A Generative Steganography Based on Attraction-Matrix-Driven Gomoku Games

1  Introduction

Steganography is a technique for covert communication [1], aiming to produce stego images imperceptible to human perception and statistical analysis, thereby enhancing security by concealing both the content and existence of information.

Recently, due to image steganography’s high embedding capacity and advancements in optimization algorithms and image processing techniques [2,3], image steganography has become a prominent research area within the broader field of information hiding. Traditional modification-based methods, such as Discrete Cosine Transform (DCT) [4] and Discrete Wavelet Transform (DWT) [5], embed data by altering pixel values or frequency-domain coefficients. However, these methods often generate detectable statistical artifacts, making them susceptible to modern steganalysis techniques [6,7].

To address these issues, recent advancements in generative models have led to the emergence of generative steganography. This technique embeds secret message directly into the image generation process, producing stego images from scratch. By avoiding modifications to existing images, this approach enhances information concealment and reduces statistical detectability compared to traditional methods. Depending on how the secret message is embedded [8], generative stego images are typically classified as either pixel-based or latent-space-based methods.

Pixel-based generative steganography embeds secret message by generating images with specific pixel values [9–11]. These methods show a certain degree of resistance to statistical detection. However, they heavily rely on exact pixel values, which makes them vulnerable to modification-based attacks. In contrast, latent-space-based methods embed secret message within the latent space of generative networks [12–16]. These methods enhance robustness by avoiding direct manipulation of pixel values. Although the generated carriers statistically resemble the training data, they often lack meaningful semantic content. Furthermore, repeated transmission of such synthetic samples may raise suspicion, thereby limiting their practical applicability.

Researchers have increasingly explored behavioral steganography to address the limitations of conventional generative steganography. This emerging subfield generates stego images by mimicking realistic interactive behaviors, such as gameplay or social interactions. In doing so, it aims to enhance communication plausibility and reduce the likelihood of detection. Prior studies [17–21] have shown that these techniques can effectively evade conventional statistical steganalysis. However, a comprehensive analysis [17,22] reveals a significant limitation. Existing behavior-based systems are constrained to discrete environments and depend on limited action strategies. These constraints reduce encoding diversity and lead to repetitive behavioral patterns. Consequently, these systems encounter two major challenges. First, the embedding capacity of each carrier is inherently limited. Second, unrealistic behavioral patterns elevate the risk of anomaly detection, potentially compromising communication security.

This paper introduces a novel generative steganography approach that embeds secret message through Gomoku gameplay. In the embedding phase, a Gomoku-playing model translates secret message into game moves, enabling covert communication. During the extraction phase, the gameplay is reconstructed from images and decoded using predefined move-encoding rules. The main contributions of this work are as follows:

•   A Steganographic Framework Based on Relative Move-Position Encoding: We propose a novel method that integrates steganography with Gomoku gameplay. The method embeds secret message by leveraging spatial dependencies between player moves. Unlike previous behavior-based methods [17,19], our method circumvents the constraints of fixed, discrete action spaces. Gomoku provides a significantly larger game tree, with approximately 3225 possible board states, and supports long, flexible move sequences. These characteristics substantially enhance data embedding capacity and encoding diversity.

•   A Gomoku-Playing Model with Attraction Matrix: We design a Gomoku-playing model that combines a basic strategy network with an attraction matrix to balance gameplay authenticity and data embedding requirements. To minimize spatial bias when selecting suboptimal moves, the model first constructs a two-dimensional attraction field centered on previous moves. It then applies geometric constraints to guide the placement of subsequent moves. This design enhances the naturalness of gameplay and effectively conceals the steganographic channel, thereby improving the robustness and real-world applicability of the proposed approach.

2  Related Work

2.1 Generative Steganography

Image generative steganography is an important subfield within information security. It embeds secret message using generative model. Unlike traditional steganographic techniques, this method embeds secret message directly into the input features of generative models. As a result, the models generate stego images that appear visually natural and realistic, effectively eliminating the statistical artifacts and reducing attackers’ suspicion.

According to recent studies [23], generative steganography can be broadly classified into two categories: pixel-defined generative steganography and latent-space-mapped generative steganography.

Pixel-defined generative steganography refers to methods that embed secret message directly into pixel values. Yang et al. [9] proposed a representative method based on Pixel Convolutional Neural Networks (PixelCNN). Their method models pixel dependencies by estimating the probability distribution of each pixel in the range [0, 255], conditioned on previously generated pixels. To embed data, they adopted a rejection sampling strategy. The process starts from the top-left corner of a blank image and proceeds pixel by pixel to the bottom-right. Appropriate pixel values are selected at each step based on the sampling rule, ultimately forming a complete stego image.

