Precise Authentication Watermarking Algorithm Based on Multiple Sorting Mechanisms for Vector Geographic Data

Abstract

Symmetry-breaking in security mechanisms can create vulnerabilities which attackers may exploit to gain unauthorized access or cause data leakage, ultimately compromising the integrity and security of vector geographic data. How to achieve tamper localization remains a challenging task in the field of data authentication research. We propose a precise authentication watermarking algorithm for vector geographic data based on multiple sorting mechanisms. During the watermark embedding process, a sequence of points is initially extracted from the original data, followed by embedding watermarks into each coordinate point. The embedded watermark information consists of the self-identification and ordering information of each coordinate point. Ordering information is crucial for establishing relationships among points and enhancing tamper localization. During the authentication phase, the extracted watermark information is compared with the newly generated watermark information. Self-identification information is used to authenticate addition attacks, while ordering information is used to authenticate deletion attacks. Experimental results demonstrate that the proposed algorithm achieves high precision in detecting and localizing both addition and deletion attacks, significantly outperforming the comparison method.

1. Introduction

Digital geospatial information products are increasingly integrated into our daily lives. Vector geographic data, characterized by its high precision, rich informational content, ease of storage, and suitability for automated processing, is widely utilized in various fields, including urban planning, environmental monitoring, and disaster management [1,2,3,4]. However, alongside its extensive application in the geographic information industry, data leakage and infringement incidents have become increasingly frequent, drawing attention to critical vulnerabilities in the security mechanisms protecting these data [5,6,7,8,9]. A major concern is the potential for symmetry-breaking within these security mechanisms, which can introduce exploitable vulnerabilities, allowing attackers to alter the data undetected. For example, interactions within federated authentication systems can propagate such vulnerabilities if dependency systems adopt weaker security measures, highlighting challenges in indirect authentication [10]. This can severely compromise data integrity, leading to erroneous decision-making and, in some cases, jeopardizing national security. Consequently, data authentication for vector geographic data has consistently remained a focal point of research due to its critical role in ensuring data integrity and reliability [11,12,13,14].

Existing data authentication algorithms can be divided into two categories. The first is the global authentication method. This type of method is generally inspired by cryptographic research findings and is foundational to many data security practices. Numerous cryptographic hash functions [15,16,17] are employed to create message digests, such as the family of secure hash algorithms [18,19,20]. These methods are easy to implement and have high accuracy, making them popular choices for various applications. However, the data and the hash values used for comparison are stored separately, leading to vulnerabilities that introduce a risk of tampering with the hash values used for comparison [21,22]. That is, an attacker could modify the data and then change the comparison hash value to match the hash of the modified data, effectively bypassing the authentication process.

The second category is the regional authentication method, which typically employs fragile watermarking [23,24,25,26,27]. It involves dividing the data into spatial blocks and embedding fragile watermarks in each block, thereby enabling precise block-level data authentication. The fragile watermark in these algorithms integrates seamlessly with the data, addressing the separation issue of data and hash values in the first category. For instance, the literature [28] divides vector geographic data into blocks based on the number of data points and embeds watermarks in them. This method verifies the integrity of vector data and locates tampering down to the block level. Reference [29], combining chaotic mapping techniques with vector data characteristics in GIS (Geographic Information System), proposed a fragile watermarking-based integrity authentication method for vector data. This method also effectively verifies the integrity of vector data and can determine the location and extent of tampering, but it is limited to block-level localization. Reference [30] proposed a point-constrained, block-based precise verification algorithm for vector geographic data. In the authentication process, it uses point constraints to divide vector geographic data into blocks and sorts each block to establish relationships between features. However, the block division results are highly susceptible to deletion attacks. It is apparent that the main limitation of these algorithms is that they can only localize blocks, which is not precise enough.

