仿射与陀螺仪双图像加密新方法：非线性矩阵操作与变换研究
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本文探讨了一种新颖的非线性双图像加密方法，该方法主要依赖于仿射变换和陀螺仪变换（Gyrator Transform）。这项研究发表在《光学学报》(Journal of Optics)上，由Aimin Yan、Jiabin Dong和Yongfang Li三位作者共同完成。他们的研究旨在提供一种增强信息安全性的图像加密方案，适用于数字图像处理领域。
首先，两位原始图像独立地进行陀螺仪变换，这是一种特殊的几何变换，能够在保持图像边缘完整性的同时改变其内部结构。随后，经过这一变换后的图像分别进行相位和幅度截断，将它们转换成两个幅度图像和两个相位密钥。这种处理方式增加了加密的复杂性和不可预测性。
进一步的加密步骤是采用下采样和上采样技术，即对两个幅度图像进行逐像素的分解，将其分割为六个部分：L1、L2、U1、U2以及两个不同的子图Pi（i=1,2）。这种方法确保了图像信息的分散，使得每个部分都包含原始图像的部分特征，但单独看是无法还原完整图像的。
在这个加密过程中，矩阵操作起到了关键作用，通过复杂的数学运算，如矩阵乘法和旋转变换，增强了数据的安全性。与传统的图像加密算法相比，这种非线性加密方案能够抵抗多种攻击，包括频率分析和统计分析，因为其加密过程包含了更多的随机性和非线性元素。
文章的接收日期为2018年5月4日，经过修订后于同年9月21日被接受，并于10月16日正式发布。通过这种新颖的双图像加密方法，研究人员旨在为网络安全提供一个更为坚固的保护层，特别是在处理敏感信息和数字版权管理方面具有潜在的应用价值。
总结来说，这篇研究论文介绍了一种利用仿射变换和陀螺仪变换的非线性双图像加密技术，通过结合矩阵操作和多步处理，提高了图像加密的强度和安全性，对于现代信息技术的保密通信具有重要意义。
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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