3480 Y.-G. Yang et al.
Fig. 1 Encoding procedure
For two dimensional images, the location information encoded in the position qubit
|j includes two parts; the vertical and horizontal co-ordinates. In 2n-qubit systems
for preparing quantum images, or n-sized images, the vector
|j=|y|x=|y
n−1
y
n−2
...y
0
|x
n−1
x
n−2
...x
0
, x
j
, y
j
∈{0, 1}
for every j = 0, 1,...,2
2n
− 1 encodes the first n-qubit y
n−1
y
n−2
···y
0
along the
vertical location and the second n-qubit x
n−1
x
n−2
···x
0
along the horizontal axis.
3 Quantum image encryption and decryption scheme
In this section, we will first review the DRPE technique [2]. Then we will introduce
its idea into our quantum encryption and decryption strategies.
3.1 DRPE technique
The DRPE technique was proposed by Refregier and Javidi in 1995 [2]. This method
allows one to encode a primary image into a stationary white noise, which has been
receiving much interest because of its high-level data security. The encoding procedure
is given by the following steps and can be shown in Fig. 1.
Assume f (x, y) is the plaintext image and the size is M × N, ϕ(x, y) is the cipher
image. The formulas of the encoding and decoding procedures are given respectively
as follows:
ϕ(x, y) = FT
−1
{FT{ f (x, y) exp[j2π n(x, y)]}exp[j2π b(ξ, η)]}, (3)
f (x, y) = FT
−1
{FT{ϕ(x, y)}exp[−j2π b(ξ, η)]}exp[−j2πn(x, y)], (4)
where n(x, y) and b(ξ, η) are the two random-phase functions in spatial domain and
frequency domain, respectively, which are uniformly distributed in [0; 1]. FT and
FT
−1
represent the Fourier transform and its inverse Fourier transform, respectively.
f (x, y) denotes the plaintext image, which is a complex image.
The basic principle of this technique is as follows. Two unrelated random phase
marks (RPM, i.e. exp[j2π n(x, y)] and exp[j2πb(ξ, η)]) act as the keys to be applied
on the input plane and Fourier spectrum plane respectively, as shown in Fig. 1.The
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