# CNT-PUFs: Highly Robust and Heat-Tolerant Carbon-Nanotube-Based Physical Unclonable Functions +
## 3 Preliminaries: PUF Metrics  
To assess the quality of the examined PUFs, several metrics are available:  
* **Uniformity** describes how the bits of an individual PUF response are distributed. It is favourable to achieve an equal distribution of zeros and ones. A bias towards one of the values leads to higher predictability of the responses, compromising the overall security of the resulting response. Uniformity is evaluated by the Hamming weight \(HW(R_{i}^{x}(C))\), which measures the number of ones in a binary PUF response \(R_{i}(C)\) of PUF instance \(i\) to challenge \(C\) that has been received at normalised time \(x\). The normalised average \(\mathcal{HV}_{i}\) for a PUF \(i\) is calculated as follows: \[\mathcal{HV}_{i}=\frac{1}{I}\cdot\sum_{x=1}^{I}\frac{HW(R_{i}^{x}(C))}{|PUF|}\,\] (1)where \(l\) is the number of responses collected for a challenge \(C\) at all normalised time values and \(|PUF|\) the length of the PUF responses of PUF \(i\). The ideal value of \(0.5=50\%\) signifies, on average, a perfectly balanced distribution of zeros and ones within the PUF responses of PUF \(i\). * **Uniqueness** measures the independence of responses originating from different PUFs for the same challenge \(C\). This property is typically evaluated by the average fractional inter-device Hamming distance \(\mathcal{HD}_{\mathit{inter}}\), which specifies the normalised average number of bit positions that differ in the responses of all different PUF instances, with response \(R_{i}^{x}(C)\) being the response of PUF \(i\) to challenge \(C\) at normalised time \(x\) and response \(R_{j}^{y}(C)\) being the response of PUF \(j\) to challenge \(C\) at normalised time \(y\). Essentially, the Hamming distance between \(R_{i}^{x}(C)\) and \(R_{j}^{y}(C)\), denoted by \(HD(R_{i}^{x}(C),R_{j}^{y}(C))\), is equal to \(sum(R_{i}^{x}(C)\ \oplus\ R_{j}^{y}(C))\), counting the number of different bit positions when comparing the two binary vectors \(R_{i}^{x}(C)\) and \(R_{j}^{y}(C)\). For PUF responses of equal length \(|PUF|\) (if the PUF responses differ in their length, the length of the shorter response is chosen for their comparison, and the number of bits remaining from the longer response is added to the comparison result, to calculate the overall \(\mathcal{HD}_{\mathit{inter}}\)), with \(k\) being the number of devices measured, \(m\) the number of responses collected from device \(i\) for a response \(C\) at all normalised times, and \(n\) the number of responses collected from device \(j\) for the response \(C\) at all normalised times; \(\mathcal{HD}_{\mathit{inter}}\) is given by: \[\mathcal{HD}_{\mathit{inter}}=\frac{2}{k\cdot(k-1)}\cdot\sum_{i=1}^{k-1}\sum_{ j=i+1}^{k}\left(\frac{1}{m\cdot n}\cdot\sum_{x=1}^{m}\sum_{y=1}^{n}\frac{HD(R_{i}^{ x}(C),R_{j}^{y}(C))}{|PUF|}\right).\] (2) A value of \(\mathcal{HD}_{\mathit{inter}}=0.5=50\%\), meaning that, on average, half of the bits of the response of one PUF instance have a different value from the bits of all other instances, indicates the highest possible degree of uniqueness. * The property of **robustness** describes the stability of PUF responses originating from the same PUF, for a given challenge \(C\), under repeated PUF measurements. This property is evaluated using the average fractional intra-device Hamming distance \(\mathcal{HD}_{\mathit{intra}}\), which measures the normalised average number of bit positions that differ between each two responses \(R_{i}^{x}(C)\) and \(R_{i}^{y}(C)\) stemming from the same device \(i\) produced using the same challenge \(C\) and captured at different normalised times \(x\) and \(y\). \(\mathcal{HD}_{\mathit{intra}}\) is given by: \[\mathcal{HD}_{\mathit{intra},i}=\frac{2}{l\cdot(l-1)}\cdot\sum_{x=1}^{l-1} \sum_{y=x+1}^{l}\frac{HD(R_{i}^{x}(C),R_{i}^{y}(C))}{|PUF|}\,,\] (3) where \(l\) is the number of responses collected for a challenge \(C\) at all normalised time values and \(|PUF|\) the length of the PUF responses of PUF \(i\).