WO2014069984A1

WO2014069984A1 - Multi-user steganography based on greenberger-horne-zeilinger states

Info

Publication number: WO2014069984A1
Application number: PCT/MY2013/000183
Authority: WO
Inventors: Ridza Wahiddin Mohamed; Su Fong Chien; Ab Manan Jamalul-Lail; Aly Aly El-Oriny Faisal
Original assignee: Mimos Berhad
Priority date: 2012-11-05
Filing date: 2013-10-22
Publication date: 2014-05-08
Also published as: MY179060A

Abstract

The invention provides a method (100) of multi-user steganography. Firstly, the method involves establishing secure channels between at least three users including the steps of: each of the users randomly selecting a sequence of ordered Greenberger-Horne-Zeilinger (GHZ) states having at least three particles (110); each of the users taking one particle from each of their respective GHZ states to each form at least three subsequences (120); each of the users keeping an initial subsequence in their own site and transmitting a respective one of the remaining subsequences to each of the other users (130); and each of the users checking the security of established channels by measuring the number of particles on either side, as agreed between the users (140). Next, the method involves the users transmitting messages between each other by performing the steps of: each of the users encoding a cover message by applying a unitary transformation on the subsequences sent by the other users (150); each of the users transmitting the subsequences encoded with their cover message to the other users (160); each of the users retrieving their respective subsequence in which the cover messages of the other users are encoded (170); each of the users performing GHZs-based measurements on the subsequence having the cover messages of the other users encoded thereon, and comparing measurement outputs with the initial subsequence to obtain the encoded messages (180); and each of the users checking the security of established channels by measuring the number of particles on either side, as agreed between the users (190). After this process, the users exchange stenographic messages (200).

Description

MULTI-USER STEGANOGRAPHY BASED ON GREENBERGER-HORNE-ZEILINGER

STATES

FIELD OF INVENTION

The present invention relates to a method for multi-user steganography based on Greenberger-Horne-Zeilinger states.

BACKGROUND ART

Steganography is the process of hiding secret information by embedding it in an innocent message and communicating the resulting data over a communication channel or by a courier so that the steganographic message is readable only by the intended receiver. Steganography is different to cryptography since the goal of steganography is render the secret information undetectable, as opposed to maintaining secrecy in cryptography. The advantage of steganography over cryptography alone, therefore, is that the secret messages do not attract attention to themselves.

Steganographic systems need a priori existence of a cover message, in which the steganography encoder hides the secret message. If the stego-data closely resembles the cover data, it is generally difficult for eavesdroppers to detect the existence of the secret message. Most steganographic systems to date use image or audio files as the cover data. In general, no classical steganographic system can be perfectly secure if its cover data is the result of a measurement and the distribution of the cover data is unknown. This is because any modification of the cover data may distort the original cover distribution. However, under the same conditions, one can construct perfectly secure quantum steganography protocol (QSP). This generally means that the QSP is more secure than classical steganographic systems.

Quantum cryptography is one of the most fruitful applications of quantum information theory and it is envisaged that this could become widely used in the near future. Quantum cryptography allows the completion of various cryptographic tasks that may otherwise be impossible using only classical (i.e. non-quantum) systems since quantum cryptography follows quantum laws, for example the Heisenberg uncertainty principle, no-cloning theorem and quantum correlations. This, of course, protects the cryptographic messages, where legitimate users can easily detect eavesdroppers.

In quantum cryptography there are three main protocols, namely BB84 protocol, B92 protocol and EPR protocol. Some of these protocols have been experimentally implemented by different means. QSP may be effectively generated from the quantum cryptographic systems.

