## 4 Experimental Results and Analysis
### Security Evaluation  
In order to evaluate the security of the proposed method, in this section we evaluate the randomness of the encrypted map. If the randomness of the encrypted map is high, it will also be difficult to attack. The randomness of the encrypted map is measured by its entropy. Thus, we calculate the entropy of the encrypted map to evaluate the security of the proposed method. From Section 3, we can see that the entropy of the encrypted object is dependent on the secret key \(\mathbf{K}\) and the value of \(|\mathbf{P}_{i}|\). Both \(\mathbf{K}\) and \(|\mathbf{P}_{i}|\) are discrete random variables. Thus, the entropy \(H_{\mathbf{P}_{i}}\) of the encrypted object \(\mathbf{E}_{i}\) is the sum of the entropies of the random variables:  
\[H_{\mathbf{P}_{i}}=H(\mathbf{K})+H(|\mathbf{P}_{i}|)= |\mathbf{K}|.\log_{2}{(\frac{1}{|\mathbf{K}|})}+|\mathbf{P}_{i}log_{2}{( \frac{1}{|\mathbf{P}_{i}|})} \tag{12}\]  
Furthermore, the entropy \(H_{\mathbf{L}}\) of the encrypted map from the original map layer \(\mathbf{L}\) will be the sum of the entropies of the encrypted objects:  
\[H_{\mathbf{L}}=\sum_{i=1}^{|\mathbf{L}|}H_{\mathbf{P}_{i}} \tag{13}\]  
Clearly, \(H_{\mathbf{L}}\) is dependent on the values of \(|\mathbf{L}|\) and \(\mathbf{K}\); the key value \(\mathbf{K}\) is a random variable dependent on a user's key input. If the value of \(\mathbf{K}\) is fixed, \(H_{\mathbf{L}}\) is only dependent on the value of \(|\mathbf{L}|\). As a result, if \(|\mathbf{L}|\) is high, the entropy is high. For example, the railway map of Scotland has \(|\mathbf{L}|\)= 9724 and \(H_{\mathbf{L}}\) = 128,817 \(dB\), but the natural map of Scotland has \(|\mathbf{L}|\) = 99,835 and \(H_{\mathbf{L}}=1.66\times 10^{6}\)\(dB\). In the vector maps of Scotland in Table 1, the entropy of the proposed method ranges from \(1.3\times 10^{5}\)\(dB\) to \(6.88\times 10^{6}\)\(dB\) with \(|\mathbf{L}|\in[9724,\ 372,\ 138]\).  
Figure 9: (**a**) Original road map, and (**b**) encrypted road map.  
The method of [PERSON] [10] only uses the AES-256 cipher operator to encrypt the data stream of a shape-file. Thus, the entropy of this method is dependent on the length of the secret key and the length of the data stream. In the work of [PERSON] [12], about 70% of polylines/polygons are selected for the encryption process. [PERSON] used a common secret key to encrypt all DC values in a vector map in the DWT domain. The length of the secret key was 512 bits. Thus, the entropy of this method is dependent on the length of the secret key. We applied the methods of both [PERSON] and [PERSON] for the GIS vector maps of Scotland for comparison. The entropy of our method was higher than that of [PERSON] (\(1.2\times 10^{5}\) vs. \(6.6\times 10^{6}\)\(dB\)), while the entropy of [PERSON]'s method was fixed at 4608 \(dB\) for every GIS vector map (Table 2). In conclusion, our method offers more security than previous methods. Figure 10 shows the difference between the entropy of our method and the entropy of the methods of [PERSON] and [PERSON] according to the number of objects. The entropy of our method is much higher than that of [PERSON] or [PERSON].