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3) Decryption Procedure:
The receiver can decrypt each ciphered text block C to retrieve
the original plaintext P by utilizing the following formula
[3, 14, 38]:
P = Cdmod n (2)
C. Chaotic Generators
1) Duffing Map:
This chaotic map is a two-dimensional discrete system that
can be expressed mathematically as:
x(n+1) = yn, y(n+1) = -bxn + ayn - yn3 (3)
where a and b represent the system constants which generate
the chaotic behavior at the values 2.75 and 0.2, respectively
[19].
2) Lu¨ System:
Lu¨ chaotic system is a three-dimensional continuous system
which is considered a transitional state between Chen and
Lorenz systems. This system is described mathematically via
a set of ordinary nonlinear differential equations given by:
x? = a(y - x), y? = cy - xz, z? = xy - bz (4)
where a, b and c denote to the control system parameters and
they produce chaotic status when their values are 36, 3 and
20, respectively [39, 40].
III. TIC ALGORITHM STRUCTURE
The block diagram of the suggested encryption image scheme
is exhibited in Fig. 2. The major stages of TIC involve zigzag
scan, RSA algorithm and chaotic systems. In the first stage,
the plaintext image is divided into 8 × 8 non-overlapping
blocks. The direction of image splitting into blocks starts
from top to bottom till it reaches to the last block. Then,
these blocks are separated to odd and even blocks according
to their indices. The pixels in the odd blocks are scrambled
by applying zigzag scan with a variable starting point at first
direction (Fig. 1b ), whereas the pixels in the even blocks
are scrambled by applying zigzag scan with a variable start-
ing point at second direction (Fig. 1c). The starting point is
determined by the generated two Lu¨ chaotic sequences. The
specific sequences are quantized to have values in the range
[1-8], which represent the row and column numbers of the
zigzag operation starting pixel. Therefore, in each block, the
zigzag scan starts at a pixel location which is different from
those of other blocks. This step is the most significant step in elements.
this work because it helps to lessen the correlation between
the original image pixels thereby increasing the value of en-
tropy in the ciphered image. Next, the output vectors from
Yousif, Hameed & Al-Zuhairi
this stage are merged together in a new permutated matrix.
In the second stage, RSA coding technique is exploited to
cipher the permutated image from the first stage by utiliz-
ing the public key (e, n). Eventually, primeval processes in
cryptography like confusion/diffusion are performed using
chaotic Duffing and Lu¨ systems. These two systems produce
the confidential keys for the suggested work by the means
of their control parameters/initial conditions. Confusion is
executed for randomly scrambling image pixel locations; this
process is implemented via 2D Duffing map. On the other
hand, diffusion is executed for varying image pixel values;
this operation is performed via 3D Lu¨ system. The details
of encryption and decryption procedures are outlined in the
following subsections.
A. Encryption Procedure
The specific steps for encryption process according to Fig. 2
are illustrated as follows:
Step 1: The input image A(m, m) is divided into (8 × 8) non-
overlapping blocks.
Step 2: The odd blocks, for instance (block1, block3, block5,. . . ,
block(k-1) are joined in one matrix M(8 × 8 × k/2), where k
represents the total blocks number and it is assumed as an
even value in this work.
Step 3: The even blocks, for instance (block2,block4, block6,. . . ,
blockk) are joined in one matrix N(8 × 8 × k/2).
Step 4: Produce three chaotic sequences u, v and w with m2
elements for each sequence, by using (4) after setting the ini-
tial conditions/system parameters values (x0, y0, z0, a, b, c).
Step 5: Choose the first k elements of the quantized v and w
sequences to determine the starting pixel position, where each
pair of (vi, wi), i = 1, 2, . . . , k, represents the starting pixel row
and column in the ith block, respectively.
Step 6: Depending on the selected starting pixel, apply zigzag
scan according to the direction in Fig. 1b on the matrix M to
acquire the vector Y1 of 64k/2 elements.
Step 7: Depending on the next selected starting pixel, apply
zigzag scan according to the direction in Fig. 1c on the matrix
N to acquire the vector Y2 of 64k/2 elements.
Step 8: Concatenate Y1 and Y2 in a new vector Y ' with 64k
Step 9: Reshape Y ' into 2D matrix Y1' with the same size of
plain image A.
Step 10: Perform the RSA encryption algorithm after deter-
mining the value of public key (n, e) on the matrix Y1' via (1)
to obtain F as:
F (i, j) = ' (i, j)]emod n (5)
[Y1
Step 11: Produce two chaotic sequences x and y with m2
elements for each sequence, by using (3) for Duffing map