It is well recognized that security is vital for reliable operation of vehicular ad-hoc networks (VANETs). Location privacy is one of the main security challenges in VANETs, which is concerned with preventing an attacker from tracking a specific vehicle. In this paper, we propose a novel location privacy preservation scheme for VANETs using random encryption periods (REP). REP is based on a privacy preserving group communication protocol, which has a conditional full statelessness property. In addition, REP ensures that the requirements to track a vehicle are always violated. By conducting detailed analysis and simulation, REP is demonstrated to be reliable, efficient, and scalable. Keywords VANETs · security · location privacy