For large image, traditional image encryption algorithm such as data encryption standard (DES) is not suitable because of the weakness of low-level efficiency. The chaos-based encryption has suggested a new and efficient way to deal with the intractable problem of fast and highly secure image encryption. 1-D chaotic systems such as logistic map are widely used, with the advantages of high-level efficiency and simplicity. But their weakness, such as small key space and weak securities, is also disturbing. During decryption the exact decryption is not possible in this method as pixels value will be floating point by thus algorithm and because of that some data loss occurs. Because of exponential sensitivity of chaotic maps, it makes a proper decryption of the original image impossible. A more frequently used 2D chaotic map is Arnolds chaotic map. Although it has proved itself to a good efficiency yet security is always a concern. The key generated for encryption of image can be hacked or guessed by intruder. It should also show more key sensitivity. A small change in key should affect the encrypted image.
During this work logistic map will be improved to make it non linear so that encryption key generated by this is unpredictable to hackers and it shows very less periodicity. This modified logistic key is diffused into arnolds cat map scrambled image.
Logistic mapping is the chaos mapping and in 1 dimensional problem it is defined as:
Here r is the system parameter which lies in between (0,4].When 1<, the solution of the system is fixed point. When 3 the formula starts the transition state, when 3.5699456, the system enters a chaotic state. As is often the case in dynamical systems theory, the action of the logistic map can not only be represented algebraically, as in above equation, but also geometrically. Given a point xn, the graph of the logistic map provides y = f(xn). To use y as the starting point of the next iteration, we must find the corresponding location in the x space, which is done simply by drawing the line from the point [xn, f(xn)] to the diagonal y = x. This simple construction is then repeated, as illustrated in Figure.
Figure:(a) Graph of the logistic map for = 2. (b) Graphical representation of the iteration of equation.
Since it is only dependent on r, so stationary and chaotic behaviors are shown by logistic map as in figure 2.
Figure 2:(a): stationary regime, r= 0.5 (stable behavior) (b):periodic regime of period 5 (chaotic behavior)
Limitations of logistic map
- The Logistic map does not satisfy uniform distribution property. When r= [0, 3.6) the points concentrate on several values and could not be used for encryption purpose.
- Cryptosystems based on Logistic map has small key space and weak security
Modified Logistic Map
In this non linear function has been adopted to change the value of key continuously for security enhancement. So the modified logistic map is defined as the tangent function of xn as
The ranges of parameters r, a and b will be discussed as follows. Firstly, they are positive. Secondly, the absolute value of the slope of the curve at fixed point should not be less than 1, and xn+1 > xn when xn = 1/(1 + b), therefore r may be defined as
Finally, parameter l is obtained by experimental analysis ; as a result, . So the NCA map is defined as follows:
where xn (0, 1), a (0, 1.4], b [5, 43], or xn (0, 1), a (1.4, 1.5], b [9, 38], or xn (0, 1), a (1.5, 1.57], b [3, 15]. The ranges of a and b are obtained by iteration experimental analysis. The bifurcation map for modified logistic map is shown in figure.
Figure : Modified Logistic map
Since this logistic map is one dimensional and one dimensional chaotic map always suffer from weak security and low key space even after sufficient improvement. So in my work logistic map is combined with arnolds 2D cat map to overcome these weakness. The flow chart of whole pipeline is shown in figure.
Figure:Flow Chart of Image encryption by modified logistic map and arnolds cat map.
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