Penumbra is a fully private proof-of-stake network and decentralized exchange for the Cosmos ecosystem.

Penumbra integrates privacy with proof-of-stake through a novel private delegation mechanism that provides staking derivatives, tax-efficient staking, and on-chain governance with private voting. Penumbra connects to the Cosmos ecosystem via IBC, acting as an ecosystem-wide shielded pool and allowing private transactions in any IBC-compatible asset. Users can also swap these assets using ZSwap, a private decentralized exchange supporting sealed-bid batch auctions and Uniswap-v3-style concentrated liquidity. Sealed-bid batch auctions prevent frontrunning, provide better execution, and reveal only the net flow across a pair of assets in each block, and liquidity positions are created anonymously, allowing traders to approximate their desired trading function without revealing their individual beliefs about prices.

This website renders the work-in-progress protocol specification for Penumbra.

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## Private Transactions

Penumbra records all value in a single multi-asset shielded pool based on the Zcash Sapling design, but allows private transactions in any kind of IBC asset. Inbound IBC transfers shield value as it moves into the zone, while outbound IBC transfers unshield value.

Unlike Zcash, Penumbra has no notion of transparent transactions or a transparent value pool; instead, inbound IBC transfers are analogous to t2z Zcash transactions, outbound IBC transfers are analogous to z2t Zcash transactions, and the entire Cosmos ecosystem functions analogously to Zcash’s transparent pool.

Unlike the Cosmos Hub or many other chains built on the Cosmos SDK, Penumbra has no notion of accounts. Only validators have any kind of long-term identity, and this identity is only used (in the context of transactions) for spending the validator’s commission.

## Private Staking

In a proof-of-stake system like the Cosmos Hub, stakeholders delegate staking tokens by bonding them to validators. Validators participate in Tendermint consensus with voting power determined by their delegation size, and delegators receive staking rewards in exchange for taking on the risk of being penalized for validator misbehavior (slashing).

Integrating privacy and proof of stake poses significant challenges. If delegations are public, holders of the staking token must choose between privacy on the one hand and staking rewards and participation in consensus on the other hand. Because the majority of the stake will be bonded to validators, privacy becomes an uncommon, opt-in case. But if delegations are private, issuing staking rewards becomes very difficult, because the chain no longer knows the amount and duration of each address’ delegations.

Penumbra sidesteps this problem using a new mechanism that eliminates staking rewards entirely, treating unbonded and bonded stake as separate assets, with an epoch-varying exchange rate that prices in what would be a staking reward in other systems. This mechanism ensures that all delegations to a particular validator are fungible, and can be represented by a single delegation token representing a share of that validator’s delegation pool, in effect a first-class staking derivative. Although delegation fungibility is key to enabling privacy, as a side effect, delegators do not realize any income while their stake is bonded, only a capital gain (or loss) on unbonding.

The total amount of stake bonded to each validator is part of the public chain state and determines consensus weight, but the delegation tokens themselves are each just another token to be recorded in a multi-asset shielded pool. This provides accountability for validators and privacy and flexibility for delegators, who can trade and transact with their delegation tokens just like they can with any other token.

It also provides an alternate perspective on the debate between fixed-supply and inflation-based rewards. Choosing the unbonded token as the numéraire, delegators are rewarded by inflation for taking on the risk of validator misbehavior, and the token supply grows over time. Choosing (a basket of) delegation tokens as the numéraire, non-delegators are punished by depreciation for not taking on any risk of misbehavior, and the token supply is fixed.

## Private Governance

Like the Cosmos Hub, Penumbra supports on-chain governance with delegated voting. Unlike the Cosmos Hub, Penumbra’s governance mechanism supports secret ballots. Penumbra users can anonymously propose votes by escrowing a deposit of bonded stake. Stakeholders vote by proving ownership of their bonded stake prior to the beginning of the voting period and encrypting their votes to a threshold key controlled by the validator set. Validators sum encrypted votes and decrypt only the per-epoch totals.

## Private DEX

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.

# Concepts and Mechanisms

This section provides an overview of the concepts involved in Penumbra’s mechanism design.

# Validators

Validators in Penumbra undergo various transitions depending on chain activity.

                                       ┌ ─ ─ ─ ─ ┐
Genesis
│Validator│
─ ─ ─ ─ ─
│
│
┌ ─ ─ ─ ─ ─ ─ ─ ─                           ▼
Validator    │      ┏━━━━━━━━┓      ╔══════╗      ┏━━━━━━━┓
│   Definition    ─────▶┃Inactive┃─────▶║Active║─────▶┃Slashed┃
(in transaction)│      ┗━━━━━━━━┛      ╚══════╝      ┗━━━━━━━┛
└ ─ ─ ─ ─ ─ ─ ─ ─            ▲              ▲             ▲
│              │             │
│              ▼             │
│         ╔═════════╗        │
└─────────║Unbonding║────────┘
╚═════════╝


Single lines represent unbonded stake, and double lines represent bonded stake.

Validators become known to the chain either at genesis, or by means of a transaction with a ValidatorDefinition action in them. Validators transition through four states:

• Inactive, where the validator is not part of the consensus set, the stake in the validator’s delegation pool is not bonded;
• Active, where the validator is part of the consensus set, and the stake in the validator’s delegation pool is bonded;
• Unbonding, where the validator is not part of the consensus set, but the stake in the validator’s delegation pool is still bonded;
• Slashed, where the validator is not part of the consensus set, and the stake in the validator’s delegation pool is not bonded.

Validators specified in the genesis config begin in the active state, with whatever stake was allocated to their delegation pool at genesis. Otherwise, validators begin in the inactive state, with no stake in their delegation pool. At this point, the validator is known to the chain, and stake can be contributed to its delegation pool. Stake contributed to an inactive validator’s delegation pool does not earn rewards (the validator’s rates are held constant), but it is also not bonded, so undelegations are effective immediately, with no unbonding period and no output quarantine.

The chain chooses a validator limit N as a consensus parameter. When a validator’s delegation pool (a) has a nonzero balance and (b) its (voting-power-adjusted) size is in the top N validators, it moves into the active state during the next epoch transition. Active validators participate in consensus, and are communicated to Tendermint. Stake contributed to an active validator’s delegation pool earns rewards (the validator’s rates are updated at each epoch to track the rewards accruing to the pool). That stake is bonded, so undelegations have an unbonding period and an output quarantine. An active validator can exit the consensus set in two ways.

First, the validator could be slashed. This can happen in any block, not just at an epoch transition. Slashed validators are immediately removed from the consensus set. Any pending undelegations from a slashed validator are cancelled: the quarantined output notes are deleted, and the quarantined nullifiers are removed from the nullifier set. The validator’s rates are updated to price in the slashing penalty, and are then held constant. Slashed validators are jailed, and permanently prohibited from participation in consensus (though their operators can create new identity keys, if they’d like to). Stake cannot be delegated to a slashed validator. Stake already contributed to a slashed validator’s delegation pool is not bonded (the validator has already been slashed and jailed), so undelegations are effective immediately, with no unbonding period and no quarantine.

Second, the validator could be displaced from the validator set by another validator with more stake in its delegation pool. The validator is then in the unbonding state. It does not participate in consensus, and the stake in its delegation pool does not earn rewards (the validator’s rates are held constant). However, the stake in its delegation pool is still bonded. Undelegations from an unbonding validator are quarantined with an unbonding period that starts when the undelegation was performed, not when the validator began unbonding. Unbonding validators have three possible state transitions:

1. they can become active again, if new delegations boost its weight back into the top N;
2. they can be slashed, if evidence of misbehavior arises during the unbonding period;
3. they can become inactive, if neither (1) nor (2) occurs before the unbonding period passes.

If (2) occurs, the same state transitions as in regular slashing occur: all pending undelegations are cancelled, etc. If (3) occurs, all pending undelegations are immediately removed from quarantine, short-circuiting the unbonding period that began when the undelegation was performed. If (1) occurs, the validator stops unbonding, but this has no effect on pending undelegations, since they were quarantined with an unbonding period that started when the undelegation was performed (i.e., as if they were undelegations from an active validator).

# Batching Flows

Penumbra’s ledger records value as it moves between different economic roles – for instance, movement between unbonded stake and delegation tokens, movement between different assets as they are traded, etc. This creates a tension between the need to reveal the total amount of value in each role as part of the public chain state, and the desire to shield value amounts in individual transactions.

To address this tension, Penumbra provides a mechanism to aggregate value flows across a batch of transactions, revealing the only the total amount and not the value contributed by each individual transaction. This mechanism is built using an integer-valued homomorphic encryption scheme that supports threshold decryption, so that the network’s validators can jointly control a decryption key.

Transactions that contribute value to a batch contain an encryption of the amount. To flush the batch, the validators sum the ciphertexts from all relevant transactions to compute an encryption of the batch total, then jointly decrypt it and commit it to the chain.

This mechanism doesn’t require any coordination between the users whose transactions are batched, but it does require that the validators create and publish a threshold decryption key. To allow batching across block boundaries, Penumbra organizes blocks into epochs, and applies changes to the validator set only at epoch boundaries. Decryption keys live for the duration of the epoch, allowing value flows to be batched over any time interval from 1 block up to the length of an epoch. We propose epoch boundaries on the order of 1-3 days.

At the beginning of each epoch, the validator set performs distributed key generation for to produce a decryption key jointly controlled by the validators (on an approximately stake-weighted basis) and includes the encryption key in the first block of the epoch.

Because this key is only available after the first block of each epoch, some transactions cannot occur in the first block itself. Assuming a block interval similar to the Cosmos Hub, this implies an ~8-second processing delay once per day, a reasonable tradeoff against the complexity of phased setup procedures.

Value transferred on Penumbra is sent to shielded payment addresses; these addresses are derived from spending keys through a sequence of intermediate keys that represent different levels of attenuated capability:

flowchart BT
end;
subgraph DTK[Detection Key]
end;
subgraph IVK[Incoming Viewing Key]
end;
subgraph OVK[Outgoing Viewing Key]
end;
subgraph FVK[Full Viewing Key]
end;
subgraph SK[Spending Key]
end;
subgraph SeedPhrase[Seed Phrase]
end;

div{ };

SeedPhrase --> SK;
SK --> FVK;
FVK --> OVK;
FVK --> IVK;

index --> div;
IVK ----> div;


From bottom to top:

• the seed phrase is the root key material. Multiple accounts - each with separate spend authority - can be derived from this root seed phrase.
• the spending key is the capability representing spending authority for a given account;
• the full viewing key represents the capability to view all transactions related to the account;
• the outgoing viewing key represents the capability to view only outgoing transactions, and is used to recover information about previously sent transactions;
• the incoming viewing key represents the capability to view only incoming transactions, and is used to scan the block chain for incoming transactions.