In summary, leveraging cryptographic research, the global authentication algorithm is easy to implement but risks data and hash value tampering. Regional authentication algorithms using fragile watermarking technology eliminate the former’s risk, but can only localize to the block level, which may not be sufficient for scenarios requiring precise data integrity checks. The point is the smallest unit of presentation in vector geographic data. Therefore, the question of how to use fragile watermarking to establish connections between points is a scientific problem.

To address these issues, this paper introduces a novel fragile watermarking authentication algorithm based on multiple sorting mechanisms. Leveraging the IEEE 754 structure [31,32] of double-precision floating-point numbers, it uncovers more watermark embedding space, facilitating the establishment of connections between points. Section 2 presents the fundamental concepts and specifics of the algorithm. The experiments are set up in Section 3. Following that, Section 4 offers the experimental results and analysis, while Section 5 engages in further discussions. Section 6 summarizes the findings.

2. Methodology

The framework of the algorithm proposed in this paper is illustrated in Figure 1. It primarily comprises two components: (1) watermark embedding and (2) watermark extracting and authentication. The process involves extracting a sequence of points from vector geographic data and embedding watermarks into each data point. The embedded watermark information includes self-identification and ordering information of the points. The self-identification information adds a unique identifier to each point, while the ordering information establishes connections among all points. Therefore, the self-identification information can determine whether a point has been subjected to an addition attack, and the ordering information can identify regions that have suffered deletion attacks.

Targeting ESRI Shapefile as the data format, storing point coordinates as double-precision floating-point numbers that provide potential space for embedding watermark information. This choice is particularly relevant given the widespread use of Shapefiles in GIS. However, the proposed algorithm is not constrained by the Shapefile format or any specific projection system. Since this method operates directly on the floating-point representation of coordinates (x, y) following the IEEE 754 standard, it is applicable to any vector geographic dataset containing floating-point coordinates. Furthermore, a projection system maps the (x, y) values from one coordinate system to another while preserving their floating-point type. Ordering information includes the sorting information of the x coordinates and y coordinates, which can confine the area of deleted points within a series of rectangular regions, thereby improving the algorithm’s effectiveness in tamper localization.

2.1. Multiple Sorting Mechanisms

The proposed algorithm employs multiple sorting mechanisms as a fundamental approach to improve the accuracy of tampering authentication in vector geographic data. Specifically, the multiple sorting mechanisms involve sorting the data points by x coordinates and y coordinates. These sorting processes constitute the foundation of the algorithm’s capability to authenticate deletion attacks.

In detail, after sorting the data points by their x coordinates, each point is embedded with information about the preceding point in the sorted sequence. Similarly, after sorting by y coordinates, each point also embeds information about the preceding point in the sorted sequence. This process establishes two chains of relationships: one in the x direction and the other in the y direction. When a point is deleted, both chains are disrupted at that location, enabling the algorithm to accurately detect and localize deletion attacks. Moreover, because disruptions occur simultaneously in both the x and y directions, their intersection naturally identifies the affected region, making the authentication of deletion attacks more accurate. This dual relationship forms the basis for the algorithm’s precise tampering authentication and represents a novel contribution to the field of vector geographic data security. Further details on the implementation of the multiple sorting mechanisms are elaborated in the subsequent subsections.

2.2. Floating-Point Number and Watermark Embedding Position

According to the IEEE 754 standard, the double-precision floating-point number consists of the sign, the exponent, and the fraction, occupying 1 bit, 11 bits, and 52 bits, respectively. This structure is fundamental for representing a wide range of values with high precision, making it ideal for geographic coordinate storage. In conjunction with analyzing the characteristics of vector geographic data, this paper uses the 1st to 16th bits of the fraction of x and y coordinates for watermark embedding. This approach ensures that the watermarking process does not significantly alter the original data representation. The structure of the floating-point number and the selection of watermark embedding positions are depicted in Figure 2.

