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Secret Key Sharing Using Entanglement Swapping and Remote Preparation of Quantum State

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In this paper we propose a new algorithm for secret key sharing by utilizing quantum entanglement swapping and remote preparation of quantum state. This algorithm is used when two parties do not share an Einstein-Podolsky-Rosen (EPR) pair but one wishes to transmit a secret key to the other. In order to successfully accomplish this process, a third party who shares an EPR pair with both parties will help them build a new EPR pair. The new EPR pair will be used between the sender and the receiver to remotely prepare a quantum state. This process will provide a secure way to share secret keys between the two parties who do not share EPR pairs. Furthermore, the process doesn't require sending any physical quantum state, instead the sender prepares a known state and sends only one classical bit to the receiver to help build an intended quantum state.
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Secret Key Sharing Using Entanglement Swapping
and Remote Preparation of Quantum State
Muneer Alshowkan, IEEE Student Member
Department of Computer Science and Engineering
University of Bridgeport University of Bridgeport
Bridgeport, USA
malshowk@bridgeport.edu
Khaled Elleithy, IEEE Senior Member
Department of Computer Science and Engineering
University of Bridgeport University of Bridgeport
Bridgeport, USA
elleithy@bridgeport.edu
AbstractIn this paper we propose a new algorithm for secret
key sharing by utilizing quantum entanglement swapping and
remote preparation of quantum state. This algorithm is used
when two parties do not share an Einstein-Podolsky-Rosen (EPR)
pair but one wishes to transmit a secret key to the other. In order
to successfully accomplish this process, a third party who shares
an EPR pair with both parties will help them build a new EPR
pair. The new EPR pair will be used between the sender and the
receiver to remotely prepare a quantum state. This process will
provide a secure way to share secret keys between the two parties
who do not share EPR pairs. Furthermore, the process doesn’t
require sending any physical quantum state, instead the sender
prepares a known state and sends only one classical bit to the
receiver to help build an intended quantum state.
Keywords- remote preparation; quantum cryptography; EPR
pairs; entangelemt swapping; secret key sharing;
I. INTRODUCTION
Quantum computing and quantum information theory have
been providing promising solutions using quantum
parallelism, teleportation and entanglement to efficiently solve
difficult problems in classical computing [1-4]. Data and
network security are of the most challenges in classical
computing. Providing that, many quantum protocols have
been proposed based on quantum entanglement to improve
and provide more secure systems [5-10]. Moreover, quantum
teleportation depends on quantum entanglement and it is one
of the most important protocols for data transmission.
Quantum teleportation is used to transmit an arbitrary
unknown state from a sender (Alice) to receiver (Bob) with a
spatial distance between them, using a quantum entanglement
channel. However, a classical communication channel
between the sender and the receiver will be required to help in
transmitting and measuring the target state. In fact, in
quantum teleportation the quantum state gets moved to a
remote place while the original state gets eliminated because,
the no-cloning theorem states that it is impossible to copy a
quantum state [11].
Moreover, the teleportation process requires two kinds of
channels; one being the quantum channel and the other the
classical channel. Considering, an eavesdropper could try to
make malicious activities on the transmission path. For this
reason, such path might not be secure for sending and
receiving data.[12, 13]. An interesting algorithm to transmit a
known pure quantum state by taking the advantage of prior
shared entanglement is known as remote state preparation
(RSP). RSP was presented by Lo [14]. Further, RSP is
similar to teleportation as in both algorithms entanglement
state and classical channel are required to successfully send
and receive the quantum state. However, the major difference
between them is in RSP, Alice knows the state she intends to
send to Bob. And the physical state in RSP is not required to
be sent. Where in teleportation, no one knows the state being
transmitted from and the quantum channel will be required to
send the physical state. Furthermore, RSP was proved to be
more economically efficient than teleportation by Pati [15].
Because using teleportation requires Alice to send two
classical bits for each qubit she sends to Bob to help him
figure the state while in RSP it only requires one classical bit
for each qubit.
The trade-off in cost between the classical information
required entanglement and RSP was provided by Bennett et al
[16]. After that, many researchers studied and proposed
different theoretical types of RSP [17-24] . On the other
hand, Peng et al [19] have implement RSP using Nuclear
magnetic resonance and Xiang et al [21] have implemented
RSP using spontaneous parametric down-conversion, single
photon detector and linear optical elements. Additionally,
other RSP methods were proposed using different
entanglement [25].
In this paper, we will use the properties of quantum
systems to provide a secure method to create and share secret
keys between Alice and Bob. We will take the advantage of
entanglement swapping to remotely build an EPR pairs
between the two nodes who do not share a prior entangled
states by the help of the EPR generator. After that, Alice and
Bob will have an entangled EPR pair. Then, using remote
state preparation of quantum state, Alice can prepare a secret
key and send it to Bob using the classical channel by only one
classical bit for each qubit she prepared.
The organization of this paper will be as follows; Quantum
computing preliminaries will be covered in section II, then the
related work will be in section III. After that the proposal
algorithm in section IV. Finally the conclusion and the final
remarks will be covered in section V.
II. QUANTUM COMPUTING PRELIMINARIES
A. Quantum bits
Quantum computing takes the advantages of the laws of
quantum mechanics to efficiently solve the difficult problems
in classical computing. Having the bit as the fundamental
unit in classical computers to represent and store data.
Where, the name of the same unit in quantum computing is
called qubit. The difference between a bit and qubit is that a
bit represents one of two different disjointed states such as a
signal to be high or low, a switch to be on or off or logical
value true or false. However, a qubit can represent one state
or two states simultaneously such as a switch to be on and off
or logical value to be true and false at the same time. The
notation of one qubit is  for zero and  for one. When a
qubit is in both states  and  it state is called a
superposition and it can be represented as a linear combination
of both stats as:  (1)
The coefficients and the coefficient are complex
numbers in Cn and the states  and  are an orthonormal
basis in the two-dimensional vector space. The value
determination in classical and quantum computers are
different. For instance, we can easily examine a classical bit
and determine if it in state 0 or 1. However, in qubits we
examine the coefficients and instead. After measuring a
qubit the result become either 0 with probability or 1 with
probability resulting in:
(2)
Having both probabilities sums to one geometrically
indicates that the qubit state must be normalized to length one
in the two-dimensional vector space.
Two qubits in quantum systems can be represented by four
states using classical bit for instance, 00, 01, 10, 11. At the
other hand, two qubits can be represented by four basis states
denoted by ,,,. Moreover, the two qubits
can also be in a superposition by forming a linear combination
of states with their complex coefficient which often called an
amplitude.
 (3)
After the measurement of this multi qubit state, the result
will be similar to a system with only one qubit, as the
probability of having one of the four states is can be donated
by
B. Quantum gates
Classical systems depends on the wires and the logic gates
in the digital circuits to carry and manipulate the information.
For instance, the NOT gate in classical system perform a
specific operation which is manipulating the stats 0 and 1 by
interchanging their values in which state 0 to be 1 and state 1
to be 0. Similarly, the NOT gate in quantum systems
interchange state  to state  and state  to stae .
 (4)
Moreover, another convenient way to represent quantum
gates is in matrix form. For instance, quantum gates I, X, and
H which represent the Identity, NOT and Hadamard gates
respectively can be represented in term of matrices as:
 
