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Despite two decades of intensive research, it remains a challenge to design a practical anonymous two-factor authentication scheme, for the designers are confronted with an impressive list of security requirements (e.g., resistance to smart card loss attack) and desirable attributes (e.g., local password update). Numerous solutions have been proposed, yet most of them are shortly found either unable to satisfy some critical security requirements or short of a few important features. To overcome this unsatisfactory situation, researchers often work around it in hopes of a new proposal (but no one has succeeded so far), while paying little attention to the fundamental question: whether or not there are inherent limitations that prevent us from designing an “ideal” scheme that satisfies all the desirable goals? In this work, we aim to provide a definite answer to this question. We first revisit two foremost proposals, i.e. Tsai et al …