Building on the previous study, Zhang et al. [24] proposed Pixel-Stega, an enhanced method for pixel-defined generative steganography. This method replaces PixelCNN with the more advanced PixelCNN++. It also incorporates a sampling strategy based on arithmetic coding. In Pixel-Stega, secret message guide the generation of pixel values. This guidance is applied jointly with the predicted probability distribution for each pixel. The integration of arithmetic coding enables adaptive bit allocation. More bits are assigned to high-entropy regions, while fewer are allocated to low-entropy regions. This adaptive embedding strategy improves the visual quality of generated images. At the same time, it maintains a fixed payload, making the method practical for controlled steganographic communication.

Notably, pixel-defined generative steganography methods generate images entirely from scratch rather than modifying existing cover images. This characteristic makes them highly resistant to steganalysis techniques that rely on detecting statistical artifacts. However, these methods tend to suffer from limited robustness. The generated images are often vulnerable to distortion-based attacks during transmission.

Natural images exhibit complex, high-dimensional distributions. Neural networks can learn mappings between these image-space distributions and simpler latent distributions, such as high-dimensional Gaussian distributions [25]. Leveraging this insight, latent-space-based generative steganography establishes a correspondence between secret message and latent vectors. The secret message is first embedded into a latent representation, which is then used to generate the stego image.

For example, Liu et al. [26] embedded secret message into latent vectors and used diffusion models to generate stego images from these vectors. Su et al. [12] proposed a distribution-preserving data modulator. This modulator embeds secret messages while maintaining the original distribution, allowing imperceptible communication. They also introduced a generic secret message extractor to ensure reliable messages recovery. Similarly, Hu et al. [27] embedded secret message into low-dimensional noise vectors. These vectors were fed into a Deep Convolutional Generative Adversarial Network (DCGAN) to produce stego images. A convolutional neural network (CNN) was then used to extract and reconstruct the original messages.

Overall, latent-space-based approaches tend to exhibit greater robustness against perturbations, as they rely on sampling from latent distributions conditioned on the secret content [28,29]. However, these methods also have inherent limitations. Image distortions can induce latent vector drift, thereby reducing the accuracy of information extraction. Moreover, their embedding capacity is generally lower than that of pixel-defined methods.

2.2 Behavioral Steganography

Behavioral steganography is a technique that conceals secret message by designing specific behaviors or leveraging social attributes. Its primary goal is to produce covert actions that appear indistinguishable from normal behavior, thereby evading steganalysis. Depending on the application scenario, existing research in this field can be broadly classified into two categories: methods based on social network behavior and those based on game behavior. Both categories generate behavior sequences that conform to contextual rules. This method avoids directly modifying existing carriers and enhances the concealment of the secret message.

Steganography based on social network behaviors facilitates covert communication by encoding secret message within behavioral features. For instance, Zhang [30] introduced an innovative steganographic approach wherein participants exchange secret messages through the act of “liking” each other’s posts. Similarly, Wu et al. [31] employed undirected graphs to embed secret messages and directed graphs to represent network topology. Gilbert et al. [32] developed steganographic content that closely resembled authentic online conversations to conceal secret messages. These techniques leverage the high frequency and variability inherent in social interactions. Nonetheless, their data embedding capacity is frequently limited by platform regulations and the inherent difficulties in precisely modeling user behavior.

In contrast, game behavior steganography embeds secret message into in-game actions or states governed by game mechanics. For example, in bridge [33], Go [34], and maze games [35], secret message is conveyed through move choices and path planning. In chess [19], messages are embedded through piece positions and board colors. Tetris [36] maps its seven block shapes to the digits 0–6 for messages embedding. Minesweeper [37] and puzzle games [22,38] embed data through mine placements and puzzle shapes, respectively. These techniques take advantage of games’ complex structure and behavioral randomness to improve concealment. Nevertheless, they typically offer low embedding capacity. The applicability and generalizability of specific game environments often limit their real-world deployment.

3  Proposed Method

Gomoku is a classic two-player, zero-sum board game. In the game, players place black and white stones on the intersections of a 15×15 or 13×13 grid. The goal is to be the first to align five consecutive stones horizontally, vertically, or diagonally. According to standard rules, the black player moves first. Both players are required to place stones only on unoccupied intersections.

Game-tree complexity analysis estimates Gomoku’s strategic space at approximately 3225. This value exceeds the complexity of traditional chess but remains lower than that of Go. This high complexity results in two notable properties. First, the vast state space offers substantial redundancy that can be exploited for information hiding. Second, the game’s well-defined and straightforward rules facilitate technical implementation.

Leveraging these characteristics, this study utilizes Gomoku game records as the carrier for information hiding. The overall framework of the proposed steganographic algorithm is illustrated in Fig. 1.

