## 4 Experimental Results
The total vaults in the proposed scheme will consist of \(k+1\) genuine points from them and \(r\) chaff points from random numbers, where \(r\gg k\). Other values for the chaff points in Figure 8, marked with \(X\), are made by representatives of \(r\) random numbers generated by the function \(rand()\). Figure 8 shows the distribution with 1000 random numbers (\(r=1000\)) included in the vaults, whereas Figure 9 shows a case including 200 random numbers (\(r=200\)). Although the chaff points from these random numbers are increased to 1000 in Figure 8 from 200 in Figure 9, the secret \(S\) can be successfully shared without a significant increase in execution time, even as the security to protect the secret \(S\) is enhanced. That is because this increase causes an increase in the complexity to crack the genuine points by brute force attack: to \(C(1020,9)\) from \(C(220,9)\) in these examples. the secret key. Accuracy depends on the secret \(S\) obtained from the decoding stage. The proposed scheme's accuracy is 1, and the FAR (False Accept Rate) and FRR (False Reject Rate) are 0. The complexity, which is the number of exhaustive searches for an attacker to determine \(S\), is calculated as in [PERSON] et al. [7]. In their research, the complexity is an average of \(5.3\times 1010\) evaluations for vaults containing 218 points (18 genuine, 200 chaff), and the number of elements for the polynomial reconstruction is nine. In previous studies, it was challenging to increase the degree or the number of chaff or genuine points, although doing so increased the complexity of the fuzzy vault. However, in our experiments, it is easier to increase this number. For a case with 220 points (20 genuine, 200 chaff) and nine elements of the polynomial with degree 8, there are a total of \(C(220,9)=2.8\times 1010\) combinations. Only \(C(20,9)=167,960\) of these combinations will unlock the vault and reveal the secret. Therefore, the complexity is \(C(220,9)/C(20,9)=1.7\times 1010\) evaluations for an attacker to crack the vault. For a case with 220 points and 13 polynomial elements with degree 12, the complexity is \(C(220,13)/C(20,13)=4.1\times 1015\). The complexities for degrees 8 and 12 for a case with 1020 points are \(C(1020,9)/C(20,9)=1.9\times 1016\) and \(C(1020,13)/C(20,13)=2.5\times 1024\), respectively. A higher degree or more chaff points leads to greater complexity. This greater complexity indicates the number of operations in unlocking that an attacker has to perform, and it means that an operation without the user's biometric features cannot extract a secret key from the volts within polynomial time, indicating that the security strength is high. Figure 10 shows encoding and decoding times based on the polynomial degree and the number of chaff points in the box plot. Each box shows distributions of running time in encoding and decoding among polynomial degrees 6 to 12 where chaff points are the same. The same colored box is a pair; one in encoding time and the other in decoding time. Encoding time with the lock algorithm increases with the number of chaff points for generating them, but there is no significant decoding time with the unlock algorithm based on confidence interval. Then, the number of chaff points does not affect the decoding time. Such running time between encoding and decoding shows little difference compared to the existing fuzzy vault execution time according to the increase in chaff points. Table 1 compares the performance with the latest research including execution time by [PERSON], [PERSON] et al. [38]. [PERSON]'s execution time for both encoding and decoding, like the results of other fuzzy vault schemes, increases according to the increase in the polynomial degree and the number of chaff points. The combination of evaluation for unlocking the secret key in decoding by [PERSON]'s is C(\(k+m\), \(d+1\)) in Table 2, but ours is C(\(k\), \(d+1\)) and unaffected by the number of chaff points.