基于仿射与旋转变换的新型双图象非线性加密技术
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本文主要探讨了一种新颖的基于仿射变换和回转器变换的双图像加密方法，发表在《光学学报》上。作者是Aimin Yan、Jiabin Dong和Yongfang Li，他们来自上海师范大学数学科学学院。该研究于2018年5月4日接收，9月11日进行了修订，并于同年9月21日被接受发表，最终于10月16日发布。
论文的核心思想是利用矩阵运算和两种特殊变换（仿射变换和回转器变换）来增强图像的安全性。首先，两个原始图像独立进行回转器变换，随后分别对变换后的相位和幅度信息进行截断，得到两个幅度图像和两个相位密钥。这些密钥在后续的加密过程中扮演关键角色。
进一步，通过将两个幅度图像进行下采样和上采样的交替处理，即进行下取整和上取整，将其分解成六个部分：L1、U1、L2和U2。其中，L1和U1（i=1,2）分别代表两个幅度图像的下半部分和上半部分。这一过程不仅增加了加密的复杂性，还使得攻击者难以直接解读原始信息。
然后，这些部分进行特定的组合和操作，例如可能涉及位移、混淆和扩散等加密技术，形成加密后的图像。这一步骤旨在混淆和保护图像的原始结构，使得即使对图像进行视觉检查，也无法轻易识别出原始图像的信息。
为了保证安全性和有效性，论文还可能讨论了加密算法的性能分析，如抗差分混沌分析、抗选择明文攻击（CPA）、抗统计分析以及时间复杂度和空间复杂度等方面的评估。此外，文章可能会提及与现有图像加密算法的比较，突出其独特性和优势。
这项研究提供了一种创新的双图像加密策略，它结合了仿射变换和回转器变换的独特特性，旨在提高图像数据的安全传输和存储。通过复杂的处理步骤和精心设计的加密机制，该方法在保护信息安全方面具有潜在的应用价值。
while ciphertext is essential to the decryption. In [28], Bha-
duri et al proposed a novel nonlinear optical image encryption
technique by using spiral phase transform, equal modulus
decomposition and SVD. In his method, SVD is performed on
Fresnel propagated output to get three decomposed matrices:
one diagonal matrix and two unitary matrices. Two different
spiral phase masks are used to modulate the two unitary
matrices and then, the inverse SVD is performed using the
diagonal matrix and modulated unitary matrices to get the
final encrypted image. Its optoelectronic setup for encryption
and decryption has been suggested.
In this paper, for the first time to our knowledge, we present
a novel double image encryption scheme by using PLU
decomposition and affine transform in GT domains. Two ori-
ginal images are independently gyrator transformed and then
phase- and amplitude-truncated to obtain two amplitude images
and two phase keys. PLU decomposition is performed on two
amplitude images to get six parts represented by L
i
, U
i
and P
i
(i=1, 2).Further,L
1
is combined with U
2
and L
2
is combi ned
with U
1
to form two new images, and P
i
serves as the decryption
key. The same operation performed above is executed on two
newly obtained images again, then affine transform is added in
the last step to obtain the final cipher images. The security keys
in the proposed method are the angle of GT, phase keys obtained
by amplitude-truncation, the iteration number of affine transform
and P
i
(i=1–4). SVD is replaced by PLU decomposition,
which appears in image encryption for the first time. Compar-
ison between SVD and PLU decomposition also has been pre-
sented and discussed in this paper. Two images are encrypted in
two different channels, but mutual encryption has been realized.
Unlike the methods mentioned above, in our method, multiple
cipher images are not obtained in the last step by using matrix
decomposition, while two channels run in parallel. Information
after PLU decomposition has been exchanged partly, so part of
one channel’sinformationisencryptedandhiddenintoanother
channel, thus to make attacks more difficult. Two cipher images
can be distributed to different authorized users, so the safety of
original information will be enhanced highly.
The rest of this paper is organized as follows: in section 2,
the principle of PLU decomposition, affine transform and GT
are detailedly introduced. In section 3, encryption and decryp-
tion processes are described in detail. In section 4, numerical
simulation results are presented to demonstrate the performance
of the proposed scheme and followed with security analysis. A
conclusion is drawn in section 5.
2. Basic theories
2.1. Affine transform
The formula written in equation (1) is called two-dimensional
affine transform [29]
x
y
ab
cd
x
y
e
f
,1
¢
¢
=+
⎡
⎣
⎢
⎤
⎦
⎥
⎡
⎣
⎢
⎤
⎦
⎥
⎡
⎣
⎤
⎦
⎡
⎣
⎢
⎤
⎦
⎥
()
where a, b, c and d are the parameters of affine transform, e
and f are the coefficients of translation,
xy,
(
)
and
xy,¢¢
(
)
are
the coordinate representations before and after affine trans-
form, respectively.
For a grayscale image with size n-by-n, affine transform
can be applied to exchange its pixel positions without
removing any information from the image, then grayscale
image will be changed into a stationery white noise image.
Different from the common used Arnold transform, affine
transform has inverse transform, this ensures the fast process
of decryption.
2.2. Permuted lower-and-upper (PLU) decomposition
PLU decomposition is an important numerical technique in
Linear Algebra [30]. A matrix can be expressed as a product
of three matrices by PLU decomposition. Its definition is
expressed as
APLU
AA A
AA A
AA A
L
LL
UU U
UU
U
1
1
1
1
1
1
1
1
,2
n
n
nn nn
nn
n
n
nn
1
11 12 1
21 22 2
12
21
12
11 12 1
22 2
==
=
´
-
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
()
where A represents a n-by-n non-singular square matrix, L is a
lower triangular matrix and U is an upper triangular matrix, P
is a n-by-n permutation matrix where the ‘1’ occurs in each
column, it is an important part to recover A.
If we want to encrypt an image, we can divide it into
three part by PLU decomposition. Then decomposed three
parts can be distributed to different authenticated users and
protected in different ways. Only when all three members get
together, the original image can be recovered.
Figure 1 shows the results of PLU decomposition.
Figure 1(a) is the original image ‘Baboon’ with 256×256
pixels. Figures 1(b)–(d) displays the three parts obtained by
PLU decomposition and figure 1(e) is the recovered image.
Figures 2 and 3 are the decrypted images with the wrong
three parts and wrong multiplication orders, respectively.
Figure 2 (a) is the decoded result when L part is wrong.
Figure 2(b) is the decrypted result when U part is wrong and
figure 2(c) is the recovered result when P part is wrong.
Under the wrong orders PUL, LPU, LUP, ULP and UPL, the
corresponding decrypted images are shown in figures 3(
a)–
(e), respectively.
Comparison between SVD and PLU decomposition is
displayed in figure 4. Figure 4( a ) is the decrypted result with
the wrong S part, which is a diagonal matrix with all ‘1’ in
diagonal line. This diagonal matrix is also used as the wrong
P part in the decryption, the corresponding decrypted result is
shown in figure 2(c). Compared with figure 2(c), it is easier to
get the profile of ‘Baboon’ from figure 4(a). When we replace
2
J. Opt. 20 (2018) 115702 AYanet al
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