Penumbra allows the same account to present multiple, publicly unlinkable addresses, keyed by an 16-byte address index. Each choice of address index gives a distinct shielded payment address. Because these addresses share a common incoming viewing key, the cost of scanning the blockchain does not increase with the number of addresses in use.

Finally, Penumbra also allows outsourcing probabilistic transaction detection to third parties using fuzzy message detection. Each address has a detection key; a third party can use this key to detect transactions that might be relevant to that key. Like a Bloom filter, this detection has false positives but no false negatives, so detection will find all relevant transactions, as well as some amount of unrelated cover traffic. Unlike incoming viewing keys, detection keys are not shared between diversified addresses, allowing fine-grained control of delegation.

This diagram shows only the user-visible parts of the key hierarchy. Internally, each of these keys has different components, described in detail in the Addresses and Keys section of the Cryptographic Protocol chapter.

# Assets and Amounts

Penumbra’s shielded pool can record arbitrary assets. To be precise, we define:

• an amount to be an untyped quantity of some asset;
• an asset type to be an ADR001-style denomination trace uniquely identifying a cross-chain asset and its provenance, such as:
• denom (native chain A asset)
• transfer/channelToA/denom (chain B representation of chain A asset)
• transfer/channelToB/transfer/channelToA/denom (chain C representation of chain B representation of chain A asset)
• an asset ID to be a fixed-size hash of an asset type;
• a value to be a typed quantity, i.e., an amount and an asset id.

Penumbra deviates slightly from ADR001 in the definition of the asset ID. While ADR001 defines the IBC asset ID as the SHA-256 hash of the denomination trace, Penumbra hashes to a field element, so that asset IDs can be more easily used inside of a zk-SNARK circuit.

# Notes, Nullifiers, and Trees

Transparent blockchains operate as follows: all participants maintain a copy of the (consensus-determined) application state. Transactions modify the application state directly, and participants check that the state changes are allowed by the application rules before coming to consensus on them.

On a shielded blockchain like Penumbra, however, the state is fragmented across all users of the application, as each user has a view only of their “local” portion of the application state recording their funds. Transactions update a user’s state fragments privately, and use a zero-knowledge proof to prove to all other participants that the update was allowed by the application rules.

Penumbra’s transaction model is derived from the Zcash shielded transaction design, with modifications to support multiple asset types, and several extensions to support additional functionality. The Zcash model is in turn derived from Bitcoin’s unspent transaction output (UTXO) model, in which value is recorded in transaction outputs that record the conditions under which they can be spent.

In Penumbra, value is recorded in notes, which function similarly to UTXOs. Each note specifies (either directly or indirectly) a type of value, an amount of value of that type, a spending key that authorizes spending the note’s value, and a unique nullifier derived from the note’s contents.

However, unlike UTXOs, notes are not recorded as part of the public chain state. Instead, the chain contains a state commitment tree, an incremental Merkle tree containing (public) commitments to (private) notes. Creating a note involves creating a new note commitment, and proving that it commits to a valid note. Spending a note involves proving that the spent note was previously included in the state commitment tree, using the spending key to demonstrate spend authorization, and revealing the nullifier, which prevents the same note from being spent twice.

# Transactions

Transactions describe an atomic collection of changes to the ledger state. Each transaction consist of a sequence of descriptions for various actions1. Each description adds or subtracts (typed) value from the transaction’s value balance, which must net to zero. Penumbra adapts the Spend and Output actions from Sapling, and adds many new descriptions to support additional functionality:

Penumbra adapts Sapling’s Spend, which spends a note and adds to the transaction’s value balance, and Output, which creates a new note and subtracts from the transaction’s value balance, and adds many new descriptions to support additional functionality:

#### Transfers

• Spend descriptions spend an existing note, adding its value to the transaction’s value balance;

• Output descriptions create a new note, subtracting its value from the transaction’s value balance;

• Transfer descriptions transfer value out of Penumbra by IBC, consuming value from the transaction’s value balance, and producing an ICS20 FungibleTokenPacketData for the counterparty chain;

#### Governance

• CreateProposal descriptions are used to propose measures for on-chain governance and supply a deposit, consuming bonded stake from the transaction’s value balance and producing a new note that holds the deposit in escrow;

• WithdrawProposal descriptions redeem an escrowed proposal deposit, returning it to the transaction’s value balance and immediately withdrawing the proposal.

• Vote descriptions perform private voting for on-chain governance and declare a vote. This description leaves the value balance unchanged.

• Swap descriptions perform the first phase of ZSwap, consuming tokens of one type from a transaction’s value balance, burning them, and producing a swap commitment for use in the second stage;

• Sweep descriptions perform the second phase of ZSwap, allowing a user who burned tokens of one type to mint tokens of the other type at the chain-specified clearing price, and adding the new tokens to a transaction’s value balance;

#### Market-making

• OpenPosition descriptions open concentrated liquidity positions, consuming value of the traded types from the transaction’s value balance and adding the specified position to the AMM state;

• ClosePosition descriptions close concentrated liquidity positions, removing the specified position to the AMM state and adding the value of the position, plus any accumulated fees or liquidity rewards, to the transaction’s value balance.

Each transaction also contains a fee specification, which is always transparently encoded. The value balance of all of a transactions actions, together with the fees, must net to zero.

1

Note that like Zcash Orchard, we use the term “action” to refer to one of a number of possible state updates; unlike Orchard, we do not attempt to conceal which types of state updates are performed, so our Action is an enum.

# Governance

Penumbra supports on-chain governance with delegated voting. Validators’ votes are public and act as default votes for their entire delegation pool, while delegators’ votes are private, and override the default vote provided by their validator.

Votes are the same as on the Cosmos Hub: Yes, No, NoWithVeto, and Abstain. NoWithVeto is the same as No but also votes that the proposer should lose their deposit. The intended cultural norm is that No should be used to indicate disagreement with good-faith proposals and NoWithVeto should be used to deter spam proposals.

## Proposals

Penumbra users can propose votes by escrowing a minimum amount of PEN. They do this by creating a transaction with a CreateProposal description, which consumes some amount of PEN from the transaction’s balance, and creates a new escrow note with the same amount. The note is escrowed in the sense that it is recorded seperately and is not included in the state commitment tree until voting completes.

Proposals can either be normal or emergency proposals. In either case, the voting period begins immediately, in the next block after the proposal has been committed to the chain. Normal proposals have a fixed-length voting period, while emergency proposals are accepted as soon as a 2/3 majority of the stake is reached.

Because validators provide default votes for their delegation pool, an emergency proposal can in principle be accepted immediately, without any input from delegators. This allows time-critical resolution of emergencies (e.g., deploying an 0day hotfix); the 2/3 majority of the stake required is already sufficient to arbitrarily rewrite the chain state.

Proposals can also be withdrawn by their proposer prior to the end of the voting period. This is done by creating a transaction with a WithdrawProposal description, and allows the community to iterate on proposals as the (social) governance process occurs. For instance, a chain upgrade proposal can be withdrawn and re-proposed with a different source hash if a bug is discovered while upgrade voting is underway. Withdrawn proposals cannot be accepted, even if the vote would have passed, but they can be vetoed.1

## Voting

Stakeholder votes are of the form , representing the weights for yes, no, abstain, and veto respectively. Most stakeholders would presumably set all but one weight to . Stakeholders vote by proving ownership of some amount of bonded stake (their voting power) prior to the beginning of the voting period.

To do this, they create a transaction with a Vote description. This description identifies the validator and the proposal, proves spend authority over a note recording dPEN(v), and reveals the note’s nullifier. Finally, it proves vote consistency , produces a new note with dPEN(v), and includes , an encryption of the vote weights to the validators’ decryption key.

The proof statements in a Vote description establishing spend authority over the note are almost identical to those in a Spend description. However, there are two key differences. First, rather than proving that the note was included in a recent state commitment tree state, it always uses the root of the note commitment tree at the time that voting began, establishing that the note was not created after voting began. Second, rather than checking the note’s nullifier against the global nullifier set and marking it as spent, the nullifier is checked against a snapshot of the nullifier set at the time that voting began (establishing that it was unspent then), as well as against a per-proposal nullifier set (establishing that it has not already been used for voting). In other words, instead of marking that the note has been spent in general, we only mark it as having been spent in the context of voting on a specific proposal.

This change allows multiple proposals to be voted on concurrently, at the cost of linkability. While the same note can be used to vote on multiple proposals, those votes, as well as the subsequent spend of the note, will have the same nullifier and thus be linkable to each other. However, the Vote descriptions are shielded, so an observer only learns that two opaque votes were related to each other.

We suggest that wallets roll over the note the first time it is used for voting by creating a transaction with Vote, Spend, and Output descriptions. This mitigates linkability between Vote and Spend descriptions, and means that votes on any proposals created after the first vote are unlinkable from prior votes.

At the end of each epoch, validators collect the encrypted votes from each delegation pool, aggregate the encrypted votes into encrypted tallies and decrypt the tallies. These intermediate tallies are revealed, because it is not possible to batch value flows over time intervals longer than one epoch. In practice, this provides a similar dynamic as existing (transparent) on-chain governance schemes, where tallies are public while voting is ongoing.

At the end of the voting period, the per-epoch tallies are summed. For each validator , the votes for each option are summed to determine the portion of the delegation pool that voted; the validator’s vote acts as the default vote for the rest of the delegation pool. Finally, these per-validator subtotals are multiplied by the voting power adjustment function to obtain the final vote totals.

If the vote was not vetoed, the escrowed note from the Proposal description is included in the state commitment tree, so that it can be spent by the proposer. Otherwise, it is not, and the funds are burned.

1

If withdrawing a proposal halted on-chain voting immediately, the escrow mechanism would not be effective at deterring spam, since the proposer could yank their proposal at the last minute prior to losing their deposit. However, at the UX level, withdrawn proposals can be presented as though voting were closed, since validators’ default votes are probably sufficient for spam deterrence.

# Staking and Delegation

As described in the overview, integrating privacy with proof-of-stake poses significant challenges. Penumbra sidesteps these challenges using a novel mechanism that eliminates staking rewards entirely, treating unbonded and bonded stake as separate assets, with an epoch-varying exchange rate that prices in what would be a staking reward in other systems. This mechanism ensures that all delegations to a particular validator are fungible, and can be represented by a single delegation token representing a share of that validator’s delegation pool, in effect a first-class staking derivative.

This section describes the staking and delegation mechanism in detail:

# Staking Tokens

Penumbra’s staking token, denoted PEN, represents units of unbonded stake.

Rather than treat bonded and unbonded stake as being two states of the same token, Penumbra records stake bonded with a particular validator with a delegation token, denoted dPEN. Conversion between PEN and dPEN occurs with an epoch-varying exchange rate that prices in what would be a staking reward in other systems. This ensures that all delegations to a particular validator are fungible, and can be represented by a single token representing an ownership share of that validator’s delegation pool.

