Payment channels serve as a promising solution to improving the scalability of permissionless blockchains, each of which allows two users to perform off-chain payments with almost unlimited frequency. Nevertheless, the payment channel risks encountering fund depletion, which threatens the availability of off-chain payments. The most recent method needs a cycle-based channel rebalancing procedure, which requires a fair leader and users with rebalancing demands forming directed cycles in the network. Therefore, its large-scale applications are restricted. Moreover, the privacy issue is not considered, leading to the leakage of rebalancing information. In this work, we introduce Shaduf++, a novel non-cycle off-chain rebalancing protocol that offers a new solution for users to shift coins among any channels to which they belong directly without relying on the cycle setting, and therefore could be applied to more general rebalancing scenarios. Furthermore, utilizing the cryptography primitives, including the commitment and the range proof schemes, Shaduf++ preserves the privacy for the rebalancing operations. We provide the details of Shaduf++ and formally prove its security under the Universal Composability framework. Our prototype demonstrates its feasibility and the experimental evaluation shows that Shaduf++ enhances the Lighting Network performance in payment success ratio and volume.

Mixing services provide unlinkability for blockchains by breaking the link between sender/receiver identities and are highly appreciated for their compatibility with the underlying blockchains. Many efforts have been made to provide mixing services for either non-confidential or confidential payments. For confidential payments, all the known mixing protocols are designed for confidential blockchains using homomorphic commitment. There is, however, no satisfactory solution for confidential blockchains using public key encryption (PKE), such as PGC and Zether.

Payment channel hubs (PCHs) serve as a promising solution to achieving quick off-chain payments between pairs of users. They work by using an untrusted tumbler to relay the payments between the payer and payee and enjoy the advantages of low cost and high scalability. However, the most recent privacy-preserving payment channel hub solution that supports variable payment amounts suffers from limited unlinkability, e.g., being vulnerable to the abort attack. Moreover, this solution utilizes zero-knowledge proofs, which bring huge costs on both computation time and communication overhead. Therefore, how to design PCHs that support variable amount payments and unlinkability, but reduce the use of huge-cost cryptographic tools as much as possible, is significant for the large-scale practical applications of off-chain payments.

The lack of scalability is a leading issue of blockchain. By transferring transactions to off-chain, 2-party payment channels achieve instant transaction confirmation between channel users and enhance the blockchain throughput, thereby becoming a promising solution. By extending the channel from 2-party to multi-party, richer application scenarios could be supported. Meanwhile, new and exclusive requirements emerge in the multi-party off-chain payments, including robustness and flexibility. The robustness requires that the channel operation would not be impeded by any uncooperative channel member, and the flexibility guarantees that parties could join or exit the channel dynamically. However, all the current attempts either fail to achieve the new emerging properties or sacrifice some merits of 2-party channels. In this paper, we propose a new multi-party channel construction, Magma, which has good scalability. Magma outperforms the previous solutions to the multi-party payment channel for the following reasons. By canceling the heavy reliance on the cooperation of all channel members when implementing the channel operation, Magma achieves robustness. Magma also allows parties to join or exit one channel flexibly, without violating the balance security. Meanwhile, Magma's whole transaction process is performed off-chain, thereby inheriting the instant confirmation and low-cost features of 2-party channels. To guarantee the security of Magma, we formalize the multi-party channel's functionality and prove that Magma is secure in the UC framework. Moreover, our implementation and comparison show that Magma is practical and performs better than existing solutions in providing off-chain payment services.