DAO Short Explainer

Prerequisites

There is a scheme called commit-and-reveal, where given an object $x$ (or set of objects), you generate a random blind $b$, and construct the commitment $C = \textrm{hash}(x, b)$. By publishing $C$, you are commmitted to the value $x$.

Secondly, we may wish to publish long lived objects on the blockchain, which are essentially commitments to several parameters that represent an object defining its behaviour. In lieu of a better term, we call these bullas.

`DAO::mint()`: Establishing the DAO

From darkfi/src/contract/dao/proof/dao-mint.zk:

	bulla = poseidon_hash(
		dao_proposer_limit,
		dao_quorum,
		dao_approval_ratio_quot,
		dao_approval_ratio_base,
		gov_token_id,
		dao_public_x,
		dao_public_y,
		dao_bulla_blind,
	);

Brief description of the DAO bulla params:

proposer_limit: minimum deposit required for proposals to become valid. TODO: rename to min_deposit.
quorum: minimum threshold of votes before it's allowed to pass. Normally this is implemented as min % of voting power, but we do this in absolute value
approval_ratio: proportion of winners to losers for a proposal to pass.

Currently there is no notion of veto although it could be trivially added if desired.

`DAO::propose()`: Propose the Vote

From darkfi/src/contract/dao/proof/dao-propose-main.zk:

	proposal_bulla = poseidon_hash(
		proposal_dest_x,
		proposal_dest_y,
		proposal_amount,
		proposal_token_id,
		dao_bulla,
		proposal_blind,
	);

We create a proposal which will send tokens to the dest provided. This will soon be changed to be generic. Proposals will commit to calling params or code instead.

`DAO::vote()`: Vote on a Proposal

Governance token holders each make an encrypted homomorphic commitment to their vote. The homomorphism is additive so $f(u) + f(v) = f(u + v)$. They also encrypt their vote to the DAO pubkey.

Finally once voting is completed, the holders of the DAO pubkey (which is up to DAO policy) can decrypt the votes $f(v₁), …, f(vₙ)$, sum the values $v₁ + ⋯ + vₙ$ and so have the value which can be used in ZK proofs alongside the publicly available commitment $f(v₁ + ⋯ + vₙ) = f(v₁) + ⋯ + f(vₙ)$.

`DAO::exec()`: Execute Passed Proposal

This is the key part. We produce a tx which has two contract calls: [money::transfer(), DAO::exec()]. The coins spent in money::transfer() belong to the DAO and have the condition that they can only be spent when combined with DAO::exec(). Here is what coins in money::transfer() look like:

	C = poseidon_hash(
		pub_x,
		pub_y,
		value,
		token,
		serial,
		spend_hook,
		user_data,
	);

When we send coins to the DAO treasury, we set spend_hook to the DAO contract, and user_data to the DAO bulla.

When spending the coins, they reveal the spend_hook publicly and user_data (encrypted). money::transfer() enforces that the next contract call must be the same as the spend_hook.

The contract invoked by spend_hook can then use the user_data. We use this to store the DAO bulla. DAO::exec() will then use this as our DAO, and check the proposal we are executing belongs to this DAO through the reference to the DAO bulla in the proposal params.

DAO::exec() then encodes the rules that specify there has to be a valid proposal where voting passed the threshold and so on.

Assuming both contracts validate successfully, the funds are transferred out of the DAO treasury.

Formalism

Let the $ℂ$ be the category for all sets of coins $C$ with one-way arrows $C → C'$ such that $C ⊆ C'$ and an initial object $C₀ = ∅ $. We require that arrows with the same source and target commute. $$ \begin{CD} C @>c_b>> C_b \ @VcₐVV @Vc_a'VV \ Cₐ @>cb'>> C{ab} \end{CD} $$

We define the nullifier functor $N : ℂ^{\t{op}} → ℕ$ which is an isomorphism of $ℂ$ that reverses arrows.

$$ \begin{CD} C @>>> NC \ @VcVV @AANcA \ C' @>>> NC' \end{CD} $$ We can see the action of adding $c$ to $C$ (expressed as the left downwards arrow) gets lifted to the arrow going backwards in the nullifier category. The collection of arrows in $ℂ$ and $ℕ$ then describes the coins and nullifier sets which are represented in merkle trees.

From the diagram we see that $C → C' → NC' → NC → C$ so that $Nc$ cancels $c$. Pasting diagrams together, we get

$$ \begin{CD} C₀ @>>> NC₀ \ @Vc₁VV @AANc₁A \ C₁ @>>> NC₁ \ @Vc₂VV @AANc₂A \ C₂ @>>> NC₂ \ \end{CD} $$ where all squares commute. Since all paths in $ℂ$ are one way, proving a coin $cₖ : Cₖ₋₁ → Cₖ$ exists is equivalent to being at any state $Cₖ, Cₖ₊₁, Cₖ₊₂, …$.

Lemma: If our state is $Cₖ$ then our set must contain the coins represented as arrows $c₁, …, cₖ$.

Anon Voting Mechanics

When making a proposal, we need to prove ownership of a threshold of coins. Likewise for voting. Essentially they are similar problems of proving ownership of a coin $c$ that is still valid. As showed above this reduces to the following statements:

Is $c$ in the set of all coins $C$?
If yes, then is $n(c)$ not in the set of nullifiers $N$?

