<p>Attribute-Based Encryption (ABE) is a cryptosystem that can provide flexible access control, which has important application value in the field of fine-grained access control to encrypted data. However, current ABEs only support unilateral access control from the sender to the receiver, and fail to provide data source authentication, which violates the principle of data source identification for sensitive information as mandated by the General Data Protection Regulation (GDPR). To address this problem, the notion of bilateral access control Attribute-Based Encryption (bilateral access control ABE) is introduced, but the existing schemes are constructed under the classical assumptions. In this paper, building upon the WWW+ IBS scheme, we proposed a bilateral access control ABE scheme that is provably secure in the Random Oracle Model (ROM) under the Learning With Errors (LWE) and Short Integer Solution (SIS) assumptions on lattices. In comparison to the existing ABEs based on lattice assumptions, our approach not only enables access control from sender to receiver, but also from receiver to sender. Particularly, our scheme also captures the data source authentication. Furthermore, we prove that our scheme achieves selective Indistinguishable Chosen-Plaintext-Attacks (sIND-CPA) security and authentication under the LWE and SIS assumptions. Beyond that, we also capture the attribute-hiding property (weak attribute-hiding, i.e., the scheme grants access to the attributes to receivers possessing the requisite decryption capability once a match is successful.) to obtain a (plausible) post-quantum secure Attribute-Based Matchmaking Encryption (AB-ME) scheme by using a lockable-obfuscation-based transformation. Compared to the existing AB-ME schemes, our scheme not only achieves privacy and authentication, but also is a lattice-based scheme that is post-quantum secure in the ROM.</p>