### Min-entropy

Min-entropy is the most conservative way to measure the unpredictability of a set of outcomes and is evaluated by the responses as follows^{16}:

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

(1)

where ({text{H}}_{{{text{min}}}}) denotes the min-entropy of the samples, and ({text{P}}_{{{text{max}}}}) maximum probability of 0 or 1 at each position of the response 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)

f *P*_{the Max} is close to 0.5, then the min-entropy leads to an ideal value of 1. The response patterns from the PUF with a min-entropy close to 1 become almost unpredictable. All the fabricated M13-SWNT-based PUFs had a desirably high min-entropy of 0.98, regardless of the individual PUF cell distribution, demonstrating the unpredictability of their responses.

### Randomness and uniqueness evaluation

Randomness evaluates the unpredictability of the responses and is obtained by measuring the number of ‘1 s’ or ‘0 s’ in the response string^{17}. An ideal PUF should have randomness of 50%, which contributes to strong tolerance against brute-force attacks. Uniqueness represents how different responses are expected to be when the same challenge is applied to different PUFs^{17}. 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%. The randomness was measured by applying 10,000 different challenges and extracting the 240-bit responses from each PUF. The uniqueness was also evaluated by applying the same challenges 10,000 times to the three PUFs and obtaining the 240-bit responses. The randomness of the M13-SWNT-based PUFs results were 50%, 50.5%, and 51%, all of which are close to the ideal value of 50%, as shown in Fig. 4a. Moreover, the uniqueness of PUFs also tended to the ideal value of 50%, as shown in Fig. 4b.

### Environmental variations

The PUF device is required to 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 bit flip in the response electrical outputs. 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 referred to as error correction^{18}. Therefore, our study investigated the dependencies of resistance on bending and temperature variation. When the M13-SWNT-based PUF was subjected to bending, resistance increased, with respect to strain, (Fig. 5a). Moreover, a temperature increase from 25 to 50 Â°C linearly decreased the resistance, indicated by increased current flow shown (Fig. 5b). Based on the linear correlation of resistance with these environmental variables, the bit errors induced by environmental change can be suppressed via a compensation algorithm.