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Fig. 4. Examples of the 9D-chaotic attractor for the C1 -C2
plane.
operation with RGB chaotic image, and scrambling. The pro-
cedures for these steps are as follows
i. A fusion process is applied to the input double-color im-
age, yielding two new color images IF1 and IF2. This method
is adopted according to the amount of information carried by
various binary bits of image pixels and reflects scrambling
at the bit-slice levels. Fig. 5 shows the process of the fusion optical FT can be implemented using a single lens. The lens
method used in this work.
ii. The image combiner gathers the two-color fusion images
IF1 and IF2, each of size M×N, one above the other, to produce
a single fusion image IF with a size of 2M×N.
iii. A chaotic color image IC of size, 2M×N is constructed as
an encryption key by three of the nine chaotic system outputs
(C1, C2 and C3, where each output is responsible for one of
the RGB channels. The chaotic image IC is XORing with
the color fusion image IF to get a chaotic fusion image ICF .
The chaotic gray images ICR, ICG and ICB, corresponding to
the RGB channels, are constructed based on the sequence
C1, C2 and C3, respectively. This is done after using integer
sequencing according to (2)
ICR(i) = f ix(mod(C1(i)1015, 255) (2)
ICG(i) = f ix(mod(C2(i)1015, 255)
ICB(i) = f ix(mod(C3(i)1015, 255),
where f ix(C) rounds each element of C to the nearest integer
and mod(a, m) returns the remainder after dividing a by m.
Here, a and m represent the dividend and the divisor, respec-
tively.
Muttashar & Fyath
Fig. 5. Brief description of the fusion process.
iv. Finally, a random scrambling technique is applied to the
image ICF to produce the encrypted digital image IDEnc. Note
that the scrambler arranges the array pixels randomly without
altering their values.
3) Optical Encryption Subscheme
The optical encryption subscheme is based on double chaotic
phase encoding (DCPE) implemented in the 2D optical Fourier
transform (FT) domain for each RGB channel of the received
digital encrypted image, as shown in Fig. 6. Note that the 2D
produces an FT image at the back focal plane for an image
located at the front focal plane. For each color channel, the
optical encryption (OE) consists of two cascaded FTs sup-
ported by two chaotic phase masks (CPMs). Each CPM is
generated by one output of the chaotic system. One CMP is
bonded with the primary image, and another is placed in the
Fourier domain, as shown in Fig. 7.
The mathematician framework describes the operation of
DCPE-based FT encryption, which can be adopted from af-
ter replacing the DRPE with double chaotic phase masks
(DCPMs). Consider the red channel as an example; the CPMs
can be explained as follow [27]
CPM1 R(x, y) = exp( jf1(x, y)) = exp( j2pC4(x, y)), (3)
CPM2 R(x, y) = exp( jf2(x, y)) = exp( j2pC5(x, y)), (4)
where C4 and C5 are the fourth and fifth chaotic. The same
operations are applied to green and blue channels. The first
step is to bound the input image whose electric field Ein with
CPM1, the combined function is FT, leading to An electric
field E1(u, v)
E1(u, v) = FT [Ein(x, y)CPM1 R(x, y)]
= [Ein(x, y)epx(2p jCCPM1 R(x, y)) (5)
exp(-2p j(ux + uv))]dxdy