Stake bonded with different validators is not fungible, as different validators may have different commission rates and different risk of misbehavior. Hence dPEN is a shorthand for a class of assets (one per validator), rather than a single asset. dPEN bonded to a specific validator can be denoted dPEN(v) when it is necessary to be precise.

Each flavor of dPEN is its own first-class token, and like any other token can be transferred between addresses, traded, sent over IBC, etc. Penumbra itself does not attempt to pool risk across validators, but nothing prevents third parties from building stake pools composed of these assets.

The base reward rate for bonded stake is a parameter indexed by epoch. This parameter can be thought of as a “Layer 1 Base Operating Rate”, or “L1BOR”, in that it serves as a reference rate for the entire chain. Its value is set on a per-epoch basis by a formula involving the ratio of bonded and unbonded stake, increasing when there is relatively less bonded stake and decreasing when there is relatively more. This formula should be decided and adjusted by governance.

Each validator declares a set of funding streams, which comprise both the destinations of their commission and the total commission rate . is subtracted from the base reward rate to get a validator-specific reward rate .

The base exchange rate between PEN and dPEN is given by the function which measures the cumulative depreciation of stake PEN relative to the delegation token dPEN from genesis up to epoch . However, because dPEN is not a single asset but a family of per-validator assets, this is only a base rate.

The actual exchange rate between stake PEN and validator ’s delegation token dPEN(v) accounts for commissions by substituting the validator-specific rate in place of the base rate to get

Delegating PEN to validator at epoch results in dPEN. Undelegating dPEN(v) from validator at epoch results in PEN. Thus, delegating at epoch and undelegating at epoch results in a return of i.e., the staking reward compounded only over the period during which the stake was bonded.

Discounting newly bonded stake by the cumulative depreciation of unbonded stake since genesis means that all bonded stake can be treated as if it had been bonded since genesis, which allows newly unbonded stake to always be inflated by the cumulative appreciation since genesis. This mechanism avoids the need to track the age of any particular delegation to compute its rewards, and makes all shares of each validator’s delegation pool fungible.

# Validator Rewards and Fees

Validators declare a set of funding streams that comprise the destination of all of their staking rewards. Each funding stream contains a rate and a destination address . The validator’s total commission rate is defined as , the sum of the rate of each funding stream. cannot exceed 1.

The spread between the base reward rate and the reward rate for their delegators is determined by the validator’s total commission , or equivalently .

Validator rewards are distributed in the first block of each epoch. In epoch , a validator whose delegation pool has size dPEN receives a commission of size PEN, issued to the validator’s address.

To see why this is the reward amount, suppose validator has a delegation pool of size dPEN. In epoch , the value of the pool is PEN. In epoch , the base reward rate causes the value of the pool to increase to Splitting as , this becomes

The value in the first term, , corresponds to the portion, and accrues to the delegators. Since , this is exactly , the new PEN-denominated value of the delegation pool.

The value in the second term, , corresponds to the portion, and accrues to the validator as commission. Validators can self-delegate the resulting PEN or use it to fund their operating expenses.

Transaction fees are denominated in PEN and are burned, so that the value of the fees accrues equally to all stake holders.

## TODO

• allow transaction fees in dPEN with appropriate price adjustment, but only in transactions (e.g., undelegations, voting) that otherwise reveal the flavor of dPEN, so that there is no additional distinguisher?

# Voting Power

The size of each validator’s delegation pool (i.e., the amount of dPEN of that validator’s flavor) is public information, and determines the validator’s voting power. However, these sizes cannot be used directly, because they are based on validator-specific conversion rates and are therefore incommensurate.

Voting power is calculated using the adjustment function , so that a validator whose delegation pool has dPEN in epoch has voting power .

The validator-specific reward rate adjusts the base reward rate to account for the validator’s commission. Since and the adjustment function accounts for the compounded effect of the validator’s commission on the size of the delegation pool.

# Delegation

The delegation process bonds stake to a validator, converting stake PEN to delegation tokens dPEN. Delegations may be performed in any block, but only take effect in the next epoch.

Delegations are accomplished by creating a transaction with a Delegate description. This specifies a validator , consumes PEN from the transaction’s balance, produces a new shielded note with dPEN associated to that validator, and includes an encryption of the delegation amount to the validators’ shared decryption key . Here is the index of the next epoch, when the delegation will take effect.

In the last block of epoch , the validators sum the encrypted bonding amounts from all delegate descriptions for validator in the entire epoch to obtain an encryption of the total delegation amount and decrypt to obtain the total delegation amount without revealing any individual transaction’s delegation amount . These total amounts are used to update the size of each validator’s delegation pool for the next epoch.

Revealing only the total inflows to the delegation pool in each epoch helps avoid linkability. For instance, if the size of each individual transaction’s delegation were revealed, a delegation of size followed by an undelegation of size could be correlated if an observer notices that there are some epochs so that

This risk is still present when only the total amount – the minimum disclosure required for consensus – is revealed, because there may be few (or no) other delegations to the same validator in the same epoch. Some care should be taken in client implementations and user behavior to mitigate the effects of this information disclosure, e.g., by splitting delegations into multiple transactions in different epochs involving randomized sub-portions of the stake. However, the best mitigation would simply be to have many users.

# Undelegation

The undelegation process unbonds stake from a validator, converting delegation tokens dPEN to stake PEN. Undelegations may be performed in any block, but only settle after the undelegation has exited the unbonding queue.

The unbonding queue is a FIFO queue allowing only a limited amount of stake to be unbonded in each epoch, according to an unbonding rate selected by governance. Undelegations are inserted into the unbonding queue in FIFO order. Unlike delegations, where only the total amount of newly bonded stake is revealed, undelegations reveal the precise amount of newly unbonded stake, allowing the unbonding queue to function.

Undelegations are accomplished by creating a transaction with a Undelegate description. This description has different behaviour depending on whether or not the validator was slashed.

In the unslashed case, the undelegate description spends a note with value dPEN, reveals , and produces PEN for the transaction’s balance, where is the index of the current epoch. However, the nullifiers revealed by undelegate descriptions are not immediately included in the nullifier set, and new notes created by a transaction containing an undelegate description are not immediately included in the state commitment tree. Instead, the transaction is placed into the unbonding queue to be applied later. In the first block of each epoch, transactions are applied if the corresponding validator remains unslashed, until the unbonding limit is reached.

If a validator is slashed, any undelegate transactions currently in the unbonding queue are discarded. Because the nullifiers for the notes those transactions spent were not included in the nullifier set, the notes remain spendable, allowing a user to create a new undelegation description.

Undelegations from a slashed validator are settled immediately. The undelegate description spends a note with value dPEN and produces PEN, where is the slashing penalty and is the epoch at which the validator was slashed. The remaining value, , is burned.

Because pending undelegations from a slashed validator are discarded without applying their nullifiers, those notes can be spent again in a post-slashing undelegation description. This causes linkability between the discarded undelegations and the post-slashing undelegations, but this is not a concern because slashing is a rare and unplanned event which already imposes worse losses on delegators.

# Example Staking Dynamics

To illustrate the dynamics of this system, consider a toy scenario with three delegators, Alice, Bob, and Charlie, and two validators, Victoria and William. Tendermint consensus requires at least four validators and no one party controlling more than of the stake, but this example uses only a few parties just to illustrate the dynamics.

For simplicity, the the base reward rates and commission rates are fixed over all epochs at and , . The PEN and dPEN holdings of participant are denoted by , , etc., respectively.

Alice starts with dPEN of Victoria’s delegation pool, Bob starts with dPEN of William’s delegation pool, and Charlie starts with unbonded PEN.

• At genesis, Alice, Bob, and Charlie respectively have fractions , , and of the total stake, and fractions , , of the total voting power.

• At epoch , Alice, Bob, and Charlie’s holdings remain unchanged, but their unrealized notional values have changed.

• Victoria charges zero commission, so . Alice’s dPEN(v) is now worth PEN.
• William charges commission, so . Bob’s dPEN(w) is now worth , and William receives PEN.
• William can use the commission to cover expenses, or self-delegate. In this example, we assume that validators self-delegate their entire commission, to illustrate the staking dynamics.
• William self-delegates PEN, to get dPEN in the next epoch, epoch .
• At epoch :

• Alice’s dPEN(v) is now worth PEN.
• Bob’s dPEN(w) is now worth PEN.
• William’s self-delegation of accumulated commission has resulted in dPEN(w).
• Victoria’s delegation pool remains at size dPEN(v). William’s delegation pool has increased to dPEN(w). However, their respective adjustment factors are now and , so the voting powers of their delegation pools are respectively and .
• The slight loss of voting power for William’s delegation pool occurs because William self-delegates rewards with a one epoch delay, thus missing one epoch of compounding.
• Charlie’s unbonded PEN remains unchanged, but its value relative to Alice and Bob’s bonded stake has declined.
• William’s commission transfers stake from Bob, whose voting power has slightly declined relative to Alice’s.
• The distribution of stake between Alice, Bob, Charlie, and William is now , , , respectively. The distribution of voting power is , , , respectively.
• Charlie decides to bond his stake, split evenly between Victoria and William, to get dPEN(v) and dPEN(w).
• At epoch :

• Charlie now has dPEN(v) and dPEN(w), worth PEN.
• For the same amount of unbonded stake, Charlie gets more dPEN(w) than dPEN(v), because the exchange rate prices in the cumulative effect of commission since genesis, but Charlie isn’t charged for commission during the time he didn’t delegate to William.
• William’s commission for this epoch is now PEN, up from PEN in the previous epoch.
• The distribution of stake between Alice, Bob, Charlie, and William is now , , , respectively. Because all stake is now bonded, except William’s commission for this epoch, which is insignificant, the distribution of voting power is identical to the distribution of stake.
• At epoch :

• Alice’s dPEN(v) is now worth PEN.
• Bob’s dPEN(w) is now worth PEN.
• Charlies’s dPEN(v) is now worth PEN, and his dPEN(w) is now worth PEN.
• William’s self-delegation of accumulated commission has resulted in dPEN(w), worth PEN.
• The distribution of stake and voting power between Alice, Bob, Charlie, and William is now , , , respectively.

This scenario was generated with a model in this Google Sheet.

# Fixed-Point Arithmetic for Rate Computation

To compute base reward rate, base exchange rate, validator-specific exchange rates, and total validator voting power, we need to carefully perform arithmetic to avoid issues with precision and rounding. We use explicitly specified fixed-precision arithmetic for this, with a precision of 8 digits. This allows outputs to fit in a u64, with all products fitting in the output and plenty of headroom for additions.

All integer values should be interpreted as unsigned 64-bit integers, with the exception of the validator’s commission rate, which is a u16 specified in terms of basis points (one one-hundredth of a percent, or in other words, an implicit denominator of ). All integer values, with the exception of the validator’s commission rate, have an implicit denominator of .

