In Vehicle-to-Everything (V2X) communication systems, vehicles as well as infrastructure devices can interact and exchange data with each other. This capability is used to implement intelligent transportation systems applications. Data confidentiality and integrity need to be preserved in unverified and untrusted environments. In this paper, we propose a solution that provides (a) role-based and attribute-based access control to encrypted data and (b) encrypted search over encrypted data. Vehicle Records contain sensitive information about the owners and vehicles in encrypted form with attached access control policies and policy enforcement engine. Our solution supports decentralized and distributed data exchange, which is essential in V2X systems, where a Central Authority is not required to enforce access control policies. Furthermore, we facilitate querying encrypted Vehicle Records through Structured Query Language (SQL) queries. Vehicle Records are stored in a database in untrusted V2X cloud environment that is prone to provide the attackers with a large attack surface. Big datasets, stored in cloud, can be used for data analysis, such as traffic pattern analysis. Our solution protects sensitive vehicle and owner information from curious or malicious information cloud administrators. Support of indexing improves performance of queries that are forwarded to relevant encrypted Vehicle Records, which are stored in the cloud. We measure the performance overhead of our security solution based on self-protecting Vehicle Records with encrypted search capabilities in V2X communication systems and analyze the effect of security over safety.