Re: [bitcoin-dev] Per-block non-interactive Schnorr signature aggregation

[Andrew Poelstra <apoelstra@wpsoftware.net>]

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On Tue, May 09, 2017 at 09:59:06PM -0400, Russell O'Connor via bitcoin-dev wrote:
> I'm a bit amateur at this sort of thing, but let me try to argue that this
> proposal is in fact horribly broken ;)
> 
> Suppose Alice has some UTXO with some money Bob wants to steal.  Grant me
> that the public key P0 protecting Alice's UTXO is public (say because the
> public key has been reused elsewhere).
> 
> Bob going to spend Alice's UTXO by generating random values s0, k0 and R0
> := k0*G and thus creating a random signature for it, [R0, s0].  Now clearly
> this signature isn't going to be valid by itself because it is just random.
> Bob's goal will be to make a transaction with other inputs such that, while
> the individual signatures are not valid, the aggregated signature will be
> valid.
>

If you seed the randomization with every R value (which would come for free
if you used, say, the witness root) then Wagner's attack no longer applies.

The idea is that no aggregation occurs until a miner produces a block. You
have a bunch of independent Schnorr sigs (s_i, R_i). Then the _miner_ multiples
each s_i by H(witness root || index) or whatever, sums up the s_i's, and commits
the sum somewhere where it doesn't affect the root.

Verifiers then multiply each R_i by the same multiplying factors and are able
to do a batch verification of them.


Verifiers who have seen a signature before and cached it as valid can save
themselves a bit of time by subtracting H(witness root || index)*s_i from
the summed s-value and then skipping R_i in the above step. These are scalar
operations and are extremely cheap.

They can recognize the signature given only the transaction it signs and R_i,
which uniquely determine a valid signature.


I believe this is what Tadge was referring to when he mentioned a talk of mine.
It's roughly what I've had in mind whenever I talk about non-interactive Schnorr
aggregation.



Cheers
Andrew


-- 
Andrew Poelstra
Mathematics Department, Blockstream
Email: apoelstra at wpsoftware.net
Web:   https://www.wpsoftware.net/andrew

"A goose alone, I suppose, can know the loneliness of geese
 who can never find their peace,
 whether north or south or west or east"
       --Joanna Newsom


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