A Hybrid Genetic Algorithm and Chaotic Function Model for Image Encryption

7/29/2019 A Hybrid Genetic Algorithm and Chaotic Function Model for Image Encryption

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A HYBRID GENETIC ALGORITHM AND

CHAOTIC FUNCTION MODEL FOR IMAGE

ENCRYPTION

SADIQUE NAYEEM,M.Tech.

Pondicherry University


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Introduction

A new method based on a hybrid model is proposed for

image encryption which is composed of a genetic

algorithm and a chaotic function.

In the first stage, a number of encrypted images are

constructed using the original image and the chaotic

function.

In the next stage, these encrypted images are used as

the initial population for the genetic algorithm.

In each stage of the genetic algorithm, the answer

obtained from the previous iteration is optimized to

produce the best-encrypted image ( with the highest

entropy and the lowestcorrelation coefficient among

adjacent pixels).

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Terminology

Genetic algorithm

Entropy of an image

Histogram of an image

Chaotic function Correlation coefficient among adjacent pixels

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Genetic algorithms

Genetic algorithms which mimic principle of evolution in nature,

search for final solutions among a population of potential solutions.

The best solutions are chosen in each generation.

These solutions, after mating, produce a new set of offspring.

At the beginning, the initial population is randomly formed, and thefitness function evaluates every population.

Optimum result is not met, new generations are produced through

the process of recombination and mutation.

The more suitable generations are selected to produce newgenerations until the optimum result is obtained.

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Entropy of an image

Entropy is a statistical measure of randomness that can be used to

characterize the texture of the input image.

It is one of the prominent features in randomization and can be

calculated as:

where N is the number of gray levels used in the image (for gray level

images, N is 8), and P(Si) shows the probability of having a ith gray

level in the image.

Matlab Code: E = entropy(I) In images that are produced completely randomly, N will be 8, and

this N is considered as an ideal value.

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Histogram of an image.

The histogram of an image refers to the frequency of

any gray level in it.

This feature is one of the most important statistical

features of an image.

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Chaotic function

Chaotic functions are similar to noise signals, but they

are completely certain; that is, if we have the primary

quantities and the drawn function, the exact signal can

be reproduced.

Advantages:

Sensitivity to primary conditions

Apparently accidental feature

Deterministic work

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Logistic Map Signal

r=3.999 (Completely Chaotic)


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Correlation coefficient among adjacent

pixels

The correlation coefficient matrix represents the normalized

measure of the strength of linear relationship between variables.

The variable can be the two adjacent pixel. Correlation coefficient is

given by:

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The correlation coefficients range from -1 to 1,

where

Values close to 1 suggest that there is a positive

linear relationship between the data columns.

Values close to -1 suggest that one column of

data has a negative linear relationship to another

column of data.

Values close to or equal to 0 suggest there is no

linear relationship between the data columns.


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THE PROPOSED METHOD

FORMATION OF THE INITIAL

POPULATION

GENETIC OPTIMIZATION

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Formation of the initial population

To form the initial population from the plain-image, first the plain-

image is divided into four equal parts.

The chaotic function logistic map is employed to separately

encrypt all of the pixels present in each of these four parts as

follows:1. Five pixels are selected as the encryption key for forming the

initial value of the logical map function and for encrypting that

part of the image.

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cont..

2. The initial value of the chaotic function logistic map is

determined by taking five pixels as under:

P = [P1, P2, P3, P4, P5] (gray scale).

Convert P into an ASCII code: B = [P1,1, P1,2, P1,3, . . .,

P5,7, P5,8]

The initial value of the logistic map function(U0k), which lies in the interval 01,is obtained.

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3. For each part of the plain-image, step 2 is repeated.

Therefore, in the end, there will be 4 different initial values for

each image (one value for each part of the image).


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cont

4. For encrypting the pixels in each part of the image, the initial value

of that part and Eq. is used as follows:

The symbol represents XOR.OldValue stands for the current value of the pixel.

NewValue shows the new value of the pixel after it is encrypted.

The value ofUik refers to the ith value of the chaotic function in the

kth part of the original image, determined for each step using Eq.

All of the pixels in each part, except the five pixels used as the key,

are sequentially (row-by-row) encrypted in this manner.

To build the rest of the population, steps one through four are

repeated.

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First generation of the genetic population

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Genetic optimization

Crossover:

After forming the initial population, the genetic algorithm is

used to optimize the encrypted images.

(a)&(b) are the input images.

(c)&(d) are images after crossover.

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Fitness function

The fitness function used is the correlation coefficient

between pairs of adjacent pixels of the image.

At each stage, the new generations are produced andthe previous ones are evaluated using the fitness

function.

The generation is required to have the lowest correlationcoefficient for the encrypted image to be as the final

cipher-image.

