Secure Information Embedding in Images with Hybrid Firefly Algorithm

The FA, or Firefly Algorithm, is a meta-heuristic optimization technique developed by Xin-She Yang in 2008. It is inspired by the bioluminescent display demonstrated by tropical fireflies. The standard Firefly algorithm utilizes the observable communication patterns seen in tropical fireflies and integrates the idealized behavior linked with their flashing patterns. The construction of the mathematical model in FA is based on three idealized principles.

1. 1.

   Fireflies exhibit a unisex characteristic, whereby their attraction towards other fireflies is not contingent upon their respective sexes.

2. 2.

   The level of attractiveness exhibited by an individual is directly correlated with their level of brightness. Additionally, it can be inferred that attractiveness diminishes as the distance between individuals rises. Hence, in the case of two fireflies exhibiting intermittent luminescence, it can be observed that the firefly with lower luminosity will exhibit a tendency to travel towards the firefly with higher luminosity.

3. 3.

   The luminosity of a firefly is contingent upon the topography underlying the cost function. Hence, within the framework of a maximizing problem, there exists a direct correlation between the amount of light and the value of the fitness function.

The conventional firefly algorithm encompasses two crucial aspects that necessitate consideration: the formulation of the brightness and the alteration in appeal. Initially, it is reasonable to argue that the luminosity of the firefly is contingent upon the cost function landscape. Next, we establish the measurement of light intensity variation and develop a model to quantify the corresponding alteration in attractiveness. It is well-established that the intensity of light exhibits a negative correlation with the distance separating the light source and the medium that absorbs the light. In our computer model, we use the assumption that the brightness $I$ changes exponentially and monotonically with both the distance $r$ and the absorption of light parameter $\gamma$. The aforementioned relationship is expressed as:

The symbol $I_{0}$ represents the initial light intensity emitted from the source, specifically at the point where $r$ = 0. Meanwhile, the symbol $\gamma$ denotes the coefficient that characterizes the absorption of light. Based on the aforementioned idealized principles, it can be inferred that the perceived appealing factor of a firefly is directly correlated with the intensity of its emitted light, denoted as I. The attractive coefficient $\beta$ of the firefly’s light can be defined in a manner analogous to the illumination coefficient $I$.

where $\beta_{0}$ represents the initial level of light attraction when the distance $r$ is equal to zero. The Euclidean distance is employed to compute the separation between two fireflies, denoted as $i$ and $j$, located at positions $x_{i}$ and $x_{j}$ respectively.

where $d$ represents the total number of dimensions. The displacement exhibited by firefly $i$ towards firefly $j$, which is more appealing due to its increased luminosity, is ascertained by:

The initial term in the equation represents the present position of firefly $i$. The subsequent term accounts for the lure between firefly $i$ and firefly $j$. Finally, in the last term, randomization is included by using a vector of independent variables denoted as $\epsilon_{i}$. These variables are sampled from many distributions, including the Uniform distribution, and the Gaussian distribution. The magnitude of the step size is determined by the coefficient scaling parameter $\eta$ in the third term.

This algorithm finds an optimal solution for a problem while evolving from some randomly initiated starting points using a vectorized mutation operator combined with either exponential or binomial crossover. It was proposed by [3] and there have been many other variants of this algorithm that have been proposed since. One of the most widely used variants is the DE/rand/1/bin or the classic DE variant. In the context of a particular minimization problem with $d$ dimensions, population consisting of $n$ independent result vectors is utilized. A change in vector $v_{i}$ is formally characterized as:

where $r_{1},r_{2},r_{1}\epsilon[1,n]$ and $r_{1}\neq r_{2}\neq r_{3}\neq i$ are three randomly selected solutions from the population and g is the generation index. There is also a perturbation parameter $F(F\epsilon[0,2])$ that adjusts the amplification of the difference vector $(x_{r_{2},g}-x_{r_{3},g})$.

Similarly, the binomial crossover operation is also used to generate a new trial vector from the perturbed or mutated vector $v_{i,g+1}$ and the target vector $x_{i,g}$

where the crossover constant $C_{r}\epsilon[0,1]$ and $random\epsilon[1,2,...,d]$ is a random permutation vector index that ensures at least one mutated vector is included in the trailing vector.

The selection mechanism for this algorithm is as below:

This approach has a resemblance to previous algorithms in which a greedy acceptance strategy is used, wherein an update is deemed acceptable only if it leads to an improvement in the present target being pursued.

