> which could involve proving something to a third party that has not seen 
> the communication between payer and payee.

OK - I think I follow now.  So a third-party who does not see any of the 
communication between the payer and payee only knows the HASH160.  Let's say 
the payee denies receipt of the funds....

It's easy to prove what public key it was sent to (it's the preimage), but 
you can't prove the parent of that public key. You can provide any number of 
ParentPubKey * Multiplier that could have been used, so the 3rd party is 
unconvinced by a "matching" ParentPubKey * Multiplier.

However, if you calculated the destination using: PubKeyParent * 
HMAC(Multiplier,PubKeyParent) as Timo said, now if you give the 3rd party a 
PubKeyParent and Multiplier (or Addend) that produces the destination 
address, you've proven the payment is in fact spendable by PubKeyParent, and 
they can't deny receipt. Very cool.

Sorry for "echoing" this back, it took me a little while to work it out, so 
I thought I'd write it down. Hope I got it right...

If you give {PubKey, ChainCode} you do get this feature. If you give 
{ParentPubKey, Addend} or {ParentPubKey, Addend, ChainCode} you're back to 
having plausible deniability.

If BIP32's CKD'((Kpar, cpar), i) was actually HMAC(HMAC(cpar, i), Kpar) you 
could give HMAC(cpar, i) instead of Addend, and then you would get this 
feature; a way to 'skip down' a level in the wallet hierarchy, keep the 
'chain of custody' so to speak back to the ParentPubKey intact, without 
having to disclose the ChainCode. Meh...


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