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QKD Protocol Based on Entangled States

By Trusted Third Party

Abdulbast A. Abushgra Khaled M. Elleithy

Computer Science & Engineering Department Computer Science & Engineering Department

Unversity of Bridgeport Unversity of Bridgeport

USA USA

aabushgr@my.bridgeport.edu Elleithy@bridgeport.edu

Abstract— Quantum cryptography is considered a solution

for sharing secret information in a secure mode. Establishing a

quantum security platform into an exciting system requires a

package of stable processes. One of these processes is based on

creating a Quantum Key Distribution (QKD) protocol or sharing

a secret key. This paper presents a QKD protocol that utilizes

two quantum channels to prepare a shared secret key. The first

communication channel will be initiated by entanglement states,

where the entangled photons will be emitted by a trusted third

party. The second communication channel utilizes the

superposition states that will be initiated by the one of the

communicated parties. Moreover, the protocol produces a string

of random qubits after verifying the communicated legitimate

parties during entangled state channels. The produced string will

reflect the shared secret key between the users.

Keywords- Entangled State, Superposition State, Qubits, Decoy

State, and Bell’s States.

I. INTRODUCTION

Flowing enormous data through various communication

channels causes leaks of important information through

classical communications by eavesdroppers. Classical

cryptography has several algorithms that defend against many

information attacks (these algorithms are still secure as long as

the quantum computer is conceptual). Furthermore, quantum

cryptography provides security of information with some

challenges that are determined in quantum attacks or natural

noise. In 1984, Charles Bennett and Gilles Brassard invented

[1] the most sparkling quantum key distribution protocol,

which is called BB84 protocol. Several QKD protocols then

were invented (such as B92 protocol [2], SARG04 protocol [3],

EPR protocol [4], and DPS protocol [5]).

Any quantum key distribution protocol technically uses

different channels to submit qubits (Quantum Bits) for data

transmission, with and regular bits for either confirming or

reconciling the submitted qubits. Each quantum channel is

initiated in varying environments that specifies the type of

platforms and used tools (such as transformers and detectors).

First of all, the quantum channel should utilize either Viper-

Optics or Free-Space to transfer a qubit from one side to

another; both cannot be protected totally from eavesdroppers.

The quantum mechanics is the only factor that makes quantum

communication unconditionally secure [6]. Moreover, the rules

of physics keep the whole system that is used active (as long as

no attempts to break the system). Therefore, any illegal alien

will be detected by destroying the system.

Furthermore, using multiple polarized states of a particle

and the measurement process of the same particle determine

the stability and efficiency of each QKD protocol. Fulfilling an

authentication between two or more communicators is one of

the challenges that cause an enormous leak of information if

the communicators cannot verify each other. This paper

presents a new algorithm that is designed to prove the

authentication within an entangled channel. The presented

protocol is based upon two quantum channels: one channel is

EPR channel (entangled states channel) and second channel is

quantum channel (qubits channel). The protocol will be

terminated in case the authentication between the

communicated parties is interrupted.

II. THE PROPOSED PROTOCOL MECHANISM

A. The EPR Preparation

Initiating an EPR connection should be done by submitting

EPR photons to the receiver (Bob). The source of EPR photon

would be from the sender (Alice) or a third party; but in this

proposed protocol, the third party will be confirmed. The

submitted EPR string S

EPR

contains several characters, which

are considered an open key for the whole scheme. These

characters involve a sequence of information (such as initiation

time t

1

, number of matrices n (if any), matrix size m, parity

diagonal p, state dimension s, matrix indices R, and termination

time t

2

) as in figure (1).

Fig. 1 The EPR string prepared by the sender.

The sender (Alice) is supposed to start talking with the third

party by sending a copy of the plaintext into a classical

2

channel. Next, the trusted third party will convert the plaintext

to encoded information

to be transferred into

entanglement states. Both of the communicated parties (the

sender and receiver) will receive a copy of the entangled

photons at the same time. The EPR string S

EPR

is the encoded

plaintext that will be shared between the sender and receiver.

The string contains particles of Pauli states (

,

,

)[7].

Each photon has two states|, where | should be sent to

Alice and the | will be sent to Bob. Based upon the

theoretical measurement and the fact of EPR photons, both

parties can initiate the communication in safe mode.

=

(1)

|EP

R

=1

√2(|00

±|11

)

(2)

|EP

R

=1

√2(|01

±|10

)

(3)

|=|0

±|1

(4)

Fig. 2 The communication between the third party with Alice and

Bob.

B. The Qubits Preparation

To create a secret (shared) key, Alice is supposed to know

the information that will be submitted to the other party. The

plaintext should be converted to qubits (data), and the third

party then sets up the converted plaintext into a designed

matrix. The matrix matches the length of the plaintext n as

follows:

= log

,

(5)

where DM is the size of the used matrix and n is the length

of the converted plaintext.

