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src/modules/musig/musig.md
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@ -12,7 +12,7 @@ API in ways that could lead to accidental signatures or loss of key material.
In MuSig all participants contribute key material to a single signing key, In MuSig all participants contribute key material to a single signing key,
using the equation using the equation
P = sum_i µ_i * P_i P = sum_i µ_i - P_i
where `P_i` is the public key of the `i`th participant and `µ_i` is a so-called where `P_i` is the public key of the `i`th participant and `µ_i` is a so-called
_MuSig coefficient_ computed according to the following equation _MuSig coefficient_ computed according to the following equation
@ -25,18 +25,18 @@ where H is a hash function modelled as a random oracle.
To produce a multisignature `(s, R)` on a message `m` using verification key To produce a multisignature `(s, R)` on a message `m` using verification key
`P`, signers act as follows: `P`, signers act as follows:
1. Each computes a nonce, or ephemeral keypair, `(k_i, R_i)`. Every participant 1. Each computes a nonce, or ephemeral keypair, `(k_i, R_i)`. Every participant
communinicates `H(R_i)` to every other participant. communinicates `H(R_i)` to every other participant.
2. Upon receipt of every `H(R_i)`, each participant communicates `R_i` to every 2. Upon receipt of every `H(R_i)`, each participant communicates `R_i` to every
other participant. The recipients check that each `R_i` is consistent with other participant. The recipients check that each `R_i` is consistent with
the previously-communicated hash. the previously-communicated hash.
3. Each participant computes a combined nonce 3. Each participant computes a combined nonce
R = sum_i R_i R = sum_i R_i
and shared challenge and shared challenge
e = H(R || P || m) e = H(R || P || m)
and partial signature and partial signature
s_i = k_i + µ_i*x_i*e s_i = k_i + µ_i*x_i*e
where `x_i` is the secret key corresponding to `P_i`. where `x_i` is the secret key corresponding to `P_i`.
The complete signature is then the `(s, R)` where `s = sum_i s_i` and `R = sum_i R_i`. The complete signature is then the `(s, R)` where `s = sum_i s_i` and `R = sum_i R_i`.
@ -72,59 +72,59 @@ signature process, which is also a supported mode) acts as follows.
### Signing Participant ### Signing Participant
1. The signer starts the session by calling `secp256k1_musig_session_initialize`. 1. The signer starts the session by calling `secp256k1_musig_session_initialize`.
This function outputs This function outputs
* an initialized session state in the out-pointer `session` - an initialized session state in the out-pointer `session`
* an array of initialized signer data in the out-pointer `signers` - an array of initialized signer data in the out-pointer `signers`
* a commitment `H(R_i)` to a nonce in the out-pointer `nonce_commitment32` - a commitment `H(R_i)` to a nonce in the out-pointer `nonce_commitment32`
It takes as input It takes as input
* a unique session ID `session_id32` - a unique session ID `session_id32`
* (optionally) a message to be signed `msg32` - (optionally) a message to be signed `msg32`
* the combined public key output from `secp256k1_musig_pubkey_combine` - the combined public key output from `secp256k1_musig_pubkey_combine`
* the public key hash output from `secp256k1_musig_pubkey_combine` - the public key hash output from `secp256k1_musig_pubkey_combine`
* the signer's index `i` `my_index` - the signer's index `i` `my_index`
* the signer's secret key `seckey` - the signer's secret key `seckey`
2. The signer then communicates `H(R_i)` to all other signers, and receives 2. The signer then communicates `H(R_i)` to all other signers, and receives
commitments `H(R_j)` from all other signers `j`. These hashes are simply commitments `H(R_j)` from all other signers `j`. These hashes are simply
length-32 byte arrays which can be communicated however is communicated. length-32 byte arrays which can be communicated however is communicated.
3. Once all signers nonce commitments have been received, the signer records 3. Once all signers nonce commitments have been received, the signer records
these commitments with the function `secp256k1_musig_session_get_public_nonce`. these commitments with the function `secp256k1_musig_session_get_public_nonce`.
This function updates in place This function updates in place
* the session state `session` - the session state `session`
* the array of signer data `signers` - the array of signer data `signers`
taking in as input the list of commitments `commitments` and outputting the taking in as input the list of commitments `commitments` and outputting the
signer's public nonce `R_i` in the out-pointer `nonce`. signer's public nonce `R_i` in the out-pointer `nonce`.
