Probability Distribution Tree-Based Dishonest-Participant-Resistant Visual Secret Sharing Using Linearly Polarized Shares

Abstract

With the rapid growth of data transmission and visual encryption technologies, Visual Secret Sharing (VSS) has become an important technique for image-based information protection. However, many existing VSS schemes remain vulnerable to dishonest participants who attempt to recover secret images through unauthorized stacking or manipulation of shares. To address this issue, this paper proposes a dishonest-participant-resistant VSS scheme based on linearly polarized shares and Probability Distribution Trees (PDTs). The proposed method embeds both secret and fake images into polarized shares, such that any unauthorized stacking of ordinary shares produces a visually plausible fake image or random noise, while only stacking that includes the master share under a predefined optical ordering reveals the true secret image. Binary image binarization and probability-guided polarization assignment are employed to improve computational efficiency and increase uncertainty against adaptive attacks. In addition to visual inspection and contrast analysis, peak signal-to-noise ratio (PSNR), structural similarity index (SSIM), and visual information fidelity (VIF) are used as complementary metrics to distinguish authorized reconstructions from unauthorized and partial ones. Experimental results show that authorized reconstructions achieve high visual fidelity and perceptual recognizability, whereas unauthorized and partial reconstructions yield significantly degraded or misleading outputs, demonstrating effective suppression of information leakage and strong resistance against dishonest behavior. Consequently, the proposed scheme enhances security and practical usability compared with existing polarization-based VSS approaches.

1. Introduction

With the continuous development of network technology, security issues have become crucial. Visual Secret Sharing (VSS) schemes are significant due to their relevance to data security applications. Its simple decryption method and easy operation have attracted the interest of many researchers. Naor and Shamir first proposed the visual cryptography (VC) scheme in 1994, which can decode hidden images without any encryption calculations [1]. In the encryption stage, the secret image is encrypted with pixel expansion using a designed matrix, and the generated shared image is distributed to n participants. No single participant can obtain secret information. When all participants work together, the secret image is restored. However, this scheme has limitations such as pixel expansion and poor restoration quality.

Later, the researchers proposed a threshold secret sharing scheme $(k,n)$ [2], which only requires k participants to restore the secret image. To solve the pixel expansion problem, Shyu et al. proposed a $(2,2)$-type scheme in 2007 [3]. The secret image is encrypted by random grids, grayscale images, and color images are converted into binary images by halftone technology, and then encrypted by three different algorithms. When decrypting, the random grids can be superimposed without any additional calculation, but the secret pixels cannot be completely restored. Later, several researchers identified the advantages of random grids and proposed several schemes to share secrets using random grids [4,5,6,7,8]. These schemes have the advantages of not having a code-book design (used for black-white pair wise visual cryptography for pixels) and no pixel expansion.

To improve the security of secret image transmission and increase the convenience of decryption, many researchers proposed progressive secret-sharing schemes [9,10,11,12,13,14,15,16]. As the number of participants increases, the secret image is gradually restored. However, all the above schemes can only perform the transmission of one secret image, which has the disadvantages of higher-time consumption and low transmission efficiency.

With the widespread development of telecommunication technology, the number of fraud-related attacks has increased, motivating extensive research on dishonest or cheating participants in visual secret sharing schemes. Several studies have focused on identifying or detecting dishonest participants through verification mechanisms or authentication strategies [17,18,19]. While such detection-based approaches can identify fraudulent behavior, they do not necessarily prevent unauthorized participants from attempting reconstruction or from observing degraded but informative outputs.

Research in an alternative direction introduces the use of misleading or meaningful shares. In this approach, even visually interpretable shares conceal useful information unless a qualified set is stacked correctly [20,21,22,23,24]. These schemes primarily aim to suppress information leakage by ensuring that unauthorized stacking does not expose the secret image. Boolean- and XOR-based multi-secret schemes further extend this idea by embedding cover images or multiple secrets, although at the cost of increased computational complexity [25,26].

In contrast, misleading-based approaches emphasize cheater deception, where dishonest behavior is actively discouraged by producing fake or noise-like reconstructions. This distinction motivates the present work, which focuses on enforcing deceptive reconstruction outcomes for unauthorized stacking rather than explicitly detecting dishonest participants.

Although visual cryptography utilizing polarization is not a recent concept [27], polarization-based VSS schemes offer advantages such as the elimination of pixel expansion and improved reconstruction quality. Prior work has demonstrated that reconstruction outcomes may depend on the relative orientation or optical ordering of stacked polarizers and that dishonest participants cannot regenerate the secret merely by stacking shares arbitrarily [28]. However, existing polarization-based schemes typically rely on deterministic polarization assignment rules, making the relationship between stacking behavior and reconstruction outcome predictable once the scheme structure is known. As a result, unauthorized stacking attempts may still exhibit consistent patterns or partial information leakage under repeated trials. This limitation motivates the incorporation of probabilistic polarization assignment mechanisms that increase uncertainty in unauthorized reconstructions while preserving correctness for authorized stacking.

Recent research has also explored the integration of polarization and optical devices to enhance the security and functionality of Visual Secret Sharing systems. Li et al. proposed a polarization-assisted VSS encryption method using metasurface holography [29], demonstrating that metasurface-engineered polarization states can significantly strengthen security and support more complex optical reconstructions compared to traditional random-grid VSS schemes. Their work highlights the growing interest in leveraging physical-layer optical modulation to improve secret sharing performance. Although this approach provides robust optical encoding, its dependence on metasurface fabrication makes practical deployment difficult. Our proposed scheme similarly incorporates polarization principles but avoids specialized devices, making it more accessible while still benefiting from the security strengths offered by polarization-based encoding.

In addition to single-secret systems, several studies have attempted to efficiently handle multiple secrets in a single sharing process. Liu et al. introduced a multi-secret grayscale VSS method that employs polarization to improve reconstruction quality and support multiple embedded secrets [30]. Complementing this direction, Wang et al. extended dual-message QR codes using a VSS-based framework, enabling robust multi-message embedding while preserving scannability and visual usability for practical authentication applications [31]. These works collectively indicate that multi-secret and functional share designs have become an important trend. Our proposed misleading-share framework aligns with this evolution by supporting multiple share interpretations (true vs. fake), while keeping the computational cost minimal and relying on simple stacking operations.

Alongside advances in multi-secret capacity, researchers have also investigated verification mechanisms and high-quality meaningful shares. Yang et al. proposed a verifiable variable-threshold VSS scheme that enables participants to detect dishonest behavior prior to reconstruction [32], addressing limitations of classical random-grid approaches where cheating participants may remain undetected. Meanwhile, Wang et al. introduced a meaningful secret image-sharing technique with improved visual quality [33], enhancing share naturalness to minimize suspicion and reduce the likelihood of targeted attacks. These studies indicate a clear research movement toward both verifiability and meaningful share generation.

In this paper, we propose a dishonest-participant-resistant visual secret-sharing scheme that integrates linearly polarized shares with a Probability Distribution Tree (PDT)-driven polarization assignment mechanism. Unlike prior polarization-based stacking approaches that rely on deterministic orientation rules, the proposed method introduces controlled, per-pixel probabilistic variation while enforcing a predefined optical ordering that includes a master share. Unauthorized stacking of ordinary shares, regardless of permutation or repetition, produces a visually plausible fake image or random noise, whereas authorized stacking under the predefined optical ordering reconstructs the secret with high visual fidelity. Rather than detecting dishonest behavior, the scheme focuses on cheater deception by ensuring that unauthorized actions yield misleading outcomes. This design enhances resistance to adaptive and collusive attacks while maintaining low computational complexity and practical implementability.

The outline of this paper is organized as follows. Section 2 presents some basic descriptions of preliminaries related to the proposed study. Section 3 presents the related literature, and the recent developments in this area. The proposed PDT-based VSS scheme using linearly polarized shares is presented in Section 4. The experimental results and analysis details are presented in Section 5, and concluding remarks are presented in Section 6.

2. Research Background

2.1. VSS, Binarization and Probability Distribution Tree

Visual Secret Sharing (VSS) is a cryptographic method that splits a secret image into multiple shares, allowing only authorized subsets to reconstruct the original image. Its primary goal is to enhance visual security by preventing unauthorized subsets from accessing the secret. VSS’s main advantage is its simple decoding process, which relies on the human visual system and avoids complex computations. An example of VSS is given in Figure 1.

Secondly, binarization is essential in our proposed VSS scheme as it simplifies the image processing by converting both the secret and fake images into binary form (black and white). This reduces computational complexity by eliminating the need to handle varying shades and intensities. Instead, any intensity above a certain threshold (0.125) is considered a “White” pixel, streamlining the process. While binary images may not be as visually smooth, they better preserve the texture of the image, making them ideal for maintaining the integrity of the secret while improving efficiency.

