## 5 Statistical Analysis of the Protocols Based on Subsets of Responses and ECC
### Error Management in the Cryptographic Keys
#### 5.3.3 Failure to Detect Matching Responses  
Consider the case where the errors injected in the responses of the subset are greater than threshold T. In this case, \(\rho\) responses see no matches. This type of failure creates only a limited number of possible keys K. The positions in the sequence of orderly response matching with the responses-located positions (\(\rho\)-1) and (\(\rho\)-2) are known, and only a small number of positions with state \"0\" are between the two. The number of possible keys is then also small, quickly validated with Equation (7). The method to handle the collisions and failures to detect matching responses is described below in Algorithm 9:  
* For each of the responses of the subset, at least \(\gamma_{0}\) successive responses of the full set are tested for a potential match. Assume that key K has been tested to have no more than \(\gamma_{0}\) successive 0s. This property must be validated during the random pick of K. If K has more than \(\gamma_{0}\) successive 0s, then the key can be modified by inserting a few 1s to be sure that the condition is fulfilled. We picked \(\gamma_{0}=6\) in our implementation. * Detect the responses of the subset with either zero, one, or more than one matches with the responses of the full set of responses. * List all possible keys. * Verify the authenticity of F with all possible keys with Equation (7) of Section 5.3.1. ```
1: Enter the subset of response \(\{\texttt{r}^{\prime}_{1},\ldots,\texttt{r}^{\prime}_{\texttt{j}},\ldots, \texttt{r}^{\prime}_{\texttt{i}}\}\), the set \(\{\texttt{r}_{1},\ldots,\texttt{r}_{\texttt{i}},\ldots,\texttt{r}_{\texttt{N}}\}\), \(\gamma_{0}\) and \(T\)
2: Start with \(j=1\), \(i=1\), and \(\gamma=\gamma_{0}\)
3: While \(i<N+1\): Measure hamming distance H(r\({}^{\prime}_{\texttt{j}}\), r\({}_{\texttt{i+k}}\)) for all responses r\({}_{\texttt{i+k}}\) with k \(\in\{0,\gamma-1\}\)
4: If H(r\({}^{\prime}_{\texttt{j}}\), r\({}_{\texttt{i+k}}\)) \(\leq\)T for all \(\gamma\) comparisons: Increment \(i\), \(j\) by 1, \(\gamma=2\gamma_{0}-1\)
5: [comment: to record that \(\texttt{r}^{\prime}_{\texttt{j}}\) has zero match]
6: [comment: to record that position (i+k) is matching, and that \(\texttt{r}^{\prime}_{\texttt{j}}\) has only one match]
7: [comment: to record that position (i+k) is matching, and that \(\texttt{r}^{\prime}_{\texttt{j}}\) has only one match]
8: [comment: to record that position (i+k) is matching, and that \(\texttt{r}^{\prime}_{\texttt{j}}\) has only one match]
9: [comment: to record that position (i+k) is matching, and that \(\texttt{r}^{\prime}_{\texttt{j}}\) has only one match]
10: [comment: to record that position (i+k) is matching, and that \(\texttt{r}^{\prime}_{\texttt{j}}\) has only one match]
11: [comment: to record that position (i+k) is matching, and that \(\texttt{r}^{\prime}_{\texttt{j}}\) has only one match]
12: [comment: to record that position (i+k) is matching, and that \(\texttt{r}^{\prime}_{\texttt{j}}\) has only one match]
13: [comment: to record that position (i+k) is matching, and that \(\texttt{r}^{\prime}_{\texttt{j}}\) has only one match]
14: [comment: to record the list of all positions (i+k) matching with \(\texttt{r}^{\prime}_{\texttt{j}}\)]
15: [comment: to record that position (i+k) matching with \(\texttt{r}^{\prime}_{\texttt{j}}\)]
16: [comment: to record the list of all positions (i+k) matching with \(\texttt{r}^{\prime}_{\texttt{j}}\)]
17: [comments: The positions \(\texttt{r}^{\prime}_{\texttt{j}}\) with only one matching response are considered correct. The positions \(\texttt{r}^{\prime}_{\texttt{j}}\) with zero match are caused by a failure to find the matching response. The positions \(\texttt{r}^{\prime}_{\texttt{j}}\) with two matching responses have at least 1 collision]
18: [18: 19:

## 6 Conclusions and Future Work

The suggested CRP mechanisms are directly using the digital files that must be protected and also their associated ciphertexts to generate ephemeral one-time use cryptographic keys.