Here are step-by-step description of two multi-signature schemes: MuSig: n-of-n Multi-Signature Scheme and t-of-n Threshold Signature Scheme
MuSig is effectively a multi-signature and key-aggregation scheme based on Schnorr Signatures. It provides security in plain public key model.
A working full round typescript example can be found here
Note: all codes are actually pseudocode just for illustrations purpose. Available functions can be found in typescript example
A step-by-step description is as follows:
User generates public-private keypair and sends public key to broker
private_key, public_key = generate_keypair();
for i in len(all_users)
send_to_user(public_key, all_users[i])
end for
Once the receives other parties' public keys, user computes and stores aggregated public key
let all_pubkeys = [pubkey_1, .., pubkey_n]
let aggregated_pubkey = compute_aggregated_pubkey(all_pubkeys)
// store aggregated_pubkey
User generates crytographically secure nonce
nonce = generate_nonce()
User commits his nonce
commitment = compute_commitment(nonce)
User hashes commitment
pre_commitment = compute_pre_commitment(commitment)
User sends pre-commitment to other users through broker
for i in len(all_users)
send_to_user(pre_commitment, all_users[i])
end for
Once user receive the pre_commitment, user reveals his commitment to other users through broker
for i in len(all_users)
send_to_user(commitment, all_users[i])
end for
Once user receives commitments from other parties, user computes aggregated commitment and stores it
let all_commitments = [commitment from user 1, ... commitment from user n]
let aggregated_commitment = compute_aggregated_commitment(all_commitments)
Note: By using Pre-Shared Commitments, number of rounds can be decreased by one
Initiator prepares transaction and sends it to other users through broker
let transaction = {from: 0xAA, to: 0xBB, ..}
let encoded_transaction = encode_transaction(transaction)
for i in len(all_users)
send_to_user(encoded_transaction, all_users[i])
end for
Once the user receives transaction signing message from initiator, user signs transaction with his private key and nonce and sends signature share to other users through broker
let transaction = message.transaction
let signature_share = sign(private_key, nonce, all_pubkeys, commitment, aggregated_commitment)
let signing_result = {
signature_shares: signature_share,
commitment: commitment,
}
for i in len(all_users)
send_to_user(signing_result, all_users[i])
end for
Once the user receives all signing results from each user, user aggregates signature shares
let all_signature_shares = [signature share from user 1, .. , signature share from user n]
let aggregated_signature = compute_aggregated_signature(all_signature_shares)
let final_signature = {
r: aggregated_commitment,
s: aggregated_signature
}
User sends final signature to the broker
send_to_broker(final_signature)
Broker submits transaction to the network if each received signature equal and valid
let all_final_signatures = [final signature from user 1, .. , final signature from user n]
for i in len(all_final_signatures)
if i == 0
continue
end if
if all_final_signatures[i-1] != all_final_signatures[i]
exit
end if
end for
if !is_signature_valid(all_final_signatures[0]) // verification of single sig is enough
exit
end if
submit_transaction(transaction, final_signature)
The threshold signing algorithm is an signature aggregation scheme based on MuSig Schnorr signatures and the EdDSA signing procedure. Signatures are non-deterministic though as they include random participant commitments.
In order to achieve threshold, original private key essentially splitted into n secret shares and each user needs to have t permuatation of them.
For example, for a 3-of-4 multisig (t=3, N=4), the permuations of size t-1=2 are:
[1, 2], [1, 3], [1, 4], [2, 3], [2, 4], [3, 4]
Every user computes the key distribution as follows:
- For each permutation, a new private key share must be assigned to all users who are not part of this permutation:
[1, 2]: A => 3, 4
[1, 3]: B => 2, 4
[1, 4]: C => 2, 3
[2, 3]: D => 1, 4
[2, 4]: E => 1, 3
[3, 4]: F => 1, 2
- For every secret share, one user from the user subset sharing it (the one with the lowest position number) must generate it and share with the others to whom it should be assigned. In our example:
Share A: generated by user 3, shared with user 4.
Share B: generated by user 2, shared with user 4.
Share C: generated by user 2, shared with user 3.
Share D: generated by user 1, shared with user 4.
Share E: generated by user 1, shared with user 3.
Share F: generated by user 1, shared with user 2.
- So, each user has following secret shares
user 1 has [D, E, F]
user 2 has [B, C, F]
user 3 has [A, C, E]
user 4 has [A, B, D]
- At the end any combination of three users can construct original private key
user 1 + user 2 + user 3 = [A, B, C, C, D, E, E, F, F]
user 1 + user 3 + user 4 = [A, A, B, C, D, D, E, E, F]
user 2 + user 3 + user 4 = [A, A, B, B, C, C, D, E, F]
- Finally any combination of two users can not construct original private key
user 1 + user 2 = [B, C, D, E, F, F] missing [A]
user 1 + user 3 = [A, C, D, E, E, F] missing [B]
user 1 + user 4 = [A, B, D, D, E, F] missing [C]
user 2 + user 3 = [A, B, C, C, E, F] missing [D]
user 2 + user 4 = [A, B, B, C, D, F] missing [E]
user 3 + user 4 = [A, A, B, C, D, E] missing [F]
A full round example for t-of-n scheme will be implemented in typescript
Every user contributes computation of the key distribution as follows:
User generates the list of all possible permutations of users of size t-1 which he needs to generate the share.