Lastly, our proposed method, Probability Distribution Tree (PDT), is employed to determine how the shares are distributed. A PDT is a structure consisting of nodes and branches, where each node represents a specific probability distribution. The tree helps to decide the distribution of polarized planes for each pixel in the shares. By using multiple distribution types, the PDT enhances security by making it harder for dishonest participants to reconstruct the secret. Additionally, the PDT optimizes light transmission and contrast, improving the visual quality of the image which PDT is discussed elaborately in Section 2.2.

Achieving a balance among security, contrast, and visual recognizability is crucial for our system. These parameters are governed by the volume of light transmitted through stacked shares. According to Malus’ law, when polarized light passes through two polarizers with a relative orientation angle $\theta$, the transmitted intensity is expressed as:

$$I=I_0{cos}^2\left(\theta \right)$$

(1)

where $I_0$ denotes the initial light intensity. Figure 2 shows demonstration of non-polarized light and uni-polarized light traveling through liner polarizers.

2.2. PDT Construction and Selection Rule

The Probability Distribution Tree (PDT) is a probabilistic decision structure used to guide the assignment of polarization orientations to individual share pixels in a controlled yet non-deterministic manner. The PDT operates at the pixel level and is designed to ensure correct reconstruction under authorized stacking while suppressing information leakage under unauthorized stacking attempts.

From a probabilistic perspective, the polarization orientation assigned to each pixel can be modeled as a discrete random variable. Let ${\mathrm{\Theta }}_{i,j}^{\left(l\right)}\in \{\pi ,\pi /4,\pi /2,3\pi /4\}$ denote the polarization orientation selected for pixel $(i,j)$ at tree level l. The Probability Distribution Tree (PDT) defines a conditional probability structure such that the orientation at level l is sampled according to a conditional distribution

$$P\left({\mathrm{\Theta }}_{i,j}^{\left(l\right)}\mid {\mathrm{\Theta }}_{i,j}^{(l-1)}\right)$$

(2)

where the probability mass function is updated based on the parent node orientation and the desired reconstruction behavior (authorized or unauthorized stacking). This local dependency induces a tree-structured stochastic process in which polarization assignments are conditionally dependent across levels but independent across pixels.

It is emphasized that the PDT is not intended as a full Markov model or an analytically optimal stochastic process. Rather, it provides a controlled probabilistic mechanism for introducing per-pixel randomness into polarization assignment while preserving correctness under authorized optical ordering and enforcing deception under unauthorized stacking.

Formally, a PDT is a rooted, level-wise probabilistic tree in which each level corresponds to one share in the predefined optical ordering, and each node represents a candidate polarization orientation selected from a finite orientation set. A complete path from the root to a leaf node represents a full polarization assignment stream for all shares at a single pixel location. A separate PDT is constructed independently for each pixel location $(i,j)$, and no global coupling is imposed across pixels. This per-pixel construction improves unpredictability and prevents inference through spatial correlation.

Each PDT is parameterized by the predefined optical ordering of shares (including the master share position), the polarization orientation set

$$\mathrm{\Theta }=\left\{0,{\displaystyle \frac{\pi }{4}},{\displaystyle \frac{\pi }{2}},{\displaystyle \frac{3\pi }{4}}\right\}$$

(3)

the target pixel pair $\left(F\right(i,j),S(i,j\left)\right)$ of the fake and secret images, and intensity thresholds derived from Malus’ law. The optical position of the master share can be chosen arbitrarily; however, it must be predefined prior to PDT generation. All PDT construction, probability updates, and intensity evaluations are performed consistently under this fixed optical ordering to ensure reproducible modeling of polarization interaction during stacking.

At the root level of PDT, polarization orientations are initialized using a uniform prior distribution over the set $\mathrm{\Theta }$, ensuring no deterministic bias at the initial selection stage. At subsequent levels, conditional probability distributions are updated based on the polarization orientation selected at the parent node and the corresponding pixel case $(F,S)$. These updates bias orientation selection toward configurations that satisfy the desired reconstruction behavior: correct transmission or suppression of light under authorized stacking, and misleading or noise-like outcomes under unauthorized stacking. The case-dependent update rules are implemented through conditional probability adjustment functions, as detailed in Algorithms 1–4.

Algorithm 1 Generate tree sequences for $F(i,j)=1$, $S(i,j)=1$
1:	Input: Total number of shares to be created
2:	nodes ← empty list
3:	polarizer_types $\leftarrow \{\pi ,\pi /4,\pi /2,3\pi /4\}$
4:	probability_map $\leftarrow \{\pi :0.25,\pi /4:0.25,\pi /2:0.25,3\pi /4:0.25\}$
5:	for level $=0$ to steps $-1$ do
6:	level_node_count $\leftarrow 2^{level}$
7:	for $i=0$ to level_node_count $-1$ do
8:	index $\leftarrow (2^{level}-1)+i$
9:	if level $=0$ then
10:	random_char ← select from polarizer_types using probability_map
11:	else
12:	parent_index $\leftarrow (index-1)÷2$
13:	parent_char ← nodes[parent_index]
14:	adjust_probabilities(probability_map, parent_char)
15:	random_char ← select from polarizer_types using updated probability_map
16:	end if
17:	nodes[index] ← random_char
18:	end for
19:	end for
20:	return nodes
21:	Function: adjust_probabilities(probability_map, parent_char)
22:	if parent_char $=\pi$ then
23:	probability_map $\leftarrow \{\pi :0.5,\pi /4:0.25,\pi /2:0,3\pi /4:0.25\}$
24:	else if parent_char $=\pi /4$ then
25:	probability_map $\leftarrow \{\pi :0.25,\pi /4:0.5,\pi /2:0.25,3\pi /4:0\}$
26:	else if parent_char $=\pi /2$    then
27:	probability_map $\leftarrow \{\pi :0,\pi /4:0.25,\pi /2:0.5,3\pi /4:0.25\}$
28:	else if parent_char $=3\pi /4$ then
29:	probability_map $\leftarrow \{\pi :0.25,\pi /4:0,\pi /2:0.25,3\pi /4:0.5\}$
30:	end if
	Note: All evaluations assume the predefined optical ordering that includes the master share. Bias polarization selection to promote light transmission under both stacking conditions.

For each candidate root-to-leaf stream generated by the PDT, the expected transmitted light intensity is evaluated according to Malus’ law. Let ${\theta }_k$ denote the polarization orientation assigned to the k-th share in the predefined optical ordering. The transmitted intensity for a given stream is computed as

$$I_{\mathrm{out}}=I_0\prod _{k=1}^N{cos}^2({\theta }_k-{\theta }_{k-1})$$

(4)

where $I_0$ denotes the incident light intensity and ${\theta }_0$ represents the reference polarization state. Intensity thresholds are applied to classify the resulting output as a black or white pixel under both authorized and unauthorized stacking conditions.

The stream selection objective is to maximize the contrast separation between authorized and unauthorized reconstruction outcomes rather than to optimize a global image-level metric. A candidate stream is considered valid if it yields the desired secret pixel value under the predefined optical ordering that includes the master share, while producing a fake or noise-like pixel value under any stacking configuration that excludes the master share. If multiple candidate streams satisfy these criteria, one stream is selected uniformly at random. This tie-breaking strategy preserves stochastic variability and prevents deterministic inference through repeated or adaptive stacking trials.

By combining per-pixel probabilistic polarization assignment with intensity-based stream validation, the PDT functions as a lightweight probabilistic control mechanism. It enhances resistance against dishonest participants by ensuring that authorized reconstructions remain visually recognizable while unauthorized stacking attempts do not converge toward partial disclosure, without introducing significant computational or optical complexity.

Algorithm 2 Generate tree sequences for $F(i,j)=0$, $S(i,j)=0$
1:	Function: adjust_probabilities(probability_map, parent_char)
2:	if parent_char = $\pi$ then
3:	probability_map ← { $\pi$: 0, $\pi /4$: 0.2, $\pi /2$: 0.6, $3\pi /4$: 0.2 }
4:	else if parent_char = $\pi /4$ then
5:	probability_map ← { $\pi$: 0.2, $\pi /4$: 0, $\pi /2$: 0.2, $3\pi /4$: 0.6 }
6:	else if parent_char = $\pi /2$ then
7:	probability_map ← { $\pi$: 0.6, $\pi /4$: 0.2, $\pi /2$: 0, $3\pi /4$: 0.2 }
8:	else if parent_char = $3\pi /4$ then
9:	probability_map ← { $\pi$: 0.2, $\pi /4$: 0.6, $\pi /2$: 0.2, $3\pi /4$: 0 }
10:	end if
	Note: All evaluations assume the predefined optical ordering that includes the master share. Bias polarization selection to suppress light transmission under authorized stacking.