The first QSP was composed by building a hidden channel within existing quantum technology via the BB84 protocol. The author analyzed the imperceptibility and the security in detail and calculated accurately the capacity of this type of hidden channel. More recently, by means of quantum 3 teleportation, a new QSP has been proposed that employs digital colour images. In particular, three-dimensional qubits are used to represent RGB (Red, Green and Blue) pixels for transmitting quantum secret messages. A QSP based on quantum secure direct communication has also been proposed. In this protocol, the hidden channel was built up within an improved ping-pong protocol. This increased the capacity of the hidden channel to four bits. Recently, a QSP with a relatively large payload has been developed where one can transmit a two bits secret message by transforming one qubit and consuming one Bell state.

Entanglement is one of the main ingredients in quantum information theory. Based on this property, various protocols have been developed. One of the well-known entangled states is the Greenberger-Horne-Zeilinger state (GHZs), which has been involved in quantum cryptography. GHZs are distinguished by a large Hilbert space compared to other entangled states, such as the Bell state. GHZs have also been experimentally implemented by various means, for example (i) using entanglement swapping starting from three down converters, (ii) using two pairs of entangled photons, (iii) based on dipole-induced transparency in a cavity-waveguide system, (v) in the framework of superconducting circuits and (vi) in nuclear magnetic resonance. All of the QSPs mentioned above have been restricted to two users. In that regard, it is also worth mentioning that four particle GHZs have been used quite recently in quantum steganography, but for two users only. Therefore, it would be advantageous if a multiuser version could be developed.

The subject matter claimed herein is not limited to embodiments that solve any disadvantages or that operate only in environments such as those described above. Rather, this background is only provided to illustrate one exemplary technology area where some embodiments described herein may be practice.

SUMMARY OF INVENTION

The task of developing a multi-user QSPs cannot be achieved in the classical world, it is necessary to compose the system in the framework of quantum information theory. The protocol depends on the entanglement property of the quantum systems. For this endeavour, the invention employs GHZs which can be generated from different physical processes. In particular, in the protocol of the invention quantum key distribution (QKD) is carried out and then steganography performed within this quantum channel. The particles/bits are exchanged among legitimate users in blocks through a two-step process. Generally, the invention provides two levels of data communication which are utilised successively in such a way that the first level serves as a carrier for the second level which hides the important data.

One aspect of the present invention provides a method (100) of multi-user steganography comprising establishing secure channels between at least three users by performing the steps of establishing secure channels between at least three users; wherein a first user, a second user and a third user perform steps of each of said users randomly selecting a sequence of ordered Greenberger-Horne-Zeilinger (GHZ) states having at least three particles (110); each of said users taking one particle from each of their respective GHZ states to each form at least three subsequences (120); each of said users keeping an initial subsequence in their own site and transmitting a respective one of the remaining subsequences to each of the other users (130); and each of said users checking the security of established channels by measuring the number of particles d on either side, as agreed between said users (140); said users transmitting messages between each other by performing the steps of each of said users encoding a cover message by applying a unitary transformation on the subsequences sent by the other users (150); each of said users transmitting the subsequences encoded with their cover message to the other users (160); each of said users retrieving their respective subsequence in which the cover messages of the other users are encoded (170); each of said users performing GHZs-based measurements on said subsequence having the cover messages of the other users encoded thereon, and comparing measurement outputs with said initial subsequence to obtain the encoded messages (180); and each of said users checking the security of established channels by measuring the number of particles d' on either side, as agreed between said users (190); and said users exchanging stenographic messages (200).

Another aspect of the invention a method wherein the step of exchanging stenographic messages (200) comprises each of the users selecting a particular number of check bits at random, the number of check bits being known to the users, the last of the check bits being the pointer for the steganography data; and each of the users announcing their positions publically for the other users.

In a further aspect the invention provides a method wherein each of said users randomly selecting a sequence of ordered Greenberger-Horne-Zeilinger (GHZ) states having at least three particles (110) further comprises steps of said first user, A, prepares a sequence of ordered GHZ triplets ( 10).

