How to Design a Private Automated Market Maker (AMM)

A private AMM extends the standard constant product formula x * y = k with cryptographic privacy. The primary goal is to hide the specific trade amounts and the identities of liquidity providers and traders from the public blockchain state. Unlike transparent AMMs like Uniswap V2, where all reserves and transactions are visible, a private AMM uses zero-knowledge proofs (ZKPs) to validate that trades and liquidity operations are correct without revealing the underlying values. This requires designing a system where the on-chain contract only stores encrypted or committed values of the reserves, while a prover generates a ZKP for every state transition.

The core design involves three main components: a commitment scheme, a zero-knowledge proof circuit, and a state management contract. First, you use a cryptographic commitment, such as a Pedersen commitment, to represent the reserve amounts Comm(x) and Comm(y). These commitments are posted on-chain. When a user submits a private swap, they construct a ZKP off-chain. This proof demonstrates, for example, that the new reserve commitments Comm(x') and Comm(y') correctly satisfy the constant product invariant x' * y' = k' given the old commitments, and that the user possesses sufficient funds, all without revealing x, y, or the trade delta.

For implementation, you would write the business logic as a zk-SNARK circuit using a framework like Circom or Halo2. The circuit takes private inputs (the actual reserve values, the trade amount) and public inputs (the old and new commitments, a public fee). It outputs a proof that the swap math is correct and that all transitions are valid. An on-chain verifier contract, written in Solidity or similar, stores the current state commitments and checks the submitted proofs. Only if the proof verifies does the contract update its stored commitments to the new state, finalizing the trade.

Key challenges include managing the nullifier pattern to prevent double-spends of private notes and designing an efficient relayer system. Since users submit encrypted transactions, a relayer is often needed to pay gas fees on their behalf. The design must also account for liquidity provision and withdrawal, which similarly require proofs that the new total supply of liquidity pool tokens matches the updated reserves. Projects like zk.money (Aztec) and Manta Network have pioneered similar models for private DeFi, offering practical references for circuit design and fee mechanism.

Designing a private Automated Market Maker (AMM) is a complex endeavor that sits at the intersection of several advanced fields. Before writing a single line of code, you must be proficient in zero-knowledge proof (ZKP) systems like zk-SNARKs (e.g., Groth16, Plonk) or zk-STARKs. These cryptographic primitives are the core engine for privacy, allowing users to prove the validity of a trade—such as sufficient balance or correct computation—without revealing the underlying asset amounts or wallet addresses. Familiarity with a ZK circuit development framework, such as Circom or Halo2, is non-negotiable.

Beyond cryptography, you need a deep understanding of traditional AMM mechanics. This includes the constant product formula x * y = k used by Uniswap V2, concentrated liquidity models from Uniswap V3, and the fee structures that incentivize liquidity providers. Your private AMM will need to replicate this logic within a ZK circuit, which operates over finite fields and cannot natively handle floating-point numbers or negative values. This requires redesigning mathematical operations for a constrained computational environment.

Smart contract expertise is equally critical. The system typically involves a verifier contract deployed on a public blockchain (like Ethereum or a Layer 2). This contract does not execute the swap logic directly but verifies the ZK proof submitted by a user, ensuring the private computation was correct. You must design a secure state transition mechanism where the contract updates its commitment to the private pool's state (e.g., a Merkle root of balances) based solely on valid proofs, a pattern used by protocols like zk.money (Tornado Cash) and Aztec Connect.

Finally, consider the trust assumptions and operational model. Will your AMM use a validator or sequencer to aggregate private transactions and post proofs, introducing a potential liveness dependency? Or will it be fully permissionless? Understanding the trade-offs between privacy, scalability, and decentralization is essential. The initial design choices in these areas will define the security and usability of your entire system.

A private AMM extends the standard constant product formula x * y = k with cryptographic privacy layers. The primary goal is to conceal the amounts being traded and the liquidity provider positions from public view, while maintaining the core invariants of the pool. This requires a shift from transparent, on-chain state to encrypted or committed state. Key architectural decisions involve choosing a privacy framework—such as zero-knowledge proofs (ZKPs), fully homomorphic encryption (FHE), or trusted execution environments (TEEs)—and integrating it with the AMM's swap and liquidity management logic.

The system architecture typically separates the privacy core from the application logic. The privacy core handles state commitments (e.g., Merkle trees of note commitments for shielded assets) and proof generation/verification. The application layer, comprising smart contracts, manages pool parameters, fee collection, and orchestrates interactions with the privacy layer. For a ZK-based design like ZK-SNARKs, every valid swap or liquidity action must generate a proof attesting that the new encrypted pool reserves satisfy the AMM curve and that the user's inputs/outputs are valid, without revealing their values.

A critical component is the shielded pool or commitment tree that holds encrypted representations of user balances and liquidity provider shares. When User A swaps token X for token Y, they submit a transaction with a zero-knowledge proof. This proof convinces the verifier contract that: 1) User A owns a committed input amount, 2) the new committed pool reserves x' and y' satisfy x' * y' = k' (where k' is updated correctly), and 3) the user's output commitment is correctly calculated. The actual token amounts remain hidden within the proofs.

