# Unpredictability

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

# One-Way Property

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.*

# Feasibility

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).*

# Tamper-Evident

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.*