If you are already a US citizen, then you would need to file a petition to change your name with your local court, as that is no longer an immigration matter. If you have not a US citizen, you can change your name as part of the naturalization procedure.
A Certificate of Naturalization is granted to green card holders after their become U.S. citizens though naturalization.
A Certificate of Citizenship is granted to a person born outside the U.S. to one or more U.S. citizen parents. Such a certificate means that this person is a U.S. citizen at birth.
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