## 4 Hardware Security
### Physical Unclonable Functions (PUFs)
#### 4.1.3 PUF Performance Indicators  
The quality of a PUF is evaluated by metrics like uniqueness, reliability, randomness, correctness, strict avalanche condition (SAC), etc., that verify its applicability to a specific application. PUF metrics are measured by collecting response bits against a set of challenges to the PUF. A specific application has unique sets of requirements; hence, all metrics are not equally important [254; 255].  
**Uniqueness**: It is a PUF characteristic representing its ability to generate a unique response against a similar set of challenges subjected to each die in a lot [256]. Uniqueness is the average inter-chip Hamming Distance (HD) of the responses collected from a group of chips. The uniqueness value of an ideal PUF is about 50%, meaning half of the bits in the responses of the PUFs should be different [254]. For example, in an FPGA-based k n-bit, PUF responses are \(P_{1},P_{2},\cdots,P_{k}\), then the average Hamming distance given by Equation (2), is the measure of uniqueness [78],  
\[u=\frac{2}{k(k-1)}{\sum_{i=1}^{k-1}}{\sum_{j=i+1}^{k}}\frac{HD\big{(}P_{i},P_{ j}\big{)}}{n}\times 100\% \tag{2}\]  
**Reliability**: The PUF and CRP under noisy and variable environmental conditions are measured by their reliability, i.e., the PUF outputs the same response under variable operating conditions. However, numerous environmental conditions like temperature, voltage, and aging of the devices are responsible for variations in the PUF signatures. The ideal value for reliability is 100% and it can be estimated using Equation (3).  
\[u=\frac{1}{x}\sum_{y=1}^{x}\frac{HD(R_{i},R_{i},y)}{n}\times 100\% \tag{3}\]where x represents the times of sampling; n is the number of bits of a signature generated by a PUF; \(R_{i,y}\) is the \(y\)th sampling of \(R_{i}\).  
**Randomness**: It is a measure of the PUF's ability to generate 0 or 1 in its response bits with equal probabilities. The randomness of a PUF should be 100% in an ideal case. PUF-based authentication protocols rely heavily on random physical imperfections that occur during the semiconductor manufacturing process, thus creating static randomness. However, the identification (ID) extraction from the PUF becomes corrupted due to dynamic randomness sources like noise which reduces the PUF's reliability [257].  
\[Randomness=1-|P_{r}(ID=0)-P_{r}(ID=1)| \tag{4}\]  
For \(2^{M}\) challenges, the probabilities to obtain an ID at 0 and 1 can be given as  
\[Randomness=1-\left|\text{erf}\left(\frac{E(D_{R})}{\sigma\sqrt{2\cdot M}} \right)\right| \tag{5}\]  
where \(D_{R}\) is the pdf of  
\[\sum_{i=1}^{M}d_{c_{i}}^{i} \tag{6}\]  
For a variance of \(Mo^{2}\), the randomness expression Equation (7) is given by [258]  
\[\begin{split} P_{R}(ID=0)=1-P_{R}(ID=1)\\ =P_{r}\Big{(}\sum_{i=1}^{M}d_{c_{i}}^{i}<0\Big{)}\end{split} \tag{7}\]  
Correctness, Bit Aliasing, Uniformity, and Steadiness are additional PUF performance metrics discussed in the literature [78; 254; 256].  
Table 7 presents a comparison of PUF performance metrics mentioned in the previous section. It is inferred from the table that the Uniqueness and Uniformity performance metrics of Lattice PUF remain closer to ideal values whereas RC-PUF is the lowest-performing one.