Wireless broadcast transmission enables Inter-vehicle or Vehicle-to-Vehicle (V2V) communication among nearby vehicles. This communication supports latency-critical applications for improved safety and maybe optimized traffic. However, V2V communication is vulnerable to cyber attacks involving message manipulation. Mechanisms are required to ensure both authenticity and integrity of broadcast data, while maintaining drivers privacy against surveillance. Considering the limited computational resources of vehicles and the possibility of high traffic density scenarios, authentication processes should have low computational overhead. Prior research has produced multiple authentication protocol proposals based on digital signatures, hash functions, or Message Authentication Codes (MACs). To date, there is no computationally efficient secure broadcast authentication scheme tolerable by the vehicles resource-constrained On-Board Units (OBUs) for latency-critical applications in heavy traffic conditions. This paper provides a new secure, efficient, and privacy-preserving scheme proposing Anonymous Lightweight Inter-vehicle (ALI) broadcast authentication with encryption. ALI provides a high level of anonymity by combining a message authentication scheme with beacon encryption. The cryptographic overhead for V2V communication in the ALI scheme is only 149 bytes, and can handle authentication of approximately 700 broadcast messages every 100 milliseconds. This demonstrates the suitability of the ALI scheme in heavy traffic scenarios. We show the security and efficiency of our proposal by conducting security proof and performance analysis.