CNTPUFs: Highly Robust and HeatTolerant CarbonNanotubeBased Physical Unclonable Functions^{ †}
Abstract
:1. Introduction
1.1. Contributions
 We conducted an additional temperature analysis of our CNTPUFs, extending the application area of the CNTPUFs to include even harsh environments. The investigated CNTPUFs demonstrated very high stability with at most 2% unstable cells in the temperature range from $23{}^{\circ}\mathrm{C}$ to $120{}^{\circ}\mathrm{C}$.
 We performed a moreextensive security analysis of the raw CNTPUF measurements. An analysis of the unpredictability of the raw PUF responses through an evaluation of the measurements by the statistical randomness tests of the National Institute of Standards and Technology (NIST) SP 80022 test suite [24] was conducted. Our PUFs passed all applicable statistical randomness tests of the NIST SP 80022 test suite [24], confirming the ability to use our CNTPUFs as strong cryptographic primitives within cryptographic protocols.
 Finally, we present a very lightweight and efficient postprocessing method that ranks all cells according to their stability and selects the moststable cells for further use. By using this method, our CNTPUFs are able to produce fully stable PUF responses.
1.2. Paper Organisation
2. Related Work
3. Preliminaries: PUF Metrics
 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 $HW\left({R}_{i}^{x}\left(C\right)\right)$, which measures the number of ones in a binary PUF response ${R}_{i}\left(C\right)$ of PUF instance i to challenge C that has been received at normalised time x. The normalised average ${\mathcal{HW}}_{i}$ for a PUF i is calculated as follows:$${\mathcal{HW}}_{i}={\displaystyle \frac{1}{l}}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\sum _{x=1}^{l}{\displaystyle \frac{HW\left({R}_{i}^{x}\left(C\right)\right)}{\leftPUF\right}}\phantom{\rule{0.166667em}{0ex}},$$
 Uniqueness measures the independence of responses originating from different PUFs for the same challenge C. This property is typically evaluated by the average fractional interdevice Hamming distance ${\mathcal{HD}}_{inter}$, which specifies the normalised average number of bit positions that differ in the responses of all different PUF instances, with response ${R}_{i}^{x}\left(C\right)$ being the response of PUF i to challenge C at normalised time x and response ${R}_{j}^{y}\left(C\right)$ being the response of PUF j to challenge C at normalised time y. Essentially, the Hamming distance between ${R}_{i}^{x}\left(C\right)$ and ${R}_{j}^{y}\left(C\right)$, denoted by $HD({R}_{i}^{x}\left(C\right),{R}_{j}^{y}\left(C\right))$, is equal to $sum({R}_{i}^{x}\left(C\right)\phantom{\rule{0.277778em}{0ex}}\oplus \phantom{\rule{0.277778em}{0ex}}{R}_{j}^{y}\left(C\right))$, counting the number of different bit positions when comparing the two binary vectors ${R}_{i}^{x}\left(C\right)$ and ${R}_{j}^{y}\left(C\right)$. For PUF responses of equal length $\leftPUF\right$ (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 ${\mathcal{HD}}_{inter}$), 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; ${\mathcal{HD}}_{inter}$ is given by:$${\mathcal{HD}}_{inter}={\displaystyle \frac{2}{k\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}(k1)}}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\sum _{i=1}^{k1}\sum _{j=i+1}^{k}\left({\displaystyle \frac{1}{m\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}n}}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\sum _{x=1}^{m}\sum _{y=1}^{n}{\displaystyle \frac{HD({R}_{i}^{x}\left(C\right),{R}_{j}^{y}\left(C\right))}{\leftPUF\right}}\right)\phantom{\rule{0.166667em}{0ex}}.$$A value of ${\mathcal{HD}}_{inter}=0.5=50\%$, 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 intradevice Hamming distance ${\mathcal{HD}}_{intra}$, which measures the normalised average number of bit positions that differ between each two responses ${R}_{i}^{x}\left(C\right)$ and ${R}_{i}^{y}\left(C\right)$ stemming from the same device i produced using the same challenge C and captured at different normalised times x and y. ${\mathcal{HD}}_{intra}$ is given by:$${\mathcal{HD}}_{intra,i}={\displaystyle \frac{2}{l\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}(l1)}}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\sum _{x=1}^{l1}\sum _{y=x+1}^{l}{\displaystyle \frac{HD({R}_{i}^{x}\left(C\right),{R}_{i}^{y}\left(C\right))}{\leftPUF\right}}\phantom{\rule{0.166667em}{0ex}},$$
 Another important property is the unpredictability of the generated responses. One precondition for high unpredictability is a uniform distribution of logical ones and zeros as explained previously. In addition, single bits of the PUF response should be free of correlations. Unpredictability is typically evaluated by a collection of statistical tests for randomness performed on the PUF responses. This paper used the wellknown NIST SP 80022 test suite [24]. This test suite consists of various statistical tests applied to presumably random strings, to check if they meet the entropy requirements to be used as primitives in cryptographic protocols.
4. Fabrication and Implementation of CNTPUFs
4.1. WaferLevel Fabrication Process
4.2. CNTPUF Characterisation
4.3. CNTPUF Quantisation
5. Evaluation of the Fabricated CNTPUFs
5.1. CNTPUF Design with Highly Distinguishable Responses
5.2. Evaluation of CNTPUFs under Normal Conditions
5.2.1. Uniformity
5.2.2. Uniqueness
5.2.3. Unpredictability
5.2.4. Robustness
5.3. HeatTolerant CNTPUFs
6. Stable Key Extraction from CNTPUFs
6.1. Enrolment
Algorithm 1 Retrieving a challenge as a list of cell indices of stable cells of a CNTPUF p. 

