2.1 2D Sine-Infinite Collapse Map (2D-SICM)
To enhance the randomness of the encryption process, 2D-SICM is used to generate the random matrix, which is better than 1D system. This makes the encryption system better able to withstand attacks. The 2D Sine-Infinite Collapse Map (2D-SICM) is described in this section and its chaotic behavior is investigated.
1D sine map is defined as
$${x_{n+1}}=\mu \sin (\pi {x_n}),$$
1
where n represents the number of iterations, xn is the n-th chaotic value,\(\mu\)is a chaotic factor, and \(\mu \in\) [0, 1][21]. The bifurcation diagram of Sine map is shown in Fig. 1. When the bifurcation parameter is \(\mu \in\)(0.87, 0.93) and \(\mu \in\)(0.95, 1), the Sine map is in a completely chaotic state and the trajectory of the equation is chaotic in this interval.
The infinite collapse map is defined as [22]
$${x_{n+1}}=\sin (\frac{a}{{{x_n}}}),$$
2
where the control parameter a is real number, a\(\in\) [3, 6], xn is the n-th chaotic value. Bifurcation diagram of ICM for a \(\in\) [3, 6] is shown in Fig. 2.
Traditional one-dimensional chaotic mapping has a straightforward structure, making it simple to estimate its trajectory [23]. By nesting a logical map and a one-dimensional infinite collapse map, a two-dimensional logical infinite collapse map (2D-SICM) is proposed to address this issue, which is defined by
$$\left\{ {\begin{array}{*{20}{c}} {{x_{n+1}}=\sin (a/\mu \sin (\pi {x_n}+{y_n})),} \\ {{y_{n+1}}=\sin (b/\mu \sin (\pi {y_n}+{x_n})),} \end{array}} \right.$$
3
where a and b are initial parameters, a ≠ 0, b ≠ 0, \(\mu \in\)(0, 1). Bifurcation diagram of 2D-SICM for a\(\in\)[19, 20] is shown in Fig. 3.
2.2 Attractors
A group of points occupying a section of two-dimensional phase space can be thought of as the attractors of a two-dimensional chaotic map [24]. Chaotic maps with better chaotic properties usually have attractors that are geometrically complex and occupy a large area in phase space. This paper chooses (0.6, 0.4) as the starting point and iterates 40000 times to intuitively depict the chaotic mapping attractors. The attractors of the chaotic map are then represented by a plot of these 40000 points in 2D space. Figure 4 depicts the attractors of 2D-SICM. This indicates that 2D-SICM entirely occupies the 2D phase space with a range of (− 1,1). It can be more ergodic or competitive as well as provide results that are more unpredictable.
2.3 8-bit binary plane decomposition
Since the gray value of each pixel of the digital image can be expressed as a binary number, the image can be decomposed into several bit planes from the microscopic point of view of each bit of binary. For example, the pixel x = x8 x7 x6 x5 x4 x3 x2 x1. The Least Significant Bits (LSBs) binary plane is represented by x1, x2, x3, x4, the Most Significant Bits (MSBs) binary plane by x5, x6, x7, x8 [25]. Figure 5 displays the plaintext image. Figure 6 displays the eight 8-bit planes of the Peppers image. The high-bit plane mainly holds the crucial information of the image, while the low-bit plane makes it difficult to distinguish between the original image data. Therefore, image encryption can be realized by using the decomposition characteristic of the bit-plane.