feat(docs): add partial notes doc (#8192)

Adds a doc on partial notes as a concept.

closes: AztecProtocol/dev-rel#363

---------

Co-authored-by: benesjan <[email protected]>

cspell.json

@@ -117,6 +117,7 @@
     "hasher",
     "headstart",
     "herskind",
+    "homomorphic",
     "ierc",
     "indexeddb",
     "initialise",

docs/docs/aztec/concepts/storage/partial_notes.md

@@ -0,0 +1,143 @@
+---
+title: Partial Notes
+description: Describes how partial notes are used in Aztec
+tags: [notes, storage]
+---
+
+Partial notes are a concept that allows users to commit to an encrypted value, and allows a counterparty to update that value without knowing the specific details of the encrypted value.
+
+## Use cases
+
+Why is this useful?
+
+Consider the case where a user wants to pay for a transaction fee, using a [fee-payment contract](../../../protocol-specs/gas-and-fees/index.md) and they want to do this privately. They can't be certain what the transaction fee will be because the state of the network will have progressed by the time the transaction is processed by the sequencer, and transaction fees are dynamic. So the user can commit to a value for the transaction fee, publicly post this commitment, the fee payer can update the public commitment, deducting the final cost of the transaction from the commitment and returning the unused value to the user.
+
+So, in general, the user is:
+
+- doing some computation in private
+- encrypting/compressing that computation with a point
+- passing that point as an argument to a public function
+
+And the fee payer is:
+
+- updating that point in public
+- treating/emitting the result(s) as a note hash(es)
+
+The idea of committing to a value and allowing a counterparty to update that value without knowing the specific details of the encrypted value is a powerful concept that can be used in many different applications. For example, this could be used for updating timestamp values in private, without revealing the exact timestamp, which could be useful for many defi applications.
+
+To do this, we leverage the following properties of elliptic curve operations:
+
+1. `x_1 * G + x_2 * G` equals `(x_1 + x_2) * G` and
+2. `f(x) = x * G` being a one-way function.
+
+Property 1 allows us to be continually adding to a point on elliptic curve and property 2 allows us to pass the point to a public realm without revealing anything about the point preimage.
+
+Before getting to partial notes let's recap what is the flow of standard notes.
+
+## Note lifecycle recap
+
+The standard note flow is as follows:
+
+1. Create a note in your contract,
+2. compute the note hash,
+3. emit the note hash,
+4. emit the note (note hash preimage) as an encrypted note log,
+5. sequencer picks up the transaction, includes it in a block (note hash gets included in a note hash tree) and submits the block on-chain,
+6. nodes and PXEs following the network pick up the new block, update its internal state and if they have accounts attached they search for relevant encrypted note logs,
+7. if a users PXE finds a log it stores the note in its database,
+8. later on when we want to spend a note, a contract obtains it via oracle and stores a note hash read request within the function context (note hash read request contains a newly computed note hash),
+9. based on the note and a nullifier secret key a nullifier is computed and emitted,
+10. protocol circuits check that the note is a valid note by checking that the note hash read request corresponds to a real note in the note hash tree and that the new nullifier does not yet exist in the nullifier tree,
+11. if the conditions in point 10. are satisfied the nullifier is inserted into the nullifier tree and the note is at the end of its life.
+
+Now let's do the same for partial notes.
+
+## Partial notes life cycle
+
+1. Create a partial/unfinished note in a private function of your contract --> partial here means that the values within the note are not yet considered finalized (e.g. `amount` in a `TokenNote`),
+2. compute a note hiding point of the partial note using a multi scalar multiplication on an elliptic curve. For `TokenNote` this would be done as `G_amt * amount0 + G_npk * npk_m_hash + G_rnd * randomness + G_slot * slot`, where each `G_` is a generator point for a specific field in the note,
+3. pass the note hiding point to a public function,
+4. in a public function determine the value you want to add to the note (e.g. adding a value to an amount) and add it to the note hiding point (e.g. `NOTE_HIDING_POINT + G_amt * amount`),
+5. get the note hash by finalizing the note hiding point (the note hash is the x coordinate of the point),
+6. emit the note hash,
+7. manually construct the note in your application and add it to your node (PXE) --> this currently has to be done manually and not automatically via encrypted note logs because we have not yet implemented partial notes delivery (tracked in [issue #8238](https://github.com/AztecProtocol/aztec-packages/issues/8238))
