An anonymous identity-based encryption with tracing identities (AIBET) system enjoys that same strong privacy for receivers as a normal anonymous identity-based encryption system. Additionally, an AIBET system offers an identity tracing mechanism, which allows a tracer, who has an identity-associated tracing key, to uncover the recipient’s identity from the ciphertext. In this paper, we present an AIBET system based on plain lattices by exploiting a hierarchical power of lattice trapdoors in a novel way. We prove the security of the system under the conservative learning-with-errors assumption in the standard model. This is the first AIBET system provably secure under quantum-resistant assumptions. Our construction’s efficiency is comparable to the state-of-the-art lattice anonymous identity-based encryption system.