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Edge Computing is a new computing paradigm where applications operate at the network edge, providing low-latency services with augmented user and data privacy. A desirable goal for edge computing is pervasiveness, that is, enabling any capable and authorized entity at the edge to provide desired edge services--pervasive edge computing (PEC). However, efficient access control of users receiving services and edge servers handling user data, without sacrificing performance is a challenge. Current solutions, based on "always-on" authentication servers in the cloud, negate the latency benefits of services at the edge and also do not preserve user and data privacy. In this paper, we present APECS, an advanced access control framework for PEC, which allows legitimate users to utilize any available edge services without need for communication beyond the network edge. The APECS framework leverages multi-authority attribute-based encryption to create a federated authority, which delegates the authentication and authorization tasks to semi-trusted edge servers, thus eliminating the need for an "always-on" authentication server in the cloud. Additionally, APECS prevents access to encrypted content by unauthorized edge servers. We analyze and prove the security of APECS in the Universal Composability framework and provide experimental results on the GENI testbed to demonstrate the scalability and effectiveness of APECS.