+8. from this point on the flow of partial notes is the same as for normal notes.
+
+### Private Fee Payment Example
+
+Alice wants to use a fee-payment contract for fee abstraction, and wants to use private balances. That is, she wants to pay the FPC (fee-payment contract) some amount in an arbitrary token privately (e.g. a stablecoin), and have the FPC pay the `transaction_fee`.
+
+Alice also wants to get her refund privately in the same token (e.g. the stablecoin).
+
+The trouble is that the FPC doesn't know if Alice is going to run public functions, in which case it doesn't know what refund is due until the end of public execution.
+
+And we can't use the normal flow to create a transaction fee refund note for Alice, since that demands we have Alice's address in public.
+
+So we define a new type of note with its `compute_note_hiding_point` defined as:
+
+$$
+\text{amount}*G_{amount} + \text{address}*G_{address} + \text{randomness}*G_{randomness} + \text{slot}*G_{slot}
+$$
+
+Suppose Alice is willing to pay up to a set amount in stablecoins for her transaction. (Note, this amount gets passed into public so that when `transaction_fee` is known the FPC can verify that it isn't losing money. Wallets are expected to choose common values here, e.g. powers of 10).
+
+Then we can subtract the set amount from Alice's balance of private stablecoins, and create a point in private like:
+
+$$
+P_a' := \text{alice address}*G_{address} + \text{rand}_a*G_{randomness} + \text{Alice note slot}*G_{slot}
+$$
+
+We also need to create a point for the owner of the FPC (whom we call Bob) to receive the transaction fee, which will also need randomness.
+
+So in the contract we compute $\text{rand}_b := h(\text{rand}_a, \text{msg_sender})$.
+
+:::warning
+We need to use different randomness for Bob's note here to avoid potential privacy leak (see [description](https://github.com/AztecProtocol/aztec-packages/blob/#include_aztec_version/noir-projects/noir-contracts/contracts/token_contract/src/main.nr#L491) of `setup_refund` function)
+:::
+
+$$
+P_b' := \text{bob address}*G_{address} + \text{rand}_b*G_{randomness} + \text{Bob note slot}*G_{slot}
+$$
+
+Here, the $P'$s "partially encode" the notes that we are _going to create_ for Alice and Bob. So we can use points as "Partial Notes".
+
+We pass these points and the funded amount to public, and at the end of public execution, we compute tx fee point $P_{fee} := (\text{transaction fee}) * G_{amount}$ and refund point $P_{refund} := (\text{funded_amount - transaction_fee}) * G_{amount}$
+
+Then, we arrive at the point that corresponds to the complete note by
+
+$$
+P_a := P_a'+P_{refund} = (\text{funded amount} - \text{transaction fee})*G_{amount} + \text{alice address}*G_{address} +\text{rand}_a*G_{randomness} + \text{Alice note slot}*G_{slot}
+$$
+
+$$
+P_b := P_b'+P_{fee} = (\text{transaction fee})*G_{amount} + \text{bob address}*G_{address} +\text{rand}_b*G_{randomness} + \text{Bob note slot}*G_{slot}
+$$
+
+Then we just emit `P_a.x` and `P_b.x` as a note hashes, and we're done!
+(Now Alice and Bob need to manually add the notes to their PXEs since [issue #8238](https://github.com/AztecProtocol/aztec-packages/issues/8238) remains to be implemented.)
+
+### Private Fee Payment Implementation
+
+[`NoteInterface.nr`](https://github.com/AztecProtocol/aztec-packages/blob/#include_aztec_version/noir-projects/aztec-nr/aztec/src/note/note_interface.nr) implements `compute_note_hiding_point`, which takes a note and computes the point "hides" it.
+
+This is implemented in the example token contract:
+
+#include_code compute_note_hiding_point noir-projects/noir-contracts/contracts/token_contract/src/types/token_note.nr rust
+
+Those `G_x` are generators that generated [here](https://github.com/AztecProtocol/aztec-packages/blob/#include_aztec_version/noir-projects/noir-projects/aztec-nr/aztec/src/generators.nr). Anyone can use them for separating different fields in a "partial note".
+
+We can see the complete implementation of creating and completing partial notes in an Aztec contract in the `setup_refund` and `complete_refund` functions.
+
+#### `setup_refund`
+
+#include_code setup_refund noir-projects/noir-contracts/contracts/token_contract/src/main.nr rust
+
+The `setup_refund` function sets the `complete_refund` function to be called at the end of the public function execution (`set_public_teardown_function`). This ensures that the partial notes will be completed and the fee payer will be paid and the user refund will be issued.
+
+#### `complete_refund`
+
+#include_code complete_refund noir-projects/noir-contracts/contracts/token_contract/src/main.nr rust
+
+## Future work
+
+This pattern of making public commitments to notes that can be modified by another party, privately, can be generalized to work with different kinds of applications. The Aztec labs team is working on adding libraries and tooling to make this easier to implement in your own contracts.