Throughout this spec representations are referred to as , where , and the value represented by representations is .

As an example, let’s take a starting value represented in our scheme (so, ) and compute its product with , also represented by our fixed point scheme, so . The product is computed as

Since both and are both representations and both have a factor of , computing their product includes an extra factor of 10^8 which we divide out. All representations are u64 in our scheme, and any fixed-point number or product of fixed point numbers with 8 digits fits well within 64 bits.

### Summary of Notation

• : the fixed-point representation of the base rate at epoch .
• : the fixed-point representation of the base exchange rate at epoch .
• : the funding rate of the validator’s funding stream at index and epoch , in basis points.
• : the sum of the validator’s commission rates, in basis points.
• : the fixed-point representation of the validator-specific reward rate for validator at epoch .
• : the fixed-point representation of the validator-specific exchange rate for validator at epoch .
• : the sum of the tokens in the validator’s delegation pool.
• : the validator’s voting power for validator at epoch .

## Base Reward Rate

The base reward is an input to the protocol, and the exact details of how this base rate is determined is not yet decided. For now, we can assume it is derived from the block header.

## Base Exchange Rate

The base exchange rate, , can be safely computed as follows:

## Commission Rate from Funding Streams

To compute the validator’s commission rate from its set of funding streams, compute where is the rate of validator ’s -th funding stream at epoch .

## Validator Reward Rate

To compute the validator’s base reward rate, we compute the following:

## Validator Exchange Rate

To compute the validator’s exchange rate, we use the same formula as for the base exchange rate, but substitute the validator-specific reward rate in place of the base reward rate:

## Validator Voting Power

Finally, to compute the validator’s voting power, take:

# IBC Protocol Implementation

Penumbra supports the IBC protocol for interoperating with other counterparty blockchains. Unlike most blockchains that currently deploy IBC, Penumbra is not based on the Cosmos SDK. IBC as a protocol supports replication of data between two communicating blockchains. It provides basic building blocks for building higher-level cross chain applications, as well as a protocol specification for the most commonly used IBC applications, the ICS-20 transfer protocol.

Penumbra implements the core IBC protocol building blocks: ICS-23 compatible state inclusion proofs, connections as well as channels and packets.

## IBC Actions

In order to support the IBC protocol, Penumbra adds a single additional Action IBCAction. an IBCAction can contain any of the IBC datagrams:

### ICS-003 Connections

• ConnOpenInit
• ConnOpenTry
• ConnOpenAck
• ConnOpenConfirm

### ICS-004 Channels and Packets

• ChanOpenInit
• ChanOpenTry
• ChanOpenAck
• ChanOpenConfirm
• ChanCloseInit
• ChanCloseConfirm
• RecvPacket
• Timeout
• Acknowledgement

These datagrams are implemented as protocol buffers, with the enclosing IBCAction type using profobuf’s OneOf directive to encapsulate all possible IBC datagram types.

## Handling Bridged Assets

Penumbra’s native state model uses notes, which contain an amount of a particular asset. Amounts in Penumbra are 128-bit unsigned integers, in order to support assets which have potentially large base denoms (such as Ethereum). When receiving an IBC transfer, if the amount being transferred is greater than u128, we return an error.

# Transfers into Penumbra

IBC transfer mechanics are specified in ICS20. The FungibleTokenPacketData packet describes the transfer:

FungibleTokenPacketData {
denomination: string,
amount: uint256,
sender: string,
}


The sender and receiver fields are used to specify the sending account on the source chain and the receiving account on the destination chain. However, for inbound transfers, the destination chain is Penumbra, which has no accounts. Instead, token transfers into Penumbra create an OutputDescription describing a new shielded note with the given amount and denomination, and insert an encoding of the description itself into the receiver field.

# ZSwap

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.

1

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.

# Sealed-Bid Batch Swaps

ZSwap’s sealed-bid batch swaps conceptually decompose into two parts: the DEX and AMM mechanism itself, and the batching procedure, which allows multiple users’ swaps to be batched into a single trade executed against the trading function. This section focuses on the batching procedure, leaving the mechanics of the trading function for later.

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. Rather than have users transact with each other, the chain permits them to transmute one asset type to another, provably updating their private state without interacting with any other users’ private state. To do this, they privately burn their input assets and encrypt the amounts to the validators. The validators aggregate the encrypted amounts and decrypt the batch total, then compute the effective (inclusive of fees) clearing prices and commit them to the chain state. In any later block, users can privately mint output funds of the new type, proving consistency with the inputs they burned.

### Swap actions

First, users create transactions with Swap actions that privately burn their input assets and encrypt the amounts to the validators. This action identifies the trading pair by asset id, consumes of types from the transaction balance, and contains an encryption of the trade inputs Rational traders will choose either or , i.e., trade one asset type for the other type, but the description provides two inputs so that different swap directions cannot be distinguished. The Swap action also consumes fee tokens from the transaction’s value balance, which are saved for use as a prepaid transaction fee when claiming the swap output.

To record the user’s contribution for later, the action mints a swap NFT. Penumbra assets are recorded as a pair of an amount (u64) and an asset id (). Usually, the asset id is the hash of a denomination string. For a swap NFT, however, the asset id is computed as where:

• is a Poseidon hash function;
• are the input amounts of types and respectively;
• is a prepaid fee amount that will be used for the swap claim;
• and are the diversified basepoint and diversified transmission key of one of the user’s addresses, used to preauthorize the swap claim.

The swap NFT is recorded like any other asset in the shielded pool. The Swap action includes a NotePayload with an encryption of a new note with the swap NFT, rather than using a separate Output action, in order to combine proof statements and skip a fixed-size memo field.

### Batching and Execution

In this description, which focuses on the state model, we treat the execution itself as a black box and focus only on the public data used for swap outputs.

Validators sum the encrypted amounts of all swaps in the batch to obtain an encryption of the combined inputs , then decrypt to obtain the batch input without revealing any individual transaction’s input . Then they execute against the trading pool, updating the pool state and obtaining the effective (inclusive of fees) clearing prices ( in terms of ) and ( in terms of ). Alternatively, the swap could fail, for instance, because there is insufficient liquidity, so the public state recording the swap results also includes a success bit that is on success and on failure.

Each user’s output amounts can be computed as which simplifies to when the batch succeeds and , or to when the batch fails and .

### Claiming Swap Outputs

In a future block, users who created transactions with Swap actions obtain assets of the new types by creating a transaction with SwapClaim actions. This action privately mints new tokens of the output type, and proves consistency with the user’s input contribution (via the swap NFT) and with the effective clearing prices (which are part of the public chain state). The SwapClaim action is carefully designed so that it is self-authenticating, and does not require any spend authorization. Any entity in posession of the full viewing key can create and submit a swap claim transaction, without further authorization by the user. This means that wallet software can automatically claim swap outputs as soon as it sees confirmation that the swap occurred.

Like a Spend action, the SwapClaim action spends a shielded note, revealing its nullifier and witnessing an authentication path from it to a recent anchor. However, it differs in several important respects:

• Rather than unlocking value from an arbitrary note, it proves that the spent note records unit of a swap NFT whose asset ID is so that the input state is available to other proof statements;

• Rather than witnessing the full authentication path from the note commitment up to a recent anchor, it reveals the block height and only witnesses the authentication path up to the block-level root, proving that the note was included in a particular block1, and allowing reference to the effective clearing prices and ;

• Rather than contributing to the transaction’s value balance, it constructs two output notes itself, one for each of and proves that the notes are sent to the address committed to by the and in the swap NFT;

The SwapClaim does not include a spend authorization signature, because it is only capable of consuming a swap NFT, not arbitrary notes, and only capable of sending the trade outputs to the address specified during construction of the original Swap action, which is signed.

Finally, the SwapClaim releases units of the fee token to the transaction’s value balance, allowing it to pay fees without an additional Spend action. The transaction claiming the swap outputs can therefore consist of a single SwapClaim action, and that action can be prepared using only a full viewing key. This design means that wallet software can automatically submit the swap claim without any explicit user intervention, even if the user’s custody setup (e.g., a hardware wallet) would otherwise require it.

Although sweep descriptions do not reveal the amounts, or which swap’s outputs they claim, they do reveal the block and trading pair, so their anonymity set is considerably smaller than an ordinary shielded value transfer. For this reason, client software should create and submit a transaction with a sweep description immediately after observing that its transaction with a swap description was included in a block, rather than waiting for some future use of the new assets. This ensures that future shielded transactions involving the new assets are not trivially linkable to the swap.

## Privacy for Market-Takers

This design reveals only the net flow across a trading pair in each batch, not the amounts of any individual swap. However, this provides no protection if the batch contains a single swap, and limited protection when there are only a few other swaps. This is likely to be an especially severe problem until the protocol has a significant base of active users, so it is worth examining the impact of amount disclosure and potential mitigations.

• TODO: on the client side, allow a “time preference” slider (immediate vs long duration), which spreads execution of randomized sub-amounts across multiple blocks at randomized intervals within some time horizon

• TODO: extract below into separate section about privacy on penumbra

Assuming that all amounts are disclosed, an attacker could attempt to deanonymize parts of the transaction graph by tracing amounts, using strategies similar to those in An Empirical Analysis of Anonymity in Zcash. That research attempted to deanonymize transactions by analysing movement between Zcash’s transparent and shielded pools, with some notable successes (e.g., identifying transactions associated to the sale of stolen NSA documents). Unlike Zcash, where opt-in privacy means the bulk of the transaction graph is exposed, Penumbra does not have a transparent pool, and the bulk of the transaction graph is hidden, but there are several potential places to try to correlate amounts:

• IBC transfers into Penumbra are analogous to t2z transactions and disclose types and amounts and accounts on the source chain;
• IBC transfers out of Penumbra are analogous to z2t transactions and disclose types and amounts and accounts on the destination chain;
• Each unbonding discloses the precise amount of newly unbonded stake and the validator;
• Each epoch discloses the net amount of newly bonded stake for each validator;
• Liquidity pool deposits disclose the precise type and amount of newly deposited reserves;
• Liquidity pool deposits disclose the precise type and amount of newly withdrawn reserves;

The existence of the swap mechanism potentially makes correlation by amount more difficult, by expanding the search space from all amounts of one type to all combinations of all amounts of any tradeable type and all historic clearing prices. However, assuming rational trades may cut this search space considerably.2

2

Thanks to Guillermo Angeris for this observation.

1

Technically, this is not quite true: by itself, all that revealing the block-level root on the authentication path proves is that the note was included in a block with that root, not the block with that root, since the block root binds all of the new note commitments produced in that block but does not explicitly bind the block height. To fix this, we can have the chain insert a dummy note whose note commitment is bound to the block height (e.g., by computing the note’s blinding factor as a hash of the height). This prevents a possible attack where an attacker who could control the exact set of note commitments included in two different blocks at heights and , both with swaps, could submit the exact same input amounts in and (without change), and then claim both outputs at whichever executed with a higher price.