  
 
 
  (5)
C. Quantum Teleportation
Quantum teleportation [7] is a technique of transferring a
quantum state from one location to another with the absence of
physical quantum channel between the sender and the
receiver[26]. However, this process of transferring the state
from one location to another doesn’t conflict with the no-
cloning which states that it is impossible to clone an exact
state without destroying the original state. That means it is
possible to move a state from one location to another but not
copying. Providing that, the teleported state will necessarily
be destroyed
Teleportation uses the EPR pairs which is also called Bell
states and Bell basis to archive its goal. Bell Basis consist of
two entangled qubits in a noncanonical basis:


(6)
The Bell basis or the noncanonical basis consists of four
entangled vectors as follow:

(7)

(8)
By using Bell basis, if Alice would like to teleport a qubit
to Bab and the qubit is in an arbitrary state.
To accomplish the teleportation process Alice perform some
operations denoted in the quantum circuit in Fig 1.
Figure 1. Quantum Teleportation Circuit
After applying the required operations Alice qubits will be
result to one of the four states  which
will indicate the state of Bob’s qubit as follows:
 (9)
 (10)
 (11)
 (12)
Alice will sends to Bob her measurement and depending
on Alice’s qubits Bob will have to fix the state in his
possession by applying one of the quantum gates. Receive
state  will require Bob to apply I gate, receiving state 
will require him to apply X gate, receiving state  will
require him to apply the Z gate and receiving state  will
require Bob to apply X and Z gates which is often called Y
gate.  
  
