A Digital Image Encryption Algorithm Based On Chaotic Logistic Maps Using A Fuzzy Controller

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					                                                       (IJCSIS) International Journal of Computer Science and Information Security,
                                                       Vol. 9, No. 3, March 2011

      A digital image encryption algorithm based on
      chaotic logistic maps using a fuzzy controller
                                  Mouad HAMRI #1 , Jilali Mikram #2 , Fouad Zinoun &3
                      Mathematics and computer science department, Science University of Rabat-Agdal
                                          4 Avenue Ibn Battouta Rabat Morocco
                       Economical sciences and management department, University of Meknes Morocco

   Abstract—In this paper we will present a digital image encryp-        and the quantum machines that can be a reality soon.
tion algorithm based on chaotic logistic maps and using fuzzy            Chaotic dynamical systems present a very important tool to
logic (FL-CM-EA). Many papers was published in the recent                build efficient and secure cryptosystems thanks to their high
years about encryption algorithm using chaotic dynamical sys-
tems thanks to the set of very interesting properties guaranteed         sensitivity to initial conditions, their ergodicity propriety, their
by these chaotic dynamical systems: high sensitivity to initial          simplicity of implementation and also the very interesting
conditions, ergodicity, simplicity of implementation..., that can        execution time that help to have a real-time applications.
be used to conceive efficient cryptosystems.                              In this paper we propose an encryption algorithms using not
The main idea of this paper is the usage of a fuzzy logic set            only one logistic map but a map of many logistic maps and
of rules to control the next iteration of our proposed iterative
mechanism using a set of logistic maps.                                  the iterations are defined using a set of fuzzy logic rules.
An introduction to chaotic dynamical systems and logistic map            The rest of this paper will be as follow: section 2 introduces
is given followed by an introduction to fuzzy logic. A complete          chaotic dynamical systems and logistic map, section 3
specification of the proposed algorithm is presented with a set           introduces fuzzy logic, section 4 presents the proposed
of security analysis tests that show the efficiency and the high          algorithm with some results, section 5 presents the security
security level of the algorithm.
                                                                         analysis tests and finally section 6 concludes this paper.
  Keywords: cryptography, logistic map, fuzzy logic, image                II. C HAOTIC DYNAMICAL SYSTEMS AND LOGISTIC MAP
encryption, security analysis, dynamical systems, chaos theory.
                                                                            Roughly speaking, a dynamical system ([1-4],[11-12]) con-
                      I. I NTRODUCTION                                   sists of two ingredients: a rule which is described by a set of
   Today the community network applications in the internet              equations and specify how the system evolves and an initial
are been used by billions of people around the world and this            condition from which the system starts. It can be defined
usage rate is growing continuously. This implies that more               also as a system of equations describing the evolution of a
and more amounts of information is being transmitted over                mathematical model where the model is fully determined by
the internet. The data being transmitted includes all kind of            a set of variables.
information format: text, audio, video, image and a lot of               The logistic map (that will be used in our algorithm) is a very
other special formats.                                                   famous discrete dynamical system used in many researches
Images are used widely in our daily life in almost our                   when dealing with dynamical systems and chaos. It is defined
communications, these communications includes military                   on the set [0, 1] and can be written:
communications, banks transactions and many other
                                                                                               xn+1 = rxn (1 − xn )
communications where the security is really mandatory.
This lead to conclude that image security is a very important            Where x0 represent the initial condition, n ∈ N and r is
topic in our internet communication world.                               positive real number.
                                                                         In reality, there is no universal definition for chaotic
Many algorithm have been proposed in the last years                      dynamical systems. The following definition tries to define a
to solve these security issues, using the classical encryption           chaotic dynamical system using three ingredients that almost
algorithms such as RSA or EL-Gamal or using the elliptic                 everyone would agree on.
curves. The problem with the previous algorithm is that their
security relies on the fact that it is not feasible with today’s         Chaotic dynamical system: Let f : X → Y a function
machines to factorize a large number or to solve the discrete            (X, Y ⊆ R).
logarithm problem but this may not be true in the near future                                    ˙
                                                                         The dynamical system x = f (x) is said to be chaotic if the
especially with the recent advances in machines performances             following proprieties are satisfied:

