Circuit Architecture Overview

The Privacy Pools protocol uses three main Circom circuits:

1. CommitmentHasher Circuit
   • Computes commitment hashes from inputs
   • Generates precommitment and nullifier hashes
   • Uses Poseidon hash for efficient ZK computation
2. LeanIMTInclusionProof Circuit
   • Verifies membership in Lean Incremental Merkle Trees
   • Computes path from leaf to root
   • Validates hashes for each tree level
   • Accommodates dynamic tree depth
3. Withdrawal Circuit
   • Combines commitment and Merkle tree proofs
   • Verifies ownership of existing commitment
   • Validates new commitment creation
   • Checks ASP root inclusion

Commitments

Commitments are cryptographic primitives that allow users to commit to values while keeping them private. In Privacy Pools:

1. Components

   • Value: The amount of assets being committed
   • Label: Unique identifier from pool scope and nonce
   • Nullifier: Secret value preventing double-spending
   • Secret: Random value proving ownership

2. Hash Construction

   nullifierHash = PoseidonHash(nullifier);
   precommitmentHash = PoseidonHash(nullifier, secret);
   commitmentHash = PoseidonHash(value, label, precommitmentHash);

Basic Proof Concepts

Privacy Pools uses Groth16 proofs with the following structure:

1. Public Inputs
   • Values visible on-chain
   • Examples: withdrawal amount, roots, context
   • Used for on-chain verification
2. Private Inputs
   • Values kept secret by the prover
   • Examples: nullifiers, secrets, siblings
   • Used to generate proofs
3. Circuit Signals
   • Internal values computed during proving
   • Enforce mathematical constraints
   • Connect public and private inputs

Verification Flow

1. Proof Generation
   • User provides private and public inputs
   • Circuit computes internal signals
   • Generates Groth16 proof elements:
     • π_A: First elliptic curve point
     • π_B: Second elliptic curve point (2x2 matrix)
     • π_C: Third elliptic curve point
2. On-chain Verification
   • Contract receives proof and public signals
   • Verifier performs pairing checks
   • Validates against verification key
   • Returns boolean indicating validity
3. Proof Integration
   • Proofs linked to protocol operations
   • Results determine state transitions
   • Failed verifications revert transactions