Affiliation-Hiding Authentication (AHA) protocols have the seemingly contradictory property of enabling users to authenticate each other as members of certain groups, without revealing their affiliation to group outsiders. Of particular interest in practice is the group-discovering variant, which handles multiple group memberships per user. Corresponding solutions were only recently introduced, and have two major drawbacks: high bandwidth consumption (typically several kilobits per user and affiliation), and only moderate performance in scenarios of practical application.While prior protocols have O(n2) time complexity, where n denotes the number of affiliations per user, we introduce a new AHA protocol running in O(nlogn) time. In addition, the bandwidth consumed is considerably reduced. We consider these advances a major step towards deployment of privacy-preserving methods in constraint devices, like mobile phones, to which the economization of these resources is priceless.