                                                                                                    ISSN 1947-5500
                                                               (IJCSIS) International Journal of Computer Science and Information Security,
                                                               Vol. 9, No. 3, March 2011

1- Sensitive dependance on initial conditions:∀β > 0,                          For TS fuzzy rules and unlike Mandany fuzzy rules, TS fuzzy
∃ε > 0 there exists a point y0 ∈ X and k > 0, such that:                       rules define the output variables as a function of the input
| x0 − y0 |< β ⇒ | xk − yk |> ε.                                               variables. If we take the same example as before, a TS fuzzy
2- Density of periodic orbits:The ensemble of periodic                         rule can be described as follow:
orbits: {x0 ∈ X, ∃k > 0, xk = x0 } is dense in X.
                                                                                   IF x1 in S1 and x2 in S1 THEN y1 = f (x1 , x2 ) and
3- Deterministic: means that the system has no random or
                                                                                                     y2 = g(x1 , x2 )
noisy inputs or parameters.
                                                                               Where f and g are two real functions of any type.
The definition above is applied to both discrete and                            In general, the steps followed to construct a fuzzy controller
continuous dynamical systems.                                                  are:
The logistic map is a chaotic dynamical system and presents                       1) Identifying and naming the fuzzy inputs and outputs.
a very high sensitivity to initial conditions for r between                       2) Creating the the fuzzy membership functions.
about 3.57 and 4 (approximatively).                                               3) Constructing the fuzzy rules (Mandany or TS rules).
Fig.1 shows the bifurcation diagram of the logistic map.                          4) Defining the defuzzification process (convert fuzzy out-
                                                                                     puts to crisp outputs).
                                                                               The figure Fig.2 shows an example of a possible fuzzy

           Fig. 1.   Bifurcation diagram of the logistic map

                        III. F UZZY LOGIC

   In the 1960s, Lotfi Zadeh invented fuzzy logic [16,17],
which combines the concepts of crisp logic and the
Lukasiewicz sets by defining graded membership. One of
Zadehs main insights was that mathematics can be used to link
language and human intelligence. Many concepts are better
defined by words than by mathematics, and fuzzy logic and its                                   Fig. 2.   Diagram of a fuzzy controller
expression in fuzzy sets provide a discipline that can construct
better models of reality.
                                                                                 In the next section, we will present our encryption algorithm
Fuzzy logic is a form of many-valued logic in the opposite of
                                                                               and we will describe all the parameters of the used fuzzy
the crisp logic which is a two-valued logic (binary logic).
Fuzzy logic involves linguistic variables with a truth value
in the interval [0, 1], it involves also fuzzy sets and fuzzy                                        IV. T HE ALGORITHM
Every fuzzy model uses fuzzy rules which are linguistic if-                       The proposed algorithm (FL-CM-EA) takes as inputs a
then statements. These rules are linking the inputs variables                  plain-image P and a 128 bits key K then generates as output
to the output variables, they simply define the control logic.                  the cipher-image C.
Two major types of fuzzy rules exist: Mandany fuzzy rules and                  The main idea of the algorithm was to use not only a simple
Takagi-Sugeno (TS) fuzzy rules. An example of a Mandany                        logistic map to generate the encryption (decryption key) but to
fuzzy rule for a fuzzy system with two inputs and two outputs                  use what we have called ”fuzzy-logistic-map”, which is also
can be described as follow:                                                    a function from the interval [0, 1] to itself, using three fuzzy
                                                                               rules and three logistic map (we can use as many logistic maps
  IF x1 in S1 and x2 in S1 THEN y1 in S3 and y2 in S4                          and fuzzy rules as we want but in this paper we will use three).

