zk-SNARKS in MACI

MACI uses zk-SNARKs to essentially hide how each person voted while still revealing the final vote result. This gives voters privacy and helps reduce bribery while still ensuring that the final results were tallied correctly off-chain.

MACI Circuits

MACI has three main zk-SNARK circuits:

1. ProcessMessages.circom, which takes a batch of encrypted messages, decrypts them, and generates a proof that the coordinator's local processing was performed correctly.
2. TallyVotes.circom, which counts votes from users' ballots, batch by batch.
3. Subsidy.circom, which implements pairwise subsidy. Please note this is an optional feature.

The rest of the circuits are utilities templates that are required for the main circuits to work correctly. These include utilities such as float math, conversion of private keys, and Poseidon hashing/encryption.

Each circuit is parameterised and it is important to set the right parameters that match your use case. For example, if you want to support up to 3125 messages, the message tree depth parameter should be set to 5 (as $5^5 = 3125$).

Background

zk-SNARKs

zk-SNARKs are a type of zero-knowledge proof which allows a "prover" to prove to a "verifier" that they know a secret without revealing the secret itself. In MACI, the prover is the coordinator. MACI uses zk-SNARKs to prove that the coordinator has correctly processed the batches of messages and that all votes have been tallied correctly. A smart contract acts as the verifier to check the proof from the coordinator. Users can also verify that the process was done correctly at any point after the proof generation.

Circom

MACI's circuits are written using Circom, a domain-specific language (DSL) used to write zk-SNARK circuits. Circom syntax resembles JavaScript, and it currently is one of the most popular DSL in use by zk developers. Please refer to their documentation to learn more about the language.

Proving system

MACI uses Groth16 as its proving system. Groth16 is a zk-SNARK proving system that allows for the generation of proofs that are small and fast to verify.

How are the circuits used?

The circuits are used by the coordinator (the prover) to prove that they have correctly processed a batch of messages and tallied the votes correctly. This happens after a Poll has completed, and the coordinator has merged the state and message trees. The coordinator then generates a proof for each batch of messages, and submits them to the contract. The contract then verifies the proofs and updates the commitments on chain.

How do the Circuits fit in a voting round?

How do the circuits work?

Message processing (processMessages)

This circuit allows the coordinator to prove that they have correctly processed each message in reverse order, in a consecutive batch of 5 ^ msgBatchDepth messages to the respective state leaf within the state tree. Coordinators would use this circuit to prove correct execution at the end of each Poll.

The processMessages circuit will try to decrypt the messages, and based on the content of the message, update within itself the trees, to generate a proof that the coordinator's off-chain processing was done correctly. In other words, the circuit takes a final state, an initial state, and the leaves (messages and user signups) - it processes these messages via the different state transitions to finally check that the expected state is correct. The pre-requisites for this circuit are:

• the related Poll has ended
• the state tree has been merged
• the message tree has been merged

This circuit requires the coordinator's private key, hence a proof for this circuit can only be generated by the coordinator. The private key is needed in order to generate the ECDH shared key used to decrypt the messages.

info

A version working with non quadratic voting (non-qv) is also available. This version is called processMessagesNonQV and is to be used when the Poll is not using the quadratic voting feature. Note that by default MACI works with quadratic voting.

Parameters

#	Parameter	Description
0	State tree depth	Allows $(5^{n})$ signups.
1	Message tree depth	Allows $(5^{n})$ votes or key-change messages.
2	Message batch tree depth	Allows $(5^{n})$ messages to be processed per batch.
3	Vote option tree depth	Allows $(5^{n})$ vote options.

Inputs

Input signal	Description
inputHash	The SHA256 hash of inputs supplied by the contract
packedVals	Described below
pollEndTimestamp	The Unix timestamp at which the poll ends
msgRoot	The root of the message tree
msgs	The batch of messages as an array of arrays
msgSubrootPathElements	Described below
coordinatorPubKeyHash	$poseidon_2([cPk_x, cPk_y])$
newSbCommitment	Described below
coordPrivKey	The coordinator's private key
coordPubKey	The coordinator's public key
encPubKeys	The public keys used to generate shared ECDH encryption keys to encrypt the messages
currentStateRoot	The state root before the commands are applied
currentStateLeaves	The state leaves upon which messages are applied
currentStateLeavesPathElements	The Merkle path to each incremental state root
currentSbCommitment	Described below
currentSbSalt	Described below
newSbCommitment	Described below
newSbSalt	Described below
currentBallotRoot	The root of the ballot tree before messages are applied
currentBallots	The ballots upon which ballots are applied
currentBallotsPathElements	The Merkle path to each incremental ballot root
currentVoteWeights	The existing vote weight for the vote option in the ballot which each command refers to
currentVoteWeightsPathElements	The Merkle path from each vote weight to the vote option root in its ballot

