Efficient Chaotic Permutations for Image Encryption Algorithms
Abir AWAD, Abdelhakim Saadane
[email protected] 1
Outlines of presentation 1. 2. 3. 4.
Introduction Chaotic maps Chaotic permutation methods Conclusions and perspectives
2
Outlines of presentation 1. 2. 3. 4.
Introduction Chaotic maps Chaotic permutation methods Conclusions and perspectives
3
Introduction
Introduction Eve Attack
Alice
Bob Transmission Channel
Encryption
Decryption
original data
Encrypted data
Encrypted data
Hello
%j$klnr
%j$klnr
Decrypted data
Hello 4
Introduction
Encryption chaotic algorithm: Descriptive Diagram Chaotic map
Substitution Clear information
Permutation SP box Encryption algorithm
Encrypted information
5
Outlines of presentation 1. Introduction
2. Chaotic maps
Chaotic signal PWLCM chaotic map Finite Precision effect Proposed perturbation technique
3. Chaotic permutation methods 5. Conclusions and perspectives 6
Chaotic maps
Chaotic signal pwlcm perturbé
autocorrelation
1 0.9 0.8
0.8 0.7
0.6
0.6 0.5
0.4
0.4 0.2
0.3 0.2
0
0.1 0
0
100
200
300
400
500
600
700
800
900
1000
-0.2
-8
-6
-4
-2
0
2
4
6
8 4
x 10
Chaotic signal is a signal like noise Sensitive to initial conditions 7
Chaotic maps
PWLCM chaotic map • A piecewise linear chaotic map (PWLCM) is a map composed of multiple linear segments. 1
x n F [ x n 1] 1 x n 1 p 1 x n 1 p 0.5 p F [1 x n 1]
0.9 0.8
if 0 x n 1 p
0.7 0.6 0.5
if p x n 1 0.5
0.4 0.3
if 0.5 x n 1 1
0.2 0.1 0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
• where the positive control parameter p є (0; 0.5) and x(i) є (0; 1). 8
Chaotic maps
Finite Precision effect Finite precision => Finite cycle length xl 1
x1
xl
x2
xl n
Transient branch
The perturbation increase the chaotic cycle length
cycle perturbation
x(n 1)
Chaotic map
x(n)
9
Chaotic maps
Proposed perturbation technique x(n 1) 0.x1 (n 1) x2 (n 1)...xN k (n 1) xN k 1 (n 1)...xN (n 1) Qk 1 (n)Qk 2 (n)...Q1 (n)Q0 (n)
F ( x(n 1)) 0.F ( x1 (n 1))F ( x2 (n 1))...F ( xN k (n 1))F ( xN k 1 (n 1))...F ( xN (n 1))
Minimal cycle length:
Tmin 2k 1
10
Outlines of presentation 1. Introduction
2. Chaotic maps 3. Chaotic permutation methods
Chaotic permutation -GRP Chaotic permutation- CROSS Socek permutation Permutation results
5. Conclusions and perspectives 11
Chaotic permutation methods
Chaotic bit permutation methods Bit permutation methods controlled by chaotic values R1= [ b1, b2, b3, b4, b5, b6, b7, b8] Permutation
R3= [ b4, b6, b7, b1, b3, b8, b2, b5] Inverse permutation
R1 = [ b1, b2, b3, b4, b5, b6, b7, b8] 12
Chaotic permutation methods
Chaotic permutation -GRP R3 = GRP (R1, R2) R2: Control bits
R1: Original bits
R3: Permuted bits
1
0
0
1
1
0
1
0
a
b
c
d
e
f
g
h
b
c
f
h
a
d
e
g
13
Chaotic permutation methods
Chaotic permutation - CROSS R3 =CROSS (m1, m2, R1, R2) R2: Control bits
1
0
0
1
1
0
1
0
R1: Original bits
a
b
c
d
e
f
g
h
m1=2
1
0
0
1
-
-
-
-
e
b
c
h
a
f
g
d
-
1
0
-
-
h
g
f
a
d
m2=1
1
0
-
R3: Permuted bits
c
b
e
14
Chaotic permutation methods
Socek Permutation R3= Socek (x, R1) x: chaotic value (control) Indices of R1 bits (to permute)
1
2
3
4
5
6
7
8
Permuted Indices
4
6
7
1
3
8
2
5
15
Chaotic permutation methods
Permutation Results
Original image 16
Chaotic permutation methods
Difference between the original and the permuted images PWLCM
Perturbed PWLCM
Grp
Cross
Socek
Grp
Cross
Socek
NPCR
79.508
87.655
99.569
76.984
80.992
98.520
UACI
