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Private on Case. Need access for the case creator who is not case owner.
Private Sharing Settings on Case object.
Case creator should be able to change a checkbox field where as case owner should not be able to do this.
You can write a small trigger on case object .. on event update/insert
1 here u can check if the checkbox valus is changed and set to true or not ...
2 if changed and if the logged in user = createdbyuser
3 if yes then dont do anything.
4 else show a error ... "you are not authrized to change the checkbox value."