Designing the liquidity provision mechanism presents unique challenges. Providers deposit two assets to receive shielded liquidity tokens (sLPs). The system must track each provider's share of the encrypted pool to enable proportional withdrawals and fee distribution. This often requires a separate, privacy-preserving accounting system, potentially using homomorphic encryption to update share balances or advanced ZK circuits that aggregate fees over multiple blocks without leaking individual earnings.

Finally, the architecture must account for practical constraints like proof generation cost, verification gas fees on the settlement layer, and blockchain data availability for state updates. Optimizing these factors is essential for usability. Projects like Penumbra and Aztec Protocol offer reference architectures for private DeFi, demonstrating how to bundle multiple actions into a single proof to amortize costs. The settlement contract's primary role reduces to verifying proofs and updating the root of the state commitment tree.

A private Automated Market Maker (AMM) requires a zero-knowledge circuit to validate trades without revealing sensitive information like trade size, wallet balances, or price impact. The core challenge is proving that a trade adheres to the AMM's constant product formula x * y = k while keeping the input amounts, output amounts, and resulting reserves secret. This involves designing a circuit where the public inputs (like the new invariant k' and a nullifier to prevent double-spends) are known, but the private inputs (pre-trade reserves x, y, input token Δx, output token Δy) are concealed. The circuit's constraints must cryptographically enforce that (x + Δx) * (y - Δy) = k' and that k' >= k to prevent reserve draining.

The circuit must also incorporate range proofs and fee calculations. Since arithmetic in zk-SNARKs (like Circom or Halo2) occurs within a finite field, we must prevent underflows and overflows. Constraints must verify that Δy < y (output doesn't exceed reserve) and that amounts are positive. Furthermore, a liquidity provider fee (e.g., 0.3%) is typically deducted from the input amount before the swap calculation. The circuit privately calculates the fee, proves its correct deduction, and validates the swap against the adjusted input. All these operations—multiplication, subtraction, comparison—must be expressed as rank-1 constraint system (R1CS) or Plonkish custom gates.

Implementing this in a framework like Circom involves creating templates for core operations. A PrivateSwap circuit would take private signals for old reserves and trade deltas, and public signals for the new invariant and a nullifier. Inside, it uses components to calculate the fee, check the constant product, and validate ranges. For example:

circomCopy
template PrivateSwap() {
  // Private signals
  signal input x, y, deltaX, deltaY;
  // Public signals
  signal output k_new, nullifier;
  // Calculate 0.3% fee
  signal fee = deltaX * 3 / 1000;
  signal deltaXAfterFee = deltaX - fee;
  // Enforce constant product formula
  (x + deltaXAfterFee) * (y - deltaY) === k_new;
  // Ensure non-negative reserves and valid trade
  deltaY < y;
}

This circuit ensures the trade is valid, but the actual reserves x and y remain hidden.

Beyond the core swap, a production system requires additional circuit components for liquidity provision and withdrawal. Adding liquidity requires proving ownership of the deposited assets and correctly calculating the pool share tokens (LP tokens) minted, which is proportional to the deposit relative to existing reserves. Withdrawal is the inverse. Each action needs a unique nullifier to prevent replay attacks. The system's state tree—a Merkle tree of private commitments to user balances and pool reserves—must be updated. The circuit proves membership of the old state in the tree and calculates the new root, enabling trustless state transitions where only the valid state root change is published on-chain.

Finally, the circuit's proof generation and verification must be efficient. Using a zk-SNARK backend like Groth16 or PLONK, the prover generates a proof off-chain that the private trade is valid. The verifier (a smart contract on Ethereum, for instance) checks this proof against the public inputs. The on-chain contract then updates the publicly known state root. This design, used by protocols like zk.money (Aztec) or Manta Network, enables fully private trading. The key trade-offs are circuit size (affecting proof time) and the requirement for a trusted setup for some SNARK systems, which projects address using perpetual powers-of-tau ceremonies or transparent setups like STARKs.

To move from theory to a working prototype, begin by implementing the core cryptographic primitives. Use established libraries like arkworks for zk-SNARKs or libsignal for secure multi-party computation (MPC) protocols. Your first milestone should be a functional commitment scheme for orders and a basic zero-knowledge proof circuit that validates a swap without revealing the input amounts. Test these components in isolation before integrating them into a larger state machine.

Next, architect the smart contract system. Design a modular structure separating the state commitment layer (e.g., a Merkle tree manager), the proof verification layer, and the liquidity management logic. For Ethereum, consider using the Verifier interface from Circom or the PlonkVerifier from Aztec. A critical design decision is choosing your data availability solution: will trade data be posted on-chain, stored off-chain with attestations, or managed via a private mempool?

For further learning, study existing private DeFi implementations. Analyze Aztec's zk.money for private rollup insights, Penumbra's shielded swap mechanism for cross-chain private AMM logic, and Tornado Cash's circuit design (for educational purposes regarding privacy pool construction). Essential research papers include "Zexe: Enabling Decentralized Private Computation" and "ZkSwap: A Practical Atomic Swap Scheme."

Key challenges you will face include gas optimization for on-chain verification, designing a front-running resistant batch auction mechanism, and ensuring liquidity bootstrapping in a private system. Prioritize security audits from firms specializing in zero-knowledge cryptography, such as Trail of Bits or Zellic, before any mainnet deployment. Your next step is to join developer communities in the ZKProof standards forum and the EthResearch privacy channel to collaborate and stay updated on cutting-edge techniques.