Next, the third party will fill up the lower and upper

triangles (the diagonal line is not included) by the converted

qubits of the plaintext. The filling scenario starts from up to

down in the lower triangle and from down to up in the upper

triangle, as shown in figure (2). The whole matrix will be filled

as a result except the diagonal line, where the third party

adjusts the diagonal cells based on the summation of each row.

If the summation of the row is odd, the third party will add (1)

to the empty cell to make the row even. On the other hand, if

the summation of the row is even, it will be added (0) bit to the

cell. Therefore, the third party prepares the whole matrix with

even row’s summation; this will be an extra protection against

PNS attacks [8], where Alice and Bob will know if the

upcoming qubits were interrupted by eavesdroppers or

environment.

Fig. 3 The prepared matrix into three sections: lower triangle, upper

triangle, and diagonal line.

III. COMMUNICATION CHANNELS

A. EPR Channel

In 1935 [9], Einstein, Podolosky, and Rosen came up with

their fabulous paper that opened a huge argument about the

wave function and incompleteness of quantum mechanics. The

main concept of EPR is a photon submission from the source

(X) to two different destinations (e

1

, e

2

). The measurement, in

the case of no interruption, will demonstrate a different state at

each side. Moreover, if Alice (one of the communicators or the

sender) received|0, then Bob (one of the communicators or

the receiver) should have |1after his measurement. The

presented algorithm is initiated by creating an EPR channel and

the protocol will be described as follows:

• Alice sends n bits of the plaintext (the length of the

plaintext) to a third party.

• The third party converts the plaintext to EPR states

(|Φ,|Ψ) based on the plaintext, and then sends the

EPR states into separate channels (where one state is

sent to Alice EPR

A

and the other state is sent to Bob

EPR

B

).

• Alice creates an unknown photon (e.g.|=

|0+

|1), which is in the superposition state.

3

• Calculating both the entangled state and

superposition state (|Ψ

⨁|

) to produce a three-

dimension particle state.

• Alice separates the three states, where | will

become||.

• The first outcome of | becomes entangled and |

is separated (or became in superposition).

• Alice submits two classical bits

(|00,|01,|10,|11) for the used gates at both

sides.

Fig. 4 The photon emits from the source, and the measurement

will be same color if one side measured.

Therefore, the authentication between the communicated

parties should either be approved to move on or to start over.

After that, Bob should have the proper quantum gates ⨁ as

well as the photon states.

Algorithm .1 QKD Protocol

1. Submit n bits to well-known third party (p) by A

2. n (|Ψ|Φ) // First loop

3.

|Ψ|0

|1 // P sent a pair to both A&B

4. if (A == 0) then (B == 1) // Second loop

5. B 1

6. else: error

7. end; // ending the loop

8. A |

9. for: 1 n //Measuring & reconciliation

10. (|Ψ⊕|) // Third loop

11. end; //use the data collected by EPR

12. B {0,1} // B gets the secret key

The proposed algorithm runs through three loops that are

involved in submitting a plaintext to a third party, initiating an

EPR connection by the third party, and the quantum

communications between the sender and receiver.

B. The Classical Communication

To ensure that Bob has the right quantum gates (as in figure

(5)) Alice initiates a communication into a classical channel.

Two bits have the needed information that Alice should send to

Bob. Each two bit has a meaning of a certain quantum gate; the

(00) bits mean using the unitary operator, (01) Z gate, (10) X

gate, and (11) X and Z gates. These gates are the only classical

operation that Alice and Bob need to use during the entire

system procedures.

Fig. 5 The three quantum gates (X, Y, and Z) used into

exchanging channel.

00−−−−→

01−−−−→

10−−−−→

11−−−−→&

Moreover, interrupting the classical communication will not

impact the protocol processes because the receiver will get

unmatched qubits during the preparation of the upcoming

qubits. Also, the decoy states (diagonal line) will show some

huge variations.

C. Quantum Channel

After an authentication proof, both parties start exchanging

qubits (data) into the quantum channel. The submitted qubits

will be in two bases (|×,|+) and four states

(|0,|45,|90,|135). Alice creates the qubits based on

the EPR

A

that was submitted by the third party; and Bob will

use Pauli-matrices with prior knowledge to measure the

upcoming qubits from Alice into the right states [10]:

=

01

10

,

(6)

=

0−

0

,

(7)

=

10

0−1

.

(8)

The physical measurements should all be correct because

Bob has already agreed on the EPR

B

confirmation. Moreover,

the mechanism of data organization into a matrix setup will

assist to protect qubits from any quantum attack. On the other

hand, Bob can realize any changes in the received qubits and he

can figure out the error by diagonal decoy states.

