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ORIGINAL RESEARCH article

Front. Phys., 19 June 2023
Sec. Quantum Engineering and Technology
This article is part of the Research Topic Multiparty Quantum Communication and Quantum Cryptography View all 7 articles

Quantitative security analysis of three-level unitary operations in quantum secret sharing without entanglement

Juan Xu,
Juan Xu1,2*Xi LiXi Li3Yunguang Han,Yunguang Han1,2Yuqian Zhou,Yuqian Zhou1,2Zhihao Liu,Zhihao Liu3,4Zhengye ZhangZhengye Zhang1Yinxiu SongYinxiu Song1
  • 1College of Computer Science and Technology, Nanjing University of Aeronautics and Astronautics, Nanjing, China
  • 2Collaborative Innovation Center of Novel Software Technology and Industrialization, Nanjing, China
  • 3School of Computer Science and Engineering, Southeast University, Nanjing, China
  • 4Key Laboratory of Computer Network and Information Integration, Ministry of Education, Southeast University, Nanjing, China

Quantum secret sharing (QSS) protocols without entanglement have showed high security by virtue of the characteristics of quantum mechanics. However, it is still a challenge to compare the security of such protocols depending on quantitative security analysis. Based on our previous security analysis work on protocols using single qubits and two-level unitary operations, QSS protocols with single qutrits and three-level unitary operations are considered in this paper. Under the Bell-state attack we propose, the quantitative security analyses according to different three-level unitary operations are provided respectively in the one-step and two-step situations. Finally, important conclusions are drawn for designing and implementing such QSS protocols. The method and results may also contribute to analyze the security of other high-level quantum cryptography schemes based on unitary operations.

1 Introduction

Quantum secret sharing (QSS) is an important branch of quantum cryptography, whose security is based on the fundamental principle of quantum mechanics. It can share both the classical message [111] and quantum information [1222]. Taking QSS protocols for classical message sharing into account, it is observed that it is a more efficient and lower cost way to use single particles instead of entangled particles. In addition, to achieve the unconditional security (in theory) and the ability of detecting eavesdropping, unitary operations are always used in such protocols. The classical secret message is encoded into quanta and/or scrambles the particles by unitary operations so that the eavesdropper cannot reliably identify each quantum state by appropriate measurement.

The quantitative security analysis of such protocols can be transformed into a quantitative calculation of unitary operation security. The feasibility of this idea has been proven by our pioneering work [23]. In [23], we proposed the substitute-Bell-state attack and the definition of minimum failure probability for the attack. Thus, the quantitative security analysis can be conducted according to the different selection methods of unitary operations. Several cases of two-level unitary operations were analyzed in [23] and more two-level cases are considered in our recent research [24].

However, these works on the security analysis of QSS are focused on the qubit system. In many scenarios, quantum communication protocols using high-dimensional states demonstrate larger capacity and better performance than the qubit system. Thus, it is worth studying the security of QSS protocols beyond the two-level system. Quantitative security analysis of QSS protocols remains a tough problem, and our pioneering idea is worth recommending. However, the research work based on this idea is not sufficient. In this paper, the security of QSS protocols using three-level unitary operations is analyzed quantitatively, and some comparisons between two-level and three-level situations are made. Finally, valuable conclusions are obtained, which can guide the designing and implementation of QSS protocols with single qutrits.

2 The security of unitary operations in quantum secret sharing protocols based on qutrits

2.1 Three-level Bell states and unitary operations

In a three-level quantum system, we call a single quantum state as a qutrit. The Z-basis of a three-level quantum system is 0,1,2, and another basis can be constructed by a quantum Fourier transform, which is called the X-basis. For an arbitrary d-level quantum system, we have

Xk=1dj=0d1e2πijk/dj=1dj=0d1ϖjkj,(1)

where 0kd1,ϖ=e2πi/d. So when d=3, the X-basis can be formed as follows:

{x0=130+1+2,x1=130+w1+w22,x2=130+w21+w2},(2)

where w=e2πi/3.

