Decentralized finance based on blockchain has experienced rapid development. To safeguard the privacy of participants, decentralized anonymous payment (DAP) systems such as ZCash and Zether have emerged. These systems employ cryptographic techniques to conceal the trader addresses and payment amounts. However, this anonymity presents challenges in terms of regulation. To address this issue, we propose the Walsh-DAP (WDAP) scheme, an efficient and generic regulation scheme for decentralized anonymous payments that strikes a balance between regulation and privacy preservation. Our scheme introduces two regulation policies: first, users who have exceeded their spending limits within a certain period will be identified during the regulation process; second, the supervisor possesses the capability to trace any anonymous transaction. To implement regulation effectively, we have designed an innovative commitment scheme, Walsh commitment, which leverages the orthogonal properties of Walsh codes to achieve the features of aggregatability and extractability. The supervisor in WDAP only needs to deal with the aggregation result of the Walsh commitments instead of the huge amount of raw transactions information, which greatly increases the efficiency. In a DAP system with 256 users, 10 transactions per second and 30 days as a regulation period, we reduced the communication cost for regulation from 14 GB to 94.20 KB, and the computing cost from$\text{1.6}\times \text{10}^{\text{5}}$s to 2.17s. Both improvement is of over five orders of magnitude. We formally discussed the security of the whole system, and verified its feasibility and practicability in the ZCash system.

A range proof serves as a protocol for the prover to prove to the verifier that a committed number lies in a specified range, such as [0,2n), without disclosing the actual value. Range proofs find extensive application in various domains. However, the efficiency of many existing schemes diminishes significantly when confronted with batch proofs encompassing multiple elements.

Outsourced Proofs of Retrievability (OPoR) with private verification enables a third party verifier to periodically check cloud data on behalf of users. However, such a scheme requires the verifier to keep a copy of the user's data and generate tags for the data like the data owner. In other words, in addition to storing the data and tags from the user, the cloud server also needs to store tags uploaded by the verifier. To overcome this limitation, we propose a concrete construction of outsourced privately verifiable PoR (OPVPoR) without requiring the additional tag storage from the verifier. Furthermore, we extend the OPVPoR scheme to the multi-user setting and build a MOPVPoR scheme, where users storing the same data to the cloud server also share the tag information to further reduce the storage cost. Finally, we implement both schemes to evaluate their performance in practice.