Can I verify a child public key derived from its parent public key without involving any private keys and child index?

Question

I have a extended private key, I'm creating multiple child private key / public key pair. If want to verify that the child public is derived from the parent public key. Can I do that? Thanks in advance!

Answer 1

If you have the parent public key and the (non hardened) index you can generate the child public key and verify that it matches. You can’t do it without the index I’m afraid since the derivation involves a one way hashing step.

Answer 2

To derive child keys, two conditions have to be met / satisfied:

1. The extended parent public key is available (public key & chain code)

2. The derivation path uses non-hardened indexes

3. (optional) know the derivation path; if you don't know it, it will take you time to iterate through the various combination (i.e. brute force), but typically they are simply incremented, so it shouldn't be costly.

Reference Public parent key → public child key in BIP-0032.