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The public key can be used to 'undo' the effects of the private key and vice versa. So, to prove possession of the private key, the user can be challenged
to decrypt a message that is encrypted with the public key
to decrypt a message that is encrypted with the private key
to encrypt a message that is encrypted with the private key
to encrypt a message that is encrypted with the public key
Since the public key is generally available, a challenger can encrypt using that public key. The private key will undo the public key, so the claimant can prove knowledge of the private key by providing the message encrypted by the public key.
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