Figure 1: Overall framework of the proposed steganographic algorithm

As shown in Fig. 1, the proposed framework transforms Gomoku gameplay into a steganographic medium through three core modules: (1) the design of a Gomoku game model; (2) the generation of steganographic game records embedding secret message; (3) the extraction of secret message from the game records.

3.1 Gomoku Game Model

Gomoku gameplay exhibits strong local spatial correlation. Effective moves are typically concentrated around existing pieces on the board. However, empirical analysis shows that even after training, game models tend to generate spatially dispersed candidate moves when the optimal move is excluded. These moves often display spatial distributions that differ significantly from those observed in professional-level strategies. For example, in a specific board configuration, the top two optimal moves may have win rates of around 0.76 and 0.19, respectively. In contrast, once these moves are removed, all remaining candidates may have win rates below 10−10. In such cases, the model might select moves far from the central cluster of pieces. This occurs because reinforcement learning models are primarily trained to identify the best move and often neglect others. As a result, they don’t explore suboptimal moves well and can produce biased results. Because of this, the move distributions generated by the model under steganographic constraints often differ significantly from the locally clustered patterns seen in expert gameplay. These deviations may reveal abnormal behaviors and increase the risk of covert communication being detected. To address this issue, we propose a Gomoku model that incorporates an attraction matrix. This matrix guides move selection and reduces unusual spatial patterns caused by steganographic constraints.

3.1.1 Basic Gomoku Game Network Construction

We trained a reinforcement learning-based neural network for Gomoku gameplay based on the project by Hu [39], capable of generating rule-compliant move strategies through extensive self-play simulations. This model serves as the foundation for enabling strategic communication via in-game actions.

The game environment is represented as a 13×13 two-dimensional matrix, where each cell corresponds to one of three possible states: empty, black stone, or white stone. The core architecture is based on a convolutional neural network (CNN), which comprises multiple convolutional layers for local feature extraction, batch normalization layers to enhance training stability, and ReLU activation functions to introduce non-linearity. The extracted features are subsequently passed through fully connected layers to generate a probability distribution over possible moves. By incorporating residual connections, the model enhances its generalization ability and mitigates training issues such as vanishing gradients.

The model is trained using self-play reinforcement learning, combining policy gradient methods are combined with Monte Carlo Tree Search (MCTS) to optimize move strategies through large-scale simulated gameplay. The overall system integrates a value network that estimates the probability of winning and a policy network that guides move selection. The model architecture is illustrated in Fig. 2.

Figure 2: Network architecture of the Gomoku game model

3.1.2 Attraction Matrix Construction

To mitigate game space anomalies induced by steganographic tasks, we introduce an attraction matrix mechanism to optimize the spatial distribution of candidate actions. The construction process comprises four sequential steps: (1) board state representation, (2) spatial influence modeling, (3) attraction value computation, and (4) candidate action optimization.

Step 1: Board State Representation

The algorithm begins by constructing a binary matrix T that captures the current game state. For a standard 13×13 Gomoku board, each position is encoded as follows:

Ti,j={1if position (i,j) is occupied,0if position (i,j) is unoccupied.(1)

Step 2: Spatial Influence Modeling

Based on the observation that effective Gomoku moves typically cluster around existing pieces, we define a spatial attraction decay function that assigns diminishing weights to positions at increasing distances from placed stones:

W(d)={w1d=1,w2d=2,w3d=3,0d>3.(2)

Here, d represents the Manhattan distance between positions, and w1>w2>w3>0 are predefined weight constants. Positions beyond distance three are considered outside the immediate influence zone.

Step 3: Attraction Value Computation

For each board position (x,y), the total attraction value Ax,y is calculated by aggregating the weighted influences from all occupied positions. Occupied positions are assigned infinite attraction values to prevent illegal move selection. The computation distinguishes between occupied and unoccupied positions:

Ax,y={∞Tx,y=1,∑i=1n∑j=1nTi,j⋅W(DM((i,j),(x,y)))Tx,y=0,(3)

where the Manhattan distance metric is defined as:

DM((i,j),(x,y))=|i−x|+|j−y|.(4)

Step 4: Candidate Action Optimization

The final step applies a two-stage optimization process to the candidate actions generated by the reinforcement learning model. First, a spatial validity filter removes any candidates corresponding to occupied positions (those with Ax,y=∞). Subsequently, the remaining valid candidates are reweighted according to their attraction values:

pk′=pk⋅exp⁡(λAx,y),(5)

where pk represents the original win probability predicted by the game model we trained, Ax,y is the attraction value at position (x,y), and λ is a tunable parameter that controls the balance between strategic optimality and spatial coherence.

