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

["Russell O'Connor" <roconnor@blockstream.io>]

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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.

To do this Bob generates a set of random public keys P1 ... P_n of the form
P_i := P0 + r_i*G, and a bunch of random k1 ... k_n with R1 := k1*G ... R_n
:= k_n*G, such that

    h(m1, R1, P1) + ... + h(m_n, R_n, P_n) = -h(m0, R0 P0) (modulo the
order of the elliptic curve)

I understand that this can be done efficiently with Wagner's Generalized
Birthday attack.

The RHS aggregated signature equation on the private side is

    k0 + k1 + ... k_n - h(m0, R0, P0)x0 - h(m1, R1, P1)(x0 + r1) - ... -
h(m_n, R_n, P_n)(x0 + r_n)

with x0 unknown to Bob.  Rearranging the terms we get

    k0 + k1 + ... k_n - [h(m0, R0, P0) + h(m1, R1, P1) + ... + h(m_n, R_n,
P_n)]*x0 - [h(m1, R1, P1)*r1 + ... + h(m_n, R_n, P_n)*r_n]

However [h(m0, R0, P0) + h(m1, R1, P1) + ... + h(m_n, R_n, P_n)] is 0 so
cancelling that we are left with

    k0 + k1 + ... k_n - [h(m1, R1, P1)*r1 + ... + h(m_n, R_n, P_n)*r_n]

which no longer depends on the unknown value x0, so that is good.  Bob
knows what this value is.

Bob creates a set UTXOs by spending to the set of public keys P1 .. P_n.
Bob don't know what the private keys are for these public keys, but that is
going to be okay.

Bob creates a final transaction that takes as input the UTXO of Alice's
funds he wants to steal, with public key P0, and also his newly created
UTXOs with public keys P1 ... P_n.
For the signature on Alice's input he uses [R0,s0].  For the rest of the
signature he picks s1 ... s_n such that

    s0 + s1 + ... + sn = k0 + k1 + ... k_n - [h(m1, R1, P1)*r1 + ... +
h(m_n, R_n, P_n)*r_n] (which is equal to k0 + k1 + ... k_n - h(m0, R0,
P0)x0 - h(m1, R1, P1)(x0 + r1) - ... - h(m_n, R_n, P_n)(x0 + r_n)).

and uses signatures [R1, s1] ... [R_n, s_n] on his other inputs.

Thus, while none of the individual signatures are valid, the aggregated
signature does validate.

One wrinkles in this argument is that Bob needs to pick m1 ... m_n before
he knows what the transaction will be.  I think this can be mitigated by
using some combination of SIGHASH_ANYONECANPAY, but I'm not sure if that
works.  Even if my argument doesn't actually work, I think it is close
enough to be pretty scary.

Thanks goes to Pieter Wuille for helping explain things to me; however any
errors above are my own.

On Sun, May 7, 2017 at 2:45 AM, adiabat via bitcoin-dev <
bitcoin-dev@lists.linuxfoundation.org> wrote:

> If / when Schnorr signatures are deployed in a future witness version, it
> may be possible to have non-interactive partial aggregation of the
> signatures on a per-block basis.  This could save quite a bit of space.  It
> *seems* not to have any security problems but this mailing list is very
> good at finding vulnerabilities so that type of feedback is the main reason
> I'm writing :) (A quick explanation of why this is horribly broken could
> save me lots of time!)
> (also sorry if this has been discussed; didn't see anything)
>
> Quick recap / context of Schnorr sigs:
>
> There are a bunch of private keys x1, x2, x3...
> multiply by generator G to get x1G = P1, x2G = P2, x3G = P3
>
> Everyone makes their sighash m1, m2, m3, and their random nonces k1, k2,
> k3.
>
> To sign, people calculate s values:
>
> s1 = k1 - h(m1, R1, P1)x1
> s2 = k2 - h(m2, R2, P2)x2
>
> (adding the P2 into the e hash value is not in most literature /
> explanations but helps with some attacks; I beleive that's the current
> thinking.  Anyway it doesn't matter for this idea)
>
> Signature 1 is [R1, s1].  Verifiers check, given P1, m1, R1, s1:
>
> s1G =? R1 - h(m1, R1, P1)P1
>
> You can *interactively* make aggregate signatures, which requires
> co-signers to build an aggregate R value by coming up with their own k
> values, sharing their R with the co-signers, adding up the R's to get a
> summed R, and using that to sign.
>
> Non-interactively though, it seems like you can aggregate half the
> signature.  The R values are unique to the [m, P] pair, but the s's can be
> summed up:
>
> s1 + s2 = k1 + k2 - h(m1, R1, P1)x1 - h(m2, R2, P2)x2
>
> (s1 + s2)G = R1 + R2 - h(m1, R1, P1)P1 - h(m2, R2, P2)P2
>
> To use this property in Bitcoin, when making transactions, wallets can
> sign in the normal way, and the signature, consisting of [R, s] goes into
> the witness stack.  When miners generate a block, they remove the s-value
> from all compatible inputs, and commit to the aggregate s-value in the
> coinbase transaction (either in a new OP_RETURN or alongside the existing
> witness commitment structure).
>
> The obvious advatage is that signatures go down to 32 bytes each, so you
> can fit more of them in a block, and they take up less disk and network
> space.  (In IBD; if a node maintains a mempool they'll need to receive all
> the separate s-values)
>
> Another advatage is that block verification is sped up.  For individual
> signatures, the computation involves:
>
> e = h(m1, R1, P1)           <- hash function, super fast
> e*P                         <- point multiplication, slowest
> R - e*P                     <- point addidion, pretty fast
> s*G                         <- base point multiplication, pretty slow
>
> with s-aggregate verification, the first three steps are still carried out
> on each signature, but the s*G operation only needs to be done once.
> Instead another point addition per signature is needed, where you have some
> accumulator and add in the left side:
> A += R - e*P
> this can be parallelized pretty well as it's commutative.
>
> The main downside I can see (assuming this actually works) is that it's
> hard to cache signatures and quickly validate a block after it has come
> in.  It might not be as bad as it first seems, as validation given chached
> signatures looks possible without any elliptic curve operations.  Keep an
> aggregate s-value (which is a scalar) for all the txs in your mempool.
> When a block comes in, subtract all the s-values for txs not included in
> the block.  If the block includes txs you weren't aware of, request them in
> the same way compact blocks works, and get the full signature for those
> txs.  It could be several thousand operations, but those are all bigInt
> modular additions / subtractions which I believe are pretty quick in
> comparison with point additions / multiplications.
>
> There may be other complications due to the fact that the witness-txids
> change when building a block.  TXIDs don't change though so should be
> possible to keep track of things OK.
>
> Also you can't "fail fast" for the signature verification; you have to add
> everything up before you can tell if it's correct.  Probably not a big deal
> as PoW check comes first, and invalid blocks are pretty uncommon and quite
> costly.
>
> Would be interested to hear if this idea looks promising.
> Andrew Polestra mentioned something like this in the context of CT /
> mimblewimble transactions a while ago, but it seems it may be applicable to
> regular bitcoin Schnorr txs.
>
> -Tadge
>
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>

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