4. The signer then communicates `R_i` to all other signers, and receives `R_j` 4. The signer then communicates `R_i` to all other signers, and receives `R_j`
from each signer `j`. On receipt of a nonce `R_j` he calls the function from each signer `j`. On receipt of a nonce `R_j` he calls the function
`secp256k1_musig_set_nonce` to record this fact. This function checks that `secp256k1_musig_set_nonce` to record this fact. This function checks that
the received nonce is consistent with the previously-received nonce and will the received nonce is consistent with the previously-received nonce and will
return 0 in this case. The signer must also call this function with his own return 0 in this case. The signer must also call this function with his own
nonce and his own index `i`. nonce and his own index `i`.
These nonces `R_i` are secp256k1 public keys; they should be serialized using These nonces `R_i` are secp256k1 public keys; they should be serialized using
`secp256k1_ec_pubkey_serialize` and parsed with `secp256k1_ec_pubkey_parse`. `secp256k1_ec_pubkey_serialize` and parsed with `secp256k1_ec_pubkey_parse`.
5. Once all nonces have been exchanged in this way, signers are able to compute 5. Once all nonces have been exchanged in this way, signers are able to compute
their partial signatures. They do so by calling `secp256k1_musig_session_combine_nonces` their partial signatures. They do so by calling `secp256k1_musig_session_combine_nonces`
which updates in place which updates in place
* the session state `session` - the session state `session`
* the array of signer data `signers` - the array of signer data `signers`
It outputs an auxilary integer `nonce_is_negated` and has an auxilary input It outputs an auxilary integer `nonce_is_negated` and has an auxilary input
`adaptor`. Both of these may be set to NULL for ordinary signing purposes. `adaptor`. Both of these may be set to NULL for ordinary signing purposes.
If the signer did not provide a message to `secp256k1_musig_session_initialize`, If the signer did not provide a message to `secp256k1_musig_session_initialize`,
a message must be provided now by calling `secp256k1_musig_session_set_msg` which a message must be provided now by calling `secp256k1_musig_session_set_msg` which
updates the session state in place. updates the session state in place.
6. The signer computes a partial signature `s_i` using the function 6. The signer computes a partial signature `s_i` using the function
`secp256k1_musig_partial_sign` which takes the session state as input and `secp256k1_musig_partial_sign` which takes the session state as input and
partial signature as output. partial signature as output.
7. The signer then communicates the partial signature `s_i` to all other signers, or 7. The signer then communicates the partial signature `s_i` to all other signers, or
to a central coordinator. These partial signatures should be serialized using to a central coordinator. These partial signatures should be serialized using
`musig_partial_signature_serialize` and parsed using `musig_partial_signature_parse`. `musig_partial_signature_serialize` and parsed using `musig_partial_signature_parse`.
8. Each signer calls `secp256k1_musig_partial_sig_verify` on the other signers' partial 8. Each signer calls `secp256k1_musig_partial_sig_verify` on the other signers' partial
signatures to verify their correctness. If only the validity of the final signature signatures to verify their correctness. If only the validity of the final signature
is important, not assigning blame, this step can be skipped. is important, not assigning blame, this step can be skipped.
9. Any signer, or central coordinator, may combine the partial signatures to obtain 9. Any signer, or central coordinator, may combine the partial signatures to obtain
a complete signature using `secp256k1_musig_partial_sig_combine`. This function takes a complete signature using `secp256k1_musig_partial_sig_combine`. This function takes
a signing session and array of MuSig partial signatures, and outputs a single a signing session and array of MuSig partial signatures, and outputs a single
Schnorr signature. Schnorr signature.
### Non-signing Participant ### Non-signing Participant
@ -132,12 +132,12 @@ A participant who wants to verify the signing process, i.e. check that nonce com
are consistent and partial signatures are correct without contributing a partial signature, are consistent and partial signatures are correct without contributing a partial signature,
may do so using the above instructions except for the following changes: may do so using the above instructions except for the following changes:
1. A signing session should be produced using `musig_session_initialize_verifier` 1. A signing session should be produced using `musig_session_initialize_verifier`
rather than `musig_session_initialize`; this function takes no secret data or rather than `musig_session_initialize`; this function takes no secret data or
signer index. signer index.
2. The participant receives nonce commitments, public nonces and partial signatures, 2. The participant receives nonce commitments, public nonces and partial signatures,
but does not produce these values. Therefore `secp256k1_musig_session_get_public_nonce` but does not produce these values. Therefore `secp256k1_musig_session_get_public_nonce`
and `secp256k1_musig_partial_sign` are not called. and `secp256k1_musig_partial_sign` are not called.
### Verifier ### Verifier
@ -149,12 +149,12 @@ public key output from `secp256k1_musig_pubkey_combine`.