Both the firefly method and differential evolution provide distinct benefits and demonstrate efficacy across a diverse array of optimization tasks. The hybrid Firefly method is derived from the FA and the DE algorithm. This hybrid approach aims to use the strengths of both algorithms, resulting in an improved optimization technique. The proposed approach integrates the attraction mechanism used in the FA with the diversification mechanism of DE algorithm. This integration aims to enhance the convergence rate while simultaneously preserving population diversity.

The use of an intensification or exploitation strategy involves directing the search towards a specific local location. This direction is based on either past knowledge or newly acquired information obtained during the search process, which suggests the potential presence of an optimal solution inside that particular region. Differential Evolution (DE) is capable of performing localized searches, particularly when converging towards local optimum solutions. Enhancing the accuracy and convergence rate of an algorithm’s solution may be achieved by effectively balancing the conflicting needs associated with these two factors. By integrating these two aspects, we may use their respective strengths to improve the efficacy of the hybrid algorithm in both exploitation and exploration.

It is vital to acknowledge that the consolidation and restructuring of individual location data takes place alone after the first iteration of concurrent FA and DE procedures. This stands in opposition to the generation of novel sites by random walks or alternative operators. The primary advantage of using this technique is to prioritize the exploration of current locations inside advantageous regions obtained during the preceding iteration, as opposed to traversing less promising places within the search space.

The hybrid approach demonstrates linear time complexity while the population size remains relatively small. Consequently, it efficiently computes solutions in terms of computational cost, mostly attributed to the evaluation of objective function values.

The primary aim of our proposed technique is to determine the optimal arrangement of pixels for the reason of incorporating PDF data into a picture, with the ultimate goal of minimizing any possible alteration in the cover photo. The proposed technique is supported by a structure of procedures that encompasses the subsequent steps:

1. 1.

   The first stage is the perusal of a confidential document intended for incorporation into the cover picture.

2. 2.

   The binary numbers derived from the PDF document will undergo encryption using the Blowfish encryption technique, as described by [47].

3. 3.

   To ensure the covert embedding of the encrypted information inside the cover picture, the 128-bit encrypted information is converted into a sequence of binary numbers.

4. 4.

   The pixels of the cover picture undergo a process of transformation whereby they are converted into their respective 8-bit counterparts.

5. 5.

   The objective function is obtained by means of a computational procedure from the multi-objective function. This metric functions as a measure of the degree of accuracy between a steganographic picture and the fundamental data. The calculation of the PSNR entails evaluating the difference between the highest achievable magnitude of the steganographic signal and the strength of the noise that impacts the accuracy of its depiction.

6. 6.

   Population initialization: The firefly population initially forms through a random initialization process in designated groups denoted as $G_{1}$ and $G_{2}$. Within the solution space $\Omega\in\mathbb{R}^{d}$, a set of fireflies, denoted as $\mathbb{F}$, is created randomly in each group $\mathbb{F}(1)$, $\mathbb{F}(2)$, $\mathbb{F}(3)$, $\ldots$, $\mathbb{F}(n)$. Here, $\mathcal{D}$ signifies the cumulative determinants. Consequently, we define a vector $\mathbb{F}(i)=\mathbb{F}(2)$, $\mathbb{F}(3)$, $\ldots$, $\mathbb{F}(n)$ to represent the values of the selection variables associated with the $i^{th}$ firefly.

7. 7.

   Rank fireflies: The fitness of every firefly within the set $\mathbb{F}(i)$ is assessed based on the value $fitness\mathbb{(F}(i))$. Subsequently, this procedure is employed to classify the population into distinct clusters. These clusters are then organized in descending order and distributed in a round-robin fashion.

8. 8.

   Execute DE algorithm: Once the population for group $G_{2}$ has been initialized and the parameters have been set, the next step is to execute mutation and crossover operations on the potential solutions.

9. 9.

   Rank fitness: The fitness of each produced vector would be assessed in a manner akin to the firefly method.

10. 10.

    Perform Search: An evaluation is carried out inside each group, wherein the proficiency of each participant in each group is assessed. The global results are then updated by considering the fittest individuals from each group. The groups are then merged and subsequently partitioned into the subsequent set of groups denoted as $G_{1}$ and $G_{2}$, using a random allocation process. Consequently, a new population is formed. The aforementioned stages are iteratively executed until either the specified termination criteria are satisfied or an optimum solution is attained.

11. 11.

    Utilizing the HFA methodology, ascertain the optimal selection of pixels for the purpose of data hiding inside the cover photo.

12. 12.

    The 1-LSB method is used to discreetly include encrypted information bits into a host image, minimizing any noticeable effects on the image’s integrity.

