Security and privacy protection are important requirements of vehicular ad hoc networks (VANETs). In recent years, many authentication schemes have been proposed to solve the security and privacy issues in VANETs. However, almost all authentication schemes do not distinguish the vehicles with different zones' access rights. This paper proposes an efficient batch authentication scheme with rule-based access control for VANETs, named BASRAC, which provides access control during the authentication phase to prevent unauthorized vehicles from accessing zones' services. At the same time, BASRAC supports batch verification, which further improves the efficiency of authentication. Moreover, BASRAC ensures data confidentiality, which prevents the semi-trusted roadside unit (RSU) from obtaining useful service information. Security analysis shows that BASRAC is secure under the random oracle model and meets the security requirements of VANETs. Performance evaluation shows that, compared with the existing schemes, BASRAC cuts down the computational cost of signing, single verification, and batch verification by 58.05%-90.84%, 29.93%-69.61%, and 59.49%-77.65%, respectively. Moreover, BASRAC is more effective in communication overhead and performs better in the average message delay and RSU serving capability.