In another aspect of the invention provides a method wherein each of said users taking one particle from each of their respective GHZ states to each form at least three subsequences (120) further comprises steps of said first user, A, takes one particle from each GHZ triplet to form three ordered sequences (120); said second and third users, B and C respectively, execute the same steps in their sites to produce further sequences respectively (120); and each of said first, second and third users exchange the sequences among each other (120) such that they keep in their sites the said sequences , and release the rest to the other users such that said first, second and third users have in their sites the respective sequences .

In another aspect of the present invention provides a method wherein each of said users checking the security of established channels by measuring the number of particles on either side, as agreed between said users (140) further comprises steps of sacrificing a portion of particles from each of the sequences to check the security of the channels; and encoding message in the qubits of the other users following the Boolean relation agreed on by said users, each of said users acting simultaneously by one operator on the two particles from the different sequences.

In yet another aspect the invention provides a method for encoding message in the qubits of the other users following the Boolean relation agreed on by said users, each of said users acting simultaneously by one operator on the two particles from the difference sequences further comprising, said first user, A, has a sequence and said second and third users have corresponding sequences.

In another aspect the invention provides a method wherein each of said users performing GHZs-based measurements on said subsequence having the cover messages of the other users encoded thereon, and comparing measurement outputs with said initial subsequence to obtain the encoded message (180) further comprising steps of each of said users thereby having three cover messages his/her own one and those of the others (180), and wherein said users subsequently use the virtue of the set of particles to establish the security of said channel.

It is envisaged that the protocol of the invention will provide several potential benefits. For certain situations, the second level can be utilised to make the steganogram unreadable even after the detection of the first level method. For example, it can carry a cryptographic key that deciphers the steganogram carried by the second level. This could possibly make the steganographic communication harder to detect. It can also be used to provide the steganogram with integrity. As described below, the protocol is flexible since it can accommodate small and large capacities per round of covert communication based on demand. The security of the protocol is achieved by the notion of the entangled state, no-cloning theorem and the laws of quantum mechanics. It is envisaged that the protocol of the invention may find application in quantum telecommunication networks in a similar way as seen in the case of classical systems.

The present invention consists of features and a combination of parts hereinafter fully described and illustrated in the accompanying drawings, it being understood that various changes in the details may be made without departing from the scope of the invention or sacrificing any of the advantages of the present invention. BRIEF DESCRIPTION OF ACCOMPANYING DRAWINGS

To further clarify various aspects of some embodiments of the present invention, a more particular description of the invention will be rendered by references to specific embodiments thereof, which are illustrated in the appended drawings. It is appreciated that these drawings depict only typical embodiments of the invention and are therefore not to be considered limiting of its scope. The invention will be described and explained with additional specificity and detail through the accompanying drawings in which:

FIGS. 1A to 1C illustrate a flowchart of a method according to an embodiment of the invention.

FIG. 2 illustrates the average probability of not detecting Eve during the control mode of the IRA. The angles and θ₂ define Eve's projective measurement direction. The angles are measured in radian.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

The present invention provides a method for multi-user steganography based on Greenberger-Horne-Zeilinger states.

Hereinafter, this specification will describe the present invention according to the preferred embodiments. It is to be understood that limiting the description to the preferred embodiments of the invention is merely to facilitate discussion of the present invention and it is envisioned without departing from the scope of the appended claims.

The following discussion provides a full description to the protocol according to embodiments of the invention. The discussion has been divided into four sections for ease of understanding. In the first section, properties of the GHZs are provided and the encoding/decoding process is illustrated. In the second section the steps of the protocol are described in detail. In the third and fourth sections the security of the protocol is examined and its applications are discussed.