The organization of Shapefile vector geographic data is fundamentally based on coordinate points, and different types of data arise from the various organizational rules for points. Each point contains at least x and y coordinates, stored as double-precision floating points. Vector geographic data have a certain accuracy tolerance when in use. Within the bounds of reasonable tolerance, the redundant space that does not affect data precision provides an opportunity for watermark embedding. Moreover, the choice of embedding within the floating-point representation minimizes the risk of noticeable distortion, ensuring that the embedded watermark remains imperceptible to end users and thus preserves the usability of the vector data for practical applications.

2.3. Watermark Information Generation

The generation of watermark information involves two primary components: self-identification information and sorting information. This paper calculates the self-identification information for each point using its own x and y coordinates. The self-identification information involves converting specific bits of the coordinates into a unique identifier for each point, which is vital for detecting addition attacks. Assuming the watermark length is N, the 17th to 52nd bits of the fraction of x and y coordinates are converted to uint64 integers, denoted as a and b, respectively. This conversion is essential for facilitating the manipulation of bits to embed watermark information effectively. The watermark information can be generated using the following formula:

$$w=\mathrm{m}\mathrm{o}\mathrm{d}\left(\mathrm{b}\mathrm{i}\mathrm{t}\mathrm{x}\mathrm{o}\mathrm{r}\right(a,b),2^N)$$

(1)

where $\mathrm{m}\mathrm{o}\mathrm{d}\left(\right)$ represents the modulo operation, and $\mathrm{b}\mathrm{i}\mathrm{t}\mathrm{x}\mathrm{o}\mathrm{r}\left(\right)$ is the bitwise XOR operation. The value range of $w$ is [0, $2^N$−1]. When generating self-identification information, the watermark length N is set to 16. When generating sorting information, the watermark length N is set to 8. The process of generating watermark information ensures that each point retains a unique identification, which is vital for subsequent authentication phases.

Regarding addition attacks, the proposed method significantly improves the accuracy in detecting added points. This improvement is achieved by utilizing 16 bits for self-identification, embedding the information directly into the binary representation of the coordinates. In contrast, traditional methods, which often operate on textual representations and embed watermark information into specific decimal places, typically have a much lower watermark capacity (e.g., 1 bit per digit). Such limited capacity makes it challenging for these methods to achieve the precision of the proposed approach, which utilizes 2¹⁶ unique possibilities for self-identification.

2.4. Watermark Information Embedding

The primary task of watermark information embedding is to embed the generated watermark information into the embedding position. For the x coordinate, its embedding position is used to embed 16-bit length self-identification information, while for the y coordinate, it is used to embed sorting information. This dual-purpose embedding approach optimizes the use of available bit space, maximizing the watermark’s effectiveness.

The embedding of sorting information involves sorting by x coordinates and y coordinates. Specifically, for the embedding position of y coordinates, the 1st to 8th bits are used to embed watermark information sorted by x, and the 9th to 16th bits are used to embed watermark information sorted by y. This embedding strategy differs significantly from traditional watermarking algorithms, which typically process coordinate values as text and embed watermark information into specific decimal places. Instead, the proposed algorithm embeds the watermark directly into the binary representation of coordinate values. This approach minimizes the impact of watermark embedding on the data’s precision. It is important to note that the watermark embedding process involves converting the watermark information into binary digits and replacing the corresponding bits in the floating-point representation of the coordinate values. This sorting mechanism is crucial, as it establishes a relational framework among the data points, enhancing the ability to detect any alterations. The embedding operation uses bit replacement. For a point (x, y), where its self-identification information is denoted as $w_1$, then

$$x^{\prime }=\mathrm{b}\mathrm{i}\mathrm{t}\mathrm{s}\mathrm{e}\mathrm{t}\left(x,1,16,w_1\right)$$

(2)

where $\mathrm{b}\mathrm{i}\mathrm{t}\mathrm{s}\mathrm{e}\mathrm{t}\left(\right)$ sets the bits at a range of specific bits, with the second and third parameters being the starting and ending positions of the bit, respectively.