                                                                                                            ISSN 1947-5500
                                                           (IJCSIS) International Journal of Computer Science and Information Security,
                                                           Vol. 9, No. 3, March 2011

                                                                                                IT ER
If we call the three logistic maps LM1 , LM2 and LM3 then                          – KFi =( l=1 F M L(i + l)2 ) ×256 mod 256.
the fuzzy rules are as follow:                                                     – Run the FLM generator and stop after IT ER.
   1) IF x IS M1 THEN FLM(x)=LM1 (x) = r1 x(1 − x)
   2) IF x IS M2 THEN FLM(x)=LM2 (x) = r2 x(1 − x)                           •   Step 3:Using the generated key, we will generate the
   3) IF x IS M3 THEN FLM(x)=LM3 (x) = r3 x(1 − x)                               image C as follow:
For the rest of this paper, we will use the following values:                      – C0 (R) = (P0 (R) + KF0 ) mod 256.
r1 = 3.95, r2 = 3.9 and r3 = 3.8.                                                  – C0 (G) = (P0 (G) + KF0 ) mod 256.
The fuzzy sets M1 , M2 and M3 membership functions f1 ,                            – C0 (B) = (P0 (B) + KF0 ) mod 256.
f2 and f3 are defined as follow:                                                  and:
                                                                                 For i in [2, n]:
                       −2x + 1             if    0≤x≤ 2
           f1 (x) =                              1                                 – Ci (R) = (Pi (R) + KFi + Ci−1 (R)) mod 256.
                       0                   if    2 ≤x≤1                            – Ci (G) = (Pi,j (G) + KFi,j + Ci−1 (G))) mod 256.
                       2x                  if    0≤x≤ 1
                                                                                   – Ci (B) = (Pi,j (B) + KFi,j + Ci−1 (B))) mod 256.
           f2 (x) =                              1
                       −2x + 2             if    2 ≤x≤1                      •   Step 4: We reverse the data of the image C :
                                                                                 For i in [1, n]:
                        0                 if    0≤x≤ 1
            f3 (x) =                            1                                  – Ci = Cn−i+1
                        2x − 1            if    2 ≤x≤1
                                                                             •   Step 5: finally we construct the cipher-image C by
  For the defuzzification process, we use a center average                        repeating the step 3 using the image C :
defuzzifier and the crisp value of FLM(x) is:
                                                                                   – C0 (R) = (C0 (R) + KF0 ) mod 256.
                                    i=1   µi LMi (x)                               – C0 (G) = (C0 (G) + KF0 ) mod 256.
                  F M L(x) =              3                                        – C0 (B) = (C0 (B) + KF0 ) mod 256.
                                          i=1   µi
Where µi represents the degree of membership of x in Mi .                        For i in [2, n]:
Before presenting the algorithm, the following notations are
                                                                                   – Ci (R) = (Ci (R) + KFi + Ci−1 (R)) mod 256.
                                                                                   – Ci (G) = (Ci,j (G) + KFi,j + Ci−1 (G))) mod 256.
      P                plain-image                                                 – Ci (B) = (Ci,j (B) + KFi,j + Ci−1 (B))) mod 256.
      K                128 bits key                                          •   End
      C                cipher-image
      Pi               ith pixel of P
                                                                           The decryption algorithm is identical to the encryption algo-
      Pi (R, GorB)     Red, Green or Blue value of the pixel i
      F LMi            fuzzy-logistic-map value after i iteration          rithm, it receives as inputs the cipher-image C and the 128
      Li (x0 , N )     Value of the logistic map i                         bits key K (the same used for the encryption) and returns as
                       starting from x0 after N iterations                 output the plain-image P.
      F                A map from the set of 32 bytes                      The only difference between the two algorithm is the step
                       numbers to the interval [0, 1]                      3 and step 5 which are defined as below for the decryption
  The encryption algorithm description can be summarized                      • Step 3:
as following:
                                                                                  – C0 (R) = (C0 (R) − KF0 ) mod 256.
                                                                                  – C0 (G) = (C0 (G) − KF0 ) mod 256.
  •   Begin:                                                                      – C0 (B) = (C0 (B) − KF0 ) mod 256.
  •   Step 1: We begin by generating an initial condition                       and:
      x0 ∈ [0, 1]:                                                              For i in [2, n]:
      x0 = F (K).
                                                                                  – Ci (R) = (Ci (R) − KFi − Ci−1 (R)) mod 256.
                                                                                  – Ci (G) = (Ci (G) − KFi − Ci−1 (G)) mod 256.
  •   Step 2: In this step we generate a key vector KF of
                                                                                  – Ci (B) = (Ci (B) − KFi − Ci−1 (B)) mod 256.
      size n where n is the number of pixels of P using the
      function getKey:                                                        • Step 5:

