In this paper, we propose an accumulation encoding scheme based on double random phase encryption (DRPE) for multipleimage transmission. The proposed scheme can be used for a lowcomplexity DRPE system due to the simple structure of the accumulation encoder and decoder. For accumulation encoding of multiple images, all of the previously encrypted data are added, and hence the accumulation encoding can improve the security of the DRPEencrypted data. We present a scheme for encryption and decryption for DRPEbased accumulation encoding, and a method for accumulation encoding and decoding. Finally, simulation results verify that the DRPEbased accumulation encoding scheme for multiple images is powerful in terms of data security.
I. INTRODUCTION
Information security is one of the most important issues in data transmission. Optical encryption has been widely investigated for information security because of its high encryption speed
[1

23]
. As a representative optical encryption technique, double random phase encryption (DRPE) has been well studied
[1

21]
and improvements have been developed, such as DRPE using the Fresnel domain
[2]
, DRPE using a fullphase processor
[3]
, DRPE using digital holography
[4
,
5]
, photoncounting DRPE
[6
,
7]
, and DRPE using fractional Fourier transform
[8

10]
.
Since DRPE uses double random phase masks for encryption, a security defect can be introduced
[11]
, so updating the phase masks is required to resolve the security problem. However, such a resolution may lead to a cost problem. Therefore, advanced DRPE techniques without phasemask updating have been introduced
[8

10
,
18

21]
. In particular, DRPE using fractional Fourier transform has been presented
[8

10]
to enhance the security of DRPE systems, where security is improved by increasing the number of encryption parameters. However, DRPE systems using fractional Fourier transform can be more complicated because of many encryption parameters. Hence, DRPE using orthogonal encoding has been presented as a lowcomplexity DRPE system in which the orthogonal encoder contains simple linear functions
[20
,
21]
.
In this paper we propose a DRPEbased accumulation encoding scheme for multipleimage transmission. It can be considered a lowcomplexity DRPE scheme because of the simple structure of its accumulation encoder and decoder. In addition, data security can be enhanced by introducing accumulation encoding into DRPE. We provide a detailed method for DRPEbased accumulation encoding and decoding, and show simulation results verifying that decryption of the image encrypted by the DRPEbased accumulation encoding scheme becomes more difficult as the number of accumulated images increases, even when the phase key used in DRPE is known.
The paper is organized as follows. Section II presents the concepts behind DRPE. The DRPEbased accumulation encoding scheme for multipleimage transmission is described in Section III. To verify this optical encryption scheme, simulation results produced by DRPE using accumulation encoding are provided in Section IV. Finally, we conclude the paper in Section V.
II. DOUBLE RANDOM PHASE ENCRYPTION
DRPE, an optical encryption technique, uses double random phase masks to encrypt data. Data encrypted by DRPE look like noise. For decryption, the key random phase mask used for encryption is required. For the sake of simplicity, in this paper we consider the encryption of onedimensional data.
Let
p
(
x
) denote the primary data in the spatial domain, and let
m_{s}
(
x
) and
m_{f}
(
w
) represent the random phases in the spatial and spatial frequency domains respectively.
m_{s}
(
x
) and
m_{f}
(
w
) are uniformly distributed over [0, 1].
Figures 1(a)
and
(b)
depict the schematic setup of encryption and decryption for DRPE respectively, where
f
is the focal length and two lenses are used for Fourier transform and inverse Fourier transform. As shown in
Fig. 1(a)
, for encryption first the primary data is multiplied by the random phase mask exp[
i
2
πm_{s}
(
x
)] in the spatial domain. After passing the signal through the first lens we obtain ℑ {
p
(
x
)exp[
i
2
π m_{s}
(
x
)]}, where ℑ denotes the Fourier transform. Next it is multiplied by the random phase mask exp[
i
2
πm_{f}
(
w
)] in the spatial frequency domain. After passing through the second lens, the data encrypted by DRPE,
s_{e}
(
x
), are obtained as follows
[6]
:
Schematic setup of (a) encryption and (b) decryption for DRPE.
where ℑ
^{−1}
represent the inverse Fourier transform. Exploiting the characteristics of a complexvalued function, the encrypted data can be expressed as amplitude and phase parts, i.e.
s_{e}
(
x
) = 
s_{e}
(
x
)exp[
iφ_{e}
(
x
)].
As shown in
Fig. 1(b)
, for decryption the encrypted data are multiplied by the complex conjugate of the key information, i.e. exp[
i
2
πm_{f}
(
w
)]. The decrypted data are then obtained as follows
[6]
:
III. DRPEBASED ACCUMULATION ENCODING AND DECODING
Figures 2(a)
and
(b)
depict the schemes for DRPEbased accumulation encoding and decoding for encryption and decryption, respectively. As shown in
Fig. 2(a)
, the
n
^{th}
primary image
p_{n}
(
x
) is encrypted and encoded for transmission. It is noted that DRPE with the same random phase mask is used for encryption of all the primary images. The data encrypted by DRPE,
s_{e,n}
(
x
), are encoded with the accumulation encoding scheme as follows:
Scheme for DRPEbased accumulation encoding: (a) encryption and (b) decryption.
where
y_{e,n}
(
x
) represents the
n
^{th}
encoded data. As seen in Eq. (3), for accumulation encoding all the previous encrypted data are added, which can enhance the security of the DRPEencrypted data. Eq. (3) can be simply rewritten as
After accumulation encoding of the
n
^{th}
encrypted data, the
n
^{th}
encoded data are normalized by the factor 1/
n
, and finally the
n
^{th}
encoded data
z_{n}
(
x
) are obtained as follows:
For perfect decryption of the accumulationencoded data, as shown in
Fig. 2(b)
, the encoded data are multiplied by
n
to scale it. Then the accumulation decoding is performed as follows:
where
s_{d,n}
(
x
) denotes the
n
^{th}
decoded data and
y_{d,n}
(
x
) =
nz_{n}
(
x
). By inserting Eq. (5) into Eq. (6), we obtain
s_{d,n}
(
x
) =
s_{e,n}
(
x
), which means that the encoded data are perfectly decoded. The
n
^{th}
decoded data are then decrypted by DRPE using the same random phase mask as for encryption. Finally the
n
^{th}
decrypted data
d_{n}
(
x
) are obtained as
d_{n}
(
x
) =
p_{n}
(
x
).
The accumulation encoder and decoder require only a simple linear operation and storage to store the data with the size of a single image, as seen in Eqs. (3) and (6). Therefore, the addition of the accumulation encoder and decoder to the DRPE system requires low cost and effort.
IV. SIMULATION RESULTS
To evaluate the performance of DRPEbased accumulation encoding we consider 20 primary images with 500(H)×500(V) pixels, as shown in
Fig. 3.
Figures 4(a)