For embedding sorting information, it is a precondition to sort the data first. After sorting by x from smallest to largest, the y of each coordinate point is embedded with the sorting information denoted as $w_2$ from the previous coordinate point.

$$y^{\prime }=\mathrm{b}\mathrm{i}\mathrm{t}\mathrm{s}\mathrm{e}\mathrm{t}\left(y,1,8,w_2\right)$$

(3)

Similarly, the sorting information of the previous point, sorted by y, is denoted as $w_3$. After sorting by y from smallest to largest, the watermarked $y^{\prime }$ can be expressed by the following equation.

$$y^{\prime }=\mathrm{b}\mathrm{i}\mathrm{t}\mathrm{s}\mathrm{e}\mathrm{t}\left(y,9,16,w_3\right)$$

(4)

2.5. Watermark Information Extraction and Authentication

Watermark extraction is the reverse process of watermark embedding. The authentication process involves comparing the extracted watermark information with the re-calculated watermark information. For a coordinate point ($x^{\prime }$, $y^{\prime }$), this paper classifies the types of attacks based on the position changes of each point in the vector geographic data into two categories: addition attacks and deletion attacks. Other types of attacks can be considered combinations of these two types. For example, a modification attack can be regarded as a combination of deletion followed by addition.

2.5.1. Addition Attack Authentication

Extract the self-identification information $w_1^{\prime }$ by the following formula.

$$w_1^{\prime }=\mathrm{b}\mathrm{i}\mathrm{t}\mathrm{g}\mathrm{e}\mathrm{t}\left(x^{\prime },1,16\right)$$

(5)

Like $\mathrm{b}\mathrm{i}\mathrm{t}\mathrm{s}\mathrm{e}\mathrm{t}( )$, $\mathrm{b}\mathrm{i}\mathrm{t}\mathrm{g}\mathrm{e}\mathrm{t}( )$ retrieves bits at a specific range from a double-precision floating-point number and interprets them as an integer.

If $w_1^{\prime }$ is equal to $w_1$, then the point ($x^{\prime }$, $y^{\prime }$) is an original point. Otherwise, ($x^{\prime }$, $y^{\prime }$) is an added point. Therefore, the authentication of addition attacks can be precise down to the individual point.

2.5.2. Deletion Attack Authentication

This mainly involves extracting the sorting information from the points, taking into account the points that have passed the addition attack authentication. For the point ($x^{\prime }$, $y^{\prime }$) sorted by x, extract the sorting information $w_2^{\prime }$ of its previous point ($x_{pre}^{\prime }$, $y_{pre}^{\prime }$) by the following equation.

$$w_2^{\prime }=\mathrm{b}\mathrm{i}\mathrm{t}\mathrm{g}\mathrm{e}\mathrm{t}\left(y^{\prime },1,8\right)$$

(6)

If $w_2^{\prime }$ is identical to $w_2$, it can be inferred that no deletion attack has occurred between the lines ${x=x}_{pre}^{\prime }$ and $x=x^{\prime }$. Conversely, any discrepancy between $w_2^{\prime }$ and $w_2$ indicates the presence of a deletion attack.

Similarly, for the sorting information $w_3^{\prime }$ that is sorted by y can extracted as below.

$$w_3^{\prime }=\mathrm{b}\mathrm{i}\mathrm{t}\mathrm{g}\mathrm{e}\mathrm{t}\left(y^{\prime },9,16\right)$$

(7)

If $w_3^{\prime }$ is identical to $w_3$, it can be concluded that no deletion attack has occurred between the lines ${y=y}_{pre}^{\prime }$ and $y=y^{\prime }$. Conversely, any discrepancy between $w_3^{\prime }$ and $w_3$ would indicate the presence of a deletion attack.

Finally, the rectangular block where points have been deleted can be pinpointed by intersecting the horizontal and vertical intervals. This allows for the precise localization of tampering within the data. As shown in Figure 3, the deletion attack occurred in the textured area. Additionally, if multiple rectangular blocks with adjacent boundaries are detected, they can be merged into a single larger rectangular block for a more comprehensive identification of the tampered area.

