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A user can create a public key by performing an operation with the private key. Typically, we do another operation on the public key to get a public address. This is what you hand out to other users when you want them to send funds to you.
If you sign a message (i.e., a transaction) using your private key, others can use your corresponding public key to verify its authenticity. They use your public key to check if the message was really signed by you, and to ensure it wasn't altered after that.
Note that it’s necessary (and perfectly safe) to share your public key with others. But you should never reveal your private key. If someone gets access to it, they’ll be able to spend your funds by signing transactions on your behalf.
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