inputHash

All inputs to this circuit are private except for inputHash. To save gas during verification, the MessageProcessor contract hashes the following values using SHA256 and uses the hash as the sole element of $ic$:

1. packedVals
2. coordinatorPubKeyHash
3. msgRoot
4. currentSbCommitment
5. newSbCommitment
6. pollEndTimestamp

The hash is computed using the sha256 Solidity function and is then subject to modulo $p$.

packedVals

packedVals is the following values represented as one field element. Consider that a field element is roughly 253 bits. The big-endian bit-representation is as such:

Bits	Value
1st 53 bits	0
2nd 50 bits	batchEndIndex
3rd 50 bits	currentMessageBatchIndex
4th 50 bits	numSignUps
5th 50 bits	maxVoteOptions

For instance, if maxVoteOptions is 25 and batchEndIndex is 5, and all other values are 0, the following is the packedVals representation in hexadecimal:

140000000000000000000000000000000000019

currentSbCommitment and newSbCommitment

The currentSbCommitment is the $poseidon_3$ hash of the state tree root, the ballot tree root, and a random salt. The purpose of the random salt, which should be unique to each batch, is to ensure that the value of currentSbCommitment always changes even if all the commands in a batch are invalid and therefore do not change the state tree or ballot tree root.

The result of applying a batch of messages to currentSbCommitment is newSbCommitment.

currentSbSalt

The salt used to produce currentSbCommitment (see above).

newSbSalt

The salt used to produce newSbCommitment (see above).

msgSubrootPathElements

The index of each message in msgs is consecutive. As such, in order to prove that each message in msgs is indeed a leaf of the message tree, we compute the subtree root of msgs, and then verify that the subtree root is indeed a subroot of msgRoot.

A simplified example using a tree of arity 2:

             r
           /  \
          s    ...
       /    \
      o     o
     / \   / \
   a   b  c  d

To prove that a...d are leaves of the tree with root r, we prove that the leaves have the subroot s with depth 2, and then prove that s is a member of r at depth 1.

The implementation for this is in the QuinBatchLeavesExists circuit in https://github.com/privacy-scaling-explorations/maci/blob/dev/circuits/circom/trees/incrementalQuinTree.circom.

This method requires fewer circuit constraints than if we verified a Merkle proof for each leaf.

Statements that the circuit proves

1. That the prover knows the preimage to inputHash (see above)
2. That the prover knows the preimage to currentSbCommitment (that is, the state root, ballot root, and currentSbSalt)
3. That maxVoteOptions <= (5 ^ voteOptionTreeDepth)
4. That numSignUps <= (5 ^ stateTreeDepth)
5. That coordPubKey is correctly derived from coordPrivKey
6. That coordPubKey is the preimage to the Poseidon hash of coordPubKey (provided by the contract)
7. That each message in msgs exists in the message tree
8. That after decrypting and applying each message, in reverse order, to the corresponding state and ballot leaves, the new state root, new ballot root, and newSbSalt are the preimage to newSbCommitment

Tally Votes (tallyVotes)

Parameters

#	Parameter	Description
0	State tree depth	Allows $(5^{n})$ signups.
1	State leaf batch depth	Allows $(5^{n})$ users' votes to be processed per batch.
2	Vote option tree depth	Allows $(5^{n})$ vote options.

info

A version working with non quadratic voting (non-qv) is also available. This version is called tallyVotesNonQv and is to be used when the Poll is not using the quadratic voting feature. Note that by default MACI works with quadratic voting.

