QKD Protocol Based on Entangled States By Trusted

QKD Protocol Based on Entangled States By Trusted Third Party Abdulbast A. Abushgra Computer Science & Engineering Department Unversity of Bridgeport USA [email protected]

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 ViperOptics 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

Khaled M. Elleithy Computer Science & Engineering Department Unversity of Bridgeport USA [email protected]

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 SEPR contains several characters, which are considered an open key for the whole scheme. These characters involve a sequence of information (such as initiation time t1, number of matrices n (if any), matrix size m, parity diagonal p, state dimension s, matrix indices R, and termination time t2) 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

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.

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 SEPR is the encoded plaintext that will be shared between the sender and receiver. , )[7]. The string contains particles of Pauli states ( , 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)

=

|EPR

=

|EPR

=

|

1 √2 1 √2

(|00

± |11

)

(2)

(|01

± |10

)

(3)

(4)

= |0 ± |1

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 (e1, e2). 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 |1 after his measurement. The presented algorithm is initiated by creating an EPR channel and the protocol will be described as follows:

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. •

2

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 EPRA and the other state is sent to Bob EPRB). Alice creates an unknown photon (e.g. | = |0 + |1 ), which is in the superposition state.

•

• • •

Calculating both the entangled state and superposition state (|Ψ ⨁ | ) to produce a threedimension 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. 5 The three quantum gates (X, Y, and Z) used into exchanging channel.

00 − − − −→



01 − − − −→ 10 − − − −→ Fig. 4 The photon emits from the source, and the measurement will be same color if one side measured.

11 − − − −→ &

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.

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

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.

Algorithm .1 QKD Protocol Submit n bits to well-known third party (p) by A n  (|Ψ |Φ ) // First loop |0 |Ψ // P sent a pair to both A&B |1 if (A == 0) then (B == 1) // Second loop B 1 else: error end; // ending the loop A| for: 1  n //Measuring & reconciliation (|Ψ ⊕ | ) // Third loop end; //use the data collected by EPR B  {0,1} // B gets the secret key

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 EPRA 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]:

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

=

0 1 , 1 0

(6)

=

0 − , 0

(7)

=

1 0 . 0 −1

(8)

The physical measurements should all be correct because Bob has already agreed on the EPRB 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.

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.

3

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. 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:

=

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 resorting received qubits; this usually is considered as an advantage to hide the core of a created secret key.

(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.

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:

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]:

=

=−

log

(11)

where Pi 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)

(10) 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 4

[3]

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

[4]

[5]

[6] [7] Fig. 8 The entropy of security measured for the proposed protocol that confirmed by a third party.

[8]

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.

[9] [10]

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. [1]

[2]

[11]

[12]

[13] [14]

REFERNCES

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[15]

5

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