On Tue, Mar 16, 2021 at 03:10:21PM +0100, Andrea via bitcoin-dev wrote:
> 
> Hi! Sorry for the OT, could you provide some references to ring signatures
> over/for/via taproot (I mean the schema or something like that)? And what is
> "Provisions" (the capital letter makes me think it's a product/technology)?
> I'm a rookie following this mailing since just a few months...
>

Thanks for posting such a positive message in an otherwise tense thread :)

Provisions is a scheme for providing proof of ownership of funds, developed
by Dagher et al in 2015 at https://eprint.iacr.org/2015/1008 . The way it
works is to collect all of the Bitcoin outputs which have exposed/known
public keys then associate to these keys a Pedersen commitment which commits
to the outputs' amounts in a homomorphic way.

Homomorphic means that even though the commitments hide what the original
amounts are, anyone can add them together (in some sense) to get a new
commitment to the sum of the original amounts.

So Provisions is essentially a zero-knowledge proof of the following statement

    1. I have a commitment to >100BTC (or whatever)...
    2. ...which is a sum of commitments of actual UTXO values...
    3. ...where these UTXOs come from the set of known-public-key UTXOs...
    4. ...and I am able to sign with the public keys associated to them.

which proves ownership of some amount of BTC, without revealing which specific
UTXOs were involved. This zero-knowledge proof can be done fairly efficiently
by exploiting the structure of EC public keys and Pedersen commitments.


Unfortunately, most unspent Bitcoin outputs do not have known public keys,
which means that you can only do a Provisions proof using a small anonymity
set. However, all Taproot outputs, by virtue of having exposed public keys
(which is the point under contention in this thread), will be in the set of
exposed-public-key UTXOs, allowing people to do Provisions proofs where
their anonymity set consists of a large proportion of active coins.


BTW, even without Provisions, there are some similar and simpler things you
can do with Taproot keys along these lines. See for example
https://twitter.com/n1ckler/status/1334240709814136833



-- 
Andrew Poelstra
Director of Research, Blockstream
Email: apoelstra at wpsoftware.net
Web:   https://www.wpsoftware.net/andrew

The sun is always shining in space
    -Justin Lewis-Webster