# On Improving Reliability of SRAM-Based Physically Unclonable Functions
## 4 Analysis of the Proposed Counter Based Design
### DFT Based on Trials to Settlement  
Label (4) quantifies the probability of success when infinite trials are applied to each cell. In reality, the UP/DOWN counter may settle in a value between the end states due to the limited number of trials. This is related to the inherent error rate of each cell. The expected number of trials needed by the UP/DOWN counter to reach a decision state [28] can be derived as  
\[d_{k}=\frac{k}{(p-q)}\left[2\left(\frac{1-\left(\frac{q}{p}\right)^{k}}{1- \left(\frac{q}{p}\right)^{2k}}\right)-1\right], \tag{5}\]  
where \(k=(2^{n}-2)/2\). This expected value for a 4-bit UP/DOWN counter is plotted in Figure 8 for different initial error rates. As illustrated, the average number of trials for reaching the saturation increases with the error rate. The exact distributions for probability of reaching the correct value under given number of trials can be derived using probability generating functions. Such explicit expressions for number of trials and probability of not reaching any state can be used to improve the design. The explicit expression permits the designers to estimate how much provision is to be made for the masked bits in the design.  
As the average number of trials to reach the end state is related to inherent error rate of the cell, this information can be used during the trials to filter out cells with high error rates. For example, consider the test setup shown in Figure 9. The PUF array has redundant cells so that, during the trials, the cells with high error rates can be discarded for operation. Hence, the aim of the test is to generate _mask information_ that indicates which of the cells in the PUF array should be considered for real-time operation. One simple way to achieve this is to set an empirical or analytical threshold based on Label (5) (or using tighter bound expressions) for the number of trials to apply to the PUF array. If a PUF cell does not reach the end state within the target number of trials, the cell is marked as invalid. As this mask does not reveal any information about the PUFs' values, it can be made public. During real-time operation, this mask value along with raw PUF response is combined to filter out responses of cells with high error rates. The resultant identifier is used as a key. This technique may also be combined with ECC or other post-processing techniques to reduce the probability of error further. This technique can also be used to determine whether a PUF chip is reliable enough for operation. For example, if the mask implies that the number of cells that are reliable in an array is less than the length  
Figure 7: Error rate reduction comparison using TMV and UDC.  
of the desired key length, the chip can be marked as unreliable and hence, rejected during testing. [PERSON] et al. have proposed similar mask generation, but by adjusting supply voltage [30].  
Hence, even though the UP/DOWN scheme has subtle changes in comparison to the TMV scheme, they offer significant improvement in error rate and also provide leverage to identify high error rate cells.