# Primitives

Penumbra uses the following cryptographic primitives, described in the following sections:

• The Proof System section describes the choice of proving curve (BLS12-377) and proof system (Groth16, and potentially PLONK in the future);

• The decaf377 section describes decaf377, a parameterization of the Decaf construction defined over the BLS12-377 scalar field, providing a prime-order group that can be used inside or outside of a circuit;

• The Poseidon for BLS12-377 section describes parameter selection for an instantiation of Poseidon, a SNARK-friendly sponge construction, over the BLS12-377 scalar field;

• The Fuzzy Message Detection section describes a construction that allows users to outsource a probabalistic detection capability, allowing a third party to scan and filter the chain on their behalf, without revealing precisely which transactions are theirs.

• The Homomorphic Threshold Decryption section describes the construction used to batch flows of value across transactions.

• The Randomizable Signatures section describes decaf377-rdsa, a variant of the Zcash RedDSA construction instantiated over decaf377, used for binding and spend authorization signatures.

• The Key Agreement section describes decaf377-ka, an instantiation of Diffie-Hellman key agreement over decaf377.

# Proving Considerations

Penumbra needs SNARK proofs. Because the choice of proving system and proving curve can’t really be cleanly separated from the rest of the system choices (e.g., the native field of the proving system informs what embedded curve is available, and how circuit programming is done), large parts of the rest of the system design block on making a choice of proving system.

## Goals

1. Near-term implementation availability. We’d like to ship a useful product first, and iterate and expand it later.

2. High performance for fixed functionality. Penumbra intends to support fixed functionality initially; programmability is a good future goal but isn’t a near-term objective. The fixed functionality should have as high performance as possible.

3. Longer-term flexibility. The choice should ideally not preclude many future choices for later functionality. More precisely, it should not impose high switching costs on future choices.

4. Recursion capability. Penumbra doesn’t currently make use of recursion, but there are a lot of interesting applications it could be used for.

Setup ceremonies are beneficial to avoid for operational reasons, but not for security reasons. A decentralized setup procedure is sufficient for security.

## Options

Proof systems:

• Groth16:
• Pros: high performance, very small proofs, mature system
• Cons: requires a setup for each proof statement
• PLONK:
• Pros: universal setup, still fairly compact proofs, seems to be a point of convergence with useful extensions (plookup, SHPLONK, etc)
• Cons: bigger proofs, worse constants than Groth16
• Halo 2
• Pros: no setup, arbitrary depth recursion
• Cons: bigger proof sizes, primary implementation for the Pallas/Vesta curves which don’t support pairings

Curve choices:

• BLS12-381:

• Pros: very mature, used by Sapling already
• Cons: no easy recursion
• BLS12-377:

• Pros: constructed as part of Zexe to support depth 1 recursion using a bigger parent curve, deployed in Celo, to be deployed in Zexe
• Cons: ?
• Pallas/Vesta:

• Pros: none other than support for Halo 2’s arbitrary recursion
• Cons: no pairings mean they cannot be used for any pairing-based SNARK

## Considerations

Although the choice of proof system (Groth16, Plonk, Halo, Pickles, …) is not completely separable from the choice of proving curve (e.g., pairing-based SNARKs require pairing-friendly curves), to the extent that it is, the choice of the proof system is relatively less important than the choice of proving curve, because it is easier to encapsulate.

The choice of proving curve determines the scalar field of the arithmetic circuit, which determines which curves are efficient to implement in the circuit, which determines which cryptographic constructions can be performed in the circuit, which determines what kind of key material the system uses, which propagates all the way upwards to user-visible details like the address format. While swapping out a proof system using the same proving curve can be encapsulated within an update to a client library, swapping out the proving curve is extremely disruptive and essentially requires all users to generate new addresses and migrate funds.

This means that, in terms of proof system flexibility, the Pallas/Vesta curves are relatively disadvantaged compared to pairing-friendly curves like BLS12-381 or BLS12-377, because they cannot be used with any pairing-based SNARK, or any other pairing-based construction. Realistically, choosing them is committing to using Halo 2.

Choosing BLS12-377 instead of BLS12-381 opens the possibility to do depth-1 recursion later, without meaningfully restricting the near-term proving choices. For this reason, BLS12-377 seems like the best choice of proving curve.

Penumbra’s approach is to first create a useful set of fixed functionality, and generalize to custom, programmable functionality only later. Compared to Sapling, there is more functionality (not just Spend and Output but Delegate, Undelegate, Vote, …), meaning that there are more proof statements. Using Groth16 means that each of these statements needs to have its own proving and verification key, generated through a decentralized setup.

So the advantage of a universal setup (as in PLONK) over per-statement setup (as in Groth16) would be:

1. The setup can be used for additional fixed functionality later;
2. Client software does not need to maintain distinct proving/verification keys for each statement.

(2) is a definite downside, but the impact is a little unclear. As a point of reference, the Sapling spend and output parameters are 48MB and 3.5MB respectively. The size of the spend circuit could be improved using a snark-friendly hash function.

With regard to (1), if functionality were being developed in many independent pieces, doing many setups would impose a large operational cost. But doing a decentralized setup for a dozen proof statements simultaneously does not seem substantially worse than doing a decentralized setup for a single proof statement. So the operational concern is related to the frequency of groups of new statements, not the number of statements in a group. Adding a later group of functionality is easy if the first group used a universal setup. But if it didn’t, the choice of per-statement setup initially doesn’t prevent the use of a universal setup later, as long as the new proof system can be implemented using the same curve.

Because Penumbra plans to have an initial set of fixed functionality, and performance is a concern, Groth16 seems like a good choice, and leaves the door open for a future universal SNARK. Using BLS12-377 opens the door to future recursion, albeit only of depth 1.

# The decaf377 group

Penumbra, like many other zero-knowledge protocols, requires a cryptographic group that can be used inside of an arithmetic circuit. This is accomplished by defining an “embedded” elliptic curve whose base field is the scalar field of the proving curve used by the proof system.

The Zexe paper, which defined BLS12-377, also defined (called in Figure 16 of the paper) a cofactor-4 Edwards curve defined over the BLS12-377 scalar field for exactly this purpose. However, non-prime-order groups are a leaky abstraction, forcing all downstream constructions to pay attention to correct handling of the cofactor. Although it is usually possible to do so safely, it requires additional care, and the optimal technique for handling the cofactor is different inside and outside of a circuit.

Instead, applying the Decaf construction to this curve gives decaf377, a clean abstraction that provides a prime-order group complete with hash-to-group functionality and whose encoding and decoding functions integrate validation. Although it imposes a modest additional cost in the circuit context, as discussed in Costs and Alternatives, the construction works the same way inside and outside of a circuit and imposes no costs for lightweight, software-only applications, making it a good choice for general-purpose applications.

## Curve Parameters

The cofactor-4 Edwards curve defined over the BLS12-377 scalar field has the following parameters:

• Base field: Integers mod prime
• Elliptic curve equation: with and
• Curve order: where

We use a conventional generator basepoint selected to have a convenient hex encoding:

0x0800000000000000000000000000000000000000000000000000000000000000


In affine coordinates this generator point has coordinates:

## Implementation

An implementation of decaf377 can be found here.

# Costs and Alternatives

Arithmetic circuits have a different cost model than software. In the software cost model, software executes machine instructions, but in the circuit cost model, relations are certified by constraints. Unfortunately, while Decaf is a clearly superior choice in the software context, in the circuit context it imposes some additional costs, which must be weighed against its benefits.

At a high level, Decaf implements a prime-order group using a non-prime-order curve by constructing a group quotient. Internally, group elements are represented by curve points, with a custom equality check so that equivalent representatives are considered equal, an encoding function that encodes equivalent representatives as identical bitstrings, and a decoding function that only accepts canonical encodings of valid representatives.

To do this, the construction defines a canonical encoding on a Jacobi quartic curve mod its 2-torsion (a subgroup of size 4) by making two independent sign choices. Then, it uses an isogeny to transport this encoding from the Jacobi quartic to a target curve that will be used to actually implement the group operations. This target curve can be an Edwards curve or a Montgomery curve. The isogenies are only used for deriving the construction. In implementations, all of these steps are collapsed into a single set of formulas that perform encoding and decoding on the target curve.

In other words, one way to think about the Decaf construction is as some machinery that transforms two sign choices into selection of a canonical representative. Ristretto adds extra machinery to handle cofactor 8 by making an additional sign choice.

In the software cost model, where software executes machine instructions, this construction is essentially free, because the cost of both the Decaf and conventional Edwards encodings are dominated by the cost of computing an inverse or an inverse square root, and the cost of the sign checks is insignificant.

However, in the circuit cost model, where relations are certified by various constraints, this is no longer the case. On the one hand, certifying a square root or an inverse just requires checking that or that , which is much cheaper than actually computing or . On the other hand, performing a sign check involves bit-constraining a field element, requiring hundreds of constraints.

## Sign checks

The definition of which finite field elements are considered nonnegative is essentially arbitrary. The Decaf paper suggests three possibilities:

• using the least significant bit, defining to be nonnegative if the least absolute residue for is even;

• using the most significant bit, defining to be nonnegative if the least absolute residue for is in the range ;

• for fields where , using the Legendre symbol, which distinguishes between square and nonsquare elements.

Using the Legendre symbol is very appealing in the circuit context, since it has an algebraic definition and, at least in the case of square elements, very efficient certification. For instance, if square elements are chosen to be nonnegative, then certifying that is nonnegative requires only one constraint, . However, the reason for the restriction to fields is that and should have different signs, which can only be the case if is nonsquare. Unfortunately, many SNARK-friendly curves, including BLS12-377, are specifically chosen so that for as large a power as possible (e.g., in the case of BLS12-377).

This leaves us with either the LSB or MSB choices. The least significant bit is potentially simpler for implementations, since it is actually the low bit of the encoding of , while the most significant bit isn’t, because it measures from , not a bit position , so it seems to require a comparison or range check to evaluate. However, these choices are basically equivalent, in the following sense:

###### Lemma.1

The most significant bit of is if and only if the least significant bit of is .

###### Proof.

The MSB of is if and only if , but this means that , which is even, is the least absolute residue, so the LSB of is also . On the other hand, the MSB of is if and only if , i.e., if , i.e., if . This means that the least absolute residue of is ; since is even and is odd, this is odd and has LSB .

This means that transforming an LSB check to an MSB check or vice versa requires multiplication by or , which costs at most one constraint.

Checking the MSB requires checking whether a value is in the range . Using Daira Hopwood’s optimized range constraints, the range check costs 2. However, the input to the range check is a bit-constrained unpacking of a field element, not a field element itself. This unpacking costs .