  (13)
III. RELATED WORK
An enhancement of multiparty quantum secret sharing
(QSS) algorithm [27] was proposed in [28]. The authors
proposed two algorithms taking advantage of the entanglement
swapping operation. The first proposed algorithm requires
the sender to release the encoded classical bits to help the
receiver to deduce intended classical bits from a qubit state.
However, in the second proposed scheme the sender and the
receiver need to physically meet and exchange the classical
bits. However, the new algorithms improve the amount of
data the original QSS protocol transmit by the reducing it
twice. Further, the new algorithms are more efficient in term
of the performance compared to the original QSS. In
addition, a reused scheme was also proposed to reuse some
qubits from previous round in new round.
A protocol for quantum authentication using entanglement
swapping was proposed in [12]. The aim in this paper is to
securely exchange messages between the participating parties.
The proposed protocol provides mutual authentication for the
sender and the receiver when using unsecure routing path.
Further, the authentication protocol depends on four sequence
numbers called Si generated by a third party with the following
functions for each number: Quantum key generation by S1,
eavesdropping detection by S2, identity identification by S3 and
message transferring using S4 . In order to obtain the secret
key, the eavesdropper on the channel need to successfully
break S3. However, the eavesdropped on the routing path
cannot break the entanglement swapping technique and cannot
have access to the controlled qubits.
Network cryptographic protocol based on entanglement
swapping key management center was proposed in [29]. The
goal was to securely distribute the secret keys between parties
with prior sharing of entanglement pairs. However, this
protocol only requires channels between the users and the key
manger center and not between the users themselves. This
protocol preserve the networks resource by only allowing the
physical communication channels between the users and the
key management center and eliminating user-to-user channels.
Also, this protocol performs well even if the users are far away
from each other’s.
Quantum direct communication (QDC) for mutual
authentication based on entanglement swapping was proposed
in [13]. There are two phases in this protocol. First phase is
used to provide mutual authentication and the second phase is
used for direct communication. The identification between
Alice and Bob can be performed by testing the Einstein-
Podolsky-Rosen (EPR) pairs. Moreover, the properties of
entanglement swapping allows Bob to decode Alice’s message
by just performing exclusive-or operation on both of Alice’s
public key and Bob’s measurement. Further, the
authentication process and the direct communication process
are proved to be secure because there is no physical
transmitting of qubits in both operations. The public key for
Alice will consist of two classical bits. Alice will have to
send it to Bob using the public classical channel. However,
that will not reveal any information about the secret key Alice
holds because they are irrelative to each other.
In [30] a study of quantum cryptography was conducted
including in details description of protocol BB84. Also,
described key reconciliation, distillation, security measure and
level of security. Security measure is a probability that
indicates if the distributed key was intercepted or not by
unauthorized third party. Two security measures were
defined as in (14) and (15) where log is the natural logarithm,
k is the number of the compared bits in the public channel and
n is the length of the key. 
(14)

(15)
In J(k) the first 20% of bits have more effect on the result
compared to the last 30% of the bits in the key. And dividing
S(k) by n gives maximum value of 0.1 which is equivalent to
37% of the bits in the key.
Travis Humble discussed securing quantum
communication in the link layer [31]. Besides, describing the
basics of quantum communications and quantum optical
communication. As well as, described the quantum seal Fig.
2 to provide integrity and monitoring to quantum
communication. As illustration, an entangled pair of photons
are created by SPDC and passed through an active and
reference fiber channels. An attempt to change a photon by
an attacker will result in destroying the correlation between
these two photons and will result in losing the entanglement.
On the other hand, Cyber-Physical security is implement using
quantum seal. Detecting any violation will be by setting
threshold stating if the communication is safe or not when the
threshold value will be the result of quantum seal process.
Figure 2. Quantum Seal [31]
Quantum determined key distribution scheme was
proposed in [32] and it is based on quantum teleportation. In
this protocol the sender and the receiver will share
predetermined key by taking the advantage of quantum
teleportation instead of random string as in the other key
distribution protocols. Moreover, because of quantum
mechanics properties, the system will be unconditionally
secure. In fact, the protocol consists of two major steps.
First step, building the shared EPR pairs. Second step,
building the secret key. In the first step Alice create EPR
pairs in state  and share them with Bob.

