# Multiple Observations for Secret-Key Binding with SRAM PUFs
## Appendix A Log-Likelihood Ratio for MO Helper Data Scheme
### Log-Likelihood Ratio for Binary SD Helper Data Scheme  
We give the derivation that leads to (39) the log-likelihood ratio for a codebit \(c\), after observing the noisy codebit \(r\) and the error probability \(\hat{\psi}_{t}\) for \(t\) enrollment observations. First, we note that  
\[1-\hat{\psi}_{t}=1-\frac{\pi_{t+1}(1+m_{t}^{\text{\tiny SD}})}{\pi_{t}(m_{t}^ {\text{\tiny SD}})}=\frac{\pi_{t}(m_{t}^{\text{\tiny SD}})-\pi_{t+1}(1+m_{t}^ {\text{\tiny SD}})}{\pi_{t}(m_{t}^{\text{\tiny SD}})}\overset{(a)}{=}\frac{ \pi_{t+1}(m_{t}^{\text{\tiny SD}})}{\pi_{t}(m_{t}^{\text{\tiny SD}})}\overset{ (b)}{=}\frac{\pi_{t+1}(1+t-m_{t}^{\text{\tiny SD}})}{\pi_{t}(m_{t}^{\text{ \tiny SD}})}\] (A13)  
in \((a)\) we used that \(\pi_{t}(m_{t}^{\text{\tiny SD}})=\pi_{t+1}(m_{t}^{\text{\tiny SD}})+\pi_{t+1} (1+m_{t}^{\text{\tiny SD}})\) and \((b)\) follows from (7). The log-likelihood ratio at the encoder, for a received value \(r\) and given an estimated error probability \(\hat{\psi}_{t}\), is  
\[\begin{array}{rcl}\text{LLR}^{\text{\tiny SD}}\big{(}r,\hat{\psi}_{t}\big{)}& \overset{\Delta}{=}&\log\frac{\Pr\big{(}R=r,\hat{\Psi}_{t}=\hat{\psi}_{t}|C=0 \big{)}}{\Pr\big{(}R=r,\hat{\Psi}_{t}=\hat{\psi}_{t}|C=1\big{)}}\\ &=&\log\frac{\Pr\big{(}\hat{U}_{t}\cap r|\hat{\Psi}_{t}=\hat{\psi}_{t}\big{)}} {\Pr\big{(}\hat{U}_{t}=r,\hat{\Psi}_{t}=\hat{\psi}_{t}\big{)}}\\ &=&\begin{cases}\log\frac{\Pr\big{(}\hat{U}_{t}\cap r|\hat{\Psi}_{t}=\hat{ \psi}_{t}\big{)}}{\Pr\big{(}\hat{U}_{t}=r,\hat{\Psi}_{t}=\hat{\psi}_{t}\big{)} }&\text{if $r=1$,}\\ \log\frac{\Pr\big{(}\hat{U}_{t}=r|\hat{\Psi}_{t}=\hat{\psi}_{t}\big{)}}{\Pr \big{(}\hat{U}_{t}\
eq r,\hat{\Psi}_{t}=\hat{\psi}_{t}\big{)}}&\text{otherwise,} \end{cases}\\ &=&\begin{cases}\log\frac{\hat{\psi}_{t}}{1-\hat{\psi}_{t}}&\text{if $r=1$,}\\ \log\frac{1-\hat{\psi}_{t}}{\hat{\psi}_{t}}&\text{otherwise.}\end{cases}\end{array}\] (A14)  
In the following, we show that the LLR of the SD scheme is equal to the LLR for the MO scheme when the same enrollment observations \((x_{1},\ldots,x_{t})\) and reconstruction observation \(y\) are generated by the SRAM-PUF.  
First, we can express the log-likelihood ratio (A14) as a function of \(m_{t}^{\text{\tiny SD}}\)  
\[\begin{array}{rcl}\text{LLR}^{\text{\tiny SD}}(r,m_{t}^{\text{\tiny SD}})& \overset{\Delta}{=}&\begin{cases}\log\frac{\pi_{t+1}(1+m_{t}^{\text{\tiny SD}}) }{\pi_{t}\big{(}1+t-m_{t}^{\text{\tiny SD}})}&\text{if $r=1$,}\\ \log\frac{\pi_{t}(m_{t}^{\text{\tiny SD}})}{\pi_{t+1}(t-m_{t}^{\text{\tiny SD}})} &\text{otherwise.}\end{cases}\\ &=&\log\frac{\pi_{t+1}(r+m_{t}^{\text{\tiny SD}})}{\pi_{t}\big{(}r+t-m_{t}^{ \text{\tiny SD}})}.\end{array}\] (A15)Furthermore,  
\[m_{t}^{\text{\tiny SD}} = \begin{cases}m_{t}&\text{if $c=0$ and $w_{H}(x_{1},\ldots,x_{t})\leq t/2$,}\\ &\text{or if $c=1$ and $w_{H}(x_{1},\ldots,x_{t})>t/2$,}\\ t-m_{t}&\text{otherwise,}\\ \end{cases} \tag{101}\] \[= \begin{cases}m_{t}&\text{if $r=y$,}\\ t-m_{t}&r=y\oplus 1,\end{cases}\]  
where in the last step we used the definition of \(\hat{u}_{t}\) (29) and the fact that \(r=c\oplus\hat{u}_{t}\oplus y\).  
Combining (100) and (101) we find that the log-likelihood ratio for the SD scheme is  
\[\text{LLR}^{\text{\tiny SD}}(r,m_{t}^{\text{\tiny SD}}) = \begin{cases}\text{LLR}^{\text{\tiny SD}}(y,m_{t})&\text{if $r=y$,}\\ \text{LLR}^{\text{\tiny SD}}(y\oplus 1,t-m_{t})&\text{if $r=y\oplus 1$,}\\ =&\begin{cases}\log\frac{\pi_{t+1}(y+m_{t})}{\pi_{t}(y+t-m_{t})}&\text{if $r=y$,}\\ \log\frac{\pi_{t+1}((y\oplus 1)+t-m_{t})}{\pi_{t}(y\oplus 1)+\pi_{t})}&\text{if $r=y \oplus 1$,}\\ =&\log\frac{\pi_{t+1}(y+m_{t})}{\pi_{t}(y+t-m_{t})}.\end{cases} \tag{102}\]  
In the last step we used (7). Now the last expression in (102) is equal to the log-likelihood ratio for the MO helper data scheme (28). Therefore, we conclude that the LLRs of both schemes are equal when the same enrollment observations \((x_{1},\ldots,x_{t})\), codebit \(c\) and reconstruction observation \(y\) are used.