Checking the LSB is no less expensive, because although the check only examines one bit, the circuit must certify that the bit-encoding is canonical. This requires checking that the value is in the range , which also costs , and as before, the unpacking costs .

In other words, checking the sign of a field element costs , or in the case where the field element is already bit-encoded for other reasons. These checks are the dominant cost for encoding and decoding, which both require two sign checks. Decoding from bits costs c. , decoding from a field element costs c. , and encoding costs c. regardless of whether the output is encoded as bits or as a field element.

For decaf377, we choose the LSB test for sign checks.

## Alternative approaches to handling cofactors

Decaf constructs a prime-order group whose encoding and decoding methods perform validation. A more conventional alternative approach is to use the underlying elliptic curve directly, restrict to its prime-order subgroup, and do subgroup validation separately from encoding and decoding. If this validation is done correctly, it provides a prime-order group. However, because validation is an additional step, rather than an integrated part of the encoding and decoding methods, this approach is necessarily more brittle, because each implementation must be sure to do both steps.

In the software cost model, there is no reason to use subgroup validation, because it is both more expensive and more brittle than Decaf or Ristretto. However, in the circuit cost model, there are cheaper alternatives, previously analyzed by Daira Hopwood in the context of Ristretto for JubJub (1, 2).

###### Multiplication by the group order.

The first validation method is to do a scalar multiplication and check that . Because the prime order is fixed, this scalar multiplication can be performed more efficiently using a hardcoded sequence of additions and doublings.

###### Cofactor preimage.

The second validation method provides a preimage in affine coordinates . Because the image of is the prime-order subgroup, checking that satisfies the curve equation and that checks that is in the prime-order subgroup.

In the software context, computing and computing cost about the same, although both are an order of magnitude more expensive than decoding. But in the circuit context, the prover can compute outside of the circuit and use only a few constraints to check the curve equation and two doublings. These costs round to zero compared to sign checks, so the validation is almost free.

The standard “compressed Edwards y” format encodes a point by the -coordinate and a sign bit indicating whether is nonnegative. In software, the cost of encoding and decoding are about the same, and dominated by taking an inverse square root. In circuits, the costs of encoding and decoding are also about the same, but they are instead dominated by a sign check that matches the sign of the recovered -coordinate with the supplied sign bit. This costs c. as above.

## Comparison and discussion

This table considers only approximate costs.

Operationdecaf377Compressed Ed + Preimage
Decode (from bits)400C400C
Decode (from )750C325C
Encode (to bits)750C750C
Encode (to )750C325C

When decoding from or encoding to field elements, the marginal cost of Decaf compared to the compressed Edwards + cofactor preimage is an extra bit-unpacking and range check. While this effectively doubles the number of constraints, the marginal cost of c. is still small relative to other operations like a scalar multiplication, which at 6 constraints per bit is approximately .

When decoding from or encoding to bits, the marginal cost of Decaf disappears. When the input is already bit-constrained, Decaf’s first sign check can reuse the bit constraints, saving c. , but the compressed Edwards encoding must range-check the bits (which Decaf already does), costing c. extra. Similarly, in encoding, Decaf’s second sign check produces bit-constrained variables for free, while the compressed Edwards encoding spends c. bit-constraining and range-checking them.

However, in the software context, the prime-order validation check costs approximately 10x more than the cost of either encoding. Many applications require use of the embedded group both inside and outside of the circuit, and uses outside of the circuit may have additional resource constraints (for instance, a hardware token creating a signature authorizing delegated proving, etc.).

Performing validation as an additional, optional step also poses additional risks. While a specification may require it to be performed, implementations that skip the check will appear to work fine, and the history of invalid-point attacks (where implementations should, but don’t, check that point coordinates satisfy the curve equation) suggests that structuring validation as an integral part of encoding and decoding is a safer design. This may not be a concern for a specific application with a single, high-quality implementation that doesn’t make mistakes, but it’s less desirable for a general-purpose construction.

In summary, Decaf provides a construction that works the same way inside and outside of a circuit and integrates validation with the encoding, imposing only a modest cost for use in circuits and no costs for lightweight, software-only applications, making it a good choice for general-purpose constructions.

1

I have no idea whether this is common knowledge; I learned of this fact from its use in Mike Hamburg’s Ed448-Goldilocks implementation.

2

The value 73 is computed as:

from itertools import groupby

def cost(k):
return min(k-1, 2)

def constraints(bound):
costs = [cost(len(list(g))+1) for (c, g) in groupby(bound.bits()[:-1]) if c == 1]
return sum(costs)

constraints(ZZ((q-1)/2))


as here.

# Inverse Square Roots

As in the internet-draft, the decaf377 functions are defined in terms of the following function, which computes the square root of a ratio of field elements, with the special behavior that if the input is nonsquare, it returns the square root of a related field element, to allow reuse of the computation in the hash-to-group setting.

Define as a non-square in the field and sqrt_ratio_zeta(N,D) as:

• (True, ) if and are nonzero, and is square;
• (True, ) if is zero;
• (False, ) if is zero and is non-zero;
• (False, ) if and are nonzero, and is nonsquare.

Since is nonsquare, if is nonsquare, is square. Note that unlike the similar function in the ristretto255/decaf448 internet-draft, this function does not make any claims about the sign of its output.

To compute sqrt_ratio_zeta we use a table-based method adapted from Sarkar 2020 and zcash-pasta, which is described in the remainder of this section.

## Constants

We set (the 2-adicity of the field) and odd such that . For the BLS12-377 scalar field, and .

We define where is a non-square as described above. We fix as 2841681278031794617739547238867782961338435681360110683443920362658525667816.

We then define a and such that . We also define a parameter where . For decaf377 we choose:

## Precomputation

Lookup tables are needed which can be precomputed as they depend only on the 2-adicity and the choice of above.

### lookup table:

We compute for and , indexed on and :

This table lets us look up powers of .

### lookup table:

We compute for , indexed on :

We use this table in the procedure that follows to find (they are the values) in order to compute .

## Procedure

In the following procedure, let . We use the following relations from Sarkar 2020:

• Equation 1: and for and
• Lemma 3: for
• Equation 2:

In these expressions, and are field elements. are unsigned -bit integers. At each , the algorithm first computes , then and (from the previous step’s and ), then finally and , in each case such that they satisfy the above expressions. Note that in the algorithm .

### Step 1: Compute

We compute . This corresponds to line 2 of the findSqRoot Algorithm 1 in Sarkar 2020.

Substituting :

Applying Fermat’s Little Theorem (i.e. ):

Substituting and rearranging:

We compute using a quantity we define as:

We also define:

And substitute and into which gives us:

We now can use in the computation for and :

### Step 2: Compute

Compute using and as calculated in the prior step. This corresponds to line 4 of the findSqRoot Algorithm 1 in Sarkar 2020.

### Step 3: Compute

We next compute for . This corresponds to line 5 of the findSqRoot Algorithm 1 in Sarkar 2020. This gives us the following components:

### Step 4: Compute and

Next, we loop over . This corresponds to lines 6-9 of the findSqRoot Algorithm 1 in Sarkar 2020.

#### For

Using Lemma 3:

Substituting the definition of from equation 1:

Rearranging and substituting in (initial condition):

Substituting in equation 2:

This is almost in a form where we can look up in our s lookup table to get and thus . If we define we get:

Which we can use with our s lookup table to get . Expressing in terms of , we get .

#### For

First we compute using equation 1:

Next, similar to the first iteration, we use lemma 3 and substitute in and to yield:

In this expression we can compute the quantities on the left hand side, and the right hand side is in the form we expect for the s lookup table, yielding us . Note that here too we define such that the s lookup table can be used. Expressing in terms of , we get .

#### For

The remaining iterations proceed similarly, yielding the following expressions:

Note for and the remaining iterations we do not require a trick (i.e. where ) to get in a form where it can be used with the s lookup table. In the following expressions for , is always even, and so the high bit of each value is unchanged when adding .

At the end of this step, we have found and for .

### Step 5: Return result

Next, we can use equation 1 to compute using and from the previous step:

This matches the expression from Lemma 4 in Sarkar 2020.

Next, to compute , we lookup entries in the g lookup table. To do so, we can decompose into:

then is computed as:

Multiplying in from step 1, we compute:

This corresponds to line 10 of the findSqRoot Algorithm 1 in Sarkar 2020.

In the non-square case, will be odd, and will be odd. We will have computed and multiply by a correction to get our desired output.

We can use the result of this computation to determine whether or not the square exists, recalling from Step 1 that :

• If is square, then , and
• If is non-square, then and .

# Decoding

Decoding to a point works as follows where and are the curve parameters as described here.

1. Decode s_bytes to a field element . We interpret these bytes as unsigned little-endian bytes. We check if the length has 32 bytes, where the top 3 bits of the last byte are 0. The 32 bytes are verified to be canonical, and rejected if not (if the input is already a field element in the circuit case, skip this step).

2. Check that is nonnegative, or reject (sign check 1).

3. .

4. .

5. (was_square, v) = sqrt_ratio_zeta(1, u_2 * u_1^2), rejecting if was_square is false.

6. if is negative (sign check 2)1.

7. .

The resulting coordinates are the affine Edwards coordinates of an internal representative of the group element.

1

Note this differs from the Decaf paper in Appendix A.2, but implementations of decaf377 should follow the convention described here.

# Encoding

Given a representative in extended coordinates , encoding works as follows where and are the curve parameters as described here.

1. .

2. (_ignored, v) = sqrt_ratio_zeta(1, u_1 * (a - d) * X^2).

3. (sign check 1).

4. .

5. .

6. Set s_bytes to be the canonical unsigned little-endian encoding of , which is an integer mod . s_bytes has extra 0x00 bytes appended to reach a length of 32 bytes.

# Group Hash

Elligator can be applied to map a field element to a curve point. The map can be applied once to derive a curve point suitable for use with computational Diffie-Hellman (CDH) challenges, and twice to derive a curve point indistinguishable from random.

In the following section, and are the curve parameters as described here. is a constant and sqrt_ratio_zeta(v_1,v_2) is a function, both defined in the Inverse Square Roots section.

The Elligator map is applied as follows to a field element :

1. .

2. .

3. .

4. sqrt_ratio_zeta where is a boolean indicating whether or not a square root exists for the provided input.

5. If a square root for does not exist, then and . Else, and is unchanged.

6. .

7. .

8. If ( and is true) or ( and is false) then .

The Jacobi quartic representation of the resulting point is given by . The resulting point can be converted from its Jacobi quartic representation to extended projective coordinates via:

For single-width hash-to-group (encode_to_curve), we apply the above map once. For double-width (hash_to_curve) we apply the map to two field elements and add the resulting curve points.