A. Properties of the GHZs

In quantum information theory, GHZ is a type of entangled quantum state which involves at least three subsystems (particles). In particular, the GHZ state is an entangled quantum state of N > 2 subsystems, which reads:

GHZ is a quantum superposition of all subsystems being in the state 0 with all of them being in the state 1 (states 0 and 1 of a single subsystem are fully distinguishable). Physically, the Boolean 0 and 1 are related to electron spin or photon polarization. In other words, the state l°) <W) denotes the ground atomic state or the vacuum state (i.e. the excited atomic state or the single photon state). Whilst there is no standard measure to the multi-partite entanglement, many measures define the GHZ to be maximally entangled. These states are closed under the action of any one of the four unitary (Pauli) operators * «#* , where the superscript j stands for the yth-particle and the notations have the same standard meaning in the literature. This forms the basis for using |GHZ> in quantum information theory, such as in quantum cryptography and communication complexity tasks. The first laboratory observation of GHZ correlations was by the group of Anton Zeilinger (1998). Subsequently, many more accurate observations were made. The action of the Pauli operators on the states ^¹^ can be summarized as:

/(G) = |0), /|1} = |1),

¾[0} = -|0>, *,|1>.= |1),

A_x|0> = jl), ¾|i) = jo>,

i¾W = ii), .¾li} = Ho).

Throughout the following description of the steganography protocol of the invention, without limiting the invention thereto, the discussion will focus on the three-particle GHZs. Thus, it is necessary to shed some light on this set of states. This set includes eight elements, which form a complete orthonormal basis as follows: W = (i°∞> + hW = (t«io> - !«»>)♦

|^) = £(|100) - |011))_t

],fe) = {|010> + |101)), f^> = £(|010) - |101))_t

I*0 = (|HO> + |WI», hh) = £(ino) - |ooi)).

If .one measures one of the subsystems \ ύ in such a way that the measurement distinguishes between the states 0 and 1, this will leave behind unentangled pure states of either I⁰⁰) ⁰¹ i¹³ⁱ> . This is unlike the W state, which leaves the bipartite entanglements even when one of its subsystems is measured.

In the following section, the protocol based on the given properties for the |GHZ). _will be described. To achieve this goal, the users have to agree, in advance, the following Boolean relations: B. Protocol description

Multi-user steganography is a new concept for the hidden communication within a channel using different means. In quantum theory, users do not use photos or video files to hide messages, but rather just improve the quantum cryptography protocols to communicate the steganograms. Therefore, the carrier is the quantum key distribution (QKD) itself, which makes the QSP much more secure than classical examples. To clarify certain aspects of the present invention, the following multi-key generation is provided in terms of the GHZs, including a discussion of its extension to the QSP. The analysis is restricted to the three users, without wishing the scope of the invention to be so restricted, where the generalization to the multi-user is a straightforward matter.

Referring to Figure 1, in the following discussion it is assumed there are three users, namely Alice, Bob and Charlie. It is also assumed that there is an agreement amongst the users about the encoding scheme (i.e. the relation (4) recited above), and the scenario by which they can extract the hidden messages and/or steganograms. The protocol (100) is in accordance with the following steps.

Step 1 : Alice prepares a sequence ^A of w +d+^'J of ordered GHZ triplets (110), each of which has the form ("ϊ·"-!·"!)* k = i,2,...,n + d+ df _where the superscript k denotes the order of the triplet in the sequence A. These triplets are randomly chosen from the set (3) above, and are already known to Alice herself. Alice takes one particle <*i from each GHZ triplet to form three ordered sequences

Aj =

1,2,3; ty = Oor 1.

Step 2: Bob and Charlie execute the same steps in their sites. For example, Bob and Charlie have the sequences ¾ = ί¾,¾,---,^} ^ ¾ = ^_ί --·^^} ^* ί = ¹.².^3« respectively (120).

Step 3: The users exchange the sequences among each other (120). For instance, Alice, Bob and Charlie keep in their sites the sequences -A,,J¾ and irrespectively, and release the rest to the other users. After the completion of this process, Alice, Bob and Charlie have in their sites the sequences and {A_SrB₃,Cs\, respectively. The transmission of the particles occurs in blocks, and the orders of the sequences are not known to the receivers, but are known to the senders.

Step 4: The users check the security of the channels (140) to see whether eavesdroppers are on line or not in the standard way. For this task the users sacrifice a portion d of particles from each of the sequences.

