2016 International Conference on Mathematical, Computational and Statistical Sciences and Engineering (MCSSE 2016) ISBN: 978-1-60595-396-0
Improvement of Quantum Steganography Protocol
Based on the Tensor Product of Bell States
Li LI
1,
Bassem Abd-El-Atty
2, Mohamed Amin
2and Ahmed A. Abd El-Latif
2,*1
Shenzhen Institute of Information Technology, Shenzhen, 518172, China 2
Department of Mathematics, Faculty of Science, Menoufia University, Shebin El-Koom 32511, Egypt
*corresponding author
Keywords: Quantum information, Quantum steganography, Tensor product, Bell state, Unitary transformations.
Abstract. Quantum steganography is the art of hiding secret information in an insecure quantum
channel. Recently, it is proposed a quantum steganography protocol based on the tensor product of Bell states to transmit classical 4-bit secret message per round. However, it has permits a small hidden capacity. In this work, an improvement of that quantum steganography protocol to transmit 8-bit secret message per round is proposed. The improved protocol uses tensor product of Bell states and unitary transformations with two groups of sixteen tensor products of two Bell states rather than 256 elements of tensor product of four Bell states. Simulations conducted shows that the improved protocol has superior performance in terms of imperceptibility as well as high security compared with the competitive protocol.
Introduction
Quantum information processing has great deal of attention from both engineers and quantum scientists. It is a discipline devoted to the development of novel quantum algorithms. In recent years, a small yet research has focused on quantum information processing such as quantum teleportation [1], quantum steganography and quantum cryptography [2], among many others.
Quantum steganography could be classified according to the embedding methods into four categories: quantum data hiding (QDH) [3], quantum error-correcting code (QECC) [4], quantum image steganography and quantum communications protocols [5, 6].
This paper shed the light on the recently proposed quantum steganography protocols. In [7] Qu et al. presented a quantum steganography protocol with a large payload to transmit 4-bit secret message. In [8] Xu et al. proposed a quantum steganography scheme to transmit 4-bit secret message without consuming any auxiliary quantum states.
It is noted, however, that most of the quantum steganography protocols have the drawback of small hidden capacity. The hidden capacity of [4] is only one bit, the hidden capacity of [5, 6] is also only one bit and the hidden capacity of [7] is four bits per round covert communication. Although the capacity of the hidden channel in [8] can achieve four bits per round covert communication, we still think that it should be further improved. Therefore, in this paper, an improvement of that quantum steganography protocol to transmit 8-bit secret message per round is proposed. It uses tensor product of Bell states and unitary transformations with two groups of sixteen tensor products of two Bell states rather than 256 elements of tensor product of four Bell states. Both simulations and security tests ascertains the efficiency of the improved protocol than the former protocol in [8] in terms of imperceptibility as well as high security.
The Hidden Rule
The hidden rule takes advantage of a pair of quantum subsystems |φ and |ψ in an entangled state. John Bell proved that for a 2 qubit quantum system, there are only four possible entangled states, called the Bell states. There are four Bell states are defined as stated in Eq 1.
1 00 11 2 1 01 10 2 (1)The hidden rule is based on the tensor product of Bell states and unitary transformations. There are four unitary transformations can be written as follows:
0 1
2 3
1 0 0 1
0 0 1 1 0 1 1 0
0 1 1 0
1 0 0 1
0 0 1 1 0 1 1 0
0 1 1 0
(2)
Suppose that the two particles of a Bell state belong to Alice and Bob respectively. Only applying a unitary operation σ0, σ1, σ2 or σ3 on the first particle, Alice can transform the initial state into one of the four Bell states [9]. The relationship between the initial, final Bell states and the corresponding unitary operator when Alice applies on her particle is showed in Table 1.
Table 1. The relationship between the initial Bell states and the final Bell states when performing a local unitary operation on the first particle of the initial Bell state.
σ0 σ1 σ2 σ3
|φ + |φ + |ψ + |φ - |ψ -
|φ - |φ - |ψ - |φ + |ψ +
|ψ + |ψ + |φ + |ψ - |φ -
|ψ - |ψ - |φ - |ψ + |φ +
Alice has four particles labeled as 1, 3, 5 and 7 that in an entangled state with the four particles of Bob labeled as 2, 4, 6 and 8 respectively. There are 256 tensor products of any four Bell states. Instead of using a set with 256 elements of tensor products of any four Bell states, we use two sets, the first set is B and the second set is C with 16 elements tensor products of any two Bell states for each set, so that we divided the particles into two groups. The first group has particles 1, 2, 3 and 4 and the second group has particles 5, 6, 7 and 8.