# Test Vectors

## Small generator multiples

This table has hex-encodings of small multiples of the generator :

ElementHex encoding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## Hash-to-group

This table has input field elements along with the affine coordinates of the output point after applying the elligator map once:

Input field element output point
2873166235834220037104482467644394559952202754715866736878534498814378075613(1267955849280145133999011095767946180059440909377398529682813961428156596086, 5356565093348124788258444273601808083900527100008973995409157974880178412098)
7664634080946480262422274939177258683377350652451958930279692300451854076695(1502379126429822955521756759528876454108853047288874182661923263559139887582, 7074060208122316523843780248565740332109149189893811936352820920606931717751)
707087697291448463178823336344479808196630248514167087002061771344499604401(2943006201157313879823661217587757631000260143892726691725524748591717287835, 4988568968545687084099497807398918406354768651099165603393269329811556860241)
4040687156656275865790182426684295234932961916167736272791705576788972921292(2893226299356126359042735859950249532894422276065676168505232431940642875576, 5540423804567408742733533031617546054084724133604190833318816134173899774745)
6012393175004325154204026250961812614679561282637871416475605431319079196219(2950911977149336430054248283274523588551527495862004038190631992225597951816, 4487595759841081228081250163499667279979722963517149877172642608282938805393)
7255180635786717958849398836099816771666363291918359850790043721721417277258(3318574188155535806336376903248065799756521242795466350457330678746659358665, 7706453242502782485686954136003233626318476373744684895503194201695334921001)
6609366864829739556945402594963920739176902000316365292959221199804402230199(3753408652523927772367064460787503971543824818235418436841486337042861871179, 2820605049615187268236268737743168629279853653807906481532750947771625104256)
6875465950337820928985371259904709015074922314668494500948688901607284806973(7803875556376973796629423752730968724982795310878526731231718944925551226171, 7033839813997913565841973681083930410776455889380940679209912201081069572111)

# Poseidon for BLS12-377

The Poseidon hash function is a cryptographic hash function that operates natively over prime fields. This allows the hash function to be used efficiently in the context of a SNARK. In the sections that follow we describe our instantiation of Poseidon over BLS12-377.

# Overview of the Poseidon Permutation

This section describes the Poseidon permutation. It consists of rounds, where each round has the following steps:

• AddRoundConstants: where constants (denoted by arc in the code) are added to the internal state,
• SubWords: where the S-box is applied to the internal state,
• MixLayer: where a matrix is multiplied with the internal state.

The total number of rounds we denote by . There are two types of round in the Poseidon construction, partial and full. We denote the number of partial and full rounds by and respectively.

In a full round in the SubWords layer the S-box is applied to each element of the internal state, as shown in the diagram below:

  ┌───────────────────────────────────────────────────────────┐
│                                                           │
│                                                           │
└────────────┬──────────┬──────────┬──────────┬─────────────┘
│          │          │          │
┌─▼─┐      ┌─▼─┐      ┌─▼─┐      ┌─▼─┐
│ S │      │ S │      │ S │      │ S │
└─┬─┘      └─┬─┘      └─┬─┘      └─┬─┘
│          │          │          │
┌────────────▼──────────▼──────────▼──────────▼─────────────┐
│                                                           │
│                         MixLayer                          │
│                                                           │
└────────────┬──────────┬──────────┬──────────┬─────────────┘
│          │          │          │
▼          ▼          ▼          ▼


In a partial round, in the SubWords layer we apply the S-box only to one element of the internal state, as shown in the diagram below:

               │          │          │          │
│          │          │          │
┌────────────▼──────────▼──────────▼──────────▼─────────────┐
│                                                           │
│                                                           │
└────────────┬──────────────────────────────────────────────┘
│
┌─▼─┐
│ S │
└─┬─┘
│
┌────────────▼──────────────────────────────────────────────┐
│                                                           │
│                         MixLayer                          │
│                                                           │
└────────────┬──────────┬──────────┬──────────┬─────────────┘
│          │          │          │
▼          ▼          ▼          ▼


We apply half the full rounds () first, then we apply the partial rounds, then the rest of the full rounds. This is called the HADES design strategy in the literature.

# Poseidon Parameter Generation

The problem of Poseidon parameter generation is to pick secure choices for the parameters used in the permutation given the field, desired security level in bits, as well as the width of the hash function one wants to instantiate (i.e. 1:1 hash, 2:1 hash, etc.).

Poseidon parameters consist of:

• Choice of S-box: choosing the exponent for the S-box layer where ,
• Round numbers: the numbers of partial and full rounds,
• Round constants: the constants to be added in the AddRoundConstants step,
• MDS Matrix: generating a Maximum Distance Separable (MDS) matrix to use in the linear layer, where we multiply this matrix by the internal state.

Appendix B of the Poseidon paper provides sample implementations of both the Poseidon permutation as well as parameter generation. There is a Python script called calc_round_numbers.py which provides the round numbers given the security level , the width of the hash function , as well as the choice of used in the S-box step. There is also a Sage script, which generates the round numbers, constants, and MDS matrix, given the security level , the width of the hash function , as well as the choice of used in the S-box step.

Since the publication of the Poseidon paper, others have edited these scripts, resulting in a number of implementations being in use derived from these initial scripts. We elected to implement Poseidon parameter generation in Rust from the paper, checking each step, and additionally automating the S-box parameter selection step such that one can provide only the modulus of a prime field and the best will be selected.

Below we describe where we deviate from the parameter selection procedure described in the text of the Poseidon paper.

## Choice of S-Box

The Poseidon paper focuses on the cases where , as well as BLS12-381 and BN254. For a choice of positive , it must satisfy , where is the prime modulus.

For our use of Poseidon on BLS12-377, we wanted to generate a procedure for selecting the optimal for a general curve, which we describe below.

We prefer small, positive for software (non-circuit) performance, only using when we are unable to find an appropriate positive . For positive , the number of constraints in an arithmetic circuit per S-Box is equal to its depth in a tree of shortest addition chains. For a given number of constraints (i.e. at a given depth in the tree), we pick the largest at that level that meets the GCD requirement. Since larger provides more security, choosing the largest at a given depth reduces the round requirement.

The procedure in detail:

We proceed down the tree from depth 2 to depth 5 (where depth 0 is the root of 1):

1. At a given depth, proceed from largest number to smaller numbers.
2. For a given element, check if is satisfied. If yes, we choose it, else continue.

If we get through these checks to depth of 5 without finding a positive exponent for , then we pick , which is well-studied in the original Poseidon paper.

For decaf377, following this procedure we end up with .

## Round Numbers

We implement the round numbers as described in the original paper. These are the number of rounds necessary to resist known attacks in the literature, plus a security margin of +2 full rounds, and +7.5% partial rounds.

We test our round number calculations with tests from Appendices G and H from the paper which contain concrete instantiations of Poseidon for and their round numbers.

## Round Constants

We do not use the Grain LFSR for generating pseudorandom numbers as described in Appendix F of the original paper. Instead, we use a Merlin transcript to enable parameter generation to be fully deterministic and easily reproducible.

We first append the message "poseidon-paramgen" to the transcript using label dom-sep.

We then bind this transcript to the input parameter choices:

• the width of the hash function using label t,
• the security level using label M, and
• the modulus of the prime field using label p.

We then also bind the transcript to the specific instance, as done with the Grain LFSR in Appendix F, so we bind to:

• the number of full rounds using label r_F,
• the number of partial rounds using label r_P, and
• the choice of S-box exponent using label alpha.

We generate random field elements from hashes of this complete transcript of all the input parameters and the derived parameters , , and .

Each round constant is generated by obtaining challenge bytes from the Merlin transcript, derived using label round-constant. We obtain bytes where is the field size in bits. These bytes are then interpreted as an unsigned little-endian integer reduced modulo the field.

## MDS Matrix

We generate MDS matrices using the Cauchy method. However instead of randomly sampling the field as described in the text, we deterministically generate vectors and as:

Each element of the matrix is then constructed as:

where .

This deterministic matrix generation method has been verified to be safe over the base field of decaf377, using algorithms 1-3 described in Grassi, Rechberger and Schofnegger 2020 over the t range 1-100.

The parameters that are used for poseidon377 are located in the params.rs module of the source code. The parameters were generated on an Apple M1.

# Test Vectors

The following are test vectors for the poseidon377 Poseidon instantiation.

Each section is for a given rate of a fixed-width hash function, where capacity is 1. Inputs and output are elements. The domain separator used in each case are the bytes "Penumbra_TestVec" decoded to an element, where we interpret these bytes in little-endian order.

## Rate 1

Input element:

• 7553885614632219548127688026174585776320152166623257619763178041781456016062

Output element:

• 2337838243217876174544784248400816541933405738836087430664765452605435675740

## Rate 2

Input elements:

• 7553885614632219548127688026174585776320152166623257619763178041781456016062
• 2337838243217876174544784248400816541933405738836087430664765452605435675740

Output element:

• 4318449279293553393006719276941638490334729643330833590842693275258805886300

## Rate 3

Input elements:

• 7553885614632219548127688026174585776320152166623257619763178041781456016062
• 2337838243217876174544784248400816541933405738836087430664765452605435675740
• 4318449279293553393006719276941638490334729643330833590842693275258805886300

Output element:

• 2884734248868891876687246055367204388444877057000108043377667455104051576315

## Rate 4

Input elements:

• 7553885614632219548127688026174585776320152166623257619763178041781456016062
• 2337838243217876174544784248400816541933405738836087430664765452605435675740
• 4318449279293553393006719276941638490334729643330833590842693275258805886300
• 2884734248868891876687246055367204388444877057000108043377667455104051576315

Output element:

• 5235431038142849831913898188189800916077016298531443239266169457588889298166

## Rate 5

Input elements:

• 7553885614632219548127688026174585776320152166623257619763178041781456016062
• 2337838243217876174544784248400816541933405738836087430664765452605435675740
• 4318449279293553393006719276941638490334729643330833590842693275258805886300
• 2884734248868891876687246055367204388444877057000108043377667455104051576315
• 5235431038142849831913898188189800916077016298531443239266169457588889298166

Output element:

• 66948599770858083122195578203282720327054804952637730715402418442993895152

## Rate 6

Input elements:

• 7553885614632219548127688026174585776320152166623257619763178041781456016062
• 2337838243217876174544784248400816541933405738836087430664765452605435675740
• 4318449279293553393006719276941638490334729643330833590842693275258805886300
• 2884734248868891876687246055367204388444877057000108043377667455104051576315
• 5235431038142849831913898188189800916077016298531443239266169457588889298166
• 66948599770858083122195578203282720327054804952637730715402418442993895152

Output element:

• 6797655301930638258044003960605211404784492298673033525596396177265014216269

# Fuzzy Message Detection

By design, privacy-preserving blockchains like Penumbra don’t reveal metadata about the sender or receiver of a transaction. However, this means that users must scan the entire chain to determine which transactions relate to their addresses. This imposes large bandwidth and latency costs on users who do not maintain online replicas of the chain state, as they must “catch up” each time they come online by scanning all transactions that have occurred since their last activity.