To analyze the security of QSS protocols based on qutrits quantitatively, we design the Bell-state attack, which uses three-level Bell states and unitary operations. In a d×d-level double quantum system, the d-level Bell state can be denoted as

ψnm=1dj=0d1wjnjj+m,(3)

where w=e2πi/3, d2, 0n,md1, and the symbol + means module d plus. Obviously, there are d2 d-level Bell states. In addition, the vector group ψnm constitutes a complete orthogonal basis of the double quantum system. Thus, nine Bell states forming a complete orthogonal basis in a three-level quantum system can be specified as

ψ00=1300+11+22,ψ01=1301+12+20,ψ02=1302+10+21,ψ10=13w00+w211+22,ψ11=13w01+w212+20,ψ12=13w02+w210+21,ψ20=13w200+w11+22,ψ21=13w201+w12+20,ψ22=13w202+w10+21.(4)

Without loss of generality, the first Bell state ψ00=1/300+11+22 is chosen to be the initial state that the eavesdropper uses for a Bell-state attack. Furthermore, an arbitrary three-level double quantum state can be expressed as follows:

ab=x1ψ00+x2ψ10+x3ψ20+y1ψ01+y2ψ11+y3ψ21+z1ψ02+z2ψ12+z3ψ22,(5)

where xi,yi,ziCi=1,2,3.

A unitary operation is a bounded linear operation U:HH on a Hilbert space H that satisfies UU*=I, where U* is the adjoint of U and I:HH is the identity operation. In a three-level system, the unitary operations can be represented by a 3*3 matrix. For simplicity, we consider the double quantum states consisting of three basis states, that is, one of x1, x2, and x3; y1, y2, and y3; or z1, z2, and z3 is 1 (see Formula 5). Here, xi=1 is bound to yi=0 and zi=0 because the two-qutrit quantum state is obtained under a Bell-state attack and selected unitary operation (see Section 2.3). Therefore, there will be six situations showed as follows (if the unitary matrix at the left side of the arrow is supposed to be used in a Bell-state attack, three Bell states will be obtained at the right side of the arrow):

U01=131w1w11w2w21ψ00,ψ11,ψ22,U02=131w21w211ww1ψ00,ψ12,ψ21,U03=13ww11w21w211ψ10,ψ01,ψ22,U04=13w11w2w211w1ψ10,ψ02,ψ21,U05=13w2w211w1w11ψ20,ψ01,ψ12,U06=13w211ww11w21ψ20,ψ02,ψ11.(6)

Without the loss of generality, we select the first situation as an example for research.

Next, to measure the security of different unitary operations, we first provide a general QSS protocol based on qutrits, and then propose the so-called Bell-state attack and state the minimum failure probability for this attack. The quantitative computing results show how to select appropriate three-level unitary operations to protect the QSS protocols against a Bell-state attack.

2.2 A general quantum secret sharing protocol based on qutrits

A generic QSS protocol based on qutrits is shown in Figure 1 (it is supposed to be a (n, n) secret sharing threshold protocol). The procedure is as follows:

(1) First, Alice prepares qutrits all in 0. A binary string A0 is the secret to be shared. Alice performs unitary operations on qutrits according to the value of A0 and a random string B0 (to be used to change the basis).

(2) After that, the qutrit sequence, denoted as Ψ0, is sent to Bob 1.

(3) Bob 1 to Bob n perform local three-level unitary operations according to the values of A and B.

(4) Bob n sends the qutrits back to Alice. Then, Bob 1 to Bob n declare the information about individual unitary operations (not unitary operations themselves) via classical communication.

(5) Alice measures the qutrits using proper bases according to the information and publishes the result.

(6) Only when all the Bobs collaborate together, the secret can be revealed. Sample detection can be implemented by classical communication to judge if there is a wiretap.

FIGURE 1
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FIGURE 1. A general QSS protocol based on single states. Binary string A0 is the secret, and B0 is a random string. Alice prepares, transform, and finally measures the single qutrits, while Bob 1 to Bob n only do local unitary operations on these qutrits according to Ai and Bi.

In addition, unitary operations can be executed in one or two steps, which were defined in our previous work, shown as follows [23].

Definition 1. (one-step unitary operation): Bob i performs once a random local unitary operation on each quantum state before sending. All possible options are put into “{}”, called a unitary operation set.

Definition 2. (two-step unitary operation): Bob i performs twice a random local unitary operation on each quantum state before sending, and the probability of the first and second operations being the same is zero. The symbol “{; }” is used to indicate the two-step unitary operation, while the first possible options are listed before the semicolon, and the subsequent options, after the semicolon.

2.3 Bell-state attack in a three-level quantum system

Bob i is supposed to be dishonest during the execution of the three-level QSS protocol. He aims to generate a Bell-state attack to obtain the integrated encoded information from Bob i+1 to Bob j. The schematic diagram of the Bell-state attack is illustrated in Figure 2 (which is a variant of the substitute-Bell-state attack [23]), and the procedure of the attack is as follows:

(1) Bob i retains the single qutrits Ψi1 sent from Bob i-1 and does nothing on these particles. Meanwhile, he generates N three-level Bell states Φ12=k=1Nϕk12. ϕk12 can be any one of the nine Bell states given in Formula 4. Without the loss of generality, ϕk is assumed to be ψ00=1/300+11+22.