 (16)
First qubit A will belong to Alice and the second qubit B
will belong to Bob. Then, Bob measures his qubit in one of
three basis. After that Alice and Bob declare the basis they
used in their measurements and compare their results. If both
used different basis they discard the EPR pair. However, if
they find they are many disagreement when they used the
same basis, they can conclude that there is an eavesdropper on
the channel. Building the will be based on quantum
teleportation using the EPR pairs were previously built.
IV. PROPOSED ALGORITHM
Our proposed algorithm is based on two important
algorithms in quantum computing. The first algorithm is
entanglement swapping [33] and the second algorithm is the
remote state preparation [15] In this Algorithm we establish an
EPR-pair between source Alice and destination Bob where
Alice and Bob share EPR-pairs with an intermediate node
called Charlie. Charlie will be acting as a trusted generator
for EPR pair between Alice and Bob. The shared EPR pair
between Alice and Charlie is as follows:

(17)
And the shared EPR pair between Charlie and Bob is as
follows: 
(18)


(19)

 (20)
Applying CNOT to C:

 (21)
Applying Hadamard gate to C in the first EPR-pair:

 (22)
Rearrange and group C:



 (23)
Depending on the result of Charlie’s measurement, Alice and
Bob can build their entangled qubits after applying Pauli-X,
Pauli-Z, both or no gate. For the particles in Alice’s and
Bob’s possessions, the result of the process will be one of the
following EPR pairs:

 (24)

 (25)

 (26)

 (27)
After creating the EPR pair between the Alice and Bob, Alice
can remotely prepare a known quantum state and share it with
the Bob. For example, if Alice wants to transmit a qubit in
pure state:  (28)
And since Alice and Bob share an EPR pair, let’s consider it in
state  from the previous step. The EPR will as follows:

 (29)
As the particle A is related to Alice and particle B is related to
Bob. Now Alice wants to transmit a known state  to Bob.
So Alice can chose to measure the state in any qubit basis such
as which is related to basis  as:
 (30)
Or state  which is related to basis as:
 (31)
Writing the state  with these basis will result in:

 (32)
After Alice applies Von Neumann measurement on single
particle, let’s consider Alice’s particle result to be in
state. Then the total state will be as follows:

 (33)
When Alice sends the measurement result to Bob by sending
only one classical bit, Bob will find the particle in state
. However, when the measurement of Alice’s
particle is  then Bob will find it in state:
 (34)
Which is the complement to the original state.
This method works on any EPR pair result from the
entanglement swapping from the basis .
However, applying Pauli matrices (  will be
required to form the correct state based on the EPR pair used
between Alice and Bob.
I. CONCLUSION
In this paper we presented a secure algorithm based on
entanglement swapping and remote state preparation of
quantum state. Initially, Alice and Bob do not share an
entanglement EPR pairs and ask for Charlie’s help to create
one. After forming the EPR pairs Between Alice and Bob by
Charlie’s help, Alice can prepare a quantum state and then
help Bob to create it. However, before Alice will be able to
help Bob to create the intended state, Alice will need to fully
know the state by measuring it first using one of the bases as
aforementioned. Once Alice becomes fully aware of her state,
she can just send her one classical bit measurement result to
Bob and Bob will have to be able to construct Alice’s state
because it will be the complement of the original state. We
assume Charlie’s system is secure and therefore, it will be
impossible for any third party to manipulate the entanglement
between Alice and Bob because Charlie will process the
entanglement swapping there will not be transmission of any
physical quantum state. Moreover, the one classical bit that
Alice will send to Bob will not reveal any information about
the target state Alice prepared to Bob and also, Alice did not
send any physical quantum to Bob.
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... Entanglement swapping is widely used in many fields such as quantum secret sharing [10][11][12], quantum secure direct communication [13,14], and quantum signature [15]. Before introducing entanglement swapping, we agree on four Bell states as follows: ...
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