(t)
show the simulated results of DRPEbased accumulation encoding, meaning
z_{n}
(
x
) in
Fig. 2(a)
. From the figures we can see that the accumulationencoded images are perfectly encrypted because they look like noise.
Figures 5
and
6
show the decrypted images without and with accumulation decoding respectively, where it is assumed that the DRPE key information is perfectly known for decryption. When no decoding is used, the encoded data are not decoded with the accumulation decoder but are directly decrypted by DRPE decryption; thus,
s_{d,n}
(
x
) =
z_{n}
(
x
). From
Figs. 5(a)

(t)
, it is observed that when no decoding is used the image recognition becomes worse as the index of accumulation encoding
n
increases, even though perfect DRPE key informationis known for decryption. This also, means that accumulation encoding is able to improve the security of DRPEencrypted data even when the DRPE key information is known.
Figures 6(a)

(t)
indicate that the encoded images are perfectly decrypted when correct decoding and decryption are performed.
Primary images.
Simulated results of DRPEbased accumulation encoding: (a)(t) the 1^{st}20^{th} encoded images.
Simulated results of decryption of DRPEbased accumulation encoding when no decoding is used: (a)(t) 1^{st}20^{th} decrypted images.
Simulated results of decryption of DRPEbased accumulation encoding when decoding is used: (a)(t) 1^{st}20^{th} decrypted images.
To quantify the differences between the primary images in
Figs. 3(a)

(t)
and the decrypted images without decoding in
Figs. 5(a)

(t)
respectively, mean square error (MSE) results are calculated, as shown in
Fig. 7
, where we assume that the image pixel values are integers ranging from 0 to 255. In
Fig. 7
the average MSE for the
n
^{th}
accumulation encoding is obtained as follows:
MSE results for primary versus decrypted images, without decoding.
When no decoding is used, the
n
^{th}
decrypted data
d_{n}
(
x
) contain the data for the 1
^{st}
through
n
^{th}
primary images, as described in Eq. (5). Thus we show the average of MSE results between the
n
^{th}
decrypted image and the 1
^{st}
through
n
^{th}
primary images, as in Eq. (7).
Figure 7
demonstrates that the average MSE increases with the accumulation encoding index
n
. Especially when
n
≥ 11 the average MSE increases greatly because the 11
^{th}
through 20
^{th}
primary images have a different color pattern than the 1
^{st}
through 10
^{th}
primary images. From these results it can be seen that the encoded data become more secure as the accumulation encoding index rises, even though the DRPE key is known in decryption. However, for the first accumulation encoding the MSE result is zero, as the decrypted data contain only the data for the first primary image.
V. CONCLUSIONS
We propose a DRPEbased accumulation encoding technique for multipleimage transmission. In particular, we present the schemes for encryption and decryption for DRPEbased accumulation encoding and decoding, respectively, and methods for accumulation encoding and decoding. The simulation results verify that DRPEbased accumulation encoding for multipleimage transmission can enhance the security of DRPEencrypted data when the key information of DRPE is perfectly known in decryption. Furthermore, since the accumulation encoder and decoder require only a simple linear operation and enough storage to hold a dataset of a single image, the DRPE system with accumulation encoding can be implemented with little cost and effort.
Color versions of one or more of the figures in this paper are available online.
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