the sum of the integer weights tends to be far from the theoretical worst case and, sometimes, even smaller than the number of participants. While, for some protocols, our transformation requires an arbitrarily small reduction in resilience (i.e., $f_w = f_n - \epsilon$), surprisingly, for many important problems, we manage to obtain weighted solutions with the same resilience ($f_w = f_n$) as nominal ones. Notable examples include erasure-coded distributed storage and broadcast protocols, verifiable secret sharing, and asynchronous consensus.

翻译：大多数容错分布式算法在设计时都假设了名义腐败模型，即最多有 $f_n$ 比例的参与方可能被对手腐化。然而，由于臭名昭著的Sybil攻击，名义模型不足以表达开放（即无许可）环境中的信任假设。相反，无许可系统通常在加权模型中运行，其中每个参与者关联一个权重，对手最多可以腐化持有总权重 $f_w$ 比例的参与方集合。在本文中，我们提出了一种将一大类为名义模型设计的协议转换为加权模型的简单方法。为此，我们形式化并解决了三个新的优化问题，统称为权重约简问题，使我们能够将较大的实数权重映射为较小的整数权重，同时保持协议正确性所需的属性。在所有情况下，我们成功将整数权重之和控制在参与方数量的线性范围内，从而为加权模型实现了极其高效的协议。此外，我们证明，在实际出现的权重分布上，整数权重之和往往远低于理论最坏情况，有时甚至小于参与者数量。尽管对于某些协议，我们的转换需要弹性略微降低（即 $f_w = f_n - \epsilon$），但令人惊讶的是，对于许多重要问题，我们成功获得了与名义模型具有相同弹性（$f_w = f_n$）的加权解决方案。值得注意的例子包括纠删码分布式存储与广播协议、可验证秘密共享以及异步共识协议。