1. 1.

   Reduce the dimensionality of a 3-channel steganographic color picture (m x n x 3) to a 2-dimensional image (m’ x n’).

2. 2.

   Equation 10 is used to create a specialized objective function that takes into account SSIM and PSNR values.

3. 3.

   Population initialization: The initial population is generated in accordance with section 4.1

4. 4.

   After configuring all parameters, the HFA algorithm is executed in order to minimize the cost function. The result of this procedure is the generation of an ideal arrangement of pixels that could be used for the retrieval of embedded data.

5. 5.

   The encrypted data may be accessed by using the order derived from the preceding step.

6. 6.

   For decryption, Blowfish algorithm is used to transform the data into its binary representation.

7. 7.

   From the binary bits reconstruct a PDF file.

The Peak Signal-to-Noise Ratio (PSNR) is a numerical measure, expressed in decibels (dB), that enables the evaluation of the extent of degradation in the steganographic signal. The correlation between the quality of steganographic photos and the rise in PSNR value is directly proportional.

The SSIM, or Structural Similarity Index, is a quantitative metric utilized to assess the degree of resemblance between two given images. The comparison approach combines the structural characteristics and luminosity measurements of the images under scrutiny. The Structural Similarity Index (SSIM) is calculated by assessing the resemblance between two photographs based on their brightness, contrast, and general arrangement. The system produces a numerical result that is between -1 and 1. A score of 1 indicates a significant degree of resemblance between the two photos, whereas a score of -1 indicates a considerable degree of dissimilarity.

The Mean Squared Error (MSE) is a widely utilized statistical metric in academics for quantifying the average squared discrepancy between the anticipated and actual values of a designated variable. Currently, the assessment focuses on evaluating the quality of the steganographic image in relation to the cover image. The technique involves calculating the mean of the squared differences between corresponding pixels or patterns in the original and hidden images.

where $(i,j)$ represents the $i^{th}$ and $j^{th}$ pixel of the cover $\mathcal{X}_{c}(i,j)$ and steganographic $\mathcal{X}_{s}(i,j)$ image, and $|m|=|n|$ as the sum of the host’s and the steganographic photo’s pixels is the same.

The equation representing the PSNR may be stated as follows:

The variable $MAX_{C}$ denotes the highest pixel value included in the cover image. Hence, a comprehensive fitness function may be defined as

In this context, $\mathbb{X}_{s}$ and $\mathbb{X}_{c}$ denote two images that are being subjected to a comparative analysis. In the context of our study, the symbol $\mathbb{X}_{s}$ is used to denote a steganographic image, whereas $\mathbb{X}_{c}$ is used to represent a cover image. Since the cover image $\mathbb{X}_{c}$ remains unchanged during the embedding process, the fitness function may be expressed as

Here the value of $\eta$ is taken as 0.5.

The Structural Similarity Index (SSIM) value gives a consistent metric for evaluating the quality of a picture. Mathematically, it is defined as:

The window size of the steganographic image is denoted by variables $u$ and $v$, while its average is denoted by variables $\hat{u}$ and $\hat{v}$. The variance of variables $u$ and $v$ is denoted as $\sigma^{u}_{2}$ and $\sigma^{v}_{2}$, respectively, whereas their covariance is denoted as $\sigma_{uv}$.

This statistic considers the influence of the Human Visual System (HVS). The model distortions of the HVS consist of three primary elements: luminance distortion, loss of correction, and contrast distortion. The mathematical models used to represent them are as follows:

The primary component has a linear relationship and encompasses a comprehensive range of values ranging from -1 to 1. The second element demonstrates a dynamic range including the values of 0 and 1, inclusively. The aforementioned range is used for the purpose of representing the luminance values of variables $u$ and $v$. The third component exhibits a contrast similarity measure that spans from 0 to 1. Hence, the variable $Q$ is constrained inside the interval [-1, 1]. It is essential to recognize that this condition is true only in cases when the denominator is not equal to zero. When the denominator term in the quotient being examined approaches zero, the expression demonstrates instability. To address this concern, the use of the SSIM is often employed as a means of quantifying the level of similarity between two pictures. The metric is derived by assessing the given equation.

Take the window sizes of the original and steganographic images to be $u$ and $v$, respectively. Let the median values of $u$ and $v$ also stand for attention to detail. The symbols $\sigma^{u}_{2}$ and $\sigma^{v}_{2}$ represent the variances of the variables $u$ and $v$, respectively. On the other hand, $\sigma_{u}v$ denotes the covariance between $u$ and $v$.