Resilient Delegation Revocation with Precedence for Predecessors is NP-Complete

In ownership-based access control frameworks with the possibility of delegating permissions and administrative rights, chains of delegated accesses will form. There are different ways to treat these delegation chains when revoking rights, which give rise to different revocation schemes. One possibility studied in the literature is to revoke rights by issuing negative authorizations, meant to ensure that the revocation is resilient to a later reissuing of the rights, and to resolve conflicts between principals by giving precedence to predecessors, i.e.\ principals that come earlier in the delegation chain.
However, the effects of negative authorizations have been defined differently by different authors. Having identified three definitions of this effect from the literature, the first contribution of this paper is to point out that two of these three definitions pose a security threat. However, avoiding this security threat comes at a price: We prove that with the safe definition of the effect of negative authorizations, deciding whether a principal does have access to a resource is an NP-complete decision problem. We discuss two limitations that can be imposed on an access-control system in order to reduce the complexity of the problem back to a polynomial complexity: Limiting the length of delegation chains to an integer m reduces the runtime complexity of determining access to O(n^m), and requiring that principals form a hierarchy that graph-theoretically forms a rooted tree makes this decision problem solvable in quadratic runtime. Finally we discuss an approach that can mitigate the complexity problem in practice without fully getting rid of NP-completeness.