      KF = getKey(x0 ).                                                           – P0 (R) = (C0 (R) − KF0 ) mod 256.
      The function getKey is defined as bellow:                                    – P0 (G) = (C0 (G) − KF0 ) mod 256.
                                                                                  – P0 (B) = (C0 (B) − KF0 ) mod 256.
          Run the FLM generator and stop after IT ER                            and:
          iterations (the initial value is x0 and IT ER is an                   For i in [2, n]:
          iteration parameter).                                                   – Pi (R) = (Ci (R) − KFi − Ci−1 (R)) mod 256.
          For i in [1, n]:                                                        – Pi (G) = (Ci (G) − KFi − Ci−1 (G)) mod 256.
                                                                                  – Pi (B) = (Ci (B) − KFi − Ci−1 (B)) mod 256.

                                                                                                        ISSN 1947-5500
                                                       (IJCSIS) International Journal of Computer Science and Information Security,
                                                       Vol. 9, No. 3, March 2011

   In the next section, we will present the security analysis             going to be this time :(C1 (i, j),C2 (i, j)).
tests performed on our algorithm.                                         Other measures are going to be used to compare two images
                                                                          C1 and C2 as the Number of Pixels Change Rate (NPCR)
                   V. S ECURITY ANALYSIS
                                                                          defined as below:
   In this section we will discuss the security analysis of
                                                                                                         i,j D(i, j)
our algorithm such as key space analysis, sensitivity analysis                            N CP R =                   × 100%
(with respect to both the key and the plain-image) and finally
                                                                          Where n is the images size (number of pixels) and:D(i, j) = 0
statistical analysis as any robust encryption algorithm should
                                                                          if C1 (i, j) = C2 (i, j) and D(i, j) = 1 otherwise.
resist these attacks.
                                                                          The Unified Average Changing Intensity (UACI) will be used
The computation was done using a PC with the following
                                                                          as well and it is defined as:
characteristics: 1,8GHz Core(TM) 2 Duo, 1.00 Go RAM and
120 Go hard-disk capacity.                                                                      1     C1 (i, j) − C2 (i, j)
                                                                                    U ACI =                                 × 100%
                                                                                                n i,j          255
A. Key space analysis
   The used key for our algorithm is a 128 bits key which                 Here C1 (i, j) and C1 (i, j) are grey-scale values of the images
means that we have 2128 possibilities to generate a secret key.           pixels.
With such large key space, the encryption algorithm can be
                                                                             1) Key sensitivity analysis: Key sensitivity is a required
considered secured. In addition to that, the chaotic system that
                                                                          property to ensure the security of any image encryption
we are using to generate the cipher-image is highly sensitive
                                                                          algorithm against some brute-force attacks.
to initial condition which will guarantee that having this large
                                                                          To test the key sensitivity of the proposed algorithm,
key space both key and plain-image attacks will not affect the
                                                                          we have generated randomly an encryption key:
security of the algorithm as we will see in the next sections.
                                                                          ”0CDA03C2D734F06C48A33ECBE3178632”                 then     we
B. Sensitivity analysis                                                   encrypted an original image P using this key to obtain the
   An efficient image encryption algorithm should be highly                image C1.
sensitive to the secret key and to the plaint-image, which                We       then     slightly    modified      the      key     by
means that a single bit change in the encryption key will lead            changing      the    most    significant    bit   to     obtain:
to a very different cipher-image from the initial cipher-image            ”8CDA03C2D734F06C48A33ECBE3178632”, and using
and similarly, only a pixel change in the plaint-image should             this key we’ve encrypted the same original P message the
lead to a very different cipher-image from the initial cipher-            obtain image C2.
image.                                                                    Finally, we did the same as the last operation but
We will present in this section the results obtained by changing          changing the least significant bit to obtain the key:
one bit in the encryption key and one pixel in the plain-image            ”0CDA03C2D734F06C48A33ECBE3178633” and using
and we will see the effects on the cipher-image.                          this last key we encrypted the original image P to obtain the
Before starting our analysis, we will introduce some famous               image C3 (see figure Fig.3).
statistical measures that we will use in the next sections.
The first measure that we will talk about is the statistical corre-
lation between two vertically adjacent pixels, two horizontally
adjacent pixels and two diagonally adjacent pixels.
To compute this measure, we first take randomly a set of
adjacent pixels (vertically, horizontally or diagonally) from the
image (let’s say 1000 pairs) then we calculate their correlation
using the formulas:                                                          Fig. 3.   From the left to the right: original image P, C1, C2 and C3
                                cov(x, y)
                    rxy =
                               D(x) D(y)                                   We have calculated the correlation, the NCPR and the
                                                                          UACI of each two of the three cipher-images C1, C2 and
                                       N                                  C3 (Table I, II and III).
                       E[x] =               xi
                              N       i=1                                 For the obtained results we can see clearly that a negligible
                               N                                          correlation exists among the three images even if they was
                  D[x] =             (xi − E[x])2                         produced using the same original image and with a slightly
                           N   i=1                                        different keys. We can see also that the rate of change NPCR,
                                                                          the intensity of change UACI are really high,then we can
            cov(x, y) = E[(x − E[x])(y − E[y])]
                                                                          conclude that our algorithm is very sensitive to encryption
We will use also to compare two images C1 and C2 , their                  key change.
correlation defined as above but the used pairs of pixels are