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Selection

50% of the population with the minimum correlation coefficient +

10% of the remaining population are selected for the next

generations.

Non-Elites are not selected.

Therefore, 10% of the remaining population are chosen arbitrarilyto help the selected elites from 90% to produce new generations.

This process is repeated until the correlation coefficient of the best

generation produced does not significantly change in two

successive stages.

After this stage, the generation that has the lowest correlationcoefficient is encrypted as the final cipher-image.

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Example

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Plain-image Plain-image divided into four equalparts.


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Example cont..

A matrix with dimensions 12 12 is used in which the numbers are

in the interval [0, 7].

The first five pixels of the first row of each part are used in

determining the initial value of the chaotic function logistic map.

The first five pixels of the first part of the image are (3, 1, 7, 6, 2) P=[ 2, 6, 7, 1, 3 ]

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Example cont..

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First generation Second generation


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Example cont..

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Third generation after crossover Forth generation aftercrossover.


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EXPERIMENTAL

RESULTS21

1. ANALYSIS OF THE ENTROPY OF THE

IMAGE

2. HISTOGRAM ANALYSIS3. ANALYSIS OF THE CORRELATION

COEFFICIENT

4. ANALYSIS OF THE FITNESS FUNCTION

5. KEY ANALYSIS


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Test conducted

Gray level images with dimensions of 256 256.

The initial number of the population for GA = 128.

The crossover operation rate = 90%

Mutation operation rate = Zero.

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1. ANALYSIS OF THE ENTROPY OF THE

IMAGE

Entropy is a

statistical measure of

randomness of the

pixels

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2. HISTOGRAM ANALYSIS

Fig. 8. (a) Plain-image, (b) cipher-image, (c) histogram of plain-image, and (d)

histogram of cipher-image.

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2. HISTOGRAM ANALYSIS

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Fig. 9. (a) Plain-image, (b) histogram of plain-image, histogram of image after, (c) 2nd iteration, (d) 25th iteration, and

(e) 70th iteration.


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3. ANALYSIS OF THE CORRELATION

COEFFICIENT

In a good encryption algorithm, the correlation coefficient between

pairs of encrypted adjacent pixels in the horizontal, vertical, and

diagonal positions are as small as possible.

To test the correlation coefficient between two adjacent vertical

pixels, two adjacent horizontal pixels, and two adjacent diagonal

pixels in a cipher-image, the following procedure was used: first,

2500 pairs of pixels were randomly selected. Then, the

correlation coefficients were obtained using Eq.

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COEFFICIENT27

Fig:(a) Correlation analysis of plain-image and (b) correlation analysis of

cipherimage.


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COEFFICIENT

In another test, the results of the correlation coefficients of two

adjacent pixels in the vertical, horizontal, and diagonal positions

were investigated in each of the different iterations.

The results obtained for the correlation coefficients in this method

are far better than those of the methods of [22,24,25]

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4. ANALYSIS OF THE FITNESS

FUNCTION

Selection of the fitness function in genetic algorithms is one of themost important and most influential elements for achieving the

desired result.

Tests conducted showed that the correlation coefficient is a

better optimizerthan the function of the entropy of the image.

As observed in this table, when the correlation coefficient is used as

the fitness function, the degree of entropy and the correlation

coefficient of the image are better than when entropy is used as the

fitness function.

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5. KEY ANALYSIS

For a good encryption algorithm, key must be long

enough to resist brute-force attacks.

A 40-bit key is suggested which produces a key space

equivalent to 2^40.

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Fig:

(a) Plain-image,

(b and c) encrypted images using

user keys with 1-bit difference in

each part,

(d) the similarity between (b) and(c), (99.76% different.)


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KEY ANALYSIS CONT.

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Fig: Percentage of difference between encrypted images versus

iterations.


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Decryption

The decryption of encrypted images is possible if the

chaotic function logistic map and the numbers of the

population (or the numbers of the two populations) from

which the final image is composed are known.

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Conclusions

A new method has been suggested for encrypting images with a

chaotic function logistic map and a genetic algorithm.

In this method, the chaotic function is employed for initial

encryption, and the genetic algorithm is used to improve the

encryption process of the image. The main innovation in this paper is that this is the first time genetic

algorithms are used in this manner to encrypt images.

The results obtained for the correlation coefficients and the

entropies of the images proved the high efficiency of this method.

This algorithm can be used in the future with a new crossover

operatoror with another fitness function to encrypt images.

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My Idea for further enhancement

Use of RGB image

Separate the Red, green and blue component.

Use the chaotic function logistic map to encrypt the each

component. Do the crossover among each component.

Use Intensity Profile as the fitness function.

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A Hybrid Genetic Algorithm and Chaotic Function Model for Image Encryption