Algorithm 3 Generate tree sequences for $F(i,j)=1$, $S(i,j)=0$
1:	Function: adjust_probabilities(probability_map, parent_char)
2:	if parent_char = $\pi$ then
3:	probability_map ← { $\pi$: 0.5, $\pi /4$: 0.25, $\pi /2$: 0, $3\pi /4$: 0.25 }
4:	else if parent_char = $\pi /4$ then
5:	probability_map ← { $\pi$: 0.25, $\pi /4$: 0.5, $\pi /2$: 0.25, $3\pi /4$: 0 }
6:	else if parent_char = $\pi /2$ then
7:	probability_map ← { $\pi$: 0, $\pi /4$: 0.25, $\pi /2$: 0.5, $3\pi /4$: 0.25 }
8:	else if parent_char = $3\pi /4$ then
9:	probability_map ← { $\pi$: 0.25, $\pi /4$: 0, $\pi /2$: 0.25, $3\pi /4$: 0.5 }
10:	end if
11:	Function: adjust_probabilities_level(probability_map, parent_char)
12:	if parent_char = $\pi$ then
13:	probability_map ← { $\pi$: 0, $\pi /4$: 0, $\pi /2$: 1, $3\pi /4$: 0 }
14:	else if parent_char = $\pi /4$ then
15:	probability_map ← { $\pi$: 0, $\pi /4$: 0, $\pi /2$: 0, $3\pi /4$: 1 }
16:	else if parent_char = $\pi /2$ then
17:	probability_map ← { $\pi$: 1, $\pi /4$: 0, $\pi /2$: 0, $3\pi /4$: 0 }
18:	else if parent_char = $3\pi /4$ then
19:	probability_map ← { $\pi$: 0, $\pi /4$: 1, $\pi /2$: 0, $3\pi /4$: 0 }
20:	end if
	Note: All evaluations assume the predefined optical ordering that includes the master share. Enforce high intensity under unauthorized stacking and low intensity under authorized stacking.

Algorithm 4 Generate tree sequences for $F(i,j)=0$, $S(i,j)=1$
1:	Function: adjust_probabilities(probability_map, parent_char)
2:	if parent_char = $\pi$ then
3:	probability_map ← { $\pi$: 0.5, $\pi /4$: 0, $\pi /2$: 0.25, $3\pi /4$: 0.25 }
4:	else if parent_char = $\pi /4$ then
5:	probability_map ← { $\pi$: 0, $\pi /4$: 0.5, $\pi /2$: 0.25, $3\pi /4$: 0.25 }
6:	else if parent_char = $\pi /2$ then
7:	probability_map ← { $\pi$: 0.25, $\pi /4$: 0.25, $\pi /2$: 0.5, $3\pi /4$: 0 }
8:	else if parent_char = $3\pi /4$ then
9:	probability_map ← { $\pi$: 0.25, $\pi /4$: 0.25, $\pi /2$: 0, $3\pi /4$: 0.5 }
10:	end if
11:	Function: adjust_probabilities_level(probability_map, parent_char)
12:	if parent_char = $\pi$ then
13:	probability_map ← { $\pi$: 0, $\pi /4$: 0.5, $\pi /2$: 0, $3\pi /4$: 0.5 }
14:	else if parent_char = $\pi /4$ then
15:	probability_map ← { $\pi$: 0.5, $\pi /4$: 0, $\pi /2$: 0.5, $3\pi /4$: 0 }
16:	else if parent_char = $\pi /2$ then
17:	probability_map ← { $\pi$: 0, $\pi /4$: 0.5, $\pi /2$: 0, $3\pi /4$: 0.5 }
18:	else if parent_char = $3\pi /4$ then
19:	probability_map ← { $\pi$: 0.5, $\pi /4$: 0, $\pi /2$: 0.5, $3\pi /4$: 0 }
20:	end if
21:	Function: adjust_probabilities_opp(probability_map, parent_char)
22:	if parent_char = $\pi$ then
23:	probability_map ← { $\pi$: 0, $\pi /4$: 0, $\pi /2$: 1, $3\pi /4$: 0 }
24:	else if parent_char = $\pi /4$ then
25:	probability_map ← { $\pi$: 0, $\pi /4$: 0, $\pi /2$: 0, $3\pi /4$: 1 }
26:	else if parent_char = $\pi /2$ then
27:	probability_map ← { $\pi$: 1, $\pi /4$: 0, $\pi /2$: 0, $3\pi /4$: 0 }
28:	else if parent_char = $3\pi /4$ then
29:	probability_map ← { $\pi$: 0, $\pi /4$: 1, $\pi /2$: 0, $3\pi /4$: 0 }
30:	end if
	Note: All evaluations assume the predefined optical ordering that includes the master share. Enforce high intensity under authorized stacking and low intensity under unauthorized stacking.

2.3. Threat Model and Security Goals

2.3.1. Threat Model

In this work, we consider a Visual Secret Sharing (VSS) scenario involving both honest and dishonest participants. The dealer generates a total of $N+1$ shares, consisting of N ordinary shares distributed to participants and one master share retained by a trusted authority or coordinator. The threat model focuses on dishonest or unauthorized participants who attempt to recover information about the secret image without access to the master share.

The adversary model is defined as follows:

• Adversarial capability: Dishonest participants may obtain up to N ordinary shares through collusion. They can arbitrarily stack these shares in any order, perform repeated stacking trials, and visually observe the reconstructed outputs. The adversary may adaptively change stacking permutations based on previously observed results.
• Physical access assumptions: The adversary has physical access to the distributed ordinary shares and can observe their polarization behavior through stacking. However, the adversary does not possess specialized optical measurement equipment capable of precisely estimating polarizer orientation angles at the pixel level, nor can they alter the encoded polarization orientations of a share without destroying its physical structure.
• Master share restriction: The master share is never accessible to any ordinary participants. Any stacking attempt that does not include the master share is considered unauthorized.
• Knowledge assumptions: The adversary may know the general design of the scheme, including the set of allowed polarizer orientations and the use of probability-driven assignment. However, the adversary does not know the per-pixel polarization realizations selected by the PDT. The fake image embedded in the scheme is also unknown to the adversary; however, the security of the proposed method does not rely on secrecy of the fake image.

This threat model reflects realistic misuse scenarios in which insiders or colluding participants attempt to infer the secret image through unauthorized stacking, repeated observation, or adaptive manipulation of share combinations.

2.3.2. Threat Mitigation Categories

Existing approaches to dishonest participation in VSS generally fall into two distinct categories:

• Cheating detection and verification, where dishonest behavior is identified by detecting invalid or modified shares before or during reconstruction;
• Cheater deception, where unauthorized reconstruction attempts are deliberately misled to produce meaningless noise or a plausible but incorrect image.

Most prior cheating-prevention VSS schemes emphasize detection or verification, often requiring additional verification images, hash functions, or cryptographic checks. In contrast, the proposed scheme primarily targets the cheater deception category. Rather than attempting to detect or flag dishonest participants, the scheme ensures that unauthorized stacking attempts deterministically yield a fake image or random-like output, thereby preventing meaningful information leakage even when cheating goes undetected.

Detection or explicit identification of dishonest participants is not a primary design goal of this work. Instead, the objective is to suppress the incentive and effectiveness of cheating by ensuring that unauthorized behavior produces misleading outcomes.

2.3.3. Security Goals

Under the defined threat model, the proposed scheme is designed to achieve the following security goals:

• Correctness: When the master share and a qualified set of ordinary shares are stacked together according to the prescribed optical ordering, the secret image is reconstructed with sufficient visual fidelity to be correctly recognized, despite minor distortions caused by polarization overlap and optical variability.
• Confidentiality against unauthorized stacking: Any stacking attempt that excludes the master share reveals no meaningful information about the secret image. Unauthorized reconstructions should not enable reliable inference of secret pixels beyond random guessing.
• Fake-image enforcement (cheater deception): Unauthorized stacking of ordinary shares results in a visually plausible fake image or random noise, rather than a degraded or partial version of the secret. This enforces semantic misdirection rather than simple quality degradation.
• Resistance to adaptive and repeated trials: Repeated stacking attempts and adaptive permutation strategies using ordinary shares do not improve the adversary’s ability to infer the secret image, due to the probabilistic polarization assignment governed by the PDT.

These goals are evaluated experimentally through visual inspection, contrast analysis, and objective metrics that differentiate authorized reconstructions from unauthorized and partial ones.

2.3.4. Distinction from Prior Polarization-Based Stacking Methods

Prior polarization-based VSS schemes have shown that the relative orientation or stacking order of polarizers can influence reconstruction outcomes. However, such schemes typically rely on deterministic or fixed polarization assignment rules, making the relationship between stacking behavior and reconstruction outcome predictable once the scheme is known.