Alternatively, users could delegate scanning to a third party, who would monitor updates to the chain state on their behalf and forward only a subset of those updates to the user. This is possible using viewing keys, as in Zcash, but viewing keys represent the capability to view all activity related to a particular address, so they can only be delegated to trusted third parties.

Instead, it would be useful to be able to delegate only a probabilistic detection capability. Analogous to a Bloom filter, this would allow a detector to identify all transactions related to a particular address (no false negatives), while also identifying unrelated transactions with some false positive probability. Unlike viewing capability, detection capability would not include the ability to view the details of a transaction, only a probabilistic association with a particular address. This is the problem of fuzzy message detection (FMD), analyzed by Beck, Len, Miers, and Green in their paper Fuzzy Message Detection, which proposes a cryptographic definition of fuzzy message detection and three potential constructions.

This section explores how Penumbra could make use of fuzzy message detection:

• In Sender and Receiver FMD, we propose a generalization of the original definition where the false positive probability is set by the sender instead of the receiver, and discusses why this is useful.

• In Constructing S-FMD, we realize the new definition using a variant of one of the original FMD constructions, and extend it in two ways:

1. to support arbitrarily precise detection with compact, constant-size keys;
2. to support diversified detection, allowing multiple, publicly unlinkable addresses to be scanned by a single detection key.

Unfortunately, these extensions are not mutually compatible, so we only use the first one, and record the second for posterity.

## Acknowledgements

Thanks to George Tankersley and Sarah Jamie Lewis for discussions on this topic (and each independently suggesting the modifications to realize S-FMD), to Gabrielle Beck for discussions about the paper and ideas about statistical attacks, and to Guillermo Angeris for pointers on analyzing information disclosure.

The goal of detection capability is to be able to filter the global stream of state updates into a local stream of state updates that includes all updates related to a particular address, without identifying precisely which updates those are. The rate of updates on this filtered stream should be bounded below, to ensure that there is a large enough anonymity set, and bounded above, so that users processing the stream have a constant and manageable amount of work to process it and catch up with the current chain state. This means that the detection precision must be adaptive to the global message rates: if the false positive rate is too low, the filtered stream will have too few messages, and if it is too high, it will have too many messages.

However, this isn’t possible using the paper’s original definition of fuzzy message detection, because the false positive probability is chosen by the receiver, who has no way to know in advance what probability will produce a correctly sized stream of messages. One way to address this problem is to rename the original FMD definition as Receiver FMD (R-FMD), and tweak it to obtain Sender FMD (S-FMD), in which the sender chooses the detection probability.

The paper’s original definition of fuzzy message detection is as a tuple of algorithms (KeyGen, CreateClue, Extract, Examine).1 The receiver uses KeyGen to generate a root key and a clue key. A sender uses the receiver’s clue key as input to CreateClue to produce a clue. The Extract algorithm takes the root key and a false positive rate (chosen from some set of supported rates), and produces a detection key. The Examine algorithm uses a detection key to examine a clue and produce a detection result.

This scheme should satisfy certain properties, formalizations of which can be found in the paper:

###### Correctness.

Valid matches must always be detected by Examine; i.e., there are no false negatives.

###### Fuzziness.

Invalid matches should produce false positives with probability approximately , as long as the clues and detection keys were honestly generated.

###### Detection Ambiguity.

An adversarial detector must be unable to distinguish between a true positive and a false positive, as long as the clues and detection keys were honestly generated.

In this original definition, the receiver has control over how much detection precision they delegate to a third party, because they choose the false positive probability when they extract a detection key from their root key. This fits with the idea of attenuating credentials, and intuitively, it seems correct that the receiver should control how much information they reveal to their detector. But the information revealed to their detector is determined by both the false positive probability and the amount of other messages that can function as cover traffic. Without knowing the extent of other activity on the system, the receiver has no way to make a principled choice of the detection precision to delegate.

## Sender FMD

To address this problem, we generalize the original definition (now Receiver FMD) to Sender FMD, in which the false positive probability is chosen by the sender.

S-FMD consists of a tuple of algorithms (KeyGen, CreateClue, Examine). Like R-FMD, CreateClue creates a clue and Examine takes a detection key and a clue and produces a detection result. As discussed in the next section, S-FMD can be realized using tweaks to either of the R-FMD constructions in the original paper.

Unlike R-FMD, the false positive rate is set by the sender, so CreateClue takes both the false positive rate and the receiver’s clue key. Because the false postive rate is set by the sender, there is no separation of capability between the root key and a detection key, so KeyGen outputs a clue key and a detection key, and Extract disappears.

In R-FMD, flag ciphertexts are universal with respect to the false positive rate, which is applied to the detection key; in S-FMD, the false positive rate is applied to the flag ciphertext and the detection key is universal.

Unlike R-FMD, S-FMD allows detection precision to be adaptive, by having senders use a (consensus-determined) false positive parameter. This parameter should vary as the global message rates vary, so that filtered message streams have a bounded rate, and it should be the same for all users, so that messages cannot be distinguished by their false positive rate.

1

We change terminology from the FMD paper; the paper calls detection and clue keys the secret and public keys respectively, but we avoid this in favor of capability-based terminology that names keys according to the precise capability they allow. The “clue” terminology is adopted from the Oblivious Message Retrieval paper of Zeyu Liu and Eran Tromer; we CreateClue and Examine clues rather than Flag and Test flag ciphertexts.

# Constructing S-FMD

The original FMD paper provides three constructions of R-FMD. The first two realize functionality for restricted false-positive probabilities of the form ; the third supports arbitrary fractional probabilities using a much more complex and expensive construction.

The first two schemes, R-FMD1 and R-FMD2, are constructed similarly to a Bloom filter: the CreateClue procedure encrypts a number of 1 bits, and the Examine procedure uses information in a detection key to check whether some subset of them are 1, returning true if so and false otherwise. The false positive probability is controlled by extracting only a subset of the information in the root key into the detection key, so that it can only check a subset of the bits encoded in the clue. We focus on R-FMD2, which provides more compact ciphertexts and chosen-ciphertext security.

In this section, we:

• recall the construction of R-FMD2, changing notation from the paper;
• extend S-FMD2 to use deterministic derivation, allowing 32-byte flag and detection keys to support arbitrary-precision false positive probabilities;
• extend S-FMD2 to support diversified detection, allowing multiple, unlinkable flag keys to be detected by a single detection key (though, as this extension conflicts with the preceding one, we do not use it);
• summarize the resulting construction and describe how it can be integrated with a Sapling- or Orchard-style key hierarchy.

## The R-FMD2 construction

First, we recall the paper’s construction of R-FMD2, changing from multiplicative to additive notation and adjusting variable names to be consistent with future extensions.

The construction supports restricted false positive values for , a global parameter determining the minimum false positive rate. is a group of prime order and and are hash functions.

###### R-FMD2/KeyGen

Choose a generator . For , choose and compute . Return the root key and the clue key .

###### R-FMD2/Extract

On input and root key , parse and return as the detection key.

###### R-FMD2/Flag

On input , first parse , then proceed as follows:

1. Choose and compute .
2. Choose and compute .
3. For each , compute
1. a key bit ;
2. a ciphertext bit .
4. Compute .
5. Compute .

Return the clue .

###### R-FMD2/Examine

On input , , first parse , , then proceed as follows:

1. Compute .
2. Recompute as .
3. For each , compute
1. a key bit ;
2. a plaintext bit .

If all plaintext bits , return (match); otherwise, return .

To see that the Test procedure detects true positives, first note that since and , so the detector will recompute the used by the sender; then, since , the detector’s is the same as the sender’s .

On the other hand, if the clue was created with a different clue key, the resulting will be random bits, giving the desired false positive probability.

One way to understand the components of the construction is as follows: the message consists of bits of a Bloom filter, each encrypted using hashed ElGamal with a common nonce and nonce commitment . The points are included in the ElGamal hash, ensuring that any change to either will result in a random bit on decryption. The pair act as a public key and signature for a one-time signature on the ciphertext as follows: the sender constructs the point as the output of a chameleon hash with basis on input , then uses the hash trapdoor (i.e., their knowledge of the discrete log relation ) to compute a collision involving , a hash of the rest of the ciphertext. The collision acts as a one-time signature with public key .

The fact that the generator was selected at random in each key generation is not used by the construction and doesn’t seem to play a role in the security analysis; in fact, the security proof in Appendix F has the security game operate with a common generator. In what follows, we instead assume that is a global parameter.

## From R-FMD2 to S-FMD2

Changing this construction to the S-FMD model is fairly straightforward: rather than having the sender encrypt bits of the Bloom filter and only allow the detector to decrypt of them, we have the sender only encrypt bits of the Bloom filter and allow the detector to decrypt all potential bits. As noted in the previous section, this means that in the S-FMD context, there is no separation of capability between the root key and the detection key. For this reason, we skip the Extract algorithm and (conceptually) merge the root key into the detection key.

The S-FMD2 construction then works as follows:

###### S-FMD2/KeyGen

For , choose and compute . Return the detection key and the clue key .

###### S-FMD2/CreateClue

On input clue key , first parse , then proceed as follows:

1. Choose and compute .
2. Choose and compute .
3. For each , compute
1. a key bit ;
2. a ciphertext bit .
4. Compute .
5. Compute .

Return the clue . (We include explicitly rather than have it specified implicitly by to reduce the risk of implementation confusion).

###### S-FMD2/Examine

On input detection key and clue , first parse and , then proceed as follows:

1. Compute .
2. Recompute as .
3. For each , compute
1. a key bit ;
2. a plaintext bit .

If all plaintext bits , return (match); otherwise, return .

The value of is included in the ciphertext hash to ensure that the encoding of the ciphertext bits is non-malleable. Otherwise, the construction is identical.

## Compact clue and detection keys

One obstacle to FMD integration is the size of the clue keys. Clue keys are required to create clues, so senders who wish to create clues need to obtain the receiver’s clue key. Ideally, the clue keys would be bundled with other address data, so that clues can be included with all messages, rather than being an opt-in mechanism with a much more limited anonymity set.

However, the size of clue keys makes this difficult. A clue key supporting false positive probabilities down to requires group elements; for instance, supporting probabilities down to with a 256-bit group requires 448-byte flag keys. It would be much more convenient to have smaller keys.

One way to do this is to use a deterministic key derivation mechanism similar to BIP32 to derive a sequence of child keypairs from a single parent keypair, and use that sequence as the components of a flag / detection keypair. To do this, we define a hash function . Then given a parent keypair , child keys can be derived as with