Step 5: Each partner now has his own cover message or key, and he or she wants to transmit to the other users. Therefore, each one encodes his own message in the qubits of the other users following the relation (4) agreed on and cited above (150). Each user acts simultaneously by one operator on the two particles from the different sequences. For instance, suppose that Alice wants to encode the bit 1 in the yth-partiele for the other users. She should act by ^ on the jth particle in the sequences ¾ and <¾. . Throughout the encoding process the users should be aware of the information and the positions of the particles of the set <?- . The reason for this is that the users will sacrifice these particles when checking for eavesdroppers. This has to be carried out before executing the steganography in the final step.

Step 6: After the completion of the encoding process, each user transmits back the blocks of particles (message particles) to their original owners (160). For instance, Alice, after a successful transmission, has the sequence

A = {(-^^.tf.£.^.^(^ ^««S***)}. _Where the dash means that these particles are different from the original ones since they carry the messages of the other users. The other users will follow similar scenarios. It is worth mentioning that the sequences {o¾, = l, -,n + cf} and {o^,i = i,...,n+ -f} carry Bob's and Charlie's messages, respectively, while i°i.i = ^■<^η+^<ί) is the home sequence. As the users prepared these qubits initially, they know them very well. Now each user performs the GHZs-basis measurement on the ordered n+f GHZs and compares the results with the initial forms to obtain the cover messages of the other users (170). For instance, if Alice initially prepared one of the triplet in the state l^)and the measurement result is ΙΨτ)- From (3) and (4) Alice has = ^2>σ1^3,|ώι>. According to the arrangement (4) Alice knows with certainty that Bob's and Charlie's bits are 0 and 1 , respectively, and so on. At the same time Bob and Charlie perform the same procedures. At the end of this process, each user has three cover messages: his/her own one and those of the others (180). At this moment, the users can use the virtue of the set ^ to check eavesdroppers (190). After the successful completion of this process, the users switch to the steganographic channel.

Step 7: In this step the users can make simple covert tricks to exchange the steganographic messages among themselves (200). For instance, each user may select a portion of the bits (the number of the bits in the portion has to be fixed or based on the scheme they agreed on in advance) from his own message. Of course, this portion includes the information bit, or the blocks of bits he/she would like to communicate to the other users. One scenarios which may be used is that the steganographic bit/block can be transmitted by announcing the location (not the value) of each bit in the string via a classical channel, the target bits being located between the immediate left of the last announced bit 'pointer' and the previous check bit. To explain this step in more detail, the following example is provided.

Assume that Alice would like to send the block data 1011 to Bob and Charlie. For this task Alice specifies the location of this block in her own data, which is already sent to the other users in the step 6. Assume that Alice, Bob and Charlie share this string:

00101011100101011 10101001100

The under braced bits represent the information block. Alice may then start to announce the location of the bits, which the other partners have to remove from the string.

0 * 10 * *11 * 00 *^"* 01J,*01 * 1 * ot * oo

The arrow star is the last check bit announced by Alice, which is the pointer to the target block data. This is one scenario which enables Alice to carry out the steganography to multi-users. Of course the other users follow similar steps to achieve this goal. The flowchart illustrated in Figures 1A to 1C describes the evolution of the QSP. This technique can be extended to N parties in a straightforward way via state (1). In this case, the users can generate N steganograms. C. Security analysis

In this section the security of the protocol will be discussed in more detail. The protocol has two main processes, which are the QKD and steganography processes, as discussed above. For the QKD there are many possible attacks, such as intercept resend attack (IRA), double control attack (2CNOTA) and the quantum man-in- the-middle. Once the QKD proceeds peacefully, the steganography process can be carried out safely, even if another party (Eve) can watch the communication that is underway. The steganography process is performed over a classical channel.