The key novelty of the proposed approach lies in the integration of a Probability Distribution Tree (PDT) into the polarization assignment process. Unlike prior methods, the PDT introduces controlled, per-pixel probabilistic variation in polarizer orientations while preserving correctness for authorized stacking. This probabilistic mechanism ensures that:

• Identical stacking strategies across different pixels do not exhibit uniform or predictable behavior;
• Unauthorized stacking does not converge toward partial disclosure even under repeated trials;
• Fake-image enforcement emerges as a systematic outcome rather than a coincidental artifact of polarization mismatch.

Thus, the PDT does not merely enforce stacking constraints but actively increases uncertainty and suppresses inference in unauthorized reconstructions, distinguishing the proposed scheme from existing polarization-based stacking-sequence approaches.

2.4. Definitions

This subsection introduces the primary parameters used in the proposed scheme to show the functionality of the proposed scheme at the initial stage. The notations and parameters used into the proposed scheme are given in

Table 1. Notation and parameters used in the proposed scheme.

Symbol	Description
N	Number of ordinary shares distributed to participants
M	Master share (unique, not distributed to participants)
$N+1$	Total number of shares, including N ordinary shares and one master share
$n\times n$	Spatial resolution of the secret and fake images
$\mathrm{\Theta }$	Set of allowed polarization orientations $\{0,\pi /4,\pi /2,3\pi /4\}$
$F(i,j)$	Pixel value of the fake image at location $(i,j)$
$F^{\prime }(i,j)$	Pixel value of the fake image at location $(i,j)$ after stacking
$S(i,j)$	Pixel value of the secret image at location $(i,j)$
$S^{\prime }(i,j)$	Pixel value of the secret image at location $(i,j)$ after stacking
$I_0$	Incident light intensity
$I_{\mathrm{out}}$	Transmitted light intensity after stacking

.

In the remainder of this paper, N denotes the number of ordinary shares, and the term “authorized stacking” refers to stacking that includes the master share under the predefined optical ordering.

2.4.1. Average Light Transmission ($T_{avg}$)

Average light transmission represents the mean intensity of light that passes through the stacked shares. For an $n\times n$ image, it is defined as:

$$T_{avg}={\displaystyle \frac{1}{n^2}}\sum _{i=1}^n\sum _{j=1}^n{I}_{out}(i,j)$$

(5)

where $I_{out}(i,j)$ denotes the transmitted light intensity of pixel $(i,j)$ after stacking all polarized layers. For secure reconstruction, $0.1\le T_{avg}\le 0.5$, ensuring randomness and contrast stability.

2.4.2. Contrast ($\alpha$)

Contrast measures the visual distinction between black and white regions of the reconstructed image and is defined as:

$$\alpha ={\displaystyle \frac{T_{white}-T_{black}}{1+T_{black}}}$$

(6)

where $T_{white}$ and $T_{black}$ are the average light transmissions of white and black pixels, respectively. A higher $\alpha$ indicates stronger visual contrast.

2.4.3. Visual Recognizability ($V_r$)

Visual recognizability indicates the degree to which the reconstructed image can be visually identified:

$$V_r=\{\begin{array}{cc}1,\hfill & \mathrm{if}\mathrm{the}\mathrm{reconstructed}\mathrm{image}\mathrm{is}\mathrm{visually}\mathrm{identifiable};\hfill \\ 0,\hfill & \mathrm{otherwise}.\hfill \end{array}$$

(7)

In practical evaluation, $V_r>0$ implies successful recovery of the secret.

2.4.4. Security ($S_c$)

Security quantifies the inability of unauthorized subsets of shares to reveal the secret image:

$$S_c=\left\{\begin{array}{cc}1,\hfill & \mathrm{if}\mathrm{unauthorized}\mathrm{stacking}\mathrm{reveals}\mathrm{no}\mathrm{information};\hfill \\ 0,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

(8)

In the proposed scheme, $S_c=1$ when stacking only ordinary shares produces a fake or noisy image, thus preventing the exposure of secret data.

3. Related Work

3.1. Recent Developments in VSS

Secret Image Sharing (SIS) involves distributing a secret image among mutually suspicious receivers by allocating seemingly meaningless shares; the original image can only be recovered when all shares are combined [34]. As an application of SIS, VSS is critical for ensuring the secure and private exchange of sensitive information. For instance, the Multilevel Secret Image-Sharing Scheme (MLSIS) [35] provides an effective method for sharing images among participants with distinct access privileges. Furthermore, a Threshold Changeable SIS (TCSIS) scheme based on univariate and asymmetric bivariate polynomials is proposed in [36] to securely share grayscale and color images within a semi-honest model. In another approach, a hierarchical SIS scheme with multi-group joint management is introduced in [37], designed to protect secret images through distributed storage across multiple cloud platforms. Similarly, a new hierarchical VSS scheme is introduced in [38], which uses a steganographic weight matrix-based embedding method to maintain hierarchy and detect fake shares. Experimental results based on PSNR and relative entropy show the proposed scheme’s superiority in PSNR over prior schemes.

A foundational (k,n)-threshold VSS model was investigated in [39], which recovers the original image only when at least k shares are stacked. Subsequently, ref. [40] introduced a multiple-secret visual cryptography scheme (MVCS) using XOR-based VCS. Further advancements include a novel fuzzy random grid method for encrypting grayscale and color images without pixel expansion [41] and a visual encryption algorithm based on 2D compressive sensing and variable threshold secret sharing [32]. In the biometric domain, [42] proposed a VSS-based cancelable fingerprint template, where fingerprint images are encrypted into shares stored across separate databases. Additionally, a secure XOR-based visual cryptography technique for banking transactions involving joint accounts is presented in [43]. Based on the recent developments, the relevant literature can be broadly categorized into three critical domains: dishonest participation, polarization-based techniques, and probabilistic approaches.

3.2. Dishonest Participation

A critical security concern in visual cryptography is cheating, in which dishonest participants submit counterfeit shares to deceive honest participants into accepting an invalid secret [44,45]. Cheating in VCS commonly involves presenting fake shares to compromise secret reconstruction. Most schemes do not adequately address this vulnerability. To counter it, the Improved Extended Progressive Visual Cryptography Scheme (IEPVCS), a cheating-prevention mechanism that identifies fraudulent shares and prevents the recovery of an invalid secret image is proposed in [46]. The threat of dishonest participants is a key impediment to practical SIS applications. To counter this, a method that integrates polynomial-based SIS, visual cryptography, and hash functions via a screening mechanism is proposed in [20]. They argued that, this fusion ensures participant authentication—with or without a dealer—and guarantees lossless secret image reconstruction. A significant vulnerability exists during the secret reconstruction phase, as dishonest shareholders can present falsified shares, causing considerable harm to honest participants. To address this, authors in [47] proposed a secure verification method designed to detect cheating shares, thereby enabling fair reconstruction of the secret image. Their approach embeds a verification image within the original shares of an XOR-based visual cryptography scheme. Detection capability is realized via pairwise superimposition of shares, which permits honest participants to detect any malicious manipulation. A novel multi-factor cheating-prevention scheme is proposed in [48], where the proposed method leverages a hybrid codebook to conceal verification data within the share images. Detection operates on two factors of individual shares and their reconstruction to prevent adversarial cheating.

3.3. Polarization-Based Techniques

Polarization-based visual cryptography employs the polarization state of light to encrypt images, which eliminates pixel expansion and allows for lossless reconstruction of the secret image. Its limitation, however, is the generation of meaningless noise-like shares. To address this, a multi-tone meaningful visual cryptography scheme based on optical axis angle superposition (OAAS-MVC) has been proposed in [49]. In [29], the authors experimentally present a polarization-assisted secret sharing scheme as a proof of concept. This method integrates Visual Secret Sharing with phase encoding by leveraging an orthogonally polarization-multiplexed metasurface. Physical key theft presents another significant problem for conventional visual cryptography. In response, ref. [27] introduces a new field of visual cryptography based on polarization, which decodes images by stacking polarizing films to achieve superior image quality. This study aims to develop a visual cryptography system grounded in such polarization processing. The work in [50] presents a VSS scheme that employs metasurfaces featuring multiple polarization-dependent channels with minimized crosstalk. This strategy enables scalable, high-capacity encryption and can be extended to utilize additional optical degrees of freedom. The approach offers an advanced platform for secure communication, information storage, and anti-counterfeiting applications.

A summary of recent polarization-based visual cryptography schemes includes a multi-secret sharing method for grayscale images (PMSVC) [30]; a high-fidelity optical decryption technology using incoherent polarization XNOR operations [51]; and a novel (t, n)-threshold P-VSS scheme [28]. The method in [51] specifically employs the RGVSS algorithm to split a plaintext image into binary ciphertexts, which are then printed onto orthogonally polarized films.