Normally this logic is handled by transfers, but we need to additionally check it without leaking info about $c$. Since $n(c)$ is derived deterministically, leaking $n(c)$ also leaks info on $c$.

Nullifiers must be checked otherwise expired coins can be used.

Forking the Global State

The first method involves copying the coins state $C$. Every proof makes use of $C$ while revealing $n(c)$ which is checked against the current nullifier state. To avoid anonymity leaks from revealing $n(c)$, we additionally move the coin using a Money::transfer() call.

The downside is that wallets need to:

Keep track of the coins tree $C$. This will involve logic to periodically checkpoint the incremental tree in a deterministic way.
When doing any action, the wallet must move coins simultaneously. Wallets must also keep track of the unspent coin since for example it might be used in another vote (or the wallet makes a proposal and wants to vote with the same coin).

Additionally you cannot obtain a coin then vote. You must own the coin before the vote is proposed.

Forking the Global State (with SMT)

Instead of revealing the nullifier, we instead snapshot the the nullifier tree alongside $C$.

The downsides are:

More expensive for voters since SMT is expensive in ZK.
We're taking an older snapshot of the coins state. Spent coins spent after the vote are proposed will still be able to vote.

Tracking Coins with Local State

Each coin's user_data contains an SMT of all proposals they voted in. When transferring a coin, you must preserve this user_data. The spend_hook only allows modifying it with a parent call that adds proposals when voting to the SMT field in the coin.

The downside for wallets is that:

The SMT committed to is large and needs to be transferred to receivers when sending the coin.
- Alternatively coins could contain a special key (also in the user_data field), which when voting you must make a verifiable encryption. That way wallets can later scan all proposals for a DAO to find where their particular governance token voted.
It's very complex. For example, can DAOs own governance tokens? So far DAO tokens must have the spend_hook set, but with this, we now require another spend_hook which preserves the SMT when transferring coins. The mechanics for two parents of a call aren't specified, so we'd maybe have to add some concept of symlinks.

However while complex, it is the most accurate of all 3 methods reflecting the current state.

Tree States On Disk

This section is not specific to DAO or Money, but describes a generic set abstraction which you can add or remove items from.

Requirements:

Add and remove items, which is represented by two G-sets denoted coins $Cₖ$ and nullifiers $Nₖ$. Let $Rₖ$ and $Sₖ$ be commitments to them both respectively.
Given any snapshot $(Rₖ, Sₖ)$, it's possible to definitively say whether the set it represents contains an item $x$ or not.
- Be able to do this fully ZK.
Wallets can easily take any snapshot, and using delta manipulation of the state, be able to make inclusion and exclusion proofs easily.
Verify in WASM that $Rₖ$ and $Sₖ$ correspond to the same $k$.

The proposal is as follows and involves a merkle tree $𝐂$, and a SMT $𝐍$.

For the sake of clarity, we will not detail the storing of the trees themselves. For $𝐂$, the tree is stored in db_info, while $𝐍$ has a full on-disk representation. Instead the info in this section concerns the auxilliary data required for using the trees with snapshotted states.

DB Merkle Roots

This is used to quickly lookup a state commitment for $𝐂$ and figure out when it occurred.

Key or Value	Field Name	Size	Desc
k	Root	32	The current root hash $Rₖ$
v	Tx hash	32	Current block height
v	Call index	1	Index of contract call

We call get_tx_location(tx_hash) -> (block_height, tx_index), and then use the (block_height, tx_index) tuple to figure out all info about this state change (such as when it occurred).

DB SMT Roots

Just like for the merkle case, we want to quickly see whether $Rₖ$ and $Sₖ$ correspond to each other. We just compare the tx hash and call index. If they match, then they both exist in the same update() call.

Key or Value	Field Name	Size	Desc
k	Root	32	The current root hash $Sₖ$
v	Tx hash	32	Current block height
v	Call index	1	Index of contract call

DB Coins (Wallets)

This DB is maintained by the user wallet, and periodic garbage collection will remove values older than a cutoff.

Keeps track of values added to $𝐂$ or $𝐍$.

For $𝐂$ given an earlier tree checkpoint state, we can rewind, then fast forward to have a valid merkle tree for the given snapshot. Wallets should additionally periodically copy the merkle tree $𝐂$.

In the case of $𝐍$, we construct an overlay for SMT, which allows rewinding the tree so exclusion proofs can be constructed.

Key or Value	Field Name	Size	Desc
k	Block height	4	Block height for coin
k	Tx index	2	Index in block for tx containing coin
k	Call index	1	Index of contract call
k	Val index	2	Index of this coin or nullifier
v	Value	32	Coin or nullifier
v	Type	1	Single byte indicating the type

This structure for the keys in an ordered B-Tree, means it can be iterated from any point. We can start from any location from our last stored merkle tree checkpoint, and iterate forwards adding coins until we reach our desired snapshot $(Rₖ, Sₖ)$. We then have a valid merkle tree and SMT reconstructed and can create the desired inclusion or exclusion proofs.

Q: should this be 2 databases or one? If we use 2 then we can remove the type byte. Maybe more conceptually clearer?

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