4

Fig. 6 The whole mechanism for the proposed scheme in two quantum

channels.

IV. THE PROPOSED PROTOCOL SIMULATIONS

A. The Runtime-Execution

To test the simplicity of the proposed protocol, it was

simulated technically by measuring the run time execution

during the generation of a secret key by two legitimate parties.

The simulation is considered a test of the time taken from

initiation the communication to generation of the secret key.

Even the loops that were required for some function will be

included, as well as the reconciliation phase. The following

equation will simply explain the calculation of the run time

execution:

=

(9)

where P is the required loop for each function process

into the entire algorithm initiation.

The proposed protocol runs in a low time rate if there is no

error created by eavesdropper. On the other hand, applying an

error during the communications between the legal parties will

increase the rate of time taken to create a secret key.

B. The Efficiency

Based upon the measurements that were applied on the

proposed protocol, the efficiency can be approved by

measuring the Qubit Error Rate (QBER). The total of used

qubits at the beginning of the communication will be different

at the end for many reasons. The environment is one reason

that causes a qubit drop or weak light. Quantum attacks can

also cause several damages to the submitted qubits, either by

splitting the state of the photon or by interrupting and

resending a photon.

The efficiency measurement was applied by counting the

QBER, where correcting errors should be realized by the

following equation [11, 12]:

=

(10)

where n is the total of the submitted qubits, and r is the

qubits that were measured and successfully uncovered. The

results show the proposed protocol is efficient even if the

quantum attacks are applied. Therefore, there is no leaked

information even if the eavesdropper tried to use one of the

attacks scenarios.

Fig. 7 The correlation between submitted and received qubits

measured with 50 qubits.

The correlation in the figure (7) between the submitted and

received qubits reflects the difficulties of finding out the

relation between the two parties. Hence, the main point is

utilizing a matrix either in sorting submitted qubits or re-

sorting received qubits; this usually is considered as an

advantage to hide the core of a created secret key.

C. The Security

The security measurement is applied by several methods, but

this proposed protocol utilizes Shannon Entropy [13, 14] to

measure the level of security. The probability in the next

equation shows the rate of corrupted qubits of the received

qubits:

=−

log

(11)

where P

i

is the probability of the shown character (certain

qubits) in i numbers.

The security measurement can be applied into the entropy

of security in general, where it can measure the rate of

uncovered qubits .

()=−

×log(

)

(12)

where log represents the natural logarithm (the

logarithm with the base e). The constant e is called Euler’s

number and it is equal to an approximately: ≈ 2.71828

[15]. Moreover, k is the uncovered qubits that should be

5

measured by Bob and n is the total of qubits that are submitted

by Alice.

Fig. 8 The entropy of security measured for the proposed protocol

that confirmed by a third party.

The figure (8) demonstrates the S(k) function to calculate the

entropy of security, where the used key length is 32 qubits.

The rate of uncovered qubits will be approx. 0.53 qubits of the

secret key.

I. CONCLUSION

The proposed scheme presents a quantum key distribution

protocol that is essentially designed in two quantum channels.

The EPR channel (confirmation channel) uses the entangled

states rather than states in superposition, which has a low risk

and certain probability. The second channel is utilized to

transfer data from sender to receiver with the ability to detect

any interruption. Generally, the proposed scheme treats

missing authentication between legitimate parties in most

well-known quantum key distribution protocols. Also, it uses

qubit preparation in matrix (or matrices if any) by the sender,

which is considered a powerful procedure to ignore PNS and

IRA attacks. The proposed scheme has approved its stability

against the Man-In-The-Middle attack, where there is no

chance to impersonate the sender or the receiver.

V. REFERNCES

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Public Key Distribution and Coin Tossing "

International Conference on Computers, Systems &

Signal Processing, p. 5, December 10 - 12, 1984

1984.

[2] C. H. Bennett, "Quantum cryptography using any two

nonorthogonal states," Physical Review Letters, vol.

68, p. 3121, 1992.

[3] V. Scarani, A. Acin, G. Ribordy, and N. Gisin,

"Quantum cryptography protocols robust against

photon number splitting attacks for weak laser pulse

implementations," Physical Review Letters, vol. 92,

p. 057901, 2004.

[4] A. Einstein, B. Podolsky, and N. Rosen, "Can

quantum-mechanical description of physical reality

be considered complete?," Physical review, vol. 47,

p. 777, 1935.

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phase-shift quantum key distribution using coherent

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27 2003.

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IEEE Long Island Systems, Applications and

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[11] D. Gottesman, L. Hoi-Kwong, Lu, x, N. tkenhaus,

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