3.2 Information Hiding Process

This method implements information-hiding by dynamically encoding move positions within Gomoku gameplay. The core process comprises four stages: information preprocessing, action space generation, encoding mapping, and iterative embedding. Each stage utilizes an optimized Gomoku model to ensure realistic gameplay and computational efficiency. The secret message, denoted by I, is converted into a binary sequence B=g(I) through an encoding function g agreed upon by the communicating parties. The hiding process unfolds across a temporal sequence of states sti=0T∈S.

Step 1: Information Preprocessing: The raw binary stream B=(b1,…,bL)∈{0,1}L is encapsulated with control symbols to support protocol identification and error isolation. The control symbol set is defined as 𝒞= {SOH, EOT, ESC}, with the following binary representations:

SOH=00012,(6)

EOT=01002,(7)

ESC=110112.(8)

Here, SOH indicates the message start, EOT marks the end, and ESC escapes reserved sequences. The final transmission sequence is constructed as:

Bfinal=SOH∥ESC(B)∥EOT,(9)

where ∥ denotes string concatenation, and ESC(B) replaces occurrences of control symbols in B to prevent protocol collisions.

Step 2: Action Space Generation: For each game state st, a candidate action set is generated by the designed model πθ. We filter low-quality actions to ensure plausible gameplay and reliable encoding:

M^t=(ai,pi)∣pi=πθ(st,ai)i=1k,(10)

Mt=(ai,pi)∈M^t∣p1≥⋯≥pk≥ε,(11)

where M^t represents the initial action set, pi denotes the probability of selecting action ai, and ε represents a predefined threshold to filter low-confidence moves. The final action set Mt ensures diversity and strategic plausibility.

Step 3: Encoding Mapping: We use the encoding function ϕ to encode an action at position (x,y) into a binary code based on its spatial relation to the current position (xc,yc), where the offset is defined as (dx,dy)=(x−xc,y−yc). Depending on the region of the current position, the n×n board is partitioned into three distinct regions: the corner region Rc=(0,0),(0,n−1),(n−1,0),(n−1,n−1), the boundary region Rb comprising all edge positions excluding the corners, and the central region Rm which includes all remaining interior positions. The encoding strategy is region-specific:

Figure 3: Encoding layout in the central region

•   Central Region Encoding: In order to maximize the coding capacity, the central region adopts a two-tier encoding scheme, where the encoding function ϕm(dx,dy) is defined as follows:

ϕm(dx,dy)={4-bit code∈{0,1}4, if(x,y) lies in the second outer layer,3-bit code∈{0,1}3,otherwise.(12)

We first determine whether the relative move falls within the second outer layer surrounding the current position. If it does, we apply a 4-bit code. Otherwise, a 3-bit code is used, based on the eight cardinal and intercardinal directions from the current position: North (N), Northeast (NE), East (E), Southeast (SE), South (S), Southwest (SW), West (W), and Northwest (NW). The corresponding coding scheme is illustrated in Fig. 3.

•   Boundary Region Encoding: For non-corner positions on the board edges, the encoding function ϕb(dx,dy) is defined as:

ϕb(dx,dy)={00if dx⋅dy>0,01if dx⋅dy<0,10if dx=0,11if dy=0.(13)

•   Corner Region Encoding: For corner positions (x,y)∈{(0,0),(0,n),(n,0),(n,n)}, define:

ϕc(dx,dy)={0if dx⋅dy=0,1otherwise.(14)

Step 4: Iterative Embedding: Initialize Bremain(0)=B. A greedy prefix-matching algorithm maps bits to actions as follows:

at=arg⁡maxa∈𝒜t⁡{ℓ∣∃b∈{0,1}ℓ,b⊑Bremain(t)andϕ(a)=b},(15)

Bremain(t+1)=Bremain(t)∖ϕ(at),(16)

where ⊑ denotes prefix matching and ∖ represents bit removal. The iteration continues until Bremain(t)=∅ or no valid actions remain available.

Algorithm 1 summarizes the complete information-hiding process, integrating all four steps into a unified workflow.

3.3 Extraction of Stone Coordinates in Gomoku

In this paper, we integrate color space analysis, Hough circle detection, and numeric label recognition to accurately extract information from Gomoku board images. Our proposed pipeline comprises three stages. First, we employ a board localization module based on HSV color space conversion, using experimentally optimized threshold values (H: 16–22, S: 50–255, V: 128–255) to robustly isolate the board region from the background. Second, we apply the Hough circle detection algorithm to identify and segment the game pieces. It employs an adaptive radius range (20–35 pixels at 800 dpi), enabling precise detection of stone regions. Third, we utilize a convolutional neural network (CNN) with multi-resolution feature fusion to recognize the numerical labels printed on the stones.

