Penumbra provides private, sealed-bid batch swaps using ZSwap. ZSwap allows users to privately swap between any pair of assets. Individual swaps do not reveal trade amounts. Instead, all swaps in each block are executed in a single batch. Only the total amount in each batch is revealed, and only after the batch has been finalized. This prevents front-running and provides better execution, but also provides long-term privacy for individual swaps. Users can also provide liquidity by anonymously creating concentrated liquidity positions. These positions reveal the amount of liquidity and the bounds in which it is concentrated, but are not otherwise linked to any identity, so that (with some care) users can privately approximate arbitrary trading functions without revealing their specific views about prices.

Frequent batch swaps

Budish, Cramton, and Shim (2015) analyze trading in traditional financial markets using the predominant continuous-time limit order book market design, and find that high-frequency trading arises as a response to mechanical arbitrage opportunities created by flawed market design:

These findings suggest that while there is an arms race in speed, the arms race does not actually affect the size of the arbitrage prize; rather, it just continually raises the bar for how fast one has to be to capture a piece of the prize… Overall, our analysis suggests that the mechanical arbitrage opportunities and resulting arms race should be thought of as a constant of the market design, rather than as an inefficiency that is competed away over time.

— Eric Budish, Peter Cramton, John Shim, The High-Frequency Trading Arms Race: Frequent Batch Auctions as a Market Design Response

Because these mechanical arbitrage opportunities arise from the market design even in the presence of symmetrically observed public information, they do not improve prices or produce value, but create arbitrage rents that increase the cost of liquidity provision1. Instead, they suggest changing from a continuous-time model to a discrete-time model and performing frequent batch auctions, executing all orders that arrive in the same discrete time step in a single batch with a uniform price.

In the blockchain context, while projects like Uniswap have demonstrated the power and success of CFMMs for decentralized liquidity provision, they have also highlighted the mechanical arbitrage opportunities created by the mismatch between continuous-time market designs and the state update model of the underlying blockchain, which operates in discrete batches (blocks). Each prospective state update is broadcast to all participants to be queued in the mempool, but only committed as part of the next block, and while it is queued for inclusion in the consensus state, other participants can bid to manipulate its ordering relative to other state updates (for instance, front-running a trade).

This manipulation is enabled by two features:

  • Although trades are always committed in a batch (a block), they are performed individually, making them dependent on miners’ choices of the ordering of trades within a block;

  • Because trades disclose the trade amount in advance of execution, all other participants have the information necessary to manipulate them.

ZSwap addresses the first problem by executing all swaps in each block in a single batch, first aggregating the amounts in each swap and then executing it against the CFMM as a single trade.

ZSwap addresses the second problem by having users encrypt their swap amounts using a flow encryption key controlled by the validators, who aggregate the encrypted amounts and decrypt only the batch trade. This prevents front-running prior to block inclusion, and provides privacy for individual trades (up to the size of the batch) afterwards.

Users do not experience additional trading latency from the batch swap design, because the batch swaps occur in every block, which is already the minimum latency for finalized state updates. A longer batch latency could help privacy for market-takers by increasing the number of swaps in each batch, but would impair other trading and impose a worse user experience.

Private, sealed-bid batch swaps

A key challenge in the design of any private swap mechanism is that zero-knowledge proofs only allow privacy for user-specific state, not for global state, because they don’t let you prove statements about things that you don’t know. While users can prove that their user-specific state was updated correctly without revealing it, they cannot do so for other users’ state.

Instead of solving this problem, ZSwap sidesteps the need for users to do so. At a high level, swaps work as follows: users privately burn funds of one kind in a coordinated way that allows the chain to compute a per-block clearing price, and mint or burn liquidity pool reserves. Later, users privately mint funds of the other kind, proving that they previously burned funds and that the minted amount is consistent with the computed price and the burned amount. No interaction or transfer of funds between users or the liquidity pool reserves is required. Users do not transact with each other. Instead, the chain permits them to transmute one asset type to another, provably updating their private state without interacting with any other users’ private state.

This mechanism is described in more detail in the Sealed-Bid Batch Swaps section.

Concentrated Liquidity

ZSwap executes trades against concentrated liquidity positions.

Uniswap v3’s insight was that that existing constant-product market makers like Uniswap v2 allocate liquidity inefficiently, spreading it uniformly over the entire range of possible prices for a trading pair. Instead, allowing liquidity providers (LPs) to restrict their liquidity to a price range of their choosing provides a mechanism for market allocation of liquidity, concentrating it into the range of prices that the assets in the pair actually trade.