All the sixteen tensor products of any two Bell states of the first and the second group form the set B and set C respectively:
{| | | | | | | | | | | | | | | |
| | | | | | | | | | |
, , , , , , , ,
, , , , , | ,| | ,| | }
B C
There are 16 different results for each set, labeled as 0000, 0001, …, 1111, respectively. Suppose that the initial state of particles is a combination of the initial tensor product of the set B and the set C, so that the initial state of 8-particles are labeled as 00000000, 00000001,00000010, …, 11111111. Where the first four bits represent the initial state of the set B and the last four bits represents the initial state of the set C. For example, if the initial state of the particles is A37. At first we decode 37
into binary form of 8-bit 00100101, and then decode the first 4-bit into decimal form to label the initial state in set B (B2) and decode the last 4-bit into decimal form to label the initial state in set C
(C5).
Convert 4-bit to decimal form b
and take Bb as initial state of set B
Convert 4-bit to decimal form c
and take Cc as initial state of set C
f b j
B B (3)
f c k
C C (4)
Where, f , , , , b c j k {0, 1, . . . , 15},⊕ means the modulo-16 addition operation, Bb is the
initial state of set B, Cc is the initial state of set C, Bf is the final state of set B and Cf is the final
state of set C. We take two unitary transformations to transform Bb Bf and another two unitary
transformations to transform Cc Cf by this way the hidden rule can be established among the secure message, the initial and final states when respectively applying local unitary operations on the first particle of the four initial Bell states.
The Proposed Quantum Steganography Protocol
In this section, we will explain how to send 8-bit from Alice to Bob using a QSDC scheme. We use the hidden rule without using any auxiliary quantum states or any extra quantum communication besides the QSDC. In the proposed steganography protocol to transmit 8-bit, we use eight particles labeled as 1,2,3,4,5,6,7 and 8, the particles 1, 3, 5 and 7 are at Alice side, and other particles are in Bob side. The particles 1, 3, 5 and 7 are in an entangled state with particles 2, 4, 6 and 8 respectively. The unitary transformations are used to link the QSDC process and the steganography process, by which Alice and Bob can determine the final tensor product of the hidden rule.
In our protocol we use the main parts of the QSDC protocol [9] that described in [8, 9], while our purpose is to intensively present the steganography scheme to transmit 8-bit. Before performing the protocol, Alice and Bob agree that each of the four unitary transformations represents 2-bit classical information as showed in Table 2.
Table 2. Four unitary transformations represent 2-bit.
unitary
transformation σ
0 σ1 σ2 σ3
2-bit 00 01 10 11
The proposed quantum steganography protocol based on the tensor product of Bell states to transmit 8-bit is illustrated in detail as follows:
Algorithm for Encoding Secret Message
Inputs. Initial state of particles Ai, secret message 8-bit
Outputs. Stego-message 8-bit
Step 1. Alice and Bob choose an element Ai from the initial states
Step 2. Convert i to binary form 8-bit
Step 3. Alice wants to send secret message 8-bit
S1 S2 S3 S4 S5 S6 S7 S8
Step 4. Alice sends the sequence
to Bob using the protocol quantum secure direct communication with χ-type entangled states [9]
Algorithm for Decoding the Secret Message
Inputs. Initial state of particles Ai, stego-message 8-bit
Outputs. Secret message 8-bit
Step 1. By Quantum secure direct communication with χ-type entangled states protocol
Bob obtain the sequence
Step 2. Bob determine the final tensor product of set B by Bf = σp⊗σq Bb and the final tensor
product of set C by Cf = σx⊗σy Cc where
p = 2n1 + n2, q = 2n3 + n4, x = 2n5 + n6, y = 2n7 + n8
Step 3. Bob get j using the index f in Bf and index b in Bb where j=f b Similarly Bob gets k using
the index f in Cf and index c in Cc where k=f c
Step 4. Then convert j into binary form 4-bit and convert k into binary form 4-bit
To illustrate the proposed quantum steganography protocol, a simple example is given as follows. Suppose that the initial state of particles is A0 and Alice wants to send the secret message
10011011 to Bob, the binary form of 0 is 00000000, so that the initial tensor product of set B is B0
and the initial tensor product of set C is C0 the secret message divided into two parts, the first part is
1001 and the second part is 1011. The secret message encoding and decoding process is shown as:
Encoding Process. Alice first determines the final tensor product Bf B0 9 B9 based on the
secret message part one 1001 and the final tensor product Cf B0 11 C11 based on the secret message part two 1011. Then she can choose four proper unitary transformations σ2, σ3, σ3 and σ2 using Table 1. The four unitary transformations σ2, σ3, σ3
and σ2 can be used to encode the information 10111110. Alice needs to find out one of the positions of 10111110 in the information sequence and chooses it as m. Finally, Alice sends m using the QSDC channel.
