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An improved coding method of quantum key distribution protocol based on Fibonacci-valued OAM entangled states

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Hong Lai

a,∗

b

a

, Ming-Xing Luo , Cheng Zhan , Josef Pieprzyk

c ,d

, Mehmet A. Orgun

e ,f

81 82

a

83

College of Computer and Information Science, Southwest University, Chongqing 400715, China b School of Information Science and Technology, Southwest Jiaotong University, Chengdu 610031, China c School of EE&CS, Queensland University of Technology, Brisbane, Australia d Institute of Computer Science, Polish Academy of Sciences, Warsaw, Poland e Department of Computing, Macquarie University, Sydney, NSW 2109, Australia f Faculty of Information Technology, Macau University of Science and Technology, Avenida Wai Long, Taipa 999078, Macau

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a r t i c l e

i n f o

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a b s t r a c t

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Article history: Received 20 March 2017 Received in revised form 6 July 2017 Accepted 11 July 2017 Available online xxxx Communicated by A. Eisfeld

31 32 33 34 35

We propose an improved coding method of quantum key distribution protocol based on a recently proposed (QKD) protocol using Fibonacci-valued OAM entangled states. To be exact, we deﬁne a new class of Fibonacci-matrix coding and Fibonacci-matrix representation and show how they can be used to extend and improve the original protocols. Compared with the original protocols, our protocol not only greatly improves the encoding eﬃciency but has veriﬁability. © 2017 Elsevier B.V. All rights reserved.

Keywords: The encoding eﬃciency Veriﬁability Fibonacci matrices Orbital angular momenta

92 93 94 95 96 97 98 99 100 101

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38 39 40

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1. Introduction

41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59

With the fast development of Internet of things and cloud computing [1–8], the research of quantum cryptography is becoming more and more signiﬁcant. This is because quantum cryptography is the sole cryptography that has been proved to be secure [9,10]. Ekert [11] proposed an application of the entangled states for distribution of secret keys between two parties. His idea turned out to be very fruitful and many quantum key distribution protocols have been published (see [12–17]). However, in these protocols, an entangled photon can carry up to 2 classical bits only. Also, since the photon detectors are chosen at random, one in two trials has to be discarded. That is to say, the eﬃciency ratio of entangled photons is low. To improve the capacity of quantum encoding and the eﬃciency of the protocol, Simon et al. [18] propose to use entangled states whose orbital angular momentum (OAM) values always sum up to Fibonacci numbers, using a golden angle (GA) spiral array with spontaneous parametric down-conversion (SPDC) in a nonlinear crystal. Their protocols have many advantages in that

60 61 62 63 64 65 66

*

Corresponding author. E-mail address: [email protected] (H. Lai).

http://dx.doi.org/10.1016/j.physleta.2017.07.015 0375-9601/© 2017 Elsevier B.V. All rights reserved.

1 4

the coding capacity per photon is doubled and only of the detected entangled photons are discarded. Because the ratio of successive Fibonacci numbers approaches the golden ratio asymptotically, which are helpful in preparing OAM entangled states as they can greatly reduce misattribution errors. Considering the fact that preparation and processing of received entangled photons are very diﬃcult and expensive, there is an urgent need to further improve the eﬃciency of coding. The second related problem is how to reduce the cost of the authenticated classical channel during the ciphertext’s transmission. To address the two above-mentioned problems, we study Fibonacci numbers and ﬁnd that the Fibonacci numbers investigated in [18] can be directly used for generating Fibonacci matrices [19]. Note that Fibonacci matrices have the recursive property. Consequently, we can use Fibonacci matrices for quantum coding. The Fibonacci block-diagonal matrix coding can be used to improve the eﬃciency of coding in Simon et al.’s protocol [18] without changing their experimental setting. Furthermore, we observe that the determinants of Fibonacci block-diagonal matrices can be used to verify whether Alice is honest. The remainder of this paper is organised as follows. In Section 2, Fibonacci Q -Matrices are introduced. Next the Simon et al. QKD protocol and our improved QKD protocol are described in Section 3 and Section 4 respectively. Security analysis of the improved

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Fig. 1. The sketch for quantum key distribution of Simon et al.’s protocol [18], where SPDC stands for spontaneous parametric down-conversion, OAM for orbital angular momenta, and BS is short for beam splitter. The sorters L and D are two different types of OAM sorters. With sorters L and D used, the Fibonacci values carried by the entangled photon and superposition of the form √1 (| F n + | F n−2 ) can be obtained respectively. 2

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The sorter chosen by Alice

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The sorter chosen by Bob Whether the photon can be available for the key establishment

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3. Quantum key distribution protocol based on Fibonacci matrix coding

2. Fibonacci Q -matrices

In this section, we ﬁrst introduce Simon et al.’s quantum key distribution (QKD) protocol [18]. Then based on Simon et al.’s protocol and Fibonacci matrices introduced in Section 2, we propose an improved QKD protocol that achieves higher quantum encoding eﬃciency and veriﬁability.

