Adaptor signatures (AS) provide a crucial cryptographic technique that can be used to solve the problems of poor scalability and low transaction throughput cross-chain such as cryptocurrency and atomic swap. To address the limitations of existing attribute-based signature (ABS) schemes lacking conditional payment capabilities for the policy constraints of traditional adaptor signature schemes, this paper proposes the first Linear Secret Sharing Scheme (LSSS)-compatible Attribute-Based Adaptor Signature (ABAS) scheme. The ABAS scheme embeds a hard relation witness y and innovatively integrates access control matrix M with a pre-signature mechanism to unify attribute privacy preservation, which attains the correctness of both pre-signature and completed signature. The ABAS scheme provides the security of Pre-signature Adaptability, Adaptor Existential Unforgeability in the selective policy model (sP-aEUF-CMA security), Witness Extractability under the q-BDHE assumption, and Attributes Anonymity. We provide a concrete control-based atomic swap protocol that employs our proposed attribute-based adaptor signature scheme as a security primitive to implement multiple-party atomic swap for satisfying the control policy in cross-chain systems.