Input signals

Input signal	Description
inputHash	The SHA256 hash of inputs supplied by the contract
packedVals	Described below
sbCommitment	Described below
currentTallyCommitment	Described below
newTallyCommitment	Described below
stateRoot	The root of the state tree after all messages have been applied
ballotRoot	The root of the ballot tree after all messages have been applied
sbSalt	The salt used to produce sbCommitment
ballots	The ballots in the batch being tallied
ballotPathElements	The Merkle path to each ballot leaf
votes	The votes in each ballot cast per result
currentResults	The current tally of votes per vote option
currentResultsRootSalt	A random value
currentSpentVoiceCreditSubtotal	The subtotal of voice credits spent across all vote options
currentSpentVoiceCreditSubtotalSalt	A random value
currentPerVOSpentVoiceCredits	The voice credits spent on each vote option so far
currentPerVOSpentVoiceCreditsRootSalt	A random value
newResultsRootSalt	A random value
newPerVOSpentVoiceCreditsRootSalt	A random value
newSpentVoiceCreditSubtotalSalt	A random value

inputHash

All inputs to this circuit are private except for inputHash. To save gas during verification, the Tally contract hashes the following values using SHA256 and uses the hash as the sole element of $ic$:

1. packedVals
2. sbCommitment
3. currentTallyCommitment
4. newTallyCommitment

The hash is computed using the sha256 Solidity function and is then subject to modulo $p$.

packedVals

packedVals is the following values represented as one field element. Consider that a field element is roughly 253 bits. The big-endian bit-representation is as such:

Bits	Value
1st 53 bits	0
2nd 50 bits	0
3rd 50 bits	0
4th 50 bits	numSignUps
5th 50 bits	batchStartIndex

numSignUps, a value provided by the contract, is the number of users who have signed up. This is one less than the number of leaves inserted in the state tree (since the 0th state leaf is a blank state leaf). batchStartIndex is the ballot tree index at which the batch begins.

For instance, if numSignUps is 25 and the batch index is 5, and all other values are 0, the following is the packedVals representation in hexadecimal:

64000000000005

sbCommitment

The commitment to stateRoot, ballotRoot, and sbSalt:

$poseidon_3([stateRoot, ballotRoot, sbSalt])$

Proving preimage of sbCommitment is one out of the several steps required to prove that the votes were tallied correctly. By establishing that the coordinator knows ballotRoot, the coordinator can (using other parts of the circuit) prove that they know the preimage of the ballot leaves in the batch being tallied.

currentTallyCommitment and newTallyCommitment

A tally is represented by a tally commitment, which is the $poseidon_3$ hash of:

1. $tc_r$: a commitment to the votes per option
   • This is the hash of the Merkle root $r_r$ of the votes and a salt $r_s$, computed as $poseidon_2([r_r, r_s])$
2. $tc_t$: a commitment to the total spent voice credits
   • This is the hash of the total spent voice credits $t_c$ and a salt $t_s$, computed as $poseidon_2([t_c, t_s])$
3. $tc_p$: a commitment to the spent voice credits per vote option
   • This is the hash of the Merkle root of the spent voice credits per vote option $p_v$ and a salt $p_s$, computed as $poseidon_2([p_v, p_s])$

The tally commitment is computed as such:

$poseidon_3([tc_r, tc_t, tc_p])$

Statements that the circuit proves

1. That the coordinator knows the preimage of sbCommitment
2. That the coordinator knows the preimage of inputHash
3. That batchStartIndex is less than or equal to numSignUps
4. That each ballot in ballots is in a member of the ballot tree with the Merkle root ballotRoot at indices batchStartIndex to batchStartIndex + (5 ** intStateTreeDepth)
5. That each set of votes (votes[i]) has the Merkle root $blt_r$ whose value equals ballots[i][1]
6. That the tally is valid, which is:
   • That the sum of votes per vote option is correct

Subsisdy (subsidy)

This circuit is an optional feature - it's not required for MACI to function.
The subsidy circuit is used to implement pairwise subsidy. This is a technique that can be used to detect voters collusion. It currently is not optimized for production and the team will work on a more efficient implementation in the future.

Parameters

#	Parameter	Description
0	State tree depth	Allows $(5^{n})$ signups.
1	State leaf batch depth	Allows $(5^{n})$ users' votes to be processed per batch.
2	Vote option tree depth	Allows $(5^{n})$ vote options.

Utility circuits

Process Messages Input Hasher

A utility circuit used by the main processMessages circuit to hash its inputs.

