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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

Abstract—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.

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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