The technology of blockchain has propelled industry development, giving rise to blockchain platforms serving different industries, and highlighting the importance of cross-chain technology in achieving interoperability between different blockchains. Cross-chain technology facilitates operations across accounts and platforms, but it introduces complexities far surpassing those within single-blockchain platforms about transactions, thereby exacerbating risks of privacy leaks, data forgery, and payment repudiation, posing challenges for regulatory audits spanning multiple accounts and platforms. In response, this paper presents a cross-chain transaction auditing scheme grounded in bilinear pairings and cryptographic hard problems. The proposed scheme effectively verifies accuracy and completeness of transaction content, while logging audit information on the relay chain. Moreover, symmetric encryption is employed during the cross-platform transaction audit to safeguard transaction data privacy and ensure the reliability of audit results. Comprehensive theoretical analyses and experimental evaluations are conducted, proving the scheme is feasible and efficient.