Update the solution for AA-based FHE-DKSAP

Update the solution for Verifiable FHE-DKSAP

Zero-Knowledge Proofs (ZKPs) are cryptographic methods where one party (the prover) can prove to another party (the verifier) that a statement is true without revealing any specific information about the statement itself. In the context of privacy pools and blockchain, ZKPs are used to enhance privacy and scalability.
In the context of Ethereum and other smart contract platforms, ZKPs can be used to create private smart contracts where the details of the contract’s execution are hidden but can still be verified. Privacy pools are essentially smart contracts that leverage ZKPs to allow users to deposit, withdraw, and transact privately. Users deposit into the pool and receive private tokens. They can then transact these tokens without the details being public on the blockchain. When they want to exit the pool, they can withdraw, again without revealing specific transaction details.
In the interactions of FHE-DKSAP in privacy pool, we both verify the trans- fer ZKP and recive ZKP. When Alice wants to transfer an amount using stealth address and makes a private transaction within the pool, she create a zero- knowledge proof that the transaction is valid (e.g., they have the funds, the transaction does not double-spend, etc.). The pool’s smart contract then veri- fies the proof, and if it’s valid, processes the transaction. However, the specifics of the transaction (like sender, recipient, and amount) remain hidden from the public blockchain. Same for the receiver Bob. When he receives a transaction, he generates the ZK proof based on the transaction. By using this ZKP attach- ments, we can verify the validness of the transaction and reaches the target of reducing the illegal transactions. The overview of ZKP can be found in below figure.

At the heart of FHE-DKSAP’s application in Privacy Pools is the following principle: users don’t just use zero-knowledge proofs to verify that their with- drawal corresponds to an earlier deposit. Instead, they prove their membership within a specific association set. This set could encompass all prior deposits, solely the user’s individual deposit, or a combination thereof. The user identifies this set by submitting a Merkle root representing the set as a publicly accessi- ble input. Here, we design a pseudocode in Algorithm 1 to give a conceptual understanding of the process.

The above pseudocode offers more details, like cryptographic hashing and zk-SNARK proofs. In our algorithm, we assume SHA-256 for hashing and a simplified zk-SNARK system for zero-knowledge proofs. However, the specific methods for generating a zk-SNARK proof, validating signatures, or even con- structing a Merkle tree can vary based on the exact cryptographic library and requirements.