The cryptographic protocol problem of how to make a secure exchange of identifying information among communicating entities, in particular within security constraints of confidentiality and personal privacy, is here denoted as the private identification problem. We consider this to be still an open problem. Although we can find its motivation and partial solutions in some of the existing systems, this paper describes the problem in a more generalised form. What is the solution space of efficient and scalable private identification protocols within the settings of the various communication systems and models, for instance in future mobile communication systems with very low latency requirements? Possible directions and solutions are discussed, in terms of pseudonyms, temporary identifiers, computational trade-offs such as key search, and public key solutions. All existing proposals to the private identification challenge suffer from one or more limitations and weaknesses, such as computational costs and time latencies, or the security is reduced. Finally, the paper collects a considerable reference list of papers related to the problem of private identification.