The CNN architecture follows a hierarchical design, as illustrated in Fig. 4. The feature extraction module comprises three convolutional blocks with progressively increasing channel sizes (64→128→256). Each block includes convolution and max pooling operations to balance feature richness and spatial resolution. To enhance robustness to scale variations and improve generalization, we incorporate a global adaptive average pooling layer at the end of the extraction module. The classification head comprises a 128-dimensional fully connected layer followed by LayerNorm regularization, projecting to a 290-dimensional output space.

Figure 4: Architecture of the Gomoku piece recognition network

The model is trained using the Adam optimizer with a cosine annealing learning rate schedule, starting at an initial learning rate of 3×10−4. It is trained on a custom dataset comprising 28,244 images, which are divided into training, validation, and test sets in an 8:1:1 ratio. After 20 epochs, the model achieves 98% training accuracy (loss: 0.13) and 85% validation accuracy (loss: 1.10). In simulated Gomoku game scenarios, the model consistently performs well in recognizing the numerical labels on the stones.

3.4 Secret Message Extraction Process

In the secret message extraction process, we include each stone’s position (x,y), move index, and player type to compose a move sequence. According to Gomoku rules, odd-numbered indices correspond to moves by the black player, while even-numbered indices correspond to moves by the white player. In this configuration, both players alternately embed secret message in their respective moves. We compute the positional offset (dx,dy) based on the relative position between the current move at index i and the previous move at index i−1. Algorithm 2 formalizes the extraction process, distinctly handling moves from both players and adapting the extraction strategy based on the board region context of each move’s origin.

Then, we use the positional differences (dx,dy) to recover the secret message according to the previously defined encoding rules. First, determine whether the previous move position (xi−1,yi−1) falls within the middle, corner, or boundary region, and then apply the corresponding encoding function ϕm,ϕc,ϕb as specified in Eq. (17) to obtain the binary conde bi:

bi={ϕm(dx,dy)if (xi−1,yi−1) is in middle region,ϕc(dx,dy)if (xi−1,yi−1) is in corner region,ϕb(dx,dy)if (xi−1,yi−1) is in boundary region.(17)

In this formulation, (xi−1,yi−1) denotes the position of the previous move. The variables dx=xi−xi−1 and dy=yi−yi−1 represent the horizontal and vertical offsets relative to the previous move, respectively. The value bi represents the binary code assigned to the i-th move.

After obtaining all binary codes, we concatenate them according to Eq. (18):

Bcat=b1∥b2∥…∥bn(18)

where ∥ denotes string concatenation.

Next, we process Bcat by removing the start and end control symbols SOH and EOT. The resulting sequence is denoted as Bmid:

Bmid=remove(SOH,EOT;Bcat)(19)

Subsequently, the inverse escape transformation is applied to recover the original binary message:

B=ESC−1(Bmid)(20)

4  Evaluation and Comparison

The experimental framework was implemented using Python 3.10 and the PyTorch 2.0.1 deep learning library. All experiments were conducted on a Tesla P4 GPU with 8 GB of memory. The Gomoku model was trained on 13×13 game board images with a resolution of 512×512 pixels. During training, we employed a batch size of 256 and trained the model for a total of 170,000 steps. The optimization was performed using stochastic gradient descent (SGD) with an initial learning rate of 0.01, a momentum coefficient of 0.9, and an L2 regularization weight of 10−4. The learning rate was scheduled to decay by a factor of 0.1 at predefined training milestones to facilitate convergence and prevent overfitting. To enhance the model’s generalization ability, we incorporated data augmentation techniques during training, specifically horizontal and vertical flipping and random rotations applied to input images. During experimental validation, game records were synthesized using an information-hiding mechanism that generated board images embedded with covert data for constructing both training and testing datasets.

4.1 Embedding Capacity

Embedding capacity, a key metric for evaluating information-hiding techniques, quantifies how much secret message can be embedded in a carrier medium. The embedding capacity of the proposed method is computed using Eq. (21):

Size=∑i=kMNi(21)

where M denotes the total number of moves in the Gomoku game, Ni represents the encoding capacity of the i-th move, and k indicates the move at which steganographic embedding begins.

Based on our calculations, the maximum theoretical embedding capacity on a 13×13 board can reach up to 552 bits. However, since a portion of the game must be reserved to conceal the presence of secret message, the effective capacity is reduced. In our experiments, we evaluated a total payload of 1953 bits and observed that the model achieved an average embedding capacity of 99.25 bits per carrier, with values ranging from 41 to 180 bits per carrier.

Experimental results evaluating the embedding capacity of the proposed method are presented in Table 1.