This analysis begins with the IRA for the control mode of one round of Alice's bits only. Alice has to prepare her particles in one of the states (3). She keeps a particle, say particle number one, and releases particles number 2 and 3 to Bob and Charlie respectively. Mathematically, this means that to perform the IRA one has to trace over particle number one with the density matrix of the states in Alice's work station. Therefore, the transmitted particles from Alice to both Bob and Charlie take either one of the two forms:

= m + infill), 3 = cioi){i£ i + (¾

It is important to note that the one particle-reduced density matrix of the eight elements in (3) is only one of those forms given in (5). This decreases the probability of Eve obtaining information about the QKD. By the control round, Alice sends each particle of (5) to the other users, who return them back to Alice without taking out any action on them. Alice checks if Eve is on the line or not by measuring these qubits. In the IRA, Eve measures both of the travelling qubits in the forward and backward paths in the same bases. For example, suppose that Eve performs projective measurements along the orthogonal bases:

Μ =

, (6) where ⁰ - ¾≤ - The index y indicates that Eve can use one or more different bases for the measurement. In the IRA, the probability that Alice (Bob and Charlie) does not detect Eve on the forward (backward) path equals the probability of obtaining the density matrices prepared initially by Alice. These two partial probabilities are equal and each of which can be expressed as:

where j = 1 , 2 according to (5) and the subscripts "^e£^»«¾c and noE«e_Amean that the probabilities are related to Bob and Charlie (forward path) and Alice (backward path), respectively. The probability that Eve is not detected after a whole control round for the set (3) is:

After straightforward calculation one can easily obtain:

P_n^ = ^{ fees* + *ίπ⁴φ] [«*«φ + Λί ] + i (β,)*^^)}. (9)

It is evident that is symmetric with respect to and θ₂. The maximum (minimum) probability of Eve capturing information about the status of the qubits is 1/16 = 0.0625 (1/32 = 0.03125) at, for example =

This information is clear in Figure 2. From this figure we can see that Eve can always obtain information about the qubits from this attack, however, it is meaningless since it is very small.

With regard to 2CNOT attacks, referring to the literature we note that a 2CNOT attack does not disturb the channels, where li, = i,j = A,B, C,E. In a 2CNOT attack, in the forward path, Eve performs a first CNOT gate between the particles in the transit from

Alice to Bob and to Charlie (control qubits) and her ancillae (target qubits). The second CNOT gate is executed in the backward path. Doing this, Eve may be able to obtain 25% of the information. This is clear since the encoding operation could be, for example, one of the set i&t∞,6Pt& . Accordingly, this indicates that the 2CNOT attack cannot give Eve valuable information about the covert messages.

Turning to the quantum man-in-the-middle attack, generally speaking in cryptography this type of attack is a form of active eavesdropping in which the attacker makes independent connections with the legitimate users and relays messages between them, making them believe that they are talking directly to each other over a private connection, when in fact the entire conversation is controlled by the attacker. The attacker must be able to intercept all of the messages passing between the different users and forward new ones in their place. This attack can succeed only when the eavesdropper can impersonate each of the users to the satisfaction of the other. That is, it is an attack on mutual authentication. Quantum mechanics guarantees that measuring quantum data disturbs that data and this can be used to detect an adversary's interference with a message. According to the invention, Eve should not be able to impersonate Alice or Bob or Charlie since she will be detected based on the laws of quantum mechanics, such as no-cloning theorem, entanglement and so on. Therefore, the users can discover the existence of Eve in step (4). Furthermore, the authentication here is granted by the notion of the entanglement mechanism inherited in the \GHZ)^,

With regard to the security of the steganography process, according to the protocol of the invention no correlation exists between the last check bit and the information bit. Therefore, there is no way for Eve to detect the existence of the hidden channel. This is better than scenarios in which the displacement between the last check bit and the information bit is always the same. In particular, if the legitimate users wish to keep their communications hidden, they need to find a way to randomize the choice of the displacement each time bits are sent. Whilst this may sound difficult, it is important to remember that the users' channel is hidden within a quantum protocol whose sole reason for existence is the production of secret keys. D. Applications

There are many possible applications for quantum steganography, such as authentication, teleportation, communication without encryption and quantum networks. Although the application of the invention is not particularly limited, these applications will be discussed in more detail below for exemplification only.