32 33 34 35 36 37 38 39 40 41 42 43 44 45 46

In this section, we introduce Fibonacci Q -matrices in terms of the results in [19].

49

3.1. Simon et al.’s QKD protocol Fibonacci numbers F n [21] are an inﬁnite sequence of integers deﬁned by the following recursion:

(3)

The main idea behind Simon et al.’s QKD protocol [18] is the use of a Vogel spiral [22] encoding. In this arrangement, either Alice or Bob (or even a third party) can be in charge of a source of entangled Fibonacci-valued orbital angular momentum (OAM) states and their encoding. Fibonacci-valued entangled pairs then leave the spiral and enter the down-conversion crystal. The downconversion breaks each Fibonacci value into two lower OAM values. In both Alice’s and Bob’s laboratories, there is a beam splitter directing some regular proportion of the beam to two different types of the OAM sorters L and D. The beam splitters in their laboratories randomly transmit the entangled photon to either the L or D sorter. The L sorter only allows Fibonacci-valued entangled photons to arrive at the arrays of single-photon detectors. The D sorter is used for allowing “diagonal” superposition of the form √1 (| F n + | F n−2 ), and ﬁltering out any non-Fibonacci-valued en-

Therefore, for Fibonacci numbers 3, 5, 8, 13, 21, 34, 55, 89, their matching Fibonacci matrices can be obtained as follows:

tangled photon (see Fig. 1). There are four possible cases for the sorter that happens in Alice’s (or Bob’s) laboratory. The possible cases are listed in Table 1.

F n +2 = F n +1 + F n , n ≥ 0

52

Q =

55 56

Qn =

57 58 59

60 61 62

63 64 65 66

F2 F1

F1 F0

=

1 1 1 0

(2)

where det( Q 1 ) = F 0 F 2 − F 12 = −1. Using Equation (1), we can compute the nth power of the Fibonacci matrix Q as follows

53 54

(1)

where the ﬁrst two elements of the sequence are F 0 = 0 and F 1 = 1. Taking the ﬁrst three integers F 0 , F 1 , F 2 of the Fibonacci sequence, we can construct a 2 × 2 Fibonacci matrix:

50 51

90 92 93

5 3 3 2

F n +1 Fn

Fn

21 13 13 8 34 21 21 13

F n −1

4

=Q ,

8 5 5 3

5

=Q ,

13 8 8 5

144 89 89 55

=Q ,

= Q 11 .

96 97 98 99

55 34 34 21

9

=Q ,

101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122

=Q ,

89 55 55 34

=Q

123 124

(4)

8

2

3.2. Improved coding method of Simon et al.’s QKD protocol

6

= Q 7,

95

100

2.1. Fibonacci matrices

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protocol is given in Section 5. Section 6 concludes the work with a brief summary of our contributions.

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Table 1 The possible outcomes by the sorters chosen by Alice and Bob.

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10

, (5)

With the experimental setup of Simon et al.’s QKD protocol, we propose an improved coding method to enhance the information capacity of every entangled photon. Our protocol has the following four stages: initialisation, key distribution, key establishment and key storage. We assume that there exist authenticated classical channels and insecure quantum channels between Alice and Bob. We follow Simon et al.’s QKD protocol [18] and use their method to produce the Fibonacci matrices.

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Table 2 Possible classical information exchange of our protocol.

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Fibonacci obtained by Alice

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The classical bits sent by Alice

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Table 3 The possible outcomes obtained by Eve in terms of the eavesdropped classical information.

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01

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The possible outcomes obtained by Eve

3, 21, 34, 89

3, 5, 13, 21

8, 55, 89

5, 13, 34, 55

76

⎛

12 14 15

3.2.1. Initialisation In our protocol, there are two honest participants Alice and Bob deﬁned as follows.

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1) Alice is a sender, who is in charge of preparing entangled states. 2) Bob is a receiver, who detects the entangled photons obtained from Alice.

⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

23 24 25

3.2.2. Key distribution In the key distribution phase, we use the same experimental setup and the eight Fibonacci numbers (3, 5, 8, 13, 21, 34, 55, 89) as in Simon et al.’s protocol [18]. The detailed steps are as follows:

26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55

Step 1. We assume that the quantum key distribution method is the same as that in Section 3.1. Alice prepares m pairs of entangled states with a source of entangled Fibonacci-valued OAM states based on a Vogel spiral. The m pairs of entangled states are in the following form

(| F n−1 | F n−2 + | F n−2 | F n−1 ) A B .

(6)

n

The ﬁrst entangled photons from each pair as per Equation (6) constitute S A , and the second entangled photons from each pair as per Equation (6) constitute S B . Alice keeps S A and sends S B to Bob. Step 2. On receiving S B , Bob randomly chooses the sorters L and D and measures the values carried by every photons in S B . Step 3. Alice randomly chooses the sorters L and D and measures the values carried by every photons in S A . Step 4. Alice transmits a sequence of bits to Bob. The communication is illustrated in Table 2. On receiving a classical bit from Alice, Bob can determine the Alice value, because the value is one of the two values adjacent to his value. Once Bob conﬁrms that Alice’s value is even, he transmits a bit to Alice. If Alice’s value is odd, Bob takes the conjugate protocol with zeros and ones swapped. Likewise, after receiving the classical information, Alice also can conﬁrm Bob’s value. That is to say, Alice and Bob are able to convince each other about the detected values by exchanging classical information. However, for an eavesdropper Eve, the classical information is insuﬃcient to identify the detected values as she obtains ambiguous outcomes only (see Table 3).

8 5 5 3

O

58 59 60 61 62 63 64 65 66

3.2.3. Key establishment Alice and Bob can establish a common key which consists of a series of block-diagonal Fibonacci matrices, by knowing the Fibonacci values in the key distribution stage. To be exact, both Alice and Bob establish the block-diagonal Fibonacci matrices in terms of the detected results each round including eight Fibonacci numbers (3, 5, 8, 13, 21, 34, 55, 89). If the available detected Fibonacci values are 5, 13, 21, 34 in the ﬁrst round, the block-diagonal Fibonacci matrix can be constructed as follows with the seeds 5, 13, 21, 34:

O

78

O

⎟ ⎟ ⎟ ⎟ 21 13 ⎟ O O ⎟ 13 8 ⎟ ⎟ 34 21 ⎟ O O ⎟ 21 13 ⎟ ⎟ 55 34 ⎠

O O

O

O

79 80 81 82

(7)

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

5 3 3 2

85 86

34 21

87

⎞ O

O

13 8 8 5

O

O

O

O

O

O

O

O

O

88 89 90 91 92 93

O

⎟ ⎟ ⎟ ⎟ O O O ⎟ ⎟ ⎟ ⎟ 21 13 ⎟ O O ⎟ 13 8 ⎟ ⎟ ⎟ 34 21 ⎟ O O ⎟ 21 13 ⎟ 144 89 ⎠ O

83 84

where O is a matrix of the dimension 2 × 2 with zero entries. If the available detected Fibonacci values are 3, 8, 13, 21, 89 in the second round, the block-diagonal Fibonacci matrix can be constructed as follows with the seeds 3, 8, 13, 21, 89:

94 95 96 97 98

(8)

99 100 101 102 103 104

89 55

105

The same method can be used in n round to construct the block-diagonal Fibonacci matrices. Moreover, the determinants of the elements (i.e., Fibonacci matrices of the dimension 2 × 2) of the principal block-diagonal matrices are 1 or −1. Therefore, the Fibonacci block-diagonal matrices are invertible.

106 107 108 109 110 111

3.2.4. Key storage Alice and Bob can store the established key using the Fibonacci matrix representation that minimises the storage cost. The details are as follows. Fibonacci matrix representation. Let us consider the representation of a positive integers abcd · · · with Fibonacci matrices as follows:

112

119

a b c d

=

11

(9)

8 5 5 3

+ 4

114 115 116 117 118

121 122

where ai ∈ {0, 1}, i = 4, 5, · · · , 11, and b = c , d = a − b. Since there are eight different matrices that can be produced, the elements of the principal block-diagonal of Equations (7) and (8) can be recoded using Equation (9) as shown below

113

120

ai Q i

i =4

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⎞

O

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The classical information obtained by Eve

11 13

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+

5

34 21 21 13

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+ 7

55 34 34 21 8

=0· Q +1· Q +0· Q +1· Q +1· Q +1· Q + 0 · Q 10 + 0 · Q 11

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(10)

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Table 4 The comparison of features comparisons among [11,12,18] and our protocol.

