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# Bio-inspired electronic fingerprint PUF device with single-walled carbon nanotube network surface mediated by M13 bacteriophage template

### Min entropy

Min-entropy is the most conservative way of measuring the unpredictability of a set of outcomes and is evaluated by the answers as follows16:

$${\text{H}}_{{{\text{min}}}} = – \log_{2} \left( {{\text{P}}_{{{\text{max}}} } } \right),$$

(1)

Where $${\text{H}}_{{{\text{minus}}}}$$ denotes the minus entropy of the samples, and $${\text{P}}_{{{\text{max}}}}$$ maximum probability of 0 or 1 on each position of the answer to the challenges.

$$\left( {{\text{H}}_{{{\text{min}}}} } \right)_{{{\varvec{total}}}} = – \frac{1}{{ \varvec{n}}}\mathop \sum \limits_{{{\varvec{i}} = 1}}^{{\varvec{n}}} \log_{2} \left( {{\text{P }}_{{{\text{max}}}} } \right)$$

(2)

If Pmax is close to 0.5, the min-entropy leads to an ideal value of 1. The response patterns of the PUF with a min-entropy close to 1 become almost unpredictable. All fabricated M13-SWNT-based PUFs had a desirable high min-entropy of 0.98, regardless of individual PUF cell distribution, demonstrating the unpredictability of their responses.

### Randomness and uniqueness evaluation

Randomness evaluates the unpredictability of the answers and is obtained by measuring the number of ‘1s’ or ‘0s’ in the answer string17. An ideal PUF should have a randomness of 50%, which contributes to a strong tolerance to brute force attacks. Uniqueness reflects how different responses are expected when the same challenge is applied to different PUFs17. It is evaluated by measuring the hamming distance between responses of different PUFs to the same challenge, and an ideal PUF should have a uniqueness of 50%. Randomness was measured by applying 10,000 different challenges and extracting the 240-bit responses from each PUF. Uniqueness was also evaluated by applying the same challenges 10,000 times to the three PUFs and obtaining the 240-bit answers. The randomness of the M13-SWNT based PUF results was 50%, 50.5% and 51%, all of which are close to the ideal value of 50%, as shown in Figure 4a. In addition, the uniqueness of PUFs also tended towards the ideal value of 50%, as shown in Fig. 4b.

### Variations in the environment

The PUF device should behave reliably by reproducing the same responses even under environmental variations. In particular, the M13-SWNT based PUFs with a flexible substrate are easily exposed to physical and temperature variations, and these changes often cause a small reversal of the electrical output of the response. However, when the electrical changes can be linearly correlated with environmental variation, the corresponding relationship between resistance and environmental variation can be used to minimize the possibility of bit flips, in a process called error correction.18. Therefore, our research investigated the dependencies of resistance on bending and temperature variation. When the M13-SWNT based PUF was subjected to bending, the resistance increased, with respect to strain, (Fig. 5a). In addition, a temperature increase from 25 to 50 °C linearly reduced the resistance, indicated by an increased current shown (Fig. 5b). Based on the linear correlation of resistance with these environmental variables, the bit errors caused by changes in the environment can be suppressed via a compensation algorithm.

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