The first application is authentication, which generally refers to any process by which one verifies that someone is who they claim to be. If we assume that someone impersonates Bob or Charlie or Alice, this may be achieved by replacing one of the legitimate channels with a fake one. In this case, the attacker receives the particles of the other partners. We can make the scenario worse by assuming that the attacker can generate his own sequences from \GHZ and communicate with the other two legitimate users. The legitimate users cannot discover the attacker in step (4) since there is nothing extraordinary in this step. If we agree the attacker does not know anything about the encoding relation (4), the legitimate users can discover the intruder through the decoding process. Explaining this in more detail, we divide the eight elements (3) into two sets:

*+ = {|Α>.|* ,|Λ).|*τ) *- = { W.I*i>, |* ,W»». («¾

If one of Alice's particles is sent from the set *+· , after a complete round Alice expects to get (after encoding using (4)) the message element in *+- In other words, if Alice finds the element in *-. this identifies that one of the users is an eavesdropper and they should abort the protocol.

The second application referred to above is that of teleportation. In quantum teleportation a qubit can be transmitted exactly (in principle) from one location to another, without the qubit being transmitted through the intervening space. Precisely,

Alice would like to send to Bob a qubit, say, I*) - ^ai°> + ^¹)· but she does not know the values of a and b. In this scenario he needs the help of Charlie to perform this task. To explain this in more detail, assume that the users share the following state:

where the subscript 0, 1 , 2, 3 denote the qubit in Alice's work station that needs to be transferred to Bob, Alice's, Bob's and Charlie's particles, respectively. In Alice's site there are two particles, and that shared from \GKZ . |_n this case we assume that the QSP has already run and Alice has then shared with the other partners h¾)- Alice interacts HWwith her particle of the entangled triplet to obtain:

(12)

where I^ are the four standard Bell states. If Alice measures the two qubits in her work station she will obtain one of four possible results of Bell state. Alice will use the steganography to send Charlie and Bob output m. They agree to the scenario:

Charlie measures his own particle, which will be in either one of the two states | ^; _±> = i/V2(|o> ± |i»- Charlie declares the output of his measurement ™i by steganography channel "^Ι' ⁼ ί°·¹ϊ ' {t^+ U'-H-On receiving this information, Bob can perform one of the operations ^<¾^«¾-"»} on his own particle to transform it to ½)■ The results are given in the following table.

As an example, assuming Alice's measurement was 1*- )∞> ^{, e}- ^m - ⁰¹- , this means that Bob and Charlie share the state: ii¾> = i<«|^Q0> - i>Iii»2_ = |μ+}₃(α|0} + fc|i»₂ + !| _}_Ηο> - &li})₂- (a) If Charlie's output is ( W* Bob will apply the operation t on his own particle. By the completion of this process the unknown state has been transferred to Bob's site.

The third application referred to above is secure communication without encryption. Once the users establish the QKD, the procedure becomes crucial in order to secure communication without encryption. The hidden information itself is not encrypted, it is protected from eavesdropping by quantum mechanics. That is, it is protected by the fact that if an eavesdropper wants to know the value of a bit, they have to measure a qubit, thus introducing errors into the sifted bits that are detectable by the users.

The last application mentioned above is quantum communication networks. It is worth mentioning that classically network steganography is already involved in telecommunication networks, where all of the information hiding techniques have been used to exchange secret data (steganograms). This application may also be inspired by the recent development of GHZs to fully connected networks of weakly coupled qubits interacting in the Heisenberg circuit.

Unless the context requires otherwise or specifically stated to the contrary, integers, steps or elements of the invention recited herein as singular integers, steps or elements clearly encompass both singular and plural forms of the recited integers, steps or elements.

