A NEW APPLICATION OF THE FRACTIONAL LOGISTIC MAP

LAN-LAN HUANG1, DUMITRU BALEANU2,3,∗, GUO-CHENG WU1, SHENG-DA ZENG1

1Data Recovery Key Laboratory of Sichuan Province,College of Mathematics and Information Science,

Neijiang Normal University, Neijiang 641100, China2Department of Mathematics and Computer Sciences, Cankaya University,

06530 Balgat, Ankara, Turkey3Institute of Space Sciences, Magurele–Bucharest, Romania

Corresponding author∗: dumitru@cankaya.edu.tr

Received September 27, 2015

The fractional chaotic map started to be applied in physics and engineering toproperly treat some real-world phenomena. A shuffling method is proposed based onthe fractional logistic map. The fractional difference order is used as a key. An imageencryption scheme is designed by using the XOR operation and the security analysis isgiven. The obtained results demonstrate that the fractional difference order makes theencryption scheme highly secure.

Key words: Discrete fractional calculus, chaos, fractional logistic map, imageencryption.

1. INTRODUCTION

Fractional calculus describes very well the complex real-world phenomena incases when the nonlocality is present [1–11]. The image encryption techniques havebeen of increased interest for extensive applications during the past decades. Thechaotic sequences have been often used in the image encryption. Baptista [12] de-signed a chaotic block encryption that can change the length of block. However,chaos-based encryption is not always secure. Alvarez et al. [13] suggested a cryp-tosystem with the ergodicity of chaos.

Very recently, the discrete fractional chaos was reported by use of fractionalmaps [14–16]. In Ref. [17], an encryption scheme was designed based on XORoperation and the fractional logistic map. The introduced fractional parameter ν letthe chaotic system’s behaviors more complicated. Besides, the fractional maps havesimpler forms and the chaotic behaviors can be readily controlled. For these reasons,fractional chaotic signals and shuffling method are adopted to develop a novel imageencryption scheme. This paper is organized as follows. Section 2 briefly introducessome results on the fractional maps. Section 3 presents a shuffling method based onthe fractional logistic map and gives the necessary steps. In Sec. 4, the obtainedencryption results are analyzed in detail.

Rom. Journ. Phys., Vol. 61, Nos. 7-8, P. 1172–1179, Bucharest, 2016 v.1.4*2016.9.28#07b5d5c1

2 A new application of the fractional Logistic map 1173

0 100 200 300 400 500 600 700 8000

0.2

0.4

0.6

0.8

1

1.2

1.4

n

x(n

)

Fig. 1 – Chaos of the fractional discrete logistic map for x(0) = 0.2, µ=2.6, and α=0.7

2. PRELIMINARIES

The famous logistic map is a difference equation that reads

x(i+1) = µ∗x(i)(1−x(i)) (1)

It also can be rewritten in the difference form as

△x(i) = µx(i−1)(1−x(i−1)) (2)

There are several fractional difference definitions in the literature and their ap-plications in different areas, see, for instance, Refs. [18–24]. The discrete fractionalcalculus (DFC) [25–30] provides an efficient tool to fractionalize the difference equa-tions. We replaced the classical difference operator △ with the fractional Caputodifference to propose the generalized map in Ref. [14]

x(i) = x(0)+µ

Γ(α)

i∑z=1

Γ(i−z+α)

Γ(i−z+1)x(z−1)(1−x(z−1)), (3)

where α is the corresponding order.Here we need to point out that our map (3) is different from the fractional

one [18, 23, 24]. The former is a fractionalization of the classical maps on timescales. The finite memory effects are introduced in this way. The latter is a numericaldiscretization of fractional differential equation and holds infinite memory effects.They are indeed two distinct classes of fractional maps.

For α = 0.7 and x(0) = 0.2, we can obtain the chaotic sequence x(i) and weplot it in Fig. 1. In fact, using the Jacobian matrix algorithm [16], we can find theareas where chaos do exists. We derive its tangent map

a(i) = a(0)+µ

Γ(α)

i∑z=1

Γ(i−z+α)

Γ(i−z+1)a(z−1)(1−2x(z−1)),a(0) = 1 (4)

(c) RJP 61(Nos. 7-8), 1172–1179 (2016) v.1.4*2016.9.28#07b5d5c1

1174 Lan-Lan Huang et al. 3

from which we can calculate the LE as

λ(x0)≃1

iln |a(i−1)|. (5)

Thus we can distinguish the chaotic series and we list the chaotic area in thefollowing Table:

Fractional orders 0.78 0.8 0.9 1µ [2.3, 2.74] [2.38, 2.81] [2.47, 2.88] [2.57, 3]

Different fractional orders lead to different chaotic behaviors. Besides, one canobtain the chaotic sequence more readily. As a result, these are the main reasons thatwe choose the fractional maps but not the fractional differential equations.