                                                                                                         ISSN 1947-5500
                                                      (IJCSIS) International Journal of Computer Science and Information Security,
                                                      Vol. 9, No. 3, March 2011

                 Image 1    Image 2     Correlation
                   C1         C2        -0.000291
                   C1         C3         0.000004
                   C2         C3        -0.001109
                           TABLE I

                                                                                 Fig. 4.   The image P1 (left) and the image C1 (right)
                  Image 1   Image 2      NCPR
                    C1        C2        99.6037%
                    C1        C3        99.6077%
                    C2        C3        99.6039%
                          TABLE II

                  Image 1   Image 2       UACI
                    C1        C2        49.8139%                                 Fig. 5.   The image P2 (left) and the image C2 (right)
                    C1        C3        49.7397%
                    C2        C3        49.8706%
                            TABLE III

   2) Plain-image sensitivity analysis: After studying the key
sensitivity of the proposed image encryption algorithm, we                       Fig. 6.   The image P3 (left) and the image C3 (right)
will study now its plaint-image sensitivity.
The algorithm should be also sensitive to any small change in                              Image 1     Image 2     Correlation
the plaint-image which means that changing only one pixel in                                 C1          C2         -0.0840
                                                                                             C1          C3         -0.0192
the plaint-image should lead to a very different cipher-image.                               C2          C3         -0.0377
This property will guarantee the security of the algorithm
                                                                                                  TABLE IV
against plaint-image brute-force attacks.                                 C ORRELATION BETWEEN THE IMAGES C1, C2  AND C3 OBTAINED BY
To test the sensitivity to plaint-image, we will take an                         CHANGING ONLY ONE PIXEL OF THE ORIGINAL IMAGE
original image (P1) then we will encrypted it (we call
the cipher-image C1), and we will randomly change a
pixel in the original message then will encrypt the im-                                     Image 1    Image 2      NCPR
                                                                                              C1         C2        99.6825%
age again (P2) to obtain a new cipher-image C2. We re-                                        C1         C3        99.8608%
peat this a last time again to obtain a new image (P3)                                        C2         C3        99.8608%
and a third cipher-image C3 (we have used as encryp-                                              TABLE V
tion key:”0CDA03C2D734F06C48A33ECBE3178632”) (see                         NCRP OF THE IMAGES C1, C2 AND C3 OBTAINED    BY BY CHANGING
                                                                                      ONLY ONE PIXEL OF THE ORIGINAL IMAGE
Fig.4, Fig.5 and Fig.6)
As we did for the previous section, we will calculate the
correlation, the NPCR and the UACI between each two of                                      Image 1    Image 2       UACI
the three cipher-images (Tables IV, V and VI).                                                C1         C2        51.3027%
Again, the obtained results show that a negligible correlation                                C1         C3        53.2280%
                                                                                              C2         C3        46.2083%
exists between the three cipher-images and we can see also that
the rate of change (NPCR) and the intensity of change UACI                                             TABLE VI
                                                                         UACI   OF THE IMAGES C1, C2 AND C3 OBTAINED BY CHANGING ONLY
are really high. Form the previous results, we can conclude that                         ONE PIXEL OF THE ORIGINAL IMAGE
our algorithm is very sensitive also to plain-image change.