Throughout this specification, unless the context requires otherwise, the word "comprise", or variations such as "comprises" or "comprising", will be understood to imply the inclusion of a stated step or element or integer or group of steps or elements or integers, but not the exclusion of any other step or element or integer or group of steps, elements or integers. Thus, in the context of this specification, the term "comprising" is used in an inclusive sense and thus should be understood as meaning "including principally, but not necessarily solely".

It will be appreciated that the foregoing description has been given by way of illustrative example of the invention and that all such modifications and variations thereto as would be apparent to persons of skill in the art are deemed to fall within the broad scope and ambit of the invention as herein set forth.

Claims

1. A method (100) of multi-user steganography based on Greenberger-Horne- Zeilinger (GHZ) states, the method comprising steps of:

establishing secure channels between at least three users; wherein a first user, a second user and a third user perform steps of:

each of said users randomly selecting a sequence of ordered Greenberger-Horne-Zeilinger (GHZ) states having at least three particles (110);

each of said users taking one particle from each of their respective GHZ states to each form at least three subsequences ( 20);

each of said users keeping an initial subsequence in their own site and transmitting a respective one of the remaining subsequences to each of the other users (130); and

each of said users checking the security of established channels by measuring the number of particles d on either side, as agreed between said users (140);

said users transmitting messages between each other by performing the steps of:

each of said users encoding a cover message by applying a unitary transformation on the subsequences sent by the other users (150);

each of said users transmitting the subsequences encoded with their cover message to the other users (160);

each of said users retrieving their respective subsequence in which the cover messages of the other users are encoded (170); each of said users performing GHZs-based measurements on said subsequence having the cover messages of the other users encoded thereon, and comparing measurement outputs with said initial subsequence to obtain the encoded messages (180); and each of said users checking the security of established channels by measuring the number of particles d' on either side, as agreed between said users (190); and said users exchanging stenographic messages (200).

A method according to claim 1 , wherein the step of said users exchanging stenographic messages (200) further comprises steps of:

each of said users selecting a particular number of check bits at random, the number of check bits being known to said users, the last of said check bits being the pointer for the steganography data; and

each of said users announcing their positions publically for the other users.

A method according to claim 1 , wherein each of said users randomly selecting a sequence of ordered Greenberger-Horne-Zeilinger (GHZ) states having at least three particles (110) further comprises steps of:

said first user, A, prepares a sequence of ordered GHZ triplets (110),

A method according to claim 1 , wherein each of said users taking one particle from each of their respective GHZ states to each form at least three subsequences (120) further comprises steps of:

said first user, A, takes one particle from each GHZ triplet to form three ordered sequences (120);

said second and third users, B and C respectively, execute the same steps in their sites to produce further sequences respectively (120); and

each of said first, second and third users exchange the sequences among each other (120) such that they keep in their sites the said sequences respectively, and release the rest to the other users such that said first, second and third users have in their sites the respective sequences.

A method according to claim 1 , wherein each of said users checking the security of established channels by measuring the number of particles on either side, as agreed between said users (140) further comprises steps of:

sacrificing a portion of particles from each of the sequences to check the security of the channels; and encoding message in the qubits of the other users following the Boolean relation agreed on by said users, each of said users acting simultaneously by one operator on the two particles from the different sequences.

6. A method according to claim 5, wherein encoding message in the qubits of the other users following the Boolean relation agreed on by said users, each of said users acting simultaneously by one operator on the two particles from the difference sequences further comprising, said first user, A, has a sequence and said second and third users have corresponding sequences.

7. A method according to claim 1 , wherein each of said users performing GHZs- based measurements on said subsequence having the cover messages of the other users encoded thereon, and comparing measurement outputs with said initial subsequence to obtain the encoded message (180) further comprising steps of each of said users thereby having three cover messages his/her own one and those of the others (180), and wherein said users subsequently use the virtue of the set of particles to establish the security of said channel.