Liquidity providers create positions that tie a quantity of liquidity to a specific price range. Within that price range, the position acts as a constant-product market maker with larger “virtual” reserves. At each price, the pool aggregates liquidity from all positions that contain that price, and tracks which positions remain (or become) active as the price moves. By creating multiple positions, LPs can approximate arbitrary trading functions.

However, the concentrated liquidity mechanism in Uniswap v3 has a number of limitations:

  • Approximating an arbitrary trading function using a set of concentrated liquidity positions is cumbersome and difficult, because each position is a scaled and translated copy of the (non-linear) constant-product trading function;

  • Liquidity providers cannot compete on trading fees, because positions must be created with one of a limited number of fee tiers, and users cannot natively route trades across different fee tiers;

  • All liquidity positions are publicly linked to the account that created them, so that any successful LP strategy can be immediately cloned by other parties.

Zswap solves all of these problems:

  • Using linear (constant-price) concentrated liquidity positions makes approximating an arbitrary trading function much easier, and makes the DEX implementation much more efficient. Users can recover a constant-product (or any other) trading function by creating multiple positions, pushing the choice of curve out of consensus and into client software.

  • Each position includes its own fee setting. This fragments liquidity into many (potentially tens of thousands) of distinct AMMs, each with a specific fee, but this is not a problem, because Zswap can optimally route batched swaps across all of them.

  • While positions themselves are public, they are opened and closed anonymously. This means that while the aggregate of all LPs’ trading functions is public, the trading function of each individual LP is not, so LPs can approximate their desired trading functions without revealing their specific views about prices, as long as they take care to avoid linking their positions (e.g., by timing or amount).

Handling of concentrated liquidity is described in more detail in the Concentrated Liquidity section.

Liquidity Compensation

Systems that support both staking and liquidity provision need a mechanism to manage the competitive equilibrium between these two uses of the staking token. If staking rewards are too high relative to LP returns, LPs will choose to stake instead, causing liquidity to dry up. On the other hand, if LP returns are too high relative to staking rewards, stakers will choose to provide liquidity instead, weakening the chain’s economic security.

Penumbra’s staking design, which provides native delegation tokens for each validator, poses additional challenges. LPs are disincentivized to provide liquidity in the staking token, if they could otherwise provide liquidity for a delegation token and get staking rewards as well as LP returns. This is undesirable for a number of reasons: it fragments liquidity across many different delegation tokens, it undermines the security model to some extent (by making it easier to unload delegation tokens before misbehavior is detected), and it drives staking centralization (since larger validators would presumably have deeper liquidity in their delegation token, making it a more attractive asset).

To address these challenges, and allow the protocol to stabilize competition between staking and liquidity provision, ZSwap uses a mechanism called liquidity compensation, so named because it compensates LPs for the opportunity cost of not staking. Instead of issuing staking rewards only to delegators, the total issuance in each epoch is split between delegators and liquidity providers. To determine the split, the chain sets a target ratio of bonded stake to stake used for liquidity provision, and compares the actual ratio to the target ratio, similarly to the way that other systems adjust staking rewards based on the proportion of tokens staked.

At the end of each epoch, the share of issuance used for liquidity compensation is allocated to each eligible liquidity position active during that epoch, pro rata to liquidity provided. Eligible liquidity positions are any positions in trading pairs where one asset is the staking token, and the other asset is not a delegation token.

This mechanism has several nice properties:

  • Although the amount of liquidity compensation is determined on the basis of value locked, it’s allocated on the basis of liquidity provided. This incentivizes LPs to efficiently deploy their capital, since LPs who can create finer liquidity positions will receive disproportionate liquidity compensation rewards.

  • Because markets in delegation tokens are not eligible for liquidity compensation, marketmakers providing liquidity between bonded and unbonded forms of the staking token must take on the opportunity cost of not staking the unbonded side. Therefore, fees in those markets must be sufficient to cover that opportunity cost, so that there’s no free, instant withdrawal from a delegation pool.

  • The mechanism is credibly neutral, applying to any trading pair involving the staking token, rather than artificially subsidizing certain tokens via governance proposals.

This mechanism is described in more detail in the Liquidity Compensation section.


on HFT, N.B. their footnote 5:

A point of clarification: our claim is not that markets are less liquid today than before the rise of electronic trading and HFT; the empirical record is clear that trading costs are lower today than in the pre-HFT era, though most of the benefits appear to have been realized in the late 1990s and early 2000s… Rather, our claim is that markets are less liquid today than they would be under an alternative market design that eliminated sniping.