3.4. Probabilistic Approaches

There are no methods directly applied probabilistic methods along with polarization techniques for Visual Secret Sharing schemes. However, in the past researchers discussed about the application of probabilistic methods in this direction. The authors in [52] proposed VSS schemes that establish new definitions for contrast and security conditions. For their probabilistic schemes, they specifically discuss the ability to recognize small areas in the recovered image. By utilizing the frequency of white pixels, the human visual system is enabled to distinguish “black” from “white.” Notably, these schemes feature non-expansible shadow sizes while preserving the same contrast level as conventional VSS. In [53], a formal framework for probabilistic secret sharing schemes, wherein the secret can be reconstructed by any qualified subset with a predetermined probability is introduced.

This paper presents a novel method for dishonest-resistant Visual Secret Sharing by uniquely combining two prominent techniques: linearly polarized shares and Probability Distribution Trees. A novel application of Probability Distribution Trees within a linear polarization framework is introduced in this research. The study is particularly significant for applications requiring high information security, such as the sharing of healthcare images or personally identifiable information (PII), where protecting visual attributes is essential.

4. Proposed Method

This paper proposes a visual secret-sharing scheme that generates a total of N + 1 shares, utilizing polarizers with four distinct orientations: $0°\left(\pi \right),45°\left({\displaystyle \frac{\pi }{4}}\right),90°\left({\displaystyle \frac{\pi }{2}}\right),\mathrm{and}135°\left({\displaystyle \frac{3\pi }{4}}\right)$. Among these shares, N are designated as ordinary shares, while one is the master share.

Unauthorized stacking produces a fake image, while only stacking that includes the master share in the prescribed ordering reveals the secret image. The scheme embeds both fake and secret images to enhance security and visualization. It leverages polarization properties of light, where passing through identical polarizers preserves intensity, and perpendicular polarizers block light (Malus law). Since polarizers are cascaded, the optical order of shares directly affects the transmitted intensity and reconstruction outcome, enabling secure and efficient image encryption. The overall structure of the secret-sharing scheme is illustrated in Figure 3.

To reduce computational complexity and improve contrast, both the secret and fake images are first binarized. Grayscale pixels are converted to binary values using a fixed threshold, producing black $\left(0\right)$ and white $\left(1\right)$ pixels. This binary representation simplifies processing and enhances visual clarity during image reconstruction. Furthermore, PDT governs the probabilistic allocation of polarizer orientations among shares. This probabilistic mechanism increases the randomness of the reconstructed output, improving resistance against unauthorized recovery.

4.1. Step 1: Before Embedding

Before embedding, the secret image (S) and fake image (F) are binarized and resized to equal square dimensions ($n\times n$) to simplify computation and maintain pixel alignment. The binary format uses only black (0) and white (1) pixels, corresponding to light-blocking and transmitting behavior in polarizers. The total number of shares, $N+1$, includes one master share and N ordinary shares, with at least three shares ($N\ge 2$) required to form the PDT. For practical implementation, N is limited to between 2 and 10 to balance security and computational efficiency. The PDT probabilistically assigns polarization orientations so that only stacking that includes the master share in the prescribed optical ordering reveals the true secret image, while any other combination produces a fake or noisy result. Figure 4. shows binarization of the original images.

4.2. Step 2: Embedding Process

The embedding process maps each pixel of the secret (S) and fake (F) images to polarization orientations across the shares. For each pixel $(i,j)$, a binary pair $\left(F\right(i,j),S(i,j\left)\right)$ is first determined. This pair guides the choice among several embedding strategies $(0,0),(1,0),(1,1),(0,1)$, ensuring that pixels intended to be fake or secret follow distinct orientation patterns. Pixel patterns and corresponding algorithm numbers are shown in

Table 2. Single pixel stage of fake and secret image, and algorithms used for that particular pattern.

Pixel (Fake Image)	Pixel (Secret Image)	Algorithm
1	1	Algorithm 1
0	0	Algorithm 2
1	0	Algorithm 3
0	1	Algorithm 4

. The basic structure of the algorithms are represented in Algorithm 1, for different embedding strategies; adjust probabilities functions are used which differs from each other. The four embedding algorithms correspond to the four possible combinations of fake and secret pixel values. Each case imposes a distinct reconstruction requirement: the secret pixel value must be correctly reproduced under authorized stacking, while a misleading or noise-like pixel value must appear under unauthorized stacking that excludes the master share. Accordingly, each algorithm applies a different conditional probability adjustment strategy within the PDT to bias polarization orientation selection toward the desired optical behavior for that specific pixel case.

Afterwards, the master share is assigned to a fixed optical position within the stack that should be predefined prior to PDT generation, and all PDT streams are generated consistently under this assumed optical ordering. The tree branches according to the total number of shares N (practically $2\le N\le 10$) and the selected master level, with probabilities adjusted so that streams including the master produce the intended secret behavior.

For the given pixel case, candidate streams from the PDT are evaluated. Using Malus’ law $I=I_0{cos}^2\theta$, the expected transmitted intensity is computed for each stream. Intensity thresholds (e.g., $0.125I_0$ for white, $0.5I_0$ upper bound) are applied to ensure that ordinary shares alone reconstruct the fake image while inclusion of the master share reveals the secret. Streams that satisfy these criteria with respect to the predefined optical ordering that includes the master share are selected for embedding.

Finally, the chosen stream orientations are assigned to the share pixels $(S_{r1}(i,j),S_{r2}(i,j),\dots ,S_{rN}(i,j))$. The resulting shares satisfy the design property that stacking ordinary shares produces the fake image $F^{\prime }$, whereas stacking with the master share reconstructs the secret $S^{\prime }$. For physical implementation, share images encode these orientations into polarizer patches. All probability updates and intensity evaluation assume the predefined optical ordering of shares, with the master share occupying its predetermined optical position.

Before presenting the embedding algorithms, the following common variables are defined. The variable steps denotes the number of levels in the PDT and is set equal to the total number of shares participating in reconstruction under the predefined optical ordering, including the master share. The variable probability_map represents the conditional probability distribution over the polarization orientation set $\mathrm{\Theta }$ at a given PDT level. This probability map is updated at each level based on the polarization orientation selected at the parent node and the corresponding pixel case $(F,S)$. All PDT construction and stream evaluation procedures operate on a per-pixel basis and assume the predefined optical ordering of shares established prior to embedding.

4.3. Share Creation and Distribution

To produce shares, polarizers are manually arranged. A single share alone does not reveal any image; at least two shares are needed to form a visible pattern, whether meaningful or random. This arrangement ensures proper overlap of polarizers and preserves the accuracy of the scheme. Shares can initially be created manually for testing. The process generates N-1 ordinary shares and one master share. Any stacking of ordinary shares that excludes the master share, regardless of order or repetition, produces a fake image to mislead dishonest participants or random noise, while only inclusion of the master share reveals the secret. The master share must be securely held by a trusted authority, ensuring that the secret remains protected even if some participants are dishonest. This approach effectively safeguards the secret within an organization. A single share, whether an ordinary share or the master share, is designed to behave equivalently to a conventional linear polarizer, despite being composed of multiple linear polarizer elements with different orientation angles. Owing to this structured arrangement, the collective optical response of the share mimics that of a single linear polarizer, which is represented later.

The proposed scheme is position-dependent but permutation-invariant. During embedding, a predefined optical ordering is assumed, including a fixed optical position for the master share. During reconstruction, any stacking permutation that preserves this optical position yields correct reconstruction, while stacking configurations that exclude the master share result in deceptive outputs.

5. Experimental Results and Analysis

5.1. Results

Experiments were conducted to evaluate the performance and efficiency of the proposed dishonest-participant-resistant VSS scheme. Both the secret and fake images were binary patterns of size $16\times 16$ (as shown in Figure 4) used to assess algorithmic accuracy and computational cost under controlled conditions. The use of small binary patterns in the experiments is intended to isolate and validate the optical behavior and security properties of the proposed scheme at the pixel level. While higher-resolution or grayscale images are important for practical deployment, they are outside the scope of this proof-of-concept study.

The evaluation focused on the core modules: image binarization and PDT generation. Image binarization exhibited negligible variation in computational time across different resolutions, indicating minimal overhead. Since each pixel required an individual PDT, higher resolutions increased both time and memory consumption significantly, with an average generation time of approximately $1\times {10}^{-5}$ s per tree. Figure 5 shows Probability Distribution Trees created based on one pixel(0,0) of both the fake and the secret image.