3. Experiments

3.1. Datasets

To validate the effectiveness of the proposed algorithm, this section uses point data in the ESRI Shapefile format for verification. Since the fundamental storage form inside point, polyline, and polygon features is the point, the principle of processing them in the watermarking algorithm is the same. The dataset contains 1000 point features, as shown in Figure 4. Its geographic coordinate system is the China Geodetic Coordinate System 2000. The data employ the 3-degree Gauss–Krüger projection [33,34], centered at 111° E longitude, and are with the China Geodetic Coordinate System 2000 as the geographic coordinate system.

3.2. Experiment Design and Implementation

The evaluation focuses on three main aspects: imperceptibility, addition attack authentication, and deletion attack authentication. Hou’s method [26] was chosen as a benchmark for comparison due to its established use in geospatial data watermarking.

3.2.1. Comparison Algorithm

Hou’s method serves as a key benchmark for our experiments. This method embeds watermark information by altering the text-based representation of coordinates, focusing specifically on decimal places. It employs an optimized k-means clustering approach to group points into clusters, referred to as entity groups, which serve as units for watermark embedding. The watermarking process uses the LSB (Least Significant Bit) technique to modify coordinates within each cluster. The parameters q and k are critical to Hou’s method. The parameter q is the position of decimal place to be adjusted for watermark embedding. In our experiments, it is set to 4, indicating that the 4th decimal places is altered to embed the watermark. The parameter k is the number of clusters used in the k-means clustering process and is set to 10.

Each entity group receives a unique watermark generated using secure integrity features combined with chaotic mappings. During the watermark detection phase, the embedded watermarks are extracted from each group and compared to detect any tampering. Figure 5 illustrates the clustering results obtained using Hou’s method, which was applied to the experimental point dataset described in Section 3.1.

3.2.2. Imperceptibility

Imperceptibility can be seen as the perceptual similarity between the watermarked vector geographic data and the original data [35,36], that is, the invisibility of the watermark. Here, changes between the watermarked data and the original data at corresponding coordinates are statistically analyzed. The corresponding vertices are denoted as ($x_i^{\prime }$, $y_i^{\prime }$) and ($x_i,y_i$). The minimum and maximum values (absolute values) of changes in the x and y coordinates, as well as the minimum and maximum distance, are calculated. The distance $d$ can be expressed by the formula below.

$$d=\sqrt{{\left(x_i^{\prime }-x_i\right)}^2+{\left(y_i^{\prime }-y_i\right)}^2}$$

(8)

3.2.3. Addition Attacks

Regarding addition attacks, a certain number of vertices are randomly added, ranging from 1% to 10% of the total number of points, with an interval of 1%. We used Precision, Recall, and F1-score as the performance metrics for evaluating detection accuracy. The definitions and calculation formulas for these metrics are as follows:

$$\mathrm{Precision} ={\displaystyle \frac{\mathrm{T}\mathrm{P}}{\mathrm{T}\mathrm{P}+\mathrm{F}\mathrm{P}}}$$

(9)

$$\mathrm{Recall} ={\displaystyle \frac{\mathrm{T}\mathrm{P}}{\mathrm{T}\mathrm{P}+\mathrm{F}\mathrm{N}}}$$

(10)

$$\mathrm{F}1-\mathrm{score} =2\times {\displaystyle \frac{ \mathrm{Precision} \times \mathrm{Recall} }{ \mathrm{Precision} + \mathrm{Recall} }}$$

(11)

where TP is True Positive, FP is False Positive, and FN is False Negative. Precision measures the proportion of correctly detected added points among all detected points, indicating the accuracy of the algorithm. Recall measures the proportion of correctly detected added points among all actually added points, reflecting the algorithm’s sensitivity. The F1-score is the harmonic mean of Precision and Recall, ranging from 0 to 1, where 1 indicates perfect detection performance and 0 indicates a complete failure in detection.

