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 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, often 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 asynchronous consensus, verifiable secret sharing, erasure-coded distributed storage and broadcast protocols.

翻译：大多数容错分布式算法均基于名义腐败模型设计，该模型假设最多有比例为$f_n$的参与方可被敌手腐化。然而，由于臭名昭著的Sybil攻击，名义模型无法表达开放（即无许可）环境中的信任假设。相反，无许可系统通常运行在加权模型中：每个参与者关联一个权重，敌手最多可腐化持有总权重比例$f_w$的参与方集合。本文提出一种将名义模型下设计的大类协议转化为加权模型的简洁方法。为此，我们形式化并解决了三个新颖的优化问题（统称为权重削减问题），该问题可将大的实数权重映射为小的整数权重，同时保留协议正确性所需的属性。在所有情形下，我们成功将整数权重之和控制在参与者数量的线性范围内，从而为加权模型提供极致高效的协议。此外，我们证明在实际出现的权重分布中，整数权重之和通常远低于理论最坏情况，甚至常小于参与者数量。对于某些协议，我们的转换需要在弹性（即$f_w = f_n - \epsilon$）上做出任意小的牺牲；令人惊讶的是，对许多重要问题，我们实现了与名义模型相同弹性（$f_w = f_n$）的加权解决方案，典型例子包括异步共识、可验证秘密共享、纠删码分布式存储与广播协议。