The signing API supports the production of "adaptor signatures", modified partial signatures The signing API supports the production of "adaptor signatures", modified partial signatures
which are offset by an auxiliary secret known to one party. That is, which are offset by an auxiliary secret known to one party. That is,
1. One party generates a (secret) adaptor `t` with corresponding (public) adaptor `T = t*G`. 1. One party generates a (secret) adaptor `t` with corresponding (public) adaptor `T = t*G`.
2. When combining nonces, each party adds `T` to the total nonce used in the signature. 2. When combining nonces, each party adds `T` to the total nonce used in the signature.
3. The party who knows `t` must "adapt" their partial signature with `t` to complete the 3. The party who knows `t` must "adapt" their partial signature with `t` to complete the
signature. signature.
4. Any party who sees both the final signature and the original partial signatures 4. Any party who sees both the final signature and the original partial signatures
can compute `t`. can compute `t`.
Using these adaptor signatures, two 2-of-2 MuSig signing protocols can be executed in Using these adaptor signatures, two 2-of-2 MuSig signing protocols can be executed in
parallel such that one party's partial signatures are made atomic. That is, when the other parallel such that one party's partial signatures are made atomic. That is, when the other
@ -165,15 +165,15 @@ Such a protocol can be executed as follows. Consider two participants, Alice and
are simultaneously producing 2-of-2 multisignatures for two blockchains A and B. They act are simultaneously producing 2-of-2 multisignatures for two blockchains A and B. They act
as follows. as follows.
1. Before the protocol begins, Bob chooses a 32-byte auxiliary secret `t` at random and 1. Before the protocol begins, Bob chooses a 32-byte auxiliary secret `t` at random and
computes a corresponding public point `T` by calling `secp256k1_ec_pubkey_create`. computes a corresponding public point `T` by calling `secp256k1_ec_pubkey_create`.
He communicates `T` to Alice. He communicates `T` to Alice.
2. Together, the parties execute steps 1-4 of the signing protocol above. 2. Together, the parties execute steps 1-4 of the signing protocol above.
3. At step 5, when combining the two parties' public nonces, both parties call 3. At step 5, when combining the two parties' public nonces, both parties call
`secp256k1_musig_session_combine_nonces` with `adaptor` set to `T` and `nonce_is_negated` `secp256k1_musig_session_combine_nonces` with `adaptor` set to `T` and `nonce_is_negated`
set to a non-NULL pointer to int. set to a non-NULL pointer to int.
4. Steps 6 and 7 proceed as before. Step 8, verifying the partial signatures, is now 4. Steps 6 and 7 proceed as before. Step 8, verifying the partial signatures, is now
essential to the security of the protocol and must not be omitted! essential to the security of the protocol and must not be omitted!
The above steps are executed identically for both signing sessions. However, step 9 will The above steps are executed identically for both signing sessions. However, step 9 will
not work as before, since the partial signatures will not add up to a valid total signature. not work as before, since the partial signatures will not add up to a valid total signature.
@ -181,16 +181,16 @@ Additional steps must be taken, and it is at this point that the two signing ses
diverge. From here on we consider "Session A" which benefits Alice (e.g. which sends her diverge. From here on we consider "Session A" which benefits Alice (e.g. which sends her
coins) and "Session B" which benefits Bob (e.g. which sends him coins). coins) and "Session B" which benefits Bob (e.g. which sends him coins).
5. In Session B, Bob calls `secp256k1_musig_partial_sig_adapt` with his partial signature 5. In Session B, Bob calls `secp256k1_musig_partial_sig_adapt` with his partial signature
and `t`, to produce an adaptor signature. He can then call `secp256k1_musig_partial_sig_combine` and `t`, to produce an adaptor signature. He can then call `secp256k1_musig_partial_sig_combine`
with this adaptor signature and Alice's partial signature, to produce a complete with this adaptor signature and Alice's partial signature, to produce a complete
signature for blockchain B. signature for blockchain B.
6. Alice reads this signature from blockchain B. She calls `secp256k1_musig_extract_secret_adaptor`, 6. Alice reads this signature from blockchain B. She calls `secp256k1_musig_extract_secret_adaptor`,
passing the complete signature along with her and Bob's partial signatures from Session B. passing the complete signature along with her and Bob's partial signatures from Session B.
This function outputs `t`, which until this point was only known to Bob. This function outputs `t`, which until this point was only known to Bob.
7. In Session A, Alice is now able to replicate Bob's action, calling 7. In Session A, Alice is now able to replicate Bob's action, calling
`secp256k1_musig_partial_sig_adapt` with her own partial signature and `t`, ultimately `secp256k1_musig_partial_sig_adapt` with her own partial signature and `t`, ultimately
producing a complete signature on blockchain A. producing a complete signature on blockchain A.
[1] https://eprint.iacr.org/2018/068 [1] https://eprint.iacr.org/2018/068