It outputs one field element, which is the SHA256 hash of the following inputs:

1. packedVals
2. pollEndTimestamp
3. msgRoot
4. coordinatorPubKeyHash
5. newSbCommitment
6. currentSbCommitment

Tally Votes Input Hasher

A utility template that generates a sha256 hash of the provided tally inputs.

It outputs one field element, which is the SHA256 hash of the following inputs:

1. packedVals
2. sbCommitment
3. currentTallyCommitment
4. newTallyCommitment

ResultsCommitmentVerifier

A utility circuit used by the main tallyVotes circuit to verify that the results commitment is correct.

QuinCheckRoot

Utility circuit that given a quin Merkle root and a list of leaves, check if the root is the correct result of inserting all the leaves into the tree in the given order.

CalculateTotal

Utility circuit used to calculate the sum of an array of elements.

ECDH

Utility circuit used to generate a shared key from a private key and a public key.

Poseidon

Utility circuit used to generate a Poseidon hash. In this case, it supports up to 13 inputs.

MessageToCommand

Utility circuit used to convert a message into a command, this involves decrypting the message.

MessageValidator

Utility circuit used to validate a message. This performs several checks:

• stateTreeIndex is valid
• voteOptionIndex is valid
• nonce is valid
• the signature is valid
• the user signed up before poll end timestamp
• the user had enough voice credits

PrivToPubKey

Utility circuit used to generate a public key from a private key.

VerifySignature

Utility circuit used to verify a EdDSA signature

UnpackElement

Utility circuit used to unpack an input element.

QuinSelector

Utility circuit used to select one element from an array of n elements at a given index.

Splicer

Utility circuit used to insert one element in an array at position index.

QuinBatchLeavesExists

Utility circuit used to check if a batch of leaves exists in a quinary tree.

QuinGeneratePathIndices

Utility circuit used to generate the indices needed to traverse the tree until we find the leaf we are looking for.

ProcessTopup

Utility circuit used to process a topup message.

ProcessOne

Utility circuit used to process one message.

Compile circuits

Prerequisites

Before building the project, make sure you have the following dependencies installed:

• circom

Building MACI circuits

To build the main circuits of MACI, run the following command (-c postfix for c++ witness gen, and -wasm postfix for WASM witness gen only):

pnpm build-test-circuits-c
pnpm build-test-circuits-wasm

Please note that the circuits are configured with testing purpose parameters, which means it can only handle a limited amount of messages (up to 25 messages). For more information on the parameters and how to configure them, please refer to the individual circuit documentation within this page. Also, within the configure-circomkit section of the installation page, you'll see how you can update the config file with new params.

To compile a single circuit, you can run:

pnpm circom:build $CIRCUIT_NAME

Please note that the name should match one of the circuit names inside the circom.json file.

Generating zKeys

Run from the root directory to save to the cli/zkeys folder:

pnpm setup:zkeys

Run from the circuits folder with --outPath to save to a custom folder:

cd circuits && pnpm gen-zkeys --outPath ./CUSTOM_FOLDER_NAME

The larger the trees, the more time this process may take. You may also need a machine with a very large amount of memory.

Note that you will have to modify the parameters inside the ./circuits/circom/circuits.json file to match your use case. For example, if you want to support up to 3125 messages, the message tree depth parameter should be set to 5 (as $5^5 = 3125$).

Measure the circuit sizes

The size of a circuit is denoted by its number of constraints. The larger this number, the more time it takes to compile it, generate its .zkey file, and perform phase 2 contributions.

Run this command to measure a circuit:

pnpm exec snarkjs r1cs info CIRCUIT_NAME.circom

Download the .ptau file

This file should be the result of the Perpetual Powers of Tau trusted setup contribution which Hermez Network selected.

When running the setup:zkeys command, the .ptau file will be downloaded automatically.

Generating and Validating ZK Proofs

To generate and validate ZK proofs from the artifacts produced by circom, you will need snarkjs.

Testing

To test the circuits package, please use pnpm run test. This will run all of the tests inside the tests folder.

To run individual tests, you can use the following commands (for all other circuits please refer to the package.json scripts section):

• pnpm run test:processMessages to run the tests for the processMessages circuit.
• pnpm run test:tallyVotes to run the tests for the tallyVotes circuit.

More details on testing are provided in the testing section of the documentation.