let permutations = generate_permutations(n, t, user_index)
// permutations : [1,3] => [2,4], [1,4] => [2,3]
User generates private keys and corresponding public keys for each remaining permutation generated in the first step
let receiving_users = []
for p in permutations
receiving_users.push(p[1])
end for
// receiving_users: 4, 3
let private_keys = []
let public_keys = []
for user in receiving_users
private_key, public_key = generate_keypair();
private_keys.push(private_key)
public_keys.push(public_key)
end for
User sends each private key to user who needs that share
for i in len(receiving_users)
send_to_user(private_keys[i], receiving_users[i])
end for
User sends each public key to all users and broker
for i in len(all_users)
if i != user_index
send_to_user(public_keys[i], all_users[i])
end if
end for
send_to_broker(public_keys)
Once the user receives public keys, user makes a unique list of public keys and stores computed aggregated public key from that unique list.
let pubkey_list = unique(all_received_pubkeys)
aggregated_pubkey = compute_aggregated_pubkey(pubkey_list)
// store aggregated_pubkey
Once broker receives all public keys, broker computes ethereum address and creates corresponding zksync account
let ethereum_address = compute_eth_address(..) // !!
create_zksync_account(ethereum_address, aggregated_pubkey)
Having to go through the three rounds every time we want to sign a message is certainly inconvenient. Especially if the signature is time-critical and network messages are slow. So how about this, we do the first two rounds whenever it’s convenient for us. And only once there is a message to sign, we exchange partial signatures. For example, the user of multisig exchange a bunch of (k) nonce commitments and pre-commitments immediately when establishing a connection. When it comes to signing a transaction, one of the pre-shared nonces is used and only one communication round is required to complete the signature.
User generates k cryptographically secure nonces and computes commitment and pre-commitment for each nonce. (k is the number of desired transactions)
let nonces = []
let commitments = []
let pre_commitments = []
for i in range(k)
nonce = generate_nonce()
commitment = compute_commitment(nonce)
pre_commitment = compute_pre_commitment(commitment)
nonces.push(nonce)
commitments.push(commitment)
pre_commitments.push(pre_commitment)
end for
User sends each pre-commitments to all users
for user in all_users
send_to_user(pre_commitments, user)
end for
Once the user receive pre-commitments from all other parties, user stores received pre-commitments
let received_pre_commitments = [[pre_commitments from user 1],..,[pre_commitments from user n]]
store_pre_commitments(received_pre_commitments)
User sends each commitments to broker
send_to_broker(commitments)
Once the broker receives commitments, broker stores each list received from each user.
let received_commitments = [[commitments from user 1],..,[commitments from user n]]
// store received_commitments
Initiator prepares transaction and sends it to broker
let transaction = {from: 0xAA, to: 0xBB, ..}
let encoded_transaction = encode_transaction(transaction)
send_to_broker(encoded_transaction)
Once the broker receive transaction signing message from initiator, broker queries nonce of multisig account from zksync
let nonce = query_nonce_from_zksync(ethereum_address);
Broker picks next commitment from each subset
let current_commitments = []
for subset in received_commitments
current_commitments.push(subset[nonce])
end for
Broker queries latest block hash from ethereum network
let latest_block_hash = query_latest_block_hash()
Broker constructs message and sends it to all users
message = {
transaction: encoded_transaction,
latest_block_hash: latest_block_hash,
commitment_list: current_commitments,
nonce: nonce,
}
for user in all_users
send_to_user(message, user)
end for
Once the user receives transaction signing message from broker, user picks next pre-commitment from each subset
let current_pre_commitments = []
for subset in received_pre_commitments
current_pre_commitments.push(subset[nonce])
end for
User compares received commitments with previously received pre-commitments
for i in len(received_commitments)
if received_commitments[i] != current_pre_commitments[i]
exit
end if
end for
User computes aggregated commitment
let aggregated_commitment = compute_aggregated_commitment(received_commitments)
User signs transaction with each private key and sends them to the broker
let transaction = message.transaction
let latest_block_hash = message.latest_block_hash
let signature_shares = []
let public_keys = []
for private_key in private_keys
let signature_share = sign(private_key, pubkey_list, aggregated_commitment, transaction, latest_block_hash)
signature_shares.push(signature_share)
let public_key = compute_pubkey(private_key)
public_keys.push(public_key)
end for
let signing_result = {
signature_shares: signature_shares,
public_keys: public_keys,
aggregated_commitment: aggregated_commitment,
}
// send signing_result to broker
send_to_broker(signing_result)
Once the broker receives all signing results from each user, broker checks that each received aggregated commitments are same
for i in len(messages)
if i == 0
continue
end if
let previous_aggregated_commitment = messages[i-1].aggregated_commitment
let current_aggregated_commitment = messages[i].aggregated_commitment
if previous_aggregated_commitment != current_aggregated_commitment
exit
end if
end for
let aggregated_commitment = messages[0].aggregated_commitment
Broker makes a unique list of received signatures by comparing public keys
let current_public_keys = []
let current_signature_shares = []
for message in messages
for i in len(signature_shares)
if !current_public_keys.has(public_keys[i])
current_signature_shares.push(signature_shares[i])
end if
end for
end for
Broker aggregates signature shares
let aggregated_signature = compute_aggregated_signature(current_signature_shares)
let final_signature = {
r: aggregated_commitment,
s: aggregated_signature
}
Broker completes ceremony by submitting transaction to the zksync if signature is valid
if !is_signature_valid(final_signature)
exit
end if
submit_transaction(transaction, final_signature)
A working full round typescript example in a e2e fashion can be found here This example illustrates only n-of-n scheme. (It is not for t-of-n scheme!)
musig-bindings directory contains generated wasm-code. Rust sources can be found in here
cd wasm/typescript-example
yarn -D && yarn build
yarn test
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