A secured image steganography method based on ballot transform and genetic algorithm

Abstract

Rapid advances in digital technology have facilitated us to transfer a huge amount of electronic files over the internet. But in the presence of malicious attackers, the security, as well as the integrity of such important files, becomes of utmost importance. Steganography, an art of hiding the data, ensures the security of these files over the internet, and this method has been used for a long. In this paper, our purpose is two-fold. First, the Ballot transform (BaT) produces an integer polynomial sequence in coefficient form for each non-overlapping m-pixel groups (m = 2 or 3) of the cover image. Second, the Genetic Algorithm (GA) is applied to generate a k-digit password using a method called Index Value Mapping (IVM) which decides positions of the transformed coefficients where the bits of the secret data can be embedded. The key idea of using GA here is to produce an optimal output in terms of stego image quality instead of using an exhaustive search considering all combinations of k number of bits. Experimental results show that the proposed method not only offers acceptable Peak Signal-to-Noise Ratio (PSNR) values with considerable payload but also provides two-way security of the data being transmitted over the digital medium.

Appendix

1.1 An overview of Ballot Transform (BaT)

In 2007, Barry [6] introduced BaT and its inverse i.e., IBaT in the context of exponential Ridoran arrays. He proposed BaT as a composition of Catalan Transform (CT) and Binomial Transform (BT). In terms of generating functions, on the application of BaT any sequence of generating function f(z) converts to sequence of generating function CN(z)f(CN(z) − 1), where the generating function of Catalan Number is CN(z). The formulation of the Riordan matrix of BaT is thus given by.

BaT = CT ∗ BT where “*” denotes composition

$$ {\displaystyle \begin{array}{c}=\left(1, CN(z)\right)\left(\frac{1}{1-z},\frac{z}{1-z}\right)\\ {}=\left(\frac{1}{1- zCN(z)},\frac{zCN(z)}{1- zCN(z)}\right)=\left( CN(z), CN(z)-1\right)\end{array}} $$

Its inverse transform or IBaT is formulated as

$$ {\displaystyle \begin{array}{c}\begin{array}{c} IBaT={BT}^{-1}\ast {CT}^{-1}\\ {}={\left(\frac{1}{1-z},\frac{z}{1-z}\right)}^{-1}{\left(1, CN(z)\right)}^{-1}\\ {}=\left(\frac{1}{1+z},\frac{z}{1+z}\right)\left(1,z\left(1-z\right)\right)\end{array}\\ {}=\left(\frac{1}{1+z}\times 1,\frac{z}{1+z}\left(1-\frac{z}{1+z}\right)\right)\\ {}=\left(\frac{1}{1+z},\frac{z}{{\left(1+z\right)}^2}\right)\end{array}} $$

Similarly, on the application of IBaT any sequence with generating function g(z) converts to sequence with generating function $\frac{1}{1+z}g\left(\frac{t}{{\left(1+z\right)}^2}\right).$ Let us consider p₀, p₁, …, p_u be the pixel values in a given pixel group P. By applying BaT one can compute the transformed components t₀, t₁, …, t_uas shown in Eq. (6):

$$ {t}_u=\sum \limits_{v=0}^u\frac{2u}{u+v}\left(\genfrac{}{}{0pt}{}{2v+1}{u+v+1}\right){p}_v $$

(6)

where, for all u, 0 ≤ u ≤ size(P) – 1.

By using Eq. (6), BaT is applied over 3-pixel groups (i.e. size (P) = 3) to derive the transformed triplets as follows:

$$ {t}_i=\left\{\begin{array}{c}{p}_0\kern7.25em if\ i=0\\ {}{p}_0+{p}_1\kern4.75em if\ i=1\\ {}2{p}_0+3{p}_1+{p}_2\kern1.25em if\ i=2\end{array}\right. $$

Again, by applying IBaT, one can re-calculate pixel values$\kern0.5em p{\prime}_0,p{\prime}_1,\dots, {p}_u^{\prime }$as shown in Eq. (7):

$$ p{\prime}_u=\sum \limits_{v=0}^u{\left(-1\right)}^{\left(u-v\right)}\left(\genfrac{}{}{0pt}{}{u+v}{2v}\right){t}_v $$

(7)

where, for all u, 0 ≤ u ≤ size(P) – 1.

By using Eq. (7), IBaT is applied over transformed group to re-compute the 3-pixel groups as follows:

$$ p{\prime}_i=\left\{\begin{array}{c}{t}_0\kern6.5em if\ i=0\\ {}-{t}_0+{t}_1\kern3.25em if\ i=1\\ {}{t}_0-3{t}_1+{t}_2\kern1em if\ i=2\end{array}\right. $$

In case of no alteration, all re-computed pixel values are found to be the same corresponding to the pixel values used before applying BaT i.e., p′_i = p_i.

1.2 An overview of Genetic Algorithm (GA)

GA is an adaptive technique that is exploited to get solutions to various optimization and searching based problems. This technique follows the genetic processes observed in the biological structures. The main idea in GA is over the generations, any natural populations evolve in accordance with the principles of natural selection and survival of the fittest, which is mentioned by Charles Darwin in his book The Origin of Species. When encoded suitably, by imitating this process, GA can evolve the solutions of any real-world optimization problem to find the better one. GA has the following rudimentary steps as proposed by Holland [23]:

• a). In the initial population, members are the representative solutions to the problem, and an optimized solution is needed. These solutions come from the search space.

• b). An encoding of the population members is followed.

• c). The fitness function is exploited to calculate the fitness of an individual in the population so that the next generation is to be created and only the individuals suitable for survival are considered.

• d). Selection of individuals is done to mate for producing offspring.

• e). The main reproduction is called a crossover and is responsible for forming offspring. The offspring inherit the characteristics of their parents, intending to cultivate better individuals in subsequent populations.

• f). The mutation operation outlined by updating one or more genes of an individual follows the crossover stage. Mutation can assure genetic diversity within the total population.

• g). Termination conditions indicate when GA should stop. Some most widely used termination conditions are: the highest solution adaptability value is consistent in successive iterations, finding a solution that meets certain minimum standards, reaching a specified number of generations, or any combination of the above.