[dlc-dev] Oracle attestations can be simplified and be more much more efficient

[Nadav Kohen]

Hey Jesse,

I think the crux is that R is already committed to in the oracle
announcement and is used in the construction of DLCs so that we don't run
the risk of forgery as the forger must use the pre-committed R. Have you
had a chance to look at
https://github.com/LLFourn/dlc-sec/blob/master/main.pdf ? It was a fun read
:)

Also in case this is relevant, we will likely be requiring proofs of
knowledge on the pre-committed Rs in the oracle announcements as it is
required for the security of the multi-oracle scheme if I understand
correctly (and I believe Lloyd is working on formalizing that).

Best,
Nadav

On Mon, Mar 1, 2021 at 4:48 PM Jesse Posner via dlc-dev <
dlc-dev at mailmanlists.org> wrote:

> Hi Lloyd,
>
> If I'm understanding the scheme correctly, I think there might be a
> forgery issue.
>
> When c_i is the result of a hash function that includes R, then R is
> committed to prior to the computation of s. If c_i = i then this is no
> longer the case. This would allow the oracle to produce a valid s for c_i
> without knowledge of x. To do this, the oracle first chooses a random
> scalar s prior to computing R. Then the oracle computes R = (s * G) - (c_i
> * X).
>
> This will only work for a single event (e.g. c_i = 2) that relates to R.
> However, it seems like the anticipation scheme should have the security
> property that s is a proof of knowledge of x for all c_i.
>
> All the best,
>
> Jesse
>
>
>
> On Tue, Dec 29, 2020 at 1:23 PM Lloyd Fournier <lloyd.fourn at gmail.com>
> wrote:
>
> > Merry Christmas list,
> >
> > A Schnorr signature has the form (R, s) where
> > 1. R = r *G for some secret nonce r
> > 2. s = r + c*x where c = H(R || X || m) for a message m and some hash
> > function H
> >
> > The original DLC paper suggests re-using this structure but with R
> > published upfront (in what we call the *announcement*) to create a list
> of
> > anticipation points computed like P = R + c_i * X where c_i = H(R || X ||
> > m_i) where m_i indicates the i'th possible outcome of the event.
> >
> > I make the claim that c_i need not be the result of a hash function. c_i
> > can instead simply be set to i. So to attest to the 3rd outcome set c_i
> = 2
> > and the oracle reveals s = r + 2 * x.
> >
> > == Security
> >
> > The only security property required for the point anticipation scheme is
> > that (i) it reveals no information other than the list of points (zero
> > knowledge) (ii) if the oracle reveals the secrets for two points then
> they
> > lose their secret key (no equivocation).
> > Clearly (i) holds since R is still just a random point.
> > (ii) holds for the same reason as in the original -- two different c_i
> and
> > c_j reveals x = (s_i - s_j) / (c_i - c_j). The change from hash of
> message
> > to index is irrelevant.
> >
> > == Impact
> >
> > The main specification change would be to make outcome descriptors
> > describe an ordered list of outcomes rather than a set of strings. e.g.
> the
> > enumeration descriptor would simply assign the index to each outcome.
> >
> > The benefits of applying the above idea are:
> > 1. It would reduce the time taken to compute the list of anticipation
> > points to ~1/12 of the time currently taken since you can avoid doing
> any
> > ec multiplications. You can incrementally add X to each previous value
> > (starting with R) to get the next anticipation point. In the current
> > protocol you have to verify ECDSA adaptor signatures so this would still
> > lead to a time reduction of 1/2 to 1/3 since it removes 2/3 of the
> ecmults
> > (depends a lot on implementation).
> > 2. In the future you could capture the full speedup by not doing adaptor
> > signatures and just putting each outcome in its own Taproot branch and
> > using OP_CTV to enforce the outcome.
> > 3. It removes arbitrary structure from the scheme and is easier to
> specify
> > and implement (no need to talk about tagged hashes or to have a BIP340
> > implementation).
> > 4. It gets rid of all concerns about utf8 and message encoding since
> > outcomes are represented by integers.
> >
> > The downsides are mostly rhetorical:
> > 1. We couldn't talk about "signing the outcome" because that's not what
> > we're doing anymore (actually we were never really doing that but it was
> at
> > least not too much of a lie since there was at least a signature at the
> end
> > of it). I don't have an idea about the best way to talk about it yet.
> > 2. The oracle's attestation could no longer provide useful information
> by
> > itself. I liked the idea that systems who didn't care about DLCs could
> > still use the oracles to get info about the world just from the signature
> > and the message it signed. You can still use oracles in this way here
> but
> > you *must* have the announcement with the attestation data to figure out
> > what was attested to.
> >
> > Let me know your thoughts and hope you have a good break.
> >
> > Cheers,
> >
> > LL
> >
>
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