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Given a payment address, is it possible to generate or query the blockchain, using cardano-graphql, cardano-cli, plutus or by other means, so as to obtain a new payment address (unused) for the same wallet?

This is to be achieved without requiring the payment verification key of the receiving wallet.

Illustration of my use case:

Say, a third party has a wallet and I was given one of the payment addresses of that wallet, address1 = addr1apq...

Now, I want to obtain an unused payment address, address2 for this wallet using the address1 above and make payment to it. Is there a way to do that?

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No, you would need a extended public key (xPub) for that. You can grab one (account xPub) from Daedalus or derive it using cardano-addresses.

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