4.2 Robustness Analysis

Robustness is a fundamental metric for evaluating steganographic systems, reflecting their resilience to adversarial conditions based on the accuracy of covert information extraction. During transmission, stego images are frequently exposed to various perturbations, such as Gaussian noise, salt-and-pepper noise, and other forms of signal degradation. This section systematically evaluates the robustness of the proposed method through comparative experiments under a variety of attack scenarios. The proposed method demonstrates remarkable robustness by emphasizing the relative positional relationships between different operations rather than the specific pixels being modified. Furthermore, enhancing the system’s detection capabilities can further improve the robustness of the proposed method. The bit error rate (BER) serves as the primary evaluation metric and is mathematically defined as follows:

BER=en,e=∑i=1npi⊕qi(22)

where e denotes the total number of erroneous bits, n represents the total number of transmitted bits, pi indicates the i-th transmitted bit, and qi corresponds to the i-th received bit.

To comprehensively evaluate the robustness of the proposed method under adverse transmission conditions, we conducted extensive experiments involving various attack vectors, including JPEG compression, spatial filtering, geometric rotation, and additive noise injection. The comparative results, shown in Tables 2 and 3, benchmark the proposed method against representative methods: CBD [46], CBZS [47], CSD [48], RICS [40], and the methods proposed by Wu et al. [45], Sun et al. [43], Chen et al. [49], and Yang et al. [50].

The results show that the proposed method achieves superior robustness across all tested attack scenarios. Furthermore, Table 4 demonstrates its stability under benign conditions such as lossless format conversion, confirming its practical robustness for real-world transmission environments.

4.2.1 Resistance to Image Processing Attacks

To mitigate potential recognition errors inherent in automated systems, a human-assisted correction mechanism was introduced as a post-processing step following the initial computer-based extraction. This hybrid approach allows for the evaluation of both fully automated performance and human-corrected outcomes, offering a more comprehensive assessment of the method’s practical applicability.

Table 2 presents the algorithm’s performance under JPEG compression, mean filtering, and Gaussian low-pass filtering.

Table 3 extends the evaluation to include geometric and noise-based attacks, comparing the proposed method’s performance with several existing approaches.

4.2.2 Resistance to Format Conversion Attacks

The format conversion experiments evaluated the robustness of the proposed method against color space transformations, file format conversions, and scaling attacks. As shown in Table 4, the method maintains a bit error rate (BER) of 0 when converting RGB images to grayscale or binary formats, and when converting PNG files to JPG, GIF, BMP, or TIF formats. In the scaling attack tests, images were resized to 30%, 50%, and 70% of their original dimensions to evaluate the accuracy of information extraction. The proposed method achieves a BER of 0 at both 50% and 70% scaling ratios. At the 30% scaling ratio, automated recognition results in a BER of 0.615, whereas human-assisted verification enables error-free extraction of the embedded information.

4.3 Ablation Experiments

This subsection examines the influence of the attraction matrix mechanism and key neural network components on overall model performance. To evaluate the effect of the attraction matrix, we compare three configurations: purely generative gameplay, steganographic tasks with the attraction matrix, and steganographic tasks without it. The analysis considers variations in model reasonableness, embedding capacity, and the incidence of decision irregularities. Experimental results indicate that incorporating the attraction matrix substantially mitigates embedding capacity loss and reduces non-professional move patterns that arise under purely self-play conditions. Additionally, to assess the contribution of the neural network architecture, we perform ablation studies on critical structural parameters, including the number of residual blocks, the number of convolutional channels, and the size of fully connected layers, thereby confirming the role and effectiveness of each component.

4.3.1 Evaluation of Game Record Reasonableness

The reasonableness of generated game records under steganographic tasks is crucial for maintaining covert communication. In AlphaZero, the value head output represents the estimated winning probability of the current position. We use the following Eq. (23) to quantify the overall plausibility of the generated game record.

R=1N∑i=1N|ΔSi|(23)

In this context, R represents the reasonableness score of a game record, N denotes the total number of moves, and ΔSi indicates the change in winning probability following the i-th move. Specifically, if the i-th move decreases the predicted winning probability, then ΔSi<0; otherwise, ΔSi≥0. During typical gameplay, each move generally affects the current winning probability, especially in games such as Gomoku, which have relatively simple rules. Consequently, the analysis focuses on the absolute value of the change in winning probability after each move. When generating game records, a reasonable move should produce an R value close to the baseline of the purely generative model (approximately 1.8941 in this study).

To evaluate the impact of the attraction matrix strength, we embed two textual excerpts from The Old Man and the Sea as the payload. Since the outcomes of the model training process remain within a fixed range, we select a representative set of values for the parameter λ∈{−1,−3,−5,−7,−9,−12} for experimental evaluation. The results are compared against two additional configurations: steganographic tasks without the attraction matrix and purely generative gameplay without any steganographic objective. The corresponding results are presented in Table 5.

The experimental results indicate that incorporating steganographic tasks leads to a significant deviation in the reasonableness score compared to the purely generative model, suggesting that the payload disrupts the naturalness of gameplay. Introducing a moderate attraction matrix strength effectively mitigates this deviation, producing game records that more closely resemble standard gameplay. However, when the attraction matrix becomes excessively strong, the reasonableness score diverges from the baseline once again, implying that overly restrictive constraints introduce new patterns of unnatural behavior.