Decoding Process. After receiving m, Bob first decodes the information that the mth pair of
particles carries is 10111110, he determines the four unitary transformations σ2, σ3, σ3 and σ2, and then determines the final tensor product B9 and C11. According to the hidden rule, Bob computes j =
9 0 = 9 and k=11 0 = 11, then he can obtain the secret message by concatenation of binary form of j and binary form of k, so that the secret message is 10011011.
n1 n2 n3 n4 n5 n6 n7 n8
n1 n2 n3 n4 n5 n6 n7 n8
S1 S2 S3 S4 S5 S6 S7 S8
Convert 4-bit to decimal form j Bf =Bb⊕j
By looking in Table 1, Alice chooses two unitary transformations σp, σq
which satisfy Bf = σp⊗σq Bb
Represent each unitry transformation by 2-bit using table 2
n1 n2 n3 n4
Convert 4-bit to decimal form k Cf =Cc⊕k
By looking in Table 1, Alice chooses two unitary transformations σx, σy
which satisfy Cf = σx⊗σy Cc
Represent each unitry transformation by 2-bit using table 2
Performance Analysis
Capacity
Embedding efficiency describes how many bits (qubits) of a secret message can be embedded in a single qubit quantum particle for a steganography scheme. The hiding channel of our protocol can be built up within the QSDC channel to transmit a secret message without consuming any extra quantum communications or quantum states besides the QSDC. By transferring a χ-type quantum state that is encoded with 8-bit of information in the QSDC process, Alice can hide an 8-bit secret message under the cover of normal information transmission. Bob decodes the information that Alice has sent to him by decode the information gets from performing measurements on the χ-type quantum state based on FBM. So we can say that The embedding capacity of the proposed protocol depends on the source capacity of the QSDC channel.
Imperceptibility Analysis
The imperceptibility of a steganography ensures that the hidden message is hardly detected so as to prevent supervisors or attackers from damaging it. The imperceptibility of a steganography is one of the most important measures to evaluate the performance of a steganography algorithm. The imperceptibility of quantum steganography has some advantages from the quantum uncertainty principle and the quantum no-cloning theorem compared with classical steganography. The imperceptibility of the proposed quantum steganography protocol completely lies on the invisibility of the hidden channel. The secret message hiding/unhiding process has no effect on the QSDC channel because the two processes are linked with the QSDC process only by unitary transformation. Let Eve be eavesdropper. Eve fails to deduce the unitary transformations only by measuring the encoding particles 1, 3, 5 and 7 because all particles are in maximum entangled state and Eve has no access to the eight qubits simultaneously. So, the intercept-resend attack has no effect on this protocol. There is no leakage of information about the unitary transformations, so that the imperceptibility of the steganography scheme can be ensured.
Comparison
[image:5.595.55.542.557.682.2]Herein, we compare the proposed approach with the related approach [8]. The comparison results are given in Table 3. From the results, we can see that the proposed approach can be able to transmit 8-bit secret message instead of 4-bit in the previous approaches.
Table 3. Comparison between [8] and the improved protocol.
The proposed protocol QCC [8]
Number of used particles 8 4
Number of transmitted bits 8 4
Number of tensor product elements in the set to transmit 8-bit in one cycle
two groups of sixteen tensor products of any two Bell states
8-particles with 256 element of tensor products of any four Bell states
Eve access to the qubits very hard to access the 8-qubits simultaneously
hard to access the 4-qubits simultaneously
Using auxiliary quantum state without without
tools the tensor product of Bell states and
unitary transformations
the tensor product of Bell states and unitary transformations
Concluding Remarks
protocol [8] in terms of imperceptibility, high capacity, reduced complexity as well as high security.
Acknowledgments
This work is supported by Menoufia University under the project number: A-WSN2016, Guangdong Natural Science Foundation: 2015A030310172 and Natural Science Foundation of Heilongjiang Province, China: QC2014C076, JJ2016ZR1068.
References
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[5] K. Matin, Steganographic communication with quantum information, Lect Note Comput Sci. 4567 (2007) 32-49.
[6] X. Liao, Q.Y. Wen, Y. Sun, J. Zhang, Multi-party covert communication with steganography and quantum secret sharing, J Syst Softw. 83 (2010) 1801-1804.
[7] Z.G. Qu, X.B. Chen, X.X. Niu, Y.X. Yang, Novel quantum steganography with large payload, Opt Commun. 283 (2010) 4782-4786.
[8] Xu ShuJiang, X.B. Chen, X.X. Niu, Y.X. Yang, High-efficiency quantum steganography based on the tensor product of Bell states, Sci China-Phys Mech Astron. 56 (2013) 1745-1754.
![Table 3. Comparison between [8] and the improved protocol.](https://thumb-us.123doks.com/thumbv2/123dok_us/303256.1031621/5.595.55.542.557.682/table-comparison-improved-protocol.webp)
