1950 words - 8 pages

II. CHAOS AND EZW

A. CHAOS BASED SCRAMBLING ALGORITHM:

“Chaos” means “state of disorder”, in mathematics for a system to be called as chaotic system; it should satisfy the following conditions [6],

1. It should be highly sensitive to initial condition.

2. It should satisfy the mixing property.

3. Periodic orbits should be dense.

A chaotic map is a function which exhibits chaotic behavior; there are different types of chaotic maps (like 1-D maps, 2-D maps, discrete-time parameter, continuous-time parameter etc.). In this paper we are considering chaotic logistic map (1-D) which is simple to implement.

Chaotic logistic map is given by equation (1),

Xj+1 =u*Xj*(1-Xj), u [1, 4] & X (0, 1) (1)

Where, u is the bifurcation parameter, if u [3.57, 4] then the equation behaves chaotically, here we are taking ‘u’ value as 4.X0 is the initial value which can be taken randomly from (0, 1).

The Scrambling algorithm used in this paper reorders the rows and columns of the 2-D data. For scrambling rows and columns we are using two different keys.

Scrambling algorithm:

The chaotic logistic map produces chaotic sequence within the range (0, 1). A position vector ‘P’ is generated, which signifies the new positions of the row or column of a given 2-D data.

P = [ Pi, Pi+1… PN ] (2)

Where,

Pi are the position elements.

The position vector is generated based on mixing property of chaotic dynamical systems. The mixing property is given as [7]:

“For any two open intervals I and J (which can be arbitrarily small, but must have a nonzero length) one can find initial values in I which when iterated will eventually lead to points in J”.

Hence mixing property states that any initial condition at any interval can traverse intervals in the entire domain (0,1) during course of iterations. To generate the position vector [8], the domain (0, 1) is equally subdivided into small intervals and each interval numbered from 1 to N depending on length of row or column keeping other fixed.

Now to generate a new position vector a key is taken and applied to the chaotic logistic map we will obtain a value (0, 1) based on the initial condition. Now fill the first element of the position vector with the number given to the subdomain to which the obtained value belongs. Repeat this process until all the position elements are filled. If we are getting an element in position vector which is already there discard that element and iterate the logistic map using the present obtained value until a new element is obtained. Using the position vector ‘P’ as in (2) we scramble the rows of the given 2-D data. Similarly by using a new key we repeat the above scrambling process on the obtained row scrambled 2-D data for column scrambling.

In the descrambling process we will generate position vectors for column and row descrambling as said above using the same keys that are used for scrambling process. Using the generated position vector we can easily descramble the received scrambled...

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