Blockchain-based digital assets have increasingly emerged in recent years, necessitating cross-chain swaps. Hash time-lock contract (HTLC) is a widely used protocol for such swaps; however, simple hash time locks can allow attackers to analyze transaction paths, thereby causing privacy breaches and financial loss to users in some sensitive scenarios. To prevent payment path leakage, a privacy-preserving cyclic cross-chain protocol is proposed herein. This protocol primarily uses the Chameleon hash (CH) protocol to obscure the correlation between users in the path, ensuring the privacy of cross-chain swaps. The protocol is divided into pre-swap, commit, and decommit phases. The pre-swap phase is firstly executed to determine the swap order. Then, users ensure atomicity via serial asset locking in the commit phase, and each receiver obtains swap assets from the corresponding sender via CH collision in the decommit phase. The security proof under the universally composable (UC) system demonstrates the correctness and usability of the protocol. In summary, the entire protocol ensures the atomicity and privacy of cross-chain swaps, providing a new principle and method to solve the privacy leakage problem caused by transaction path analysis.