The　exploration　of　submodular　optimization　problems　on　the　integer　lattice　offers　a　more　precise　approach　to　handling　the　dynamic　interactions　among　repetitive　elements　in　practical　applications.　In　today’s　data-driven　world,　the　importance　of　efficient　and　reliable　privacy-preserving　algorithms　has　become　paramount　for　safeguarding　sensitive　information.　In　this　paper,　we　delve　into　the　DR-submodular　and　lattice　submodular　maximization　problems　subject　to　cardinality　constraints　on　the　integer　lattice,　respectively.　For　DR-submodular　functions,　we　devise　a　differential　privacy　algorithm　that　attains　a　(1-1/e-ρ)-approximation　guarantee　with　additive　error　O(rσln|N|/ϵ)　for　any　ρ＞0,　where　N　is　the　number　of　groundset,　ϵ　is　the　privacy　budget,　r　is　the　cardinality　constraint,　and　σ　is　the　sensitivity　of　a　function.　Our　algorithm　preserves　O(ϵr2)-differential　privacy.　Meanwhile,　for　lattice　submodular　functions,　we　present　a　differential　privacy　algorithm　that　achieves　a　(1-1/e-O(ρ))-approximation　guarantee　with　additive　error　O(rσln|N|/ϵ).　We　evaluate　their　effectiveness　using　instances　of　the　combinatorial　public　projects　problem　and　the　budget　allocation　problem　within　the　bipartite　influence　model.　©　The　Author(s),　under　exclusive　licence　to　Springer　Science+Business　Media,　LLC,　part　of　Springer　Nature　2024.