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The on-demand mobility market, including ridesharing, is becoming increasingly important with e-hailing fares growing at a rate of approximately 130% per annum since 2013. By increasing utilization of existing vehicles and empty seats, ridesharing can provide many benefits including reduced traffic congestion and environmental impact from vehicle usage and production. However, the safety of riders and drivers has become of paramount concern and a method for privacy-preserving identity verification between untrusted parties is essential for protecting users. To this end, we propose a novel privacy-preserving identity verification system, extending zero-knowledge proof (ZKP) and blockchain for use in ridesharing applications. We design a permissioned blockchain network to perform the ZKP verification of a driver's identity, which also acts as an immutable ledger to store ride logs and ZKP records. For the ZKP module, we design a protocol to facilitate user verification without requiring the exchange of any private information. We prototype the proposed system on the Hyperledger Fabric platform, with the Hyperledger Ursa cryptography library, and conduct extensive experimentation. To measure the prototype's performance, we utilize the Hyperledger Caliper benchmark tool to perform extensive analysis and the results show that our system is suitable for use in real-world ridesharing applications.