Runtime profiling using a tic-toc timer revealed that restructuring the embedding code into modular functions and applying conditional branching logic (if–elseif–else) substantially improved execution speed. Furthermore, an increase in the number of participants directly expanded the number of PDT branches, increasing complexity. However, by strategically positioning the master share and creating optimized child nodes, the scheme achieved a notable reduction in redundant branches, leading to faster generation and improved scalability.

Overall, the results confirm that computational efficiency depends strongly on image resolution and participant count, but careful optimization of PDT generation and master-share positioning can significantly enhance performance without compromising security.

5.1.1. Average Light Transmission

In the definition section, the total average light transmission for the image is given in Equation (1). Based on that equation, the transmitted light throughout the stacking of polarizers can be expressed as:

$$T\left[S_{(i,j)}\right]={\displaystyle \frac{{\sum }_{l=1}^K{S}_{k(i,j)}}{K}}$$

(9)

where K is the number of shares stacked over each other. If just the ordinary shares are stacked, $K=N$, and if the master share is also stacked, then $K=N+1$.

For the pixel-based light transmission, both security and visual recognizability must be taken into account. That means $\alpha$, defined in the contrast Equation (2), must be greater than 0 and less than $0.5$ to ensure non-repetition of similar orientations of polarizers over the ordinary shares, which would otherwise produce a fake image or noise for any optical order of stacking.

$$\alpha ={\displaystyle \frac{T_{max}\left[F^{\prime }\left(1\right)\right]-T_{min}\left[F^{\prime }\left(1\right)\right]}{T_{max}\left[F^{\prime }\left(1\right)\right]}}$$

(10)

where $T_{max}$ is the maximum amount of light that can pass through and $T_{min}$ is the minimum amount of light that can pass through the polarizers for a particular polarizer orientation used for the shares. The contrast equation is important primarily for producing a white pixel for either the fake or the secret image.

The difference between revealing the secret and producing a fake image is also an inevitable part of ensuring security.

5.1.2. Controlled Outcomes

The scheme ensures that overlapping polarizers produce predictable results (black or white pixels) based on their orientations. The polarizers will only obey the following regulations described in

Table 3. Summary of polarizer configurations and their properties.

Secret Pixel	Fake Pixel	Intensity After Stacking	Properties
$S^{\prime }(i,j)=0$	$F^{\prime }(i,j)=0$	0	For any random orientations of the polarizers, a black pixel is produced. Node streams with mostly random polarizers or with a higher number of opposite polarizers are suitable.
$S^{\prime }(i,j)=0$	$F^{\prime }(i,j)=1$	Max: 0.51 Min: $0.125I$ (for $F^{\prime }$)	The contrast condition $0.5\ge \alpha \ge 0.125$ should be maintained when stacking ordinary shares. Similar polarizer orientations reduce randomness and increase predictability.
$S^{\prime }(i,j)=1$	$F^{\prime }(i,j)=1$	Max: 0.51 Min: $0.125I$ (overall)	The contrast condition $0.5\ge \alpha \ge 0.125$ should be maintained when stacking all shares. Similar polarizer orientations reduce randomness and increase predictability.
$S^{\prime }(i,j)=1$	$F^{\prime }(i,j)=0$	$0.125I$ (strict) 0 (for $F^{\prime }$)	Node streams containing only one opposite polarizer and mostly similar polarizers are suitable.

to produce desired pixels.

5.1.3. Resistance to Manipulation

The system is resistant to attempts to manipulate the polarizer angles to achieve unintended outcomes. Among the ordinary shares, the master share also has to be superimposed to reveal the secret information. Manipulation among ordinary shares alone does not disclose the secret information. Throughout the simulation results shown in Figure 6, Figure 7, Figure 8 and Figure 9, where $F^{\prime }$ and $S^{\prime }$ are the images revealed after stacking shares, we can make an assumption for different pixels with and without the master share, for a total of four shares (three ordinary shares and one master share).

5.1.4. Unpredictability

The arrangement of polarizers should incorporate uncertainty. The angles can be randomly chosen with the help of a Probability Distribution Tree. Initially, a random polarization orientation is selected for the root of the tree, while subsequent polarizers are selected according to a probability map. The resulting light intensity outcome should not be easily guessed. This adds a layer of security against interception of the signal. Figure 5 and Figure 10 provide visual representations of the unpredictability in selecting polarizers for each share.

5.1.5. Redundancy and Authentication Mechanism

The system incorporates redundancy and self-authentication through interdependent share encoding. Each share contains partial polarization information that allows internal consistency verification before reconstruction. When a fake or modified share is introduced, the mismatch in its encoded polarization pattern generates noticeable distortion or brightness inconsistency during stacking. This feature serves as a lightweight authentication mechanism, enabling users to identify tampered shares visually without additional cryptographic modules. Furthermore, redundancy in pixel-level encoding provides tolerance against minor data loss or damage, ensuring that the secret image remains reconstructible while preserving integrity against unauthorized modification. This mechanism is not intended as a formal authentication or verification protocol and does not provide explicit false-positive or false-negative guarantees but rather offers an informal visual consistency cue. Simulation results of the proposed scheme are represented in Figure 11, Figure 12, Figure 13 and Figure 14. Share 1 is master share in the following simulation.

5.1.6. Peak Signal-to-Noise Ratio (PSNR) Evaluation

Peak Signal-to-Noise Ratio (PSNR) is a commonly used objective metric for quantifying pixel-level similarity between a reference image and a reconstructed image. It is derived from the mean squared error (MSE) and expressed in decibels (dB), where higher values indicate smaller pixel-wise deviations. In the context of Visual Secret Sharing, PSNR provides a numerical measure of how closely a reconstructed image resembles the original secret image on a per-pixel basis.

In the proposed polarization-based $(N+1)$ share VSS scheme, all PSNR was computed between the reconstructed images and the original secret image (with character pronounced ‘Ke’). Although the reconstructed images are not binary and arise from optical intensity modulation governed by polarization overlap, PSNR remains useful for assessing relative pixel-level agreement under different stacking configurations. In this work, PSNR is employed as a supporting quantitative indicator to distinguish authorized reconstructions from unauthorized and fake reconstructions, as well as to examine the behavior of partial reconstructions involving the master share.

Experiments were first conducted across multiple image resolutions, including $8\times 8$, $16\times 16$, $32\times 32$, and $64\times 64$, using the same secret image content. Across all resolutions, stacking sequences that included the master share at the predefined optical position consistently produced higher PSNR values than unauthorized stacking sequences composed solely of ordinary shares. As image resolution increased, the observed PSNR values for authorized reconstructions remained relatively stable, indicating that the proposed scheme scales without introducing significant additional pixel-level distortion. The histogram in Figure 15 represents the PSNR difference results between different stacking sequences for OMOO (64 × 64) orientation to produce the secret. In contrast, unauthorized reconstructions consistently yielded low PSNR values, reflecting substantial deviation from the secret image and confirming the effectiveness of fake-image or noise generation.

Table 4. PSNR-based evaluation across image resolutions (comparison with binary secret image).

Image Resolution	Authorized Stacking PSNR (dB)	Unauthorized Stacking PSNR (dB)	Remarks
$8\times 8$	6.7 to 7.6	4.4 to 5.0	Clear pixel-level separation despite small image size
$16\times 16$	7.1 to 8.0	4.3 to 4.8	Improved stability with increased resolution
$32\times 32$	6.8 to 7.8	4.2 to 4.9	Consistent behavior across scales
$64\times 64$	6.2 to 7.9	4.1 to 4.8	Scalable reconstruction quality

summarizes the PSNR behavior across different image resolutions. Authorized stacking sequences typically achieve PSNR values in the range of approximately 6 to 8 dB, whereas unauthorized stacking configurations remain clustered around 4 to 5 dB. This consistent separation demonstrates that PSNR can reliably distinguish between legitimate and illegitimate reconstruction attempts, even though it does not directly measure visual recognizability.

A second set of experiments focused on a fixed image resolution of $64\times 64$ while varying the number of ordinary shares and the optical placement of the master share. Configurations including OMO, OMOO, OOMO, and OOMOO were evaluated under authorized, partial, and unauthorized stacking sequences. As summarized in

Table 5. PSNR comparison for different stacking sequences and share configurations ($64\times 64$ Images).

Configuration	Stacking Sequence Type	Representative	PSNR (dB)	Interpretation
OMO	Authorized	$O_1$, M, $O_2$	∼7.8	Recognizable secret reconstruction
OMO	Partial Authorized	M, $O_2$, $O_1$	∼8.4 to 8.7	Pixel similarity without full structure
OMO	Unauthorized	$O_1$, $O_2$	∼4.8	Fake or noise output
OMOO	Authorized	$O_1$, M, $O_2$, $O_3$	∼6.8	Stable authorized reconstruction
OOMO	Authorized	$O_1$, $O_2$, M, $O_3$	∼6.9	Master-position invariant behavior
OOMOO	Authorized	$O_1$, $O_2$, M, $O_3$, $O_4$	∼6.2	Scalable reconstruction with more shares
Any	Unauthorized	Ordinary shares only	4.0 to 4.9	No meaningful pixel-level similarity

, authorized reconstructions across all configurations consistently produced PSNR values higher than those of unauthorized reconstructions, regardless of master share placement or collusion size.

