Data　privacy　has　become　one　of　the　most　important　concerns　in　the　big　data　era.　Because　of　its　broad　applications　in　machine　learning　and　data　analysis,　many　algorithms　and　theoretical　results　have　been　established　for　privacy　clustering　problems,　such　as　k-means　and　k-median　problems　with　privacy　protection.　However,　there　is　little　work　on　privacy　protection　in　k-center　clustering.　Our　research　focuses　on　the　k-center　problem,　its　distributed　variant,　and　the　distributed　k-center　problem　under　differential　privacy　constraints.　These　problems　model　the　concept　of　safeguarding　the　privacy　of　individual　input　elements,　with　the　integration　of　differential　privacy　aimed　at　ensuring　the　security　of　individual　information　during　data　processing　and　analysis.　We　propose　three　approximation　algorithms　for　these　problems,　respectively,　and　achieve　a　constant　factor　approximation　ratio.　©　2024,　The　Author(s),　under　exclusive　licence　to　Springer-Verlag　GmbH　Germany,　part　of　Springer　Nature.