2

5 3 3 2

+

+

13 8 8 5

144 89 89 55

+

[11]

[12]

[18]

Our protocol

Coding way Veriﬁability The Fibonacci matrix representation Anti-cheating The authenticated channel to transmit ciphertext The public channel to transmit ciphertext

Binary No No No Yes No

Binary No No No Yes No

Binary No No No Yes No

Digital matrix Yes Yes Yes No Yes

21 13 13 8

+

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+0· Q +0· Q

10

+1· Q

11

(11)

Hence, Alice and Bob store the elements of the principal blockdiagonal of Equations (7) and (8) as binary strings 01011100 and 10111001, respectively.

26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55

As we can see Fibonacci numbers that are identiﬁed in Simon et al.’s protocol, can be used as seeds to produce matching Fibonacci matrices and further as the elements of the matching principal block-diagonal matrix. The method can be used to achieve quantum digital encryption and decryption, instead of usual quantum binary encryption and decryption, which is implemented and optimised in modern computers. This greatly reduces the number of used photons. It turns out that the application of the Fibonacci matrix representation cannot increase the storage cost compared with Simon et al.’s protocol [18]. The details are given below.

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We apply Fibonacci block-diagonal matrices and the Fibonaccimatrix representation to Simon et al.’s QKD protocol, to achieve veriﬁability and eﬃcient QKD. Compared with Simon et al.’s protocol, our protocol needs much smaller number of photons, the channel to transmit ciphertext can be public, and it can verify whether Alice is honest. Consequently, our protocol is better for implementation using the current technology.

73 74 75 77 79 80 81 82 83 84 85 87

4.3. Veriﬁability In QKD protocols, the determinants of the elements (i.e. Fibonacci matrices of the dimension 2 × 2) of the principal blockdiagonal matrices are 1 or −1, which can be used to test the received information. Furthermore, it is anti-cheating. Most importantly, the channel to transmit ciphertext can be the public channel, rather than the authenticated channel in existing QKD protocols [13–18]. Table 4 compares our protocol with other QKD protocols [11,12,18].

91 92 93 94 95 97 99 100 101

Hong Lai has been supported by the Fundamental Research Funds for the Central Universities (No. XDJK2016C043), the Doctoral Program of Higher Education (No. SWU115091), and the ﬁnancial support in part by the 1000-Plan of Chongqing by Southwest University (No. SWU116007). Mingxing Luo is supported by Sichuan Youth Science & Technique Foundation (No. 2017JQ0048). Josef Pieprzyk has been supported by National Science Centre, Poland, project registration number UMO-2014/15/B/ST6/05130.

In QKD protocols, the established key needs to be stored for encryption and decryption. The Fibonacci matrix representation introduced in the paper allows us to enhances the eﬃciency of the coding without increasing storage requirements. For instance, to store matrices given by Equations (7) and (8), we also only need 8 bits as Simon’s at al.’s protocol require 39 and 26 bits respectively.

90

98

Competing Financial Interests: The authors declare no competing ﬁnancial interests.

We use Fibonacci block-diagonal matrices to encode the same Fibonacci numbers obtained through the detected entangled photons as in Simon et al.’s protocol [18]. Note that in Simon et al.’s protocol [18], every of the eight Fibonacci numbers can be used to encode 3-bit binary string from 000 to 111. That is to say, in each round, at most 24-bit message can be encrypted. While, in our proposed protocol, the message can be directly divided into blocks and constitutes digital matrices in terms of the corresponding Fibonacci block-diagonal matrices, greatly enhances the eﬃciency of the coding. 4.2. Unchanged storage cost

89

96

Additional information

Acknowledgements

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6. Conclusion

4.1. Fibonacci block-diagonal matrix coding

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4. Features of our protocol

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Our QKD protocol is based on EPR pairs, so, the proof of security is similar to those in the reported works [11,12] with entangled photons. Moreover, we are only changing the ways of coding and information storage, with the quantum part of Simon et al.’s protocols unchanged, so the security of our protocol is the same as that in Simon et al.’s protocol [23]. For the detailed analysis please refer to the work [23].

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5. Security analysis

=1· Q4+0· Q5+1· Q6+1· Q7+1· Q8 9

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Protocols

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