4.3.2 Evaluation of Decision Irregularities

Beyond average variations in winning probability, the model’s responsiveness to tactically critical positions serves as another key indicator of strategic reasonableness. In Gomoku, certain board patterns—such as “Open Three” and “Open Four”—represent critical tactical threats:

•   Open Three: A pattern where three consecutive stones of the same color are aligned with two open ends. If left unaddressed, this often leads to an immediate win.

•   Open Four: A pattern where four consecutive stones of the same color are aligned with two open ends. This configuration constitutes an immediate winning threat, as the player can win on the next move regardless of the opponent’s response.

In actual gameplay, players typically prioritize responding to such threats. Thus, frequent neglect of these patterns during steganographic encoding suggests a loss of strategic rationality and undermines trust in the generated records.

To quantify this, we introduce the metric Decision Irregularities (DI), which measures the total number of critical tactical errors made by the model during a game. After each game concludes, we retrospectively analyze the entire sequence of moves to identify scenarios where the board presents a decisive opportunity—specifically, an Open Three or Open Four situation. For each such scenario, if the model fails to select the move that would block the opponent’s imminent threat, we count this as an irregular decision. All such cases throughout the game are accumulated to compute the DI value. These counts serve as indicators of abnormal decision-making behavior under critical conditions. To further explore the trade-off between embedding capacity and move reasonableness, we also track the hidden payload capacity under identical experimental settings and analyze both metrics in parallel. The results are shown in Table 6.

Experimental results reveal that increasing attraction matrix strength significantly enhances the model’s embedding capacity. However, this comes at the cost of a higher frequency of ignored tactical patterns such as “Open Three” and “Open Four”. These findings suggest that while overly clustered moves enhance hiding performance, they impair the model’s strategic awareness and degrade overall gameplay quality.

Notably, when λ is set within a moderate range (e.g., −13≤λ≤−4), the model strikes a desirable balance between embedding capacity and move reasonableness. This range avoids implausible moves caused by weak attraction while mitigating rigid behavior induced by excessive attraction.

Collectively, these results demonstrate that the attraction matrix serves a dual regulatory function in guiding move selection during information embedding. With proper tuning of the strength parameter λ, the model can effectively balance embedding capacity with strategic plausibility.

4.3.3 Evaluation of the AlphaZero Network

To ensure the effectiveness of the steganographic approach, it is essential to maintain robust model performance. A stronger Gomoku-playing model not only generates more optimal moves but also preserves the plausibility of gameplay, thereby reducing the risk of detection. Conversely, a weaker model is more likely to make unreasonable moves, compromising strategic integrity and increasing the likelihood of revealing the hidden information.

To better understand how architectural design influences model performance, we conducted a systematic investigation of key components in the AlphaZero neural network. Specifically, we examined three critical hyperparameters: the number of residual blocks, the number of convolutional filters, and the size of the fully connected layer. Each hyperparameter was varied independently using a one-factor-at-a-time experimental design, and the resulting configurations were evaluated based on playing strength, as measured by Elo ratings, and training loss metrics.

The Elo rating system, a standardized and extensively utilized framework, serves to quantify relative skill levels in board gameand analogous competitive domains. Its principal advantage lies in offering an equitable and objective means of comparison between competitors by condensing match outcomes into a singular rating value. These ratings are dynamically updated based on cumulative game results, thereby enabling the system to monitor fluctuations in actual playing strength over time. Denoting the current Elo ratings of models A and B as RA and RB, respectively, the expected score is calculated as follows:

EA=11+10(RB−RA)/400(24)

EB=1−EA(25)

If model A’s actual score in a match is SA (win = 1, draw = 0.5, loss = 0), the Elo update rule is:

RA′=RA+K⋅(SA−EA)(26)

RB′=RB+K⋅(SB−EB)(27)

SB=1−SA(28)

Here, K serves as a learning rate parameter, dynamically determined according to the current Elo score (following USCF rules):

K={32if Elo<210024if2100≤Elo<240016if Elo≥2400(29)

Elevated Elo ratings indicate greater playing strength, thereby facilitating quantitative comparisons among various network architectures. Additionally, supplementary metrics such as policy loss and value loss were employed to assess the precision of action prediction and value estimation, respectively; lower values in these metrics correspond to improved model performance. In this study, the training dataset was generated by 20 parallel agents, each conducting 300 Monte Carlo simulations per move, with each model undergoing 30,000 training steps. The term Block Elo denotes the Elo rating for the black pieces, whereas White Elo refers to the Elo rating for the white pieces. The experimental findings are presented in Table 7.

