Good morning Lloyd,
> The real
> question is what properties does the commitment scheme need to be
> appropriate for MuSig R coin tossing?
It seems to me that what is needed for a composable MuSig is to have a
commitment scheme which is composable.
Let me define a composable commitment scheme:
Given:
* `A` and `B`, two points to be committed to.
* `c[A]` and `c[B]`, commitments to the above points respectively.
* `r[A]` and `r[B]`, openings of the above commitments respectively.
Then a composable commitment scheme must have these operations:
* `ComposeCommitments(c[A], c[B])`, which returns a commitment to the point `A
+ B`.
* `ComposeOpenings(r[A], r[B])`, which returns an opening of the above
commitment to the point `A + B`.
My multi-`R` proposal is a composable commitment scheme:
* A commitment `c[A]` is the list `{h(A)}` where `h()` is some hash function.
* `ComposeCommitments(c[A], c[B])` is the concatenation on lists of hashes of
points.
* An opening `r[A]` is the list `{A}`.
* `ComposeOpenings(r[A], r[B])` is the concatenation on lists of points.
The property we want to have, is that:
* There must not exist some operation `NegateCommitment(c[A])`, such that:
* `ComposeCommitments(ComposeCommitments(c[B], NegateCommitment(c[A])), c[A])
== c[B]`.
My multi-`R` proposal works as a composable commitment scheme appropriate for
composable MuSig because there is no way to create an input to a concatenation
operation such that the concatenation operation becomes a "search and delete"
operation.
Pedersen and ElGamal commitments, I think, cannot work here, because
commitments in those schemes are negatable in such a way that composing the
commitments allows a commitment to be cancelled.
-----
Let us now turn to signature schemes.
I conjecture that the Schnorr and ECDSA signature schemes are also commitment
schemes on points.
To create a commitment `c[A]` on the point A, such that `A = a * G`, the
committer:
* Generates random scalars `r` and `m`.
* Computes `R` as `r * G`.
* Computes `s` as `r + h(R | m) * a`.
* Gives `c[A]` as the tuple `(R, s)`.
The opening `r[A]` of the above is then the tuple `(m, A)`.
The verifier then validates that the commitment was indeed to the point `A` by
doing the below:
* Computes `S[validator]` as `R + h(R | m) * A`.
* Validates that `S[validator] == s * G`.
Now, we know that signatures can be composed in such a way that points (public
keys) cannot be cancelled, i.e. preventing the creation of a
`NegateCommitment()` operation.
Thus, a signature can be used as a composable commitment in composable MuSig
scheme.
In summary, I conjecture that:
* We need a composable commitment scheme that does not allow cancellation, and
any such commitment scheme can be "slotted" into the 3-phase MuSig framework.
Regards,
ZmnSCPxj
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