The characteristic novelty of what is generally meant by a "physical unclonable function" (PUF) is precisely defined, in order to supply a firm basis for security evaluations and the proposal of new security mechanisms. A PUF is defined as a hardware device which implements a physical function with an output value that changes with its argument. A PUF can be clonable, but a secure PUF must be unclonable. This proposed meaning of a PUF is cleanly delineated from the closely related concepts of "conventional unclonable function", "physically obfuscated key", "random-number generator", "controlled PUF" and "strong PUF". The structure of a systematic security evaluation of a PUF enabled by the proposed formal definition is outlined. Practically all current and novel physical (but not conventional) unclonable physical functions are PUFs by our definition. Thereby the proposed definition captures the existing intuition about what is a PUF and remains flexible enough to encompass further research. In a second part we quantitatively characterize two classes of PUF security mechanisms, the standard one, based on a minimum secret read-out time, and a novel one, based on challenge-dependent erasure of stored information. The new mechanism is shown to allow in principle the construction of a "quantum-PUF", that is absolutely secure while not requiring the storage of an exponentially large secret. The construction of a PUF that is mathematically and physically unclonable in principle does not contradict the laws of physics.
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