3.2.4. Deletion Attacks

For deletion attacks, points were removed starting from the data center, with deletion rates set from 1% to 10% at 1% intervals. Due to the nature of deletion attacks, where the exact coordinates of removed points cannot be directly detected, the tampering localization is constrained to an area. This experiment aimed to test the algorithm’s ability to accurately localize and identify areas where points were removed. We used two metrics:

• Detection Coverage Area Rate (DCAR): This metric quantifies the relative size of the detection area, called the Detection Coverage Area (DCA), as a proportion of the area of the minimum bounding polygon of the original dataset. A lower DCAR indicates a more precise detection;
• Detection Coverage Rate (DCR): This metric represents the proportion of the actual deleted points that fall within the DCA. A higher DCR implies that the algorithm successfully covers more of the deleted points.

Hou’s method only provides localization down to the level of clusters, as it identifies the deletion within the group of points but cannot specify the exact tampered region. To make a fair comparison, we adapted Hou’s method to use the minimum bounding polygon (convex hull) of each cluster as its DCA.

4. Results and Analyses

4.1. Results of Imperceptibility

The imperceptibility results are shown in

Table 1. Imperceptibility experimental results.

Indicators	Min Change in x	Max Change in x	Min Change in y	Max Change in y	Min Change in Distance	Max Change in Distance
Values/m	6.98 × 10−10	7.03 × 10−6	4.19 × 10−9	2.97 × 10−5	4.00 × 10−8	2.97 × 10−5

. All of the indicators show very low values. For instance, the maximum change in y and the maximum change in the distance are both 2.97 × 10⁻⁵ m, which equates to 29.7 μm and is considered negligible. These values fall within acceptable limits for practical geospatial data applications, ensuring that the watermark remains imperceptible.

4.2. Authentication Results of Addition Attacks

Figure 6 presents a visual comparison between the proposed method and Hou’s method in detecting addition attacks at various ratios. The graph shows how the performance of both methods varies as more points are added to the original dataset. Specifically, the proposed method maintains consistent accuracy across all tested addition ratios, while Hou’s method exhibits a relatively low level in precision and F1-score as the addition ratio increases.

Specifically, the proposed method accurately identifies all added points without misclassifying any original points, resulting in perfect precision and recall. The detection performance remains stable even when the addition ratio reaches 10%, highlighting the method’s effectiveness in distinguishing added points from the original dataset. On the other hand, Hou’s method suffers from a considerable number of false positives, especially at higher addition ratios. The clustering-based approach leads to several original points being incorrectly flagged as added. This limitation reduces the method’s precision and demonstrates its inherent weakness in handling dense tampering scenarios where fine-grained accuracy is crucial.

To further illustrate these findings, Figure 7 and Figure 8 provide examples of the 1% addition ratio. Figure 7 shows that the proposed method successfully detects all ten added points, marked in red, with no false positives, confirming its precise detection capabilities. In contrast, Figure 8 reveals that Hou’s method misclassifies several original points, resulting in numerous false positives. This example underscores the impact of clustering granularity on detection accuracy.

4.3. Authentication Results of Deletion Attacks

Figure 9 illustrates the overall performance of the proposed method and Hou’s method in detecting deletion attacks, comparing both DCR and DCAR. The graph shows that both methods consistently achieve a DCR of 100%, indicating that they are able to detect all deleted points effectively. However, a clear distinction emerges in their DCAR values: the proposed method exhibits a controlled and gradual increase, maintaining a relatively low DCAR, while Hou’s method experiences significant fluctuations and higher DCAR values, reflecting less precise localization.

Focusing on the details, the DCAR reveals localization accuracy. At a deletion ratio of 3%, Hou’s method’s DCAR sharply rises to 100%, indicating that deletions affect the entire clustering structure, causing a significant expansion of the detection area. This result suggests that Hou’s method is more prone to inaccuracies as deletion intensity increases. In contrast, the proposed method maintains a controlled and smaller DCAR, demonstrating better precision in localizing deleted areas, even under higher tampering levels.

