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Let's say I have this scenario:

a customer has a bitcoin wallet. He/she shares the public key with a website. The website is a payment processor which lets the customer accept bitcoin payments.

The website generates a new address (based on the customer public key and a nonce) and watches the address for incoming transactions. Is this sufficient for the customer to see a balance increase on his/her wallet?

The customer wallet never sees the nonce so probably something is wrong in my reasoning.

What am I missing here?

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Public key is point on ECDSA curve

Public key + nonce != new public key

And if the above equation works, how can you calculate its private key !

Payment processors accept payment on their addresses, and send funds to users instantly, monthly, or send an equal amount of $ to their bank accounts.

I think you're looking for BIP32 wallets, in which addresses (public key hash) can be derived from other extended public keys (xPubs)

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