# Post-Quantum Biometric Authentication Based on Homomorphic Encryption and Classic McEliece
## 4 Practical Realization of the Proposed Biometric Protection Scheme for Facial Recognition
### Experimental Results
#### 4.2.2 Recognition Performance  
For the evaluation of the recognition performance, we followed the FVC (Fingerprint Verification Competition) protocol [41]. For the genuine comparisons, each sample of the same individual is compared to the remaining samples of the same individual. For the impostor comparisons, the first sample of each individual is compared with the first sample of the remaining individuals. Symmetric comparisons are removed to avoid correlation. In the FERET database, the number of genuine comparisons was 52,708 and the number of impostor comparisons was 483,636. In the LFW database, the number of genuine comparisons was 235,254 and the number of impostor comparisons was 15,487,395.  
The recognition performance in the protected domain was evaluated by considering the Classic McEliece parameter sets that provide security levels of 128 and 256 bits. These parameter sets are mceliece348864 (with \(n=3488\), \(t=64\), \(m=12\), and \(k=2720\)) and mceliece688128 (with \(n=6688\), \(t=128\), \(m=13\), and \(k=5024\)), respectively. Embeddings of 384 bits (48 bytes) were divided into two 192-bit segments for mceliece348864 and into one 384-bit segment for mceliece6688128. Therefore, \(z_{i}=192\) for mceliece348864 and \(z_{i}=384\) for mceliece6688128. The padding was considered to have Hamming weight \(t+1\) in order to ensure that the Hamming weight of the substrings \(eb_{i}\) was greater than \(t\). In this way, paddings with 65 1s were generated for mceliece348864 and paddings with 129 1s were generated for mceliece6688128.  
The recognition performance was evaluated in terms of accuracy, defined as the ratio in percentage between the number of true recognition decisions and the total number of recognition decisions, False-Acceptance Rate (FAR) and False-Rejection Rate (FRR). Table 1 shows the recognition performance of FaceNet embeddings represented in floating-point and binary in the unprotected domain. Comparisons of floating-point embeddings were performed by using the Euclidean distance. Comparisons of binary embeddings were performed by using the Hamming distance. Although some information is lost by the binarization process, accuracy, FAR and FRR are not largely affected. Table 1 also includes the recognition performance when the proposed protection based on Classic McEliece is applied. The accuracy, FAR and FRR results prove that the recognition performance was improved. For the genuine distribution, the same transformation (same secret padding and same random positions) was applied for embeddings from different samples from the same individual. Then, the _DECODE_ algorithm could recover \(b_{i}^{R}\oplus b_{i}^{V}\) from \(cb_{i}^{R}\oplus cb_{i}^{V}\) when the Hamming weight of \(eb_{i}^{R}\oplus eb_{i}^{V}\) was smaller than or equal to \(t\). For the impostor distribution,different transformations (different secret padding and different random positions) were applied for embeddings of different individuals. Then, the _DECODE_ algorithm could not decode \(cb_{i}^{R}\oplus cb_{i}^{V}\) because \(eb_{i}^{R}\oplus eb_{i}^{V}\) had Hamming weight greater than \(t\). For the accuracy indicator, true-positive decisions were obtained from genuine comparisons that could be decoded (impostor comparisons were always true-negative decisions). In the LFW database, almost all the genuine comparisons were decoded. For this reason, the accuracy was approximately 100%. FAR was 0% because impostor comparisons generated decoding fails. FRR was the same as in the unprotected domain because all genuine comparisons could be decoded so the recognition performance was not affected.