When a user A transfers bitcoins to a user B, user A waiver of his possession of a number of bitcoins, recording the amount by the Bitcoin address B and the public key of A itself, whose private key signs all . (By using asymmetric cryptography, the private key can not be deduced from the firm that derives from it). This information to the entire P2P network is transmitted as a new transaction. Then, the other network nodes check the number of bitcoins involved and authenticity of the cryptographic signatures before accepting as valid transaction.
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