Can I get my last name changed to my fathers last name?

My parents were never married and my father was ashamed of his last name and in the late 50's or early 60's they moved from Rome NY to Syracuse NY and when they went to get there first apartment (my father told me) the building owner asked him what his name was he thought if he told him he was a Capone he would not rent the apartment to him because when he was growing up in the 30's and 40's and in the 50's in the military people looked down on and disliked people with the last name of Capone, back in the 50's and early 60's he called the social security office and told them he never had a social security number and his last name was Smith and with no questions asked they sent him a card with the smith name and a new ss number. I want to have my family name as I should have from birth.