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.