The findings indicate that various network modules exert a substantial and intricate influence on the playing strength of the AlphaZero model. Performance does not exhibit a straightforward, monotonic improvement with increasing network depth or width; rather, it is essential to identify an optimal equilibrium among model capacity, training stability, and generalization capability.

4.4 Security Evaluation

Steganographic security measures the resistance of stego images to detection by advanced steganalysis techniques. For images or videos embedding secret message, steganalysis examines pixel-level statistical features to detect the presence of concealed data. This study employs XuNet [51] and YeNet [7] to evaluate the steganalysis resistance of Gomoku board images embedded with secret message within image sequences. We prepared a cover dataset comprising 5000 distinct game images generated from multiple games without secret message, and a stego dataset consisting of 5000 board images from games containing embedded messages. These datasets were used to train XuNet and YeNet, with cover and stego images treated as separate classes.

The false alarm rate, PFA, represents the probability of incorrectly classifying a cover image as containing a secret message. Let the total number of carriers without secret message be NC, and the number correctly detected be NCT; then PFA is defined as:

PFA=NC−NCTNC(30)

The miss detection rate, PMD, denotes the probability of failing to detect a secret message in a stego-image. Let the total number of stego images containing secret message be NS, and let NST denote the number correctly detected; then PMD is defined as:

PMD=NS−NSTNS(31)

The average decision error rate, PE, evaluates overall steganalysis performance:

PE=PFA+PMD2(32)

A PE value close to 50% indicates low detectability, suggesting that stego images are nearly indistinguishable from cover images under steganalysis examination.

The proposed method embeds secret message in Gomoku move patterns rather than pixel values, resulting in minimal statistical anomalies detectable by conventional steganalysis models during training. Experimental results in Table 8 suggest that the proposed method exhibits notable resistance to steganalysis detection.

5  Conclusion

In this paper, we utilize a trained AlphaZero model to automate information hiding in Gomoku. The proposed method consists of four main components: (1) training the AlphaZero model, (2) integrating the base model with an attraction matrix, (3) generating steganographic game records, and (4) extracting secret messages based on the corresponding model rules and game records. Experimental results demonstrate that the proposed method generates game records consistent with typical human gameplay patterns, thereby reducing the likelihood of detection by adversaries. Furthermore, the method exhibits high embedding capacity, which can be further enhanced by improving the AlphaZero model’s performance. The integration of the attraction matrix module effectively mitigates the negative effects of steganographic constraints, which often cause stego game records to deviate from normal gameplay patterns. Additionally, the proposed method demonstrates strong robustness and security, indicating its potential for practical deployment.

Despite its effectiveness, the proposed method has several limitations. First, the attraction matrix functions as an external heuristic rather than a trainable component within the model. Integrating this mechanism directly into the AI agent could enable the model to optimize both embedding capacity and gameplay authenticity in a unified manner. Second, the framework has been evaluated primarily in offline settings and lacks adaptive capabilities for real-time applications. Future work should focus on developing efficient algorithms to support real-time encoding and transmission. Third, the proposed framework currently operates under the assumption of a perfect, noise-free communication channel. A crucial direction for future work is to investigate methods for enhancing the system’s robustness against channel noise, ensuring reliable data extraction even under adverse transmission conditions. These improvements would enhance the system’s applicability in dynamic, practical environments.

Acknowledgement: This paper is a collaborative effort of the five authors. We would like to express our gratitude for the fund support provided by the relevant project and thank all the experts who participated in the review process.

Funding Statement: This research was funded by the Wuxi “Taihu Light” Science and Technology Key Project (Basic Research) (K20241046); the National Natural Science Foundation of China (Grant Nos. 62102189, 62122032, 42305158); the Open Project of the National Engineering Research Center for Sensor Networks (2024YJZXKFKT02); and Wuxi University Research Start-up Fund for High-Level Talents (No. 2022r043).

Author Contributions: The authors confirm contribution to the paper as follows: study conception and design: Yi Cao, Chengsheng Yuan; data collection: Kuo Zhang, Linglong Zhu; analysis and interpretation of results: Kuo Zhang, Wentao Ge; draft manuscript preparation: Kuo Zhang, Wentao Ge, Linglong Zhu. All authors reviewed the results and approved the final version of the manuscript.

Availability of Data and Materials: The original test datasets used for XuNet and YeNet in this study are openly available via Google Drive at https://drive.google.com/file/d/16Ndnc1mS7vjYPu3wXYe4u_ZIWbUqBrGu/view?usp=drive_link (accessed on 18 September 2025). All other data supporting the findings of this study are available from the corresponding author upon reasonable request.

Ethics Approval: Not applicable.

Conflicts of Interest: The authors declare no conflicts of interest to report regarding the present study.

Abbreviations

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