Algorithm 2 Selection of random PUF cells using a Random Number Generator (RNG). 

6.2. Reconstruction
Algorithm 3 Algorithm describing the reconstruction phase of a CNTPUF p. 

7. Conclusions
Supplementary Materials
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Abbreviations
AFM  Atomic Force Microscopy 
ALD  Atomic Layer Deposition 
ASIC  ApplicationSpecific Integrated Circuit 
CMOS  Complementary Metal–Oxide–Semiconductor 
CNT  Carbon NanoTube 
CNTFET  Carbon NanoTube FieldEffect Transistor 
CNTPUF  Carbon NanoTubebased Physical Unclonable Function 
CVD  Chemical Vapour Deposition 
EBL  ElectronBeam Lithography 
FET  FieldEffect Transistor 
IoT  Internet of Things 
NIST  National Institute of Standards and Technology 
PUF  Physical Unclonable Function 
RNG  Random Number Generator 
SoC  System on a Chip 
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Statistical Test  Average pValue  Passed/Total  Input Length Recommendation Test Parameters 

Frequency (Monobit)  0.28  25/30  $n\ge $ 100 bits 
Frequency Test within a Block  0.32  28/30  $n\ge $ 100 bits M = 20 bits (block length) 
Runs Test  0.43  30/30  $n\ge $ 100 bits 
Test for the Longest Run of Ones in a Block  0.41  27/30  $n\ge $ 128 bits m = 8 bits (block length) 
Binary Matrix Rank Test  /  /  $\mathit{n}$ ≥ 38,912 bits 
Discrete Fourier Transform (Spectral) Test  /  /  $\mathit{n}$ ≥ 1000 bits 
Nonoverlapping Template Matching Test  0.86  30/30  m = 9 bits (template size) 
Overlapping Template Matching Test  /  /  $\mathit{n}$ ≥ ${\mathbf{10}}^{\mathbf{6}}$ bits 
Maurer’s “Universal Statistical” Test  /  /  $\mathit{n}$ ≥ 387,840 bits 
Linear Complexity Test  /  /  $\mathit{n}$ ≥ ${\mathbf{10}}^{\mathbf{6}}$ bits 
Serial Test  0.36  30/30  $m<\left[{log}_{2}n\right]2$ m = 4 bits (block length) 
Approximate Entropy Test  0.33  27/30  $m<\left[{log}_{2}n\right]5$ m = 2 bits (block length) 
Cumulative Sums (Cusum) Test  0.30  26/30  $n\ge $ $100$ bits 
Random Excursions Test  /  /  $\mathit{n}$ ≥ ${\mathbf{10}}^{\mathbf{6}}$ bits 
Random Excursions Variant Test  /  /  $\mathit{n}$ ≥ ${\mathbf{10}}^{\mathbf{6}}$ bits 
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Frank, F.; Böttger, S.; Mexis, N.; Anagnostopoulos, N.A.; Mohamed, A.; Hartmann, M.; Kuhn, H.; Helke, C.; Arul, T.; Katzenbeisser, S.; et al. CNTPUFs: Highly Robust and HeatTolerant CarbonNanotubeBased Physical Unclonable Functions. Nanomaterials 2023, 13, 2930. https://doi.org/10.3390/nano13222930
Frank F, Böttger S, Mexis N, Anagnostopoulos NA, Mohamed A, Hartmann M, Kuhn H, Helke C, Arul T, Katzenbeisser S, et al. CNTPUFs: Highly Robust and HeatTolerant CarbonNanotubeBased Physical Unclonable Functions. Nanomaterials. 2023; 13(22):2930. https://doi.org/10.3390/nano13222930
Chicago/Turabian StyleFrank, Florian, Simon Böttger, Nico Mexis, Nikolaos Athanasios Anagnostopoulos, Ali Mohamed, Martin Hartmann, Harald Kuhn, Christian Helke, Tolga Arul, Stefan Katzenbeisser, and et al. 2023. "CNTPUFs: Highly Robust and HeatTolerant CarbonNanotubeBased Physical Unclonable Functions" Nanomaterials 13, no. 22: 2930. https://doi.org/10.3390/nano13222930