3. FRACTIONAL CHAOTIC MAPS AND IMAGE ENCRYPTION

The shuffling method is an often used method in image encryption. Due tothe chaotic sequence randomness, the method can fully confuse the positions of thepixels and hide the information. So it can also be efficient for color images. Very re-cently, Wang and Guo [31] propose a shuffling method based on the classical logisticmap. In this study, we use the fractional map (3) to develop new encryption schemes.

3.1. A NEW METHOD

Step 1: Chaotic sequence x(i) is generated by (3), i= 0, ...,m×n. α, x(0) =x1(0), and µ are used as the initial secrete keys.

Step 2: The x(i) is changed into 0, 1 sequence as

x(i) =

{1,x(i)≥ a,

0,x(i)< a,(6)

where a= 0.5.Step 3: Assume M,N,U are the empty arrays. Begin with i= 1, add one each

time till i = m×n. For x(i) = 1, put g(i) in the array M or put g(i) in the arrayN , where g(i) is the corresponding one-dimensional array after reshaping the plainimage. Then merge M and N into the array U .

Step 4: Carry the above procedure T round. If T is an odd number, set M infront of N in the array U . Or set N in front of M .

Step 5: Re-arrange the array U into a two-dimensional matrix with m rows andn columns

G= reshape(U,m,n), (7)which is the gray scale value of the shuffled image.

(c) RJP 61(Nos. 7-8), 1172–1179 (2016) v.1.4*2016.9.28#07b5d5c1

4 A new application of the fractional Logistic map 1175

3.2. THE ALGORITHM

Select a plain image and f(m,n) is its gray-scale function, where 0≤m,n≤255. We now give the necessary steps in detail:

Step 1: Select the parameters µ, α, and x(0)= x2(0) to have a chaotic sequencex(0),x(1)...,x(65535).

Step 2: Carry out the shuffling method and derive the shuffled image g(m,n).Step 3: Using the same chaotic sequence x(i), one can obtain the integer part

of x(i)×104 and we denote the result as x∗(i). Take the modular arithmetic

y(i) = mod (x∗(i),256).

Step 4: Take the ⊕ operation

c(m,n) = f(m,n)⊕ y(i) (8)

where i= 256(m−1)+n. Use the XOR operation again. The function g(m,n) canbe obtained with which one we can have a decryption result.

4. RESULTS AND ANALYSIS

Following the above steps, we select the parameters α= 0.8, µ= 2.5, x1(0) =0.3, n = 65535 and α = 0.6, µ = 2.4, x2(0) = 0.2, n = 65535 to generate twochaotic sequences for the XOR operation and the shuffling method, respectively. Weuse “Lena” as the plain image (see Fig. 2) and the encrypted results are given in Fig.3, where the round T is set to be T = 20, respectively. The decrypted image is givenin Fig. 4.

4.1. CORRELATION ANALYSIS

We can analyse the correlations in the encrypted images. We adopt the follow-ing formula:

E(A) =1

N

N∑i=1

A(i), D(A) =1

N

N∑i=1

(A(i)−E(A))2,

cov(A,B) =1

N

N∑i=1

(A(i)−E(A))(B(i)−E(B)),

r(A,B) =cov(A,B)√D(A)D(B)

,

(9)

where A and B are the values of two adjacent pixels and N is the used pixel number.From (9), left and right panels in Figs. 5 show the correlation in the two directions,

(c) RJP 61(Nos. 7-8), 1172–1179 (2016) v.1.4*2016.9.28#07b5d5c1

1176 Lan-Lan Huang et al. 5

Fig. 2 – Plain image.

Fig. 3 – Encrypted image.

(c) RJP 61(Nos. 7-8), 1172–1179 (2016) v.1.4*2016.9.28#07b5d5c1

6 A new application of the fractional Logistic map 1177

Fig. 4 – Decrypted image.

respectively. We can observe that the correlation in the encrypted one is a stochasticrelationship from which we determine that the encryption is successful.

4.2. SECURE ANALYSIS

Let us check the encrypted results’ sensitivity. It is decrypted by two differentsecret keys. Minor changes are made in the x(0), e.g., x1(0) = 0.30000001 andx2(0) = 0.20000001. The decrypted images are shown in Fig. 6. We note that, theimage decrypted from a slightly different key is totally different from the correct one(see Fig. 4).

5. CONCLUSIONS

The chaotic systems possess several inherent characteristics favorable to infor-mation security, sensitive dependence on initial conditions as well as systems’ para-meters. This study proposes a new encryption technique via the fractional chaoticmaps. The presented image encryption scheme clearly possess the following merit:the fractional chaotic signals possess a more complicated structure since there is thefractional difference order and this increases the key spaces. As a result, this charac-teristic let the encryption results highly secure.