Interestingly, certain partial reconstruction sequences that included the master share but fewer ordinary shares occasionally produced PSNR values comparable to, or slightly higher than, those of full authorized reconstructions. This behavior arises from the averaging nature of pixel-level intensity differences and highlights a known limitation of PSNR: higher PSNR does not necessarily imply better visual recognizability. Nevertheless, unauthorized reconstructions consistently remained at low PSNR levels, confirming that pixel-level similarity to the secret image is effectively suppressed when the master share is absent.

5.1.7. Structural Similarity Index (SSIM) Evaluation

Structural similarity index (SSIM) is a perceptual image quality metric designed to measure structural resemblance between two images by jointly considering luminance, contrast, and structural information. Unlike pixel-wise error metrics, SSIM evaluates how well local image structures are preserved, making it particularly suitable for assessing visual quality and recognizability. SSIM values typically range from $-1$ to 1, where values closer to 1 indicate stronger structural similarity.

In the proposed polarization-based $(N+1)$ share VSS scheme, SSIM is employed to evaluate how effectively the structural content of the binary secret image is preserved in reconstructed grayscale images obtained through polarized share stacking. Since the reconstruction process relies on optical intensity modulation rather than direct pixel replication, SSIM provides a more meaningful indication of whether the reconstructed image retains the recognizable structure of the secret image. All SSIM evaluations in this work compare reconstructed images against the original binary secret image.

Across all experiments, authorized stacking sequences that include the master share at the predefined optical position consistently achieve substantially higher SSIM values than unauthorized stacking sequences composed solely of ordinary shares. This behavior is observed consistently across different image resolutions, stacking sequences, master-share placements, and numbers of ordinary shares. As image resolution increases from $8\times 8$ to $64\times 64$, SSIM values for authorized reconstructions remain moderate and stable, indicating that the proposed scheme preserves essential structural features of the secret image even as finer details are introduced.The histogram in Figure 16 represents the SSIM difference results between different stacking sequences for OMOO (64 × 64) orientation to produce the secret.

In contrast, unauthorized reconstructions consistently yield SSIM values close to zero or negative across all tested configurations. Negative or near-zero SSIM values indicate the absence of meaningful structural correlation with the secret image, confirming that unauthorized stacking produces visually misleading outputs rather than degraded versions of the secret. Importantly, reordering or selectively stacking ordinary shares does not improve SSIM, demonstrating resistance to adaptive or permutation-based attacks.

Partial reconstruction scenarios, where the master share is present but the number or placement of ordinary shares is insufficient, exhibit intermediate SSIM values. These values decrease gradually as fewer ordinary shares participate or as their optical proximity to the master share diminishes. This graded behavior reflects controlled partial reconstruction rather than accidental leakage and confirms that structural information is revealed only in a predictable and limited manner when the master share is involved.

Table 6. SSIM evaluation across image resolutions (comparison with binary secret image).

Image Resolution	Authorized SSIM Range	Unauthorized SSIM Range	Observation
$8\times 8$	0.57 to 0.65	−0.06 to 0.05	Structural separation visible even at low resolution
$16\times 16$	0.55 to 0.62	−0.16 to 0.03	Improved stability with increased resolution
$32\times 32$	0.49 to 0.56	−0.08 to 0.04	Clear authorized vs. unauthorized distinction
$64\times 64$	0.39 to 0.49	−0.03 to 0.02	Consistent structural preservation at higher resolution

shows the SSIM results for different size reconstructions.

At a fixed image resolution of $64\times 64$, SSIM behavior was further analyzed under different stacking sequences, master-share placements, and numbers of ordinary shares.

Table 7. SSIM comparison for different stacking sequences and share configurations ($64\times 64$ Images).

Configuration	Stacking Type	Representative Pattern	SSIM	Structural Interpretation
OMO	Authorized	$O_1$, M, $O_2$	∼0.49	Clearly recognizable structure
OMO	Partial	M, $O_2$, $O_1$	∼0.38–0.41	Partial structural preservation
OMOO	Authorized	$O_1$, M, $O_2$, $O_3$	∼0.43	Stable structural reconstruction
OOMO	Authorized	$O_1$, $O_2$, M, $O_3$	∼0.43	Master-position invariant
OOMOO	Authorized	$O_1$, $O_2$, M, $O_3$, $O_4$	∼0.39	Scalable structural fidelity
Any	Unauthorized	Ordinary shares only	≈0 or negative	No meaningful structure

summarizes authorized reconstructions across OMO, OMOO, OOMO, and OOMOO configurations consistently yield SSIM values in the range of approximately $0.39$ to $0.49$, confirming that recognizable structural information is preserved regardless of master-share position. Partial authorized reconstructions show progressively lower SSIM values as the number of contributing ordinary shares decreases, while unauthorized reconstructions remain structurally decorrelated.

5.1.8. Visual Information Fidelity (VIF) Evaluation

Visual information fidelity (VIF) is a perceptual image quality metric that quantifies how much visual information from a reference image is preserved in a distorted or reconstructed image. Unlike pixel-wise or purely structural measures, VIF is based on natural scene statistics and models the human visual system, making it particularly suitable for evaluating perceptual fidelity and recognizability. Higher VIF values indicate that a larger proportion of the original visual information is retained in the reconstructed image.

In the proposed polarization-based $(N+1)$ share VSS scheme, reconstructed images are grayscale intensity patterns generated through optical polarization stacking, while the original secret image is binary. Under these conditions, VIF provides a meaningful measure of how much perceptually relevant information from the secret image is conveyed to the viewer. All VIF evaluations in this work compare reconstructed grayscale images against the original binary secret image. The histogram in Figure 17 represents the VIF difference result between different stacking sequences for OMOO (64 × 64) orientation to produce the secret.

Across all experiments, authorized stacking sequences that include the master share consistently yield significantly higher VIF values than unauthorized stacking sequences. This behavior is observed regardless of image resolution, master-share placement, or the number of ordinary shares involved. Results are demonstrated in

Table 8. VIF evaluation across image resolutions (comparison with binary secret image).

Image Resolution	Authorized VIF Range	Unauthorized VIF Range	Observation
$8\times 8$	0.16 to 0.21	0.00 to 0.03	Limited but distinguishable perceptual information
$16\times 16$	0.22 to 0.24	0.00 to 0.02	Improved visual fidelity with resolution
$32\times 32$	∼0.24	≈0	Stable perceptual separation
$64\times 64$	0.25 to 0.29	≈0	Strong perceptual recognizability

, as image resolution increases, VIF values for authorized reconstructions show a clear upward trend, indicating improved perceptual fidelity and enhanced visual recognizability of the secret image. This trend contrasts with the behavior of unauthorized reconstructions, whose VIF values remain close to zero across all tested conditions.

At a fixed image resolution of $64\times 64$, VIF was further used to examine the effects of stacking sequence, master-share placement, and collusion size. Authorized reconstructions across OMO, OMOO, OOMO, and OOMOO configurations consistently exhibit VIF values in the range of approximately $0.25$ to $0.29$. Notably, configurations involving a larger number of ordinary shares tend to yield slightly higher VIF values, indicating that increased share participation can enhance perceptual information fidelity without compromising security.

Partial authorized reconstructions involving the master share but fewer ordinary shares produce intermediate VIF values, reflecting partial but incomplete perceptual disclosure of the secret image. These values decrease smoothly as the number of contributing ordinary shares are reduced or as their optical proximity to the master share diminishes. In contrast, unauthorized reconstructions, regardless of stacking order, permutation, or subset selection, consistently yield VIF values close to zero, confirming that they convey negligible perceptual information about the secret image. Results are given in

Table 9. VIF comparison for different stacking sequences and share configurations ($64\times 64$ Images).

Configuration	Stacking Type	Representative Pattern	VIF	Perceptual Interpretation
OMO	Authorized	$O_1$, M, $O_2$	∼0.26	Clearly recognizable
OMO	Partial	M, $O_2$, $O_1$	∼0.20	Partial visual information
OMOO	Authorized	$O_1$, M, $O_2$, $O_3$	∼0.27	Stable perceptual fidelity
OOMO	Authorized	$O_1$, $O_2$, M, $O_3$	∼0.26	Master-position invariant
OOMOO	Authorized	$O_1$, $O_2$, M, $O_3$, $O_4$	∼0.29	Enhanced perceptual fidelity
Any	Unauthorized	Ordinary shares only	≈0	No perceptual information

.

