Yes, one validation per block, but not one signature per block.
To clear up confusion, there are 3 distinct technologies involved here:
- (1) non-interactive aggregation is the ability for a third party (who does not hold any private keys) to combine multiple signatures, each with their own message and public key, into a single signature that can be verified by someone knows all messages and public keys.
- (2) interactive aggregation is the same, but when the signers need to be aware of the aggregation, and communicate with each other to jointly produce a single signature.
- (3) batch validation is the ability for a verifier to verify whether multiple (pubkey,message,signature) tuples are all valid or not, faster than verifying the individual signatures. If one or more of the tuples are invalid the verifier will not learn which ones, in this case.
Schnorr signatures (and any other known discrete logarithm based signature schemes) support (2) and (3), but not (1).
The lack of (1) means there cannot be a single signature for an entire block (*), as the miner who constructs the block is a third party who isn't participating in signature creation.
Due to (2), the best we can hope for (as long as we're restricted to DL-based signatures), is one signature per transaction. Even that requires cross-input aggregation, which has complexities beyond just the implementation of on-chain Schnorr signatures (see this post for example).
However, because of (3) it is correct that there can be a single validation per block, however not a single signature per block. The speedup that is possible through batch validation does become nontrivial, in fact. Each of the 4 lines is an optimization technique that is currently implemented in libsecp256k1, which will pick the best one based on the size of the problem and the memory constraints.
(*) There exists a form of non-interactive "half aggregation" for DL based signatures, where N signatures can be non-interactively combined into a single signature of size (1+N)/2 original signatures. This could be used for blocks, though the gains aren't that great, and there are complexities around block-wide aggregation that make it less interesting.
Just saw this today -- very nice figure, why not include into the bip-schnorr? – Mark Lundeberg – 2019-05-09T15:29:15.520
I guess I have to read up the details again but wasn't the idea of the plain public key model and the construction of musig exactly that non interactive aggregation would become possible? Or should I have asked for musig signatures? I thought musig is just the exact scheme that is currently aimed to be used when people talk about schnorr. Do not mark as correct yet as I still have open questions (which are probably my fault) – Rene Pickhardt – 2019-03-10T23:18:01.023
1MuSig allows non-interactive setup - signing is still interactive. – Pieter Wuille – 2019-03-10T23:21:36.413
And MuSig is just one of the things wallet can choose to use when Schnorr validation on chain exist (because the signatures that come out of MuSig signing are compatible with a Schnorr verifier). There are (interactive setup) constructions that let you build k-of-n with a single key for example, or even arbitrary and/or-combinations of participants. – Pieter Wuille – 2019-03-10T23:25:08.313
1"There exists a form of non-interactive "half aggregation" for DL based signatures, where N signatures can be non-interactively combined into a single signature of size (1+N)/2 original signatures." --
Couldn't this be iterated for O(log(N)) iterations? – Martin Schwarz – 2019-03-12T07:26:52.877
@Martin Schwarz What do you mean by iterated? The construction (for Schnorr) is that you combine (R1,s1), ..., (Rn,sn) into (R1,...,Rn,s) where s = H(L,1)s1 + ... + H(L,n)sn and L = H(R1,...Rn,P1,...,Pn,m1,...,mn), and Pi are the pubkeys and mi are the messages. – Pieter Wuille – 2019-03-12T17:25:17.133
Oh, I see. OK, that doesn't work. – Martin Schwarz – 2019-03-13T14:06:27.653