To illustrate these findings further, Figure 10 and Figure 11 provide a specific example of the 1% deletion ratio. Figure 10 shows that the proposed method tightly confines the detection area, covering all deleted points while minimizing unnecessary coverage. On the other hand, Figure 11 reveals that Hou’s method generates a much larger detection area, extending beyond the actual tampered regions and underscoring the limitations of its approach.

Overall, the proposed algorithm exhibits strong imperceptibility and accurately authenticates addition attacks with perfect precision. For deletion attacks, it achieves precise localization with a smaller detection area compared to Hou’s method, which suffers from significant false detection regions due to its cluster-based limitations.

5. Discussion

5.1. Limitation and Extension

The proposed algorithm employs a sorting-based approach for watermark embedding, which has both strengths and inherent limitations. One significant advantage is the algorithm’s robust capability to detect and localize deletion attacks by maintaining the ordering of data points. However, this approach is particularly sensitive to deletions that occur near the boundaries of the data’s bounding box. Deleting these critical boundary points can severely impact the reference information needed for sorting, thereby reducing the accuracy of tampering detection. However, this issue is not unique to the proposed algorithm but is a common challenge for all reference-dependent methods. Because the authentication of deletion attacks inherently relies on reference points to determine the region of tampering. Despite this, the proposed algorithm demonstrates significant improvements over existing methods by effectively reducing the authenticated area, as shown in Figure 10 and Figure 11. This limitation underscores the need for careful data management when applying the algorithm to datasets where boundary points are susceptible to change.

Moreover, to further improve the precision of tampering localization, an extension of the sorting mechanism could be considered. By incorporating additional sorting along diagonal directions, such as $y=x$ and $y=-x$, the algorithm could narrow the detection area even further. This would involve sorting points along the straight lines $y=x$ or $y=-x$ and embedding the corresponding self-identification and ordering information. However, this improvement would come at the cost of additional precision loss in the data. Thus, there is a trade-off between achieving finer localization and maintaining data accuracy. If a higher tolerance for data precision loss is acceptable, this extension could be beneficial, but it would require careful evaluation of the specific application needs.

5.2. Combination Attacks

As mentioned in Section 2.5, any attack on vector geographic data can be seen as a combination of addition and deletion attacks. Thus, we have focused on evaluating the algorithm’s performance against both addition and deletion attacks in the experiments. Here, we append a discussion on combination attacks. Its core steps involve first authenticating the addition attack, then removing the points identified as added to avoid interference with the subsequent deletion authentication, and finally authenticating the deletion attack. As shown in Figure 12, we modified 28 points by moving them from the bottom-left corner to the top-right corner. This process can be considered as first deleting these 28 points and then adding 28 new points, representing a combination attack of both addition and deletion.

The results of authenticating the addition attacks are shown in Figure 13. The algorithm successfully identified all 28 added points in the top-right corner, demonstrating its effectiveness in detecting addition attacks. After removing the authenticated added points, the algorithm was then used to detect the deletion attacks. As shown in Figure 14, the algorithm correctly identified all 28 deleted points in the bottom-left corner, further validating its capability to handle deletion attacks. In summary, the proposed algorithm is effective in authenticating combination attacks by sequentially addressing addition and deletion operations. These results confirm its effectiveness in handling complex tampering scenarios.

6. Conclusions

This paper presents a fragile watermarking algorithm for vector geographic data, utilizing multiple sorting mechanisms to embed delicate watermark information. The algorithm integrates self-identification and ordering information, ensuring that each data point has a unique identity and establishing structured relationships among all points. The experimental results demonstrate that the proposed method achieves high accuracy in detecting and precisely localizing both addition and deletion attacks, significantly outperforming the comparison method by minimizing false detections. The algorithm’s effectiveness makes it well-suited for practical applications in data monitoring, version control, and copyright protection of vector geographic datasets. Future research could explore enhancements, such as incorporating additional sorting directions, including $y=x$ and $y=-x$, to further narrow detection areas. These enhancements should carefully balance the trade-offs between localization precision and potential data accuracy loss.