noir-projects/noir-contracts/contracts/private_fpc_contract/src/main.nr

@@ -30,6 +30,7 @@ contract PrivateFPC {
         // We emit fee payer randomness as nullifier to ensure FPC admin can reconstruct their fee note - note that
         // protocol circuits will perform the siloing as was done above and hence the final nullifier will be correct
         // fee payer randomness.
+        // TODO(#8238): Implement proper note delivery
         context.push_nullifier(user_randomness);
 
         Token::at(asset).setup_refund(

noir-projects/noir-contracts/contracts/token_contract/src/main.nr

@@ -491,18 +491,23 @@ contract Token {
     /// We need to use different randomness for the user and for the fee payer notes because if the randomness values
     /// were the same we could fingerprint the user by doing the following:
     ///      1) randomness_influence = fee_payer_point - G_npk * fee_payer_npk =
-    ///                              = (G_npk * fee_payer_npk + G_rnd * randomness) - G_npk * fee_payer_npk =
+    ///                              = (G_npk * fee_payer_npk + G_rnd * randomness + G_slot * fee_payer_slot)
+    ///                                - G_npk * fee_payer_npk - G_slot * fee_payer_slot =
     ///                              = G_rnd * randomness
     ///      2) user_fingerprint = user_point - randomness_influence =
-    ///                          = (G_npk * user_npk + G_rnd * randomness) - G_rnd * randomness =
-    ///                          = G_npk * user_npk
+    ///                          = (G_npk * user_npk + G_rnd * randomness + G_slot * user_slot) - G_rnd * randomness =
+    ///                          = G_npk * user_npk + G_slot * user_slot
     ///      3) Then the second time the user would use this fee paying contract we would recover the same fingerprint
     ///         and link that the 2 transactions were made by the same user. Given that it's expected that only
     ///         a limited set of fee paying contracts will be used and they will be known, searching for fingerprints
-    ///         by trying different fee payer npk values of these known contracts is a feasible attack.
+    ///         by trying different fee payers is a feasible attack.
     ///
-    /// `fee_payer_point` and `user_point` above are public information because they are passed as args to the public
-    /// `complete_refund(...)` function.
+    /// Note 1: fee_payer_npk is publicly available in a key registry contract under a fee_payer address. So if we have
+    ///         a known set of fee payer contract addresses getting fee_payer_npk and fee_payer_slot is trivial (slot
+    ///         is derived in a `Map<...>` as a hash of balances map slot and a fee payer address).
+    /// Note 2: fee_payer_point and user_point above are public information because they are passed as args to
+    ///         the public `complete_refund(...)` function.
+    // docs:start:setup_refund
     #[aztec(private)]
     fn setup_refund(
         fee_payer: AztecAddress, // Address of the entity which will receive the fee note.
@@ -573,9 +578,11 @@ contract Token {
         ]
         );
     }
+    // docs:end:setup_refund
 
     // TODO(#7728): even though the funded_amount should be a U128, we can't have that type in a contract interface due
     // to serialization issues.
+    // docs:start:complete_refund
     #[aztec(public)]
     #[aztec(internal)]
     fn complete_refund(
@@ -614,6 +621,7 @@ contract Token {
         context.push_note_hash(user_note_hash);
         // --> Once the tx is settled user and fee recipient can add the notes to their pixies.
     }
+    // docs:end:complete_refund
 
     /// Internal ///
 

noir-projects/noir-contracts/contracts/token_contract/src/types/token_note.nr

@@ -47,6 +47,7 @@ impl NoteInterface<TOKEN_NOTE_LEN, TOKEN_NOTE_BYTES_LEN> for TokenNote {
         poseidon2_hash_with_separator([note_hash_for_nullify, secret],GENERATOR_INDEX__NOTE_NULLIFIER)
     }
 
+    // docs:start:compute_note_hiding_point
     fn compute_note_hiding_point(self) -> Point {
         // We use the unsafe version because the multi_scalar_mul will constrain the scalars.
         let amount_scalar = from_field_unsafe(self.amount.to_integer());
@@ -62,6 +63,7 @@ impl NoteInterface<TOKEN_NOTE_LEN, TOKEN_NOTE_BYTES_LEN> for TokenNote {
             [amount_scalar, npk_m_hash_scalar, randomness_scalar, slot_scalar]
         )
     }
+    // docs:end:compute_note_hiding_point
 }
 
 impl TokenNote {

yarn-project/end-to-end/src/e2e_fees/private_refunds.test.ts

@@ -93,6 +93,7 @@ describe('e2e_fees/private_refunds', () => {
     // should be able to add the note to our PXE. Just calling `pxe.addNote(...)` is enough of a check that the note
     // hash was emitted because the endpoint will compute the hash and then it will try to find it in the note hash
     // tree. If the note hash is not found in the tree, an error is thrown.
+    // TODO(#8238): Implement proper note delivery
     await t.aliceWallet.addNote(
       new ExtendedNote(
         aliceRefundNote,
@@ -112,6 +113,7 @@ describe('e2e_fees/private_refunds', () => {
     const bobFeeNote = new Note([new Fr(transactionFee!), bobNpkMHash, bobRandomness]);
 
     // 7. Once again we add the note to PXE which computes the note hash and checks that it is in the note hash tree.
+    // TODO(#8238): Implement proper note delivery
     await t.bobWallet.addNote(
       new ExtendedNote(
         bobFeeNote,