Internet platforms depend on accurately determining the geographical locations of online users to deliver targeted services (e.g., advertising). The advent of decentralized platforms (blockchains) emphasizes the importance of geographically distributed nodes, making the validation of locations more crucial. In these decentralized settings, mutually non-trusting participants need to {\em prove} their locations to each other. The incentives for claiming desired location include decentralization properties (validators of a blockchain), explicit rewards for improving coverage (physical infrastructure blockchains) and regulatory compliance -- and entice participants towards prevaricating their true location malicious via VPNs, tampering with internet delays, or compromising other parties (challengers) to misrepresent their location. Traditional delay-based geolocation methods focus on reducing the noise in measurements and are very vulnerable to wilful divergences from prescribed protocol. In this paper we use Internet delay measurements to securely prove the location of IP addresses while being immune to a large fraction of Byzantine actions. Our core methods are to endow Internet telemetry tools (e.g., ping) with cryptographic primitives (signatures and hash functions) together with Byzantine resistant data inferences subject to Euclidean geometric constraints. We introduce two new networking protocols, robust against Byzantine actions: Proof of Internet Geometry (PoIG) converts delay measurements into precise distance estimates across the Internet; Proof of Location (PoLoc) enables accurate and efficient multilateration of a specific IP address. The key algorithmic innovations are in conducting ``Byzantine fortified trigonometry" (BFT) inferences of data, endowing low rank matrix completion methods with Byzantine resistance.

翻译：互联网平台依赖于准确确定在线用户的地理位置来提供定向服务（例如广告）。去中心化平台（区块链）的出现强调了地理分布节点的重要性，使得位置验证变得更为关键。在这些去中心化环境中，互不信任的参与者需要相互“证明”其位置。声称期望位置的动机包括去中心化特性（区块链验证者）、为改进覆盖范围提供明确奖励（物理基础设施区块链）以及合规性要求——这些诱因促使参与者恶意歪曲其真实位置，手段包括使用VPN、篡改互联网延迟或破坏其他方（挑战者）以误导其位置表示。传统的基于延迟的地理定位方法侧重于减少测量噪声，但极易受到故意偏离规定协议行为的影响。在本文中，我们利用互联网延迟测量来安全地证明IP地址的位置，同时能够抵御大部分拜占庭行为。我们的核心方法是为互联网遥测工具（例如ping）赋予密码学原语（签名和哈希函数），并结合满足欧几里得几何约束的拜占庭鲁棒数据推断。我们引入了两种新的网络协议，这些协议对拜占庭行为具有鲁棒性：互联网几何证明（PoIG）将延迟测量转换为互联网上的精确距离估计；位置证明（PoLoc）则能够对特定IP地址进行精确且高效的多点定位。关键的算法创新在于对数据进行“拜占庭强化三角测量”（BFT）推断，为低秩矩阵补全方法赋予拜占庭鲁棒性。