C. Statistical analysis
  After studying the security of the proposed algorithm                 image P that will be encrypted to obtain a
against some brute-force attacks (key sensitivity and plain-            cipher-image C (we have used also as encryption
image sensitivity), we will study in this section the security          key:”0CDA03C2D734F06C48A33ECBE3178632”).                       We
against statistical attacks.                                            then compare their histograms and compute for each image
To perform this study, we will consider an original                     the values of its two vertically adjacent pixels correlation, two

                                                                                                      ISSN 1947-5500
                                                         (IJCSIS) International Journal of Computer Science and Information Security,
                                                         Vol. 9, No. 3, March 2011

horizontally adjacent pixels correlation and two diagonally                                     VI. C ONCLUSION
adjacent pixels correlation.                                                 In this paper we presented a digital image encryption
                                                                          algorithm based on chaotic logistic maps and using a fuzzy
                                                                          controller (FL-CM-EA).
                                                                          The introduction of the fuzzy controller helped to use a set
   1) Histogram comparisons: Fig.7 and Fig.8, presents the                of logistic maps instead of one logistic map and therefore
histograms of the images P and C.                                         increased the randomness of the generated inputs.
                                                                          We have tested also the robustness and efficiency of the
                                                                          proposed algorithm by performing a set of security analysis as
                                                                          the key space analysis, the key sensitivity and the plaint-image
                                                                          sensitivity analysis and some other statistical analysis as the
                                                                          histogram and the pixels adjacent correlation analysis and
                                                                          all the results demonstrated the high level of security of the
                                                                          proposed algorithm.

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lost after the encryption.                                                     ment : synthese et cryptanalyse (These) (Universite Henri Poincare -
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statistical analysis performed is the adjacent pixels correlation.        [17] Zadeh, L, Fuzzy sets, Information and Control (1965)
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pixels correlations: vertically, horizontally and diagonally.                  (Newcastle University 2008)
                                                                          [19] Hossam El-din H. Ahmed, Hamdy M. Kalash, and Ossam S. Farag
The table VII shows the obtained results.                                      Allah, An efficient Chaos-Based Feedback Stream cipher (ECBFSC) for
                                                                               Image Cryptosystems (SITIS 2006)
                                                                          [20] Mouad HAMRI, Jilali Mikram and Fouad Zinoun. Chaotic Hash Func-
          Image    H Adj Corr     V Adj Corr    D Adj Corr                     tion Based on MD5 and SHA-1 Hash Algorithms (IJCSIS Vol. 8 No. 9
                                                                               DEC 2010)
            P        0.7773         0.8895        0.7507
            C       -0.0055         0.0093       -0.0007
                             TABLE VII

   From the obtained results, we can see clearly that the pixels
of the plait-image P are strongly correlated while a negligible
correlation exists between those of the cipher-image C.
This result shows again that the proposed algorithm can be
considered as secure against statistical attacks.

                                                                                                        ISSN 1947-5500