(c) RJP 61(Nos. 7-8), 1172–1179 (2016) v.1.4*2016.9.28#07b5d5c1

1178 Lan-Lan Huang et al. 7

0 50 100 150 200 250 3000

50

100

150

200

250

300

f(m,n)

f(m

+1,n

)

0 50 100 150 200 250 3000

50

100

150

200

250

300

f(m,n)f(

m,n

+1)

Fig. 5 – Correlation analysis. Left panel: Correlation analysis along x direction; Right panel: Correla-tion analysis along y direction.

Fig. 6 – Incorrect decryption results. Left panel: The same decrypted result as in Fig. 4 except x1(0) =0.3000001; Right panel: The same decrypted result as that in Fig. 4 except x2(0) = 0.2000001.

(c) RJP 61(Nos. 7-8), 1172–1179 (2016) v.1.4*2016.9.28#07b5d5c1

8 A new application of the fractional Logistic map 1179

Acknowledgements. The first author (Lan-Lan Huang) was financially supported by a ResearchGrant No. 14CZ0026.

REFERENCES

1. A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo, Theory and applications of fractional differentialequations (Elsevier, Amsterdam, 2006).

2. I. Podlubny, Fractional differential equations (Academic Press, New York, 1999).3. A.H. Bhrawy, M.A. Zaky, D. Baleanu, and M.A. Abdelkawy, Rom. J. Phys. 60, 344–359 (2015).4. A.H. Bhrawy, M.A. Zaky, and D. Baleanu, Rom. Rep. Phys. 67, 340–349 (2015).5. A.K. Golmankhaneh, X.J. Yang, and D. Baleanu, Rom. J. Phys. 60, 22–31 (2015).6. T. Blaszczyk, Rom. Rep. Phys. 67, 350–358 (2015).7. M.A. Abdelkawy et al., Rom. Rep. Phys. 67, 773–791 (2015).8. M. Merdan, Proc. Rom. Acad. A 16, 3–10 (2015).9. A.H. Bhrawy et al., Proc. Romanian Acad. A 16, 47–54 (2015).

10. Kamel Al-Khaled, Rom. J. Phys. 60, 99–110 (2015).11. G.W. Wang and T.Z. Xu, Rom. Rep. Phys. 66, 595–602 (2014).12. M.S. Baptista, Phys. Lett. A 240, 50–54 (1989).13. E. Alvarez, A. Fernandez, P. Garcıa, J. Jimenez, and A. Marcano, Phys. Lett. A 263, 373–375

(1999).14. G.C. Wu and D. Baleanu, Nonlinear Dyn. 75, 283–287 (2014).15. G.C. Wu, D. Baleanu, and S.D. Zeng, Phys. Lett. A 378, 484–487 (2014).16. G.C. Wu and D. Baleanu, Commun. Nonlinear Sci. Numer. Simul. 22, 95–100 (2015).17. G.C. Wu, D. Baleanu, and Z.X. Lin, J. Vibr. Contr., DOI: 1077546315574649 (2015).18. M. Edelman, Commun. Nonlinear Sci. Numer. Simul. 16, 4573–4580 (2011).19. J.A.T. Machado, Fract. Calc. Appl. Anal. 4, 47–66 (2001).20. J.A.T. Machado, J. Vibr. Contr. 14, 1349–1361 (2006).21. J.A.T. Machado, V. Kiryakova, and F. Mainardi, Commun. Nonlinear Sci. Numer. Simul. 16, 1140–

1153 (2006).22. M.D. Ortigueira and J.J. Trujillo, J. Comput. Nonlinear Dyn. 6, 034501 (2001).23. V.E. Tarasov and G.M. Zaslavsky, J. Phys. A: Math. Theor. 41, 435101 (2008).24. H. Xiao, Y.T. Ma, and C.P. Li, J. Vibr. Contr. 20, 964–972 (2014).25. T. Abdeljawad, Comput. Math. Appl. 62, 1602–1611 (2011).26. F.M. Atici and P.W. Eloe, Int. J. Differ. Equat. 2, 165–176 (2007).27. F.M. Atici and P.W. Eloe, J. Differ. Equat. Appl. 17, 445–456 (2011).28. G.A. Anastassiou, About Discrete Fractional Calculus with Inequalities, in: Intelligent Mathemat-

ics: Computational Analysis, Springer, 575–585, 2011.29. C. Goodrich, Comput. Math. Appl. 59, 3489–3499 (2011).30. M.T. Holm, Comput. Math. Appl. 62: 1591–1601 (2011).31. X.Y. Wang and K. Guo, Nonlinear Dyn. 76, 1943–1950 (2014).

(c) RJP 61(Nos. 7-8), 1172–1179 (2016) v.1.4*2016.9.28#07b5d5c1