Password authentication schemes based on a geometric approach are proposed. Based on geometrics such as a plane or space

the proposed schemes can be used to fulfill the various requirements for security and achieve more efficient results. They allow a user to register with many servers once without remembering different passwords for each

and also allow a user to change the password freely and update it offline. In addition

the proposed schemes do not need to maintain a verification table and are also computationally efficient. Moreover

we propose a specific access right (AR) to verify the legitimate user who has different authorization levels on a server in a multi-server environment.