Public Key – HD Wallet: How can I detect if two extended keys have the same root private key?

Let’s say you want to derive three extended public keys.^One (normal mode):

parent’s:

Public key: 035f743ee7d73d27e8c80f6b2458e6d4e2a45f3d7dd35c4e4d84ff3d939d09f40c

chain code 71d145eec8d031b39107cb7cb9641cc1e8169b6a25b489b157a0af2197717970

children:

Public key: 0317365d2ea4aa79e731fd4f0f6d60bf0ef75a35189a647667eebc9c7323d304d0

chain code 9cb9a9fe6e57b62f9a7970649d8dc4633fd9d4b5777a1397a275bd30a88c3111

Public key: 0230cdc8f7b84c0998ec182777d9d1f35ea45fe7620e979862c7025b18162108cb

chain code 44b54106ea0720a1ac226d598b89297dfe8efd2b7abf75c88761aa6df9c2886b

If I have two of these three (child-child or parent-child (don’t know which is the parent)), is it possible to detect that the keys are related? How will it be done?

If not, what additional information do you need? child index? Do you have a signature from the shared root private key?

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