The Public Key Can Be Generated From The Private Key
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There is more to a bitcoin wallet than just the address itself. It also contains the public and private key for each of your bitcoin addresses. Your bitcoin private key is a randomly generated string (numbers and letters), allowing bitcoins to be spent. A private key is always mathematically related to the bitcoin wallet address, but is impossible to reverse engineer thanks to a strong encryption code base.

If you don’t back up your private key and you lose it, you can no longer access your bitcoin wallet to spend funds.

As mentioned, there is also a public key. This causes some confusion, as some people assume that a bitcoin wallet address and the public key are the same. That is not the case, but they are mathematically related. A bitcoin wallet address is a hashed version of your public key.

Every public key is 256 bits long — sorry, this is mathematical stuff — and the final hash (your wallet address) is 160 bits long. The public key is used to ensure you are the owner of an address that can receive funds. The public key is also mathematically derived from your private key, but using reverse mathematics to derive the private key would take the world’s most powerful supercomputer many trillion years to crack.

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On the other hand, the Private Key must remain confidential to its respective owner. Because the key pair is mathematically related, whatever is encrypted with a Public Key may only be decrypted by its corresponding Private Key and vice versa. With RSA, you cannot, with EC you can. However, the public key is usually always stored together with the private key (not the other way around, though, of course), so this is not really a problem (if you have the private key, the same file also includes the public key).

Besides these key pairs and a bitcoin wallet address, your bitcoin wallet also stores a separate log of all of your incoming and outgoing transactions. Every transaction linked to your address will be stored by the bitcoin wallet to give users an overview of their spending and receiving habits.

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Last but not least, a bitcoin wallet also stores your user preferences. However, these preferences depend on which wallet type you’re using and on which platform. The Bitcoin Core client, for example, has very few preferences to tinker around with, making it less confusing for novice users to get the hang of it.

Osx I'll ask one of them to try that and see.If you have other ideas about what to do, I'd appreciate ideas.

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Your bitcoin wallet generates a “master” file where all of the preceding details are saved. For computer users, that file is called wallet.dat. It’s saved on a Windows machine, for example, in the C:UserYournameDocumentsAppDataRoamingBitcoinfolder. Make sure to create one or multiple backups of this wallet.dat file on other storage devices, such as a USB stick or memory card. The bitcoin wallet software will let you import a wallet.dat file in case your previous file is damaged or lost, restoring your previous settings, including any funds associated with your bitcoin wallet address.

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Check out more information on importing private keys and wallet.dat files.