5.2. Comparison with Existing Visual Secret Sharing Schemes

The comparison highlights how the proposed scheme differs conceptually and functionally from most existing approaches, as shown in

Table 10. Comparison of the proposed scheme with representative VSS approaches.

Aspect	Classical/Random-Grid VSS	Polarization-Based VSS	Verifiable/Cheating-Prevention VSS	Proposed Scheme
Uses polarization	No	Yes	No or limited	Yes
Probabilistic share assignment	No	No	No	Yes (PDT-based)
Explicit master share	No	Limited or implicit	No	Yes
Authorized reconstruction requirement	Threshold-based	Orientation-dependent	Threshold + verification	Master-controlled optical ordering
Unauthorized reconstruction outcome	Noise or degraded secret	Partial information leakage possible	Rejected or flagged	Fake image or random output
Cheating mitigation approach	Passive degradation	Passive degradation	Cheater detection	Active deception
Resistance to adaptive stacking	Limited	Limited	Depends on verification	High
Additional verification overhead	No	No	Yes	No

. The comparison is presented in terms of security objectives, reconstruction behavior, handling of dishonest participants, and deployment complexity, which are the most relevant dimensions for polarization-based VSS systems. Direct numerical comparison across schemes is often impractical due to differences in threat models, reconstruction mechanisms, and evaluation settings.

5.3. Analysis

The experimental evaluation provides several important insights into the behavior, robustness, and practical implications of the proposed polarization-based Visual Secret Sharing scheme. From an implementation perspective, smaller image dimensions facilitate the manual construction of physical shares but tend to reduce the perceptual distinction between secret and fake reconstructions. As image resolution increases, the structural and perceptual separation between authorized and unauthorized outcomes becomes more pronounced, confirming that the proposed scheme scales effectively beyond toy examples.

The use of a PDT for per-pixel polarization assignment significantly enhances uncertainty against dishonest stacking attempts. Although this probabilistic mechanism increases computational cost during share generation, it plays a critical role in suppressing predictable polarization patterns and preventing adaptive inference. The predefined optical placement of the master share is central to correct reconstruction: only stacking configurations that include the master share at the intended optical position reveal meaningful information, while arbitrary combinations of ordinary shares consistently result in fake images or noise.

From a quantitative perspective, PSNR serves as a supporting metric to characterize pixel-level similarity between reconstructed images and the binary secret image. Authorized reconstructions consistently exhibit higher PSNR values than unauthorized ones across all tested resolutions, confirming that valid stacking produces lower pixel-level distortion. However, PSNR alone does not reliably reflect visual recognizability, particularly in partial reconstruction scenarios where moderate PSNR values may arise without conveying correct structural information. This observation highlights the limitation of relying solely on pixel-wise error metrics for polarization-based reconstruction.

SSIM analysis provides deeper insight into reconstruction behavior. Authorized stacking sequences consistently achieve substantially higher SSIM values, indicating that the structural content of the secret image is preserved only under valid stacking conditions. Partial reconstructions involving the master share exhibit intermediate SSIM values that degrade gradually as fewer ordinary shares participate, demonstrating controlled and predictable partial disclosure. In contrast, unauthorized reconstructions yield SSIM values close to zero or negative, confirming the absence of meaningful structural correlation with the secret image.

VIF further reinforces these findings from a perceptual standpoint. Authorized reconstructions retain a significantly higher proportion of perceptually relevant information, resulting in clearly recognizable secret images. As the number of ordinary shares increases under authorized stacking, VIF values show a modest improvement, indicating enhanced perceptual fidelity without compromising security. Unauthorized reconstructions consistently produce VIF values near zero, regardless of stacking order or collusion size, confirming that they convey negligible visual information to an observer.

Orientation-based analysis confirms that the scheme is position-dependent but permutation-invariant: authorized reconstructions remain consistent across different ordering permutations that preserve the master share’s optical placement. Conversely, excluding the master share or violating its predefined position leads to a sharp degradation across PSNR, SSIM, and VIF metrics, independent of the number of ordinary shares involved.

The combined analysis of PSNR, SSIM, and VIF demonstrates that the proposed scheme satisfies three critical objectives: (a) reliable and visually recognizable reconstruction under authorized stacking, (b) controlled and limited information disclosure in partial reconstruction scenarios, and (c) effective suppression of perceptual and structural information in unauthorized stacking attempts. These results confirm that the scheme achieves both security and practical usability in polarization-based visual secret sharing.

Lastly, it is observed that the visual appearance and structure of a single share is equivalent to any ordinary linear polarizer which is shown in Figure 18.

5.3.1. Limitations and Future Work

Despite the demonstrated effectiveness of the proposed polarization-based Visual Secret Sharing scheme, several limitations should be acknowledged. First, the current evaluation is primarily simulation-based and does not include physical experiments using real polarizer films. As a result, certain practical factors such as alignment errors, material imperfections, and illumination instability are discussed qualitatively rather than being quantitatively modeled. Although the simulation framework captures the core optical behavior governed by Malus’ law, real-world optical noise may further influence reconstruction quality.

Second, the use of a Probability Distribution Tree (PDT) introduces additional computational overhead during share generation due to per-pixel probabilistic assignment. While this overhead remains manageable for the tested image resolutions, further optimization may be required for high-resolution or real-time applications. In addition, the current implementation focuses on binary secret and fake images; extending the framework to grayscale or color images will require more complex polarization encoding strategies.

Future work will focus on addressing these limitations by incorporating physical experiments with real polarizer films, developing noise-aware reconstruction models, and extending the proposed scheme to support grayscale and color images. Further investigation into optimizing the PDT structure and reducing computational complexity will also be explored to improve scalability and practical deployment.

5.3.2. Application Scenarios and Usage

The proposed scheme is particularly suitable for application scenarios where visual information must be protected against unauthorized access while avoiding complex cryptographic verification mechanisms. One representative use case is secure multi-party image sharing, where a sensitive image is distributed among several participants, and reconstruction is permitted only when an authorized party provides the master share at the predefined optical position.

Another potential application is tamper-resistant visual authentication, such as secure document verification or offline access control, where physical stacking of polarized shares can be used to visually verify authenticity without requiring electronic computation. The fake-image enforcement mechanism further ensures that unauthorized stacking attempts yield misleading outputs, thereby discouraging dishonest behavior.

In practical deployment, the dealer generates polarized shares using the PDT-guided assignment process and distributes ordinary shares to participants while retaining the master share. During reconstruction, only stacking configurations that include the master share at its predefined optical position reveal the secret image, whereas all other combinations result in fake or noisy outputs. This makes the scheme suitable for both digital simulation environments and future physical optical implementations.

6. Conclusions

This study presented a dishonest-participant-resistant Visual Secret Sharing (VSS) scheme based on polarization modulation and Probability Distribution Trees, in which image reconstruction is strictly controlled through predefined polarization orientations and a master-share-dependent optical stacking strategy. The proposed method ensures that only authorized share combinations can correctly reveal the secret image, while unauthorized or partial combinations consistently generate fake or noisy reconstructions, thereby actively misleading dishonest participants.

Comprehensive quantitative and perceptual evaluation demonstrates the effectiveness of the proposed approach. Peak signal-to-noise ratio (PSNR) analysis confirms that authorized reconstructions exhibit significantly higher pixel-level similarity to the secret image than unauthorized and partial reconstructions, supporting the scheme’s resistance to information leakage. Structural similarity index (SSIM) results further show that the structural content of the secret image is preserved only under authorized stacking, while unauthorized reconstructions remain structurally uncorrelated. In addition, visual information fidelity (VIF) analysis indicates that perceptually meaningful information is conveyed exclusively in authorized reconstructions, whereas unauthorized stacking attempts yield negligible visual information regardless of stacking order or collusion size.

The use of Probability Distribution Trees introduces controlled randomness into polarization assignment, enhancing uncertainty against adaptive stacking attacks without imposing substantial computational overhead. The predefined optical placement of the master share provides a simple yet effective mechanism for enforcing correct reconstruction while remaining invariant to permutations that preserve this placement. Importantly, the scheme does not rely on external cryptographic verification or explicit cheater detection; instead, it suppresses dishonest behavior by ensuring that unauthorized actions result in misleading visual outcomes.

In conclusion, the proposed scheme offers an efficient and robust solution for secure polarization-based Visual Secret Sharing, combining controlled reconstruction, active deception, and perceptual security. Future work will focus on extending the framework to higher-resolution imagery, grayscale and color images, and exploring physical implementations that account for optical noise, illumination variation, and real-world deployment constraints.