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With a hardware wallet, when you enter your pin and seed onto the hardware wallet software, is that data stored on their servers, and potentially accessible to hackers?

I know that the private key is stored on the hardware wallet, but how does the hardware wallet sign transactions without disclosing the private key to the hardware wallet software and their server?

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how does the hardware wallet sign transactions without disclosing the private key to the hardware wallet software

An honestly constructed hardware wallet is designed so that the interface between it and the client software only allows for the client software to pass in an unsigned transaction and receive back a signed transaction.

The hardware wallet contains a processor and code that ensures the private key is never disclosed. The client software has no direct access to the hardware wallets internal storage.

The interface is a kind of API between two computers. That API defines a list of functions (example). The arguments and return values are determined by the API. None of those functions return a private key.

There might be more functions than just transaction signing but each would be carefully designed so that the keys are never disclosed.

A dishonestly or incompetently constructed hardware wallet might leak information (example).

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