Sum of the keys attack on taproot

We have some taproot address with private key "a" and public key "a*G", owned by Alice. Bob wants to take Alice's coins without her permission. He owns taproot address with private key "b" and public key "b*G". He knows "a*G" by exploring the chain and looking for P2TR outputs. To grab Alice's funds, he creates "(b-a)*G" taproot address and send some small amount to this address. Then, Bob can create a transaction with two inputs, taking coins from "a*G" and "(b-a)*G" addresses. All that is needed is producing a signature matching the sum of the public keys used in taproot, which is "(a+b-a)*G", reduced to "b*G", so Bob uses his "b" private key to produce Schnorr signature. Is there any protection from this attack?

On Sat, 2021-05-15 at 12:21 +0200, vjudeu via bitcoin-dev wrote:

>  All that is needed is producing a signature matching the sum of the
> public keys used in taproot, which is "(a+b-a)*G",

This is simply not true.

Taproot does not enable this, or any other form of "cross-input aggregation", i.e., spending multiple UTXOs with a single signature.

Tim

What Tim said is right. To add to that, you may also wish to read about MuSig: https://blockstream.com/2018/01/23/en-musig-key-aggregation-schnorr-signatures/

Cheers, Ruben

On Sat, May 15, 2021 at 10:32 PM Tim Ruffing via bitcoin-dev < bitcoin-dev@lists•linuxfoundation•org> wrote:

> On Sat, 2021-05-15 at 12:21 +0200, vjudeu via bitcoin-dev wrote:
>
>
> >  All that is needed is producing a signature matching the sum of the
> > public keys used in taproot, which is "(a+b-a)*G",
>
> This is simply not true.
>
> Taproot does not enable this, or any other form of "cross-input
> aggregation", i.e., spending multiple UTXOs with a single signature.
>
>
> Tim
>
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