The unpredictability property can be directly inherited from the definition of mathematical unclonability, Eq. 10.2a.
What is requested to be unpredictable for a function is the inability to create a procedure Ф that, having a certain amount of challenge-response pairs Ф for a PUF в, is able to provide the same output of в for a generic challenge c. The existence of this procedure is in direct contrast with the unclonability because Ф represents a mathematical clone that can predict the в responses. Moreover, in the degeneracy case when Ф = 0, the statement 10.4 is equivalent to the Eq. 10.2a
Formally в is a one-way function given г = в (c) it is hard to find A : А (r) = c and в (c) = r, Vc e C c C. As for hash functions, in this definition “hard” is meant in the computational theory sense, so that given one output r of a PUF в, it is very expansive, in terms of computation resources and time, to find one input c such that
в (c) = r.
Being an integrated circuit, a PUF inevitably introduces an overhead in area and time. As for the occupied area, the circuit has to extract the physical information and maybe implementing the challenge/response mechanism. As for the time overhead, the response extraction could require a significant amount of time, especially when the architecture is provided with a post-processing algorithm that has to be ran. Given в e & and c e C, в is feasible if it is not hard to evaluate в (c).
The PUF в is tamper-evident if any attempt to tamper the circuit permanently changes its CRPs set, obtaining a new PUF в' : в' (c) ^ в (c), Vc e C.