19.176
21.685
29.106
18.216
20.025
27.139
M 1 N 1
1 D(i, j ) 0
if P1 (i, j ) C1 (i, j ) else
NPCR
D(i, j) i 0 j 0
M N
100 17
Chaotic permutation methods
Difference between the original and the permuted images PWLCM
Perturbed PWLCM
Grp
Cross
Socek
Grp
Cross
Socek
NPCR
79.508
87.655
99.569
76.984
80.992
98.520
UACI
19.176
21.685
29.106
18.216
20.025
27.139
1 UACI MxN
M 1 N 1
i 0 j 0
P1 (i, j ) C1 (i, j ) 255
x100 18
Chaotic permutation methods
Correlation coefficients of intra color - components Correlation
Mandrill image
Permuted image using PWLCM to control
Grp
Cross
Socek
Permuted image using perturbed PWLCM to control Grp Cross Socek
Red (R) component Correlation
0.1911
0.0717
0.0259 0.0171 0.0458 0.0243 0.0155
Green (G) component Correlation
0.0883
0.0308
0.0120 0.0066 0.0164 0.0110 0.0055
Blue (B) component Correlation
0.0948
0.0572
0.0196 0.0152 0.0356 0.0178 0.0138
Mean value
0,1247
0.0532
19 0.0192 0.0130 0.0326 0.0177 0.0116
Chaotic permutation methods
Correlation coefficients of inter color - components Mandrill Correlation image
Permuted image using PWLCM to control
Permuted image using perturbed PWLCM to control
Grp
Grp
Cross
Socek
Cross
Socek
Correlation between R and G
0.3565
0.2776
0.1925 0.1280 0.1621 0.1147 0.0703
Correlation between G and B
0.8074
0.3722
0.2453 0.0684 0.2490 0.1893 0.0591
Correlation between B and R
0.1237
0.0571
0.0491 0.0161 0.0506 0.0484 0.0088 20
Chaotic permutation methods
Distribution of two (vertically ) adjacent pixels Original image
Cross + PWLCM
300
300
300
250
250
200
200
200
150
150
150
100
100
100
50
50
50
250
0
0
50
100
Socek + PWLCM
150
200
250
300
0
0 0
50
100
150
300
300
250
250
200
200
150
150
100
100
50
50
0
0
50
100
150
200
250
300
0
200
250
300
Cross + perturbed map
0
50
100
150
200
250
300
Socek + perturbed map 0
50
100
150
200
250
300
21
Chaotic permutation methods
Histogram analysis Original image
Grp + perturbed map
Grp + PWLCM
2000
2000
1800
1800
1800
1600
1600
1600
1400
1400
1400
1200
1200
1200
1000
1000
1000
800
800
800
600
600
600
400
400
400
200
200
200
0 -50
0
50
Socek + PWLCM
100
150
200
250
300
2000
0 -50
0
50
100
150
2000
2000
1800
1800
1600
1600
1400
1400
250
300
0 -50
0
50
100
1000
1000
800
800
600
600
400
400
150
200
250
300
Socek + perturbed map
1200
1200
200
200 0 -50
200
0
50
100
150
200
250
300
0 -50
22 0
50
100
150
200
250
300
Chaotic permutation methods
Information entropy analysis Mandrill Correlation image
Entropy
7.762
Permuted image using PWLCM to control
Permuted image using perturbed PWLCM to control
Grp
Cross
Socek
Grp
Cross
Socek
7.862
7.881
7.888
7.906
7.913
7.950
H m
2 N 1
i 0
1 p mi log 2 p mi 23
Outlines of presentation 1. 2. 3. 4.
Introduction Chaotic maps Chaotic permutation methods Conclusions and perspectives
24
Conclusions Novel chaotic permutation technique Comparative study of three bit permutation methods The proposed permutation technique is more secure and suitable for chaotic image encryption schemes This study allows choosing an efficient permutation method to construct a chaotic cryptosystem with good cryptographic properties 25
perspectives Novel chaotic encryption method using this permutation technique. The measure of the impact of this permutation method of the hole encryption algorithm. Comparative study of the choosing bit permutation method and the S box of the AES encryption method. 26
Thank you 27
