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We develop formal authorization allocation algorithms for role-based access control (RBAC). The formal approaches are based on relational structure, and relational algebra and operations. The process of user-role assignments is an important issue in RBAC because it may modify the authorization level or imply high-level confidential information to be derived while users change positions and request different roles. There are two types of problems which may arise in user-role assignment. One is related to the authorization granting process. When a role is granted to a user this role may conflict with other roles of the user or together with this role; the user may have or derive a high level of authority. Another is related to authorization revocation. When a role is revoked from a user, the user may still have the role from other roles. To solve these problems, this paper presents an authorization granting algorithm, and weak revocation and strong revocation algorithms that are based on relational algebra. The algorithms can be used to check conflicts and therefore to help allocate roles without compromising the security in RBAC. We describe how to use the new algorithms with an anonymity scalable payment scheme. Finally, comparisons with other related work are discussed.