(2) Bob i transmits the second particles Φ2 of the Bell states to Bob i+1 and retains the first particles Φ1.

(3) Bob i intercepts the single qutrits Ψi+1 sent from Bob j to Bob j +1 and replaces it with Ψi1.

(4) The particle sequences Φ1 and Ψj are combined in pairs by Bob i to form new N Bell states Φ12=k=1Nϕk12. Then, Bob i measures the new Bell states by proper bases in order to obtain integrated encoded information from Bob i+1 to Bob j.

(5) When samples are tested, Bob i claims the unitary operations that are consistent with the comprehensive effect of the unitary operations that Bob i +1, Bob i +2, . . ., and Bob j implemented, to avoid the attack from being detected.

FIGURE 2
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FIGURE 2. Schematic diagram of Bell-state attack from Bob i for the integrated encoding information from Bob i+1 to Bob j. Φ1 and Φ2 are the two particles of the Bell states Bob i generates. Bob i intercepts and captures Ψj sent from Bob j to Bob j+1, and replaces it by Ψi1. Φ1 and Ψj are combined to form new Bell states Φ12.

It should be noted that Bob j can also be Bob i+1, and in this situation, Bob i aims only to obtain the unitary operations that Bob i +1 has performed.

Now, the key question is how to measure the effect of the Bell-state attack? To determine this, we pick up the minimum failure probability formula that we put forward previously [23, 24].

Definition 3. (minimum failure probability): Assume that a QSS protocol based on single qutrits is under a Bell-state attack. The attacker has acquired N new two-qutrit states ϕk. Suppose the prior probability of ϕk is pk, and the number of quantum states to be distinguished is n. Thus, the minimum failure probability of the Bell-state attack (denoted as Fmin) is defined as follows:

Fmin=1n1ijpipjϕi|ϕj.(7)

Apparently, Fmin equals to 1 minus the maximum probability of reliably distinguishing different quantum states. The value of Fmin is between 0 and 1. When Fmin=0, the protocol is totally unsecure. In other words, all the states are mutually orthogonal; Bob i could definitely distinguish each state by proper measurement and interpret all encoded information successfully. It should be noted that Fmin1 because if Fmin=1, the states to be distinguished must be the same, and this is impossible in QSS protocols.Therefore, when Fmin is larger, the effective information that a Bell-state attack can gain is less, that is, the QSS protocol is verified to be safer and vice versa.

2.4 The security of one-step three-level unitary operations

To research the security of three-level unitary operations under a Bell-state attack, we first provide the nine unitary operations corresponding to nine Bell states (shown in Formula 4) in Formula 8. In other words, the nine Bell states can be obtained when the second qutrit of the initial state Φ12=ψ00=1/300+11+22 is affected individually by the nine unitary operations and recombined with the first qutrit.

X1=100010001,X2=001100010,X3=010001100,X4=w000w20001,X5=001w000w20,X6=0w20001w00,X7=w2000w0001,X8=001w2000w0,X9=0w0001w200,(8)

where w=e2πi/3.

Moreover, without the loss of generality, U01 (shown in Formula 6), X1, X5, and X9 are chosen as the three-level unitary operations in QSS protocols for encoding the message and/or scrambling states, relabeled as

U0=U01=131w1w11w2w21,U1=X1=100010001,U2=X5=001w000w20,U3=X9=0w0001w200.(9)

It should be noted that U0 can transform a qutrit between the X-basis and Z-basis.

Here is the explanation of why X1, X5, and X9 are chosen as U1, U2, and U3, respectively. When U01 is chosen, we get Φ1U01Φ2 = 1/3ψ00+ψ11+ψ22. So we select preferentially the unitary operations leading to the states that are not orthogonal to Φ1U01Φ2. It is easy to find that selecting X1, X5, and X9 is optimal, which leads to ψ00,ψ11,andψ22, and the maximum value of Fmin. On the contrary, if other unitary operations are selected, the denominator of Formula 9 remains unchanged, while the numerator decreases, that is, Fmin decreases and the security reduces.

Finally, there are seven sets of one-step three-level unitary operations corresponding to the aforementioned matrices, and the minimum failure probability Fmin of each set is calculated and shown in Table 1.

TABLE 1
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TABLE 1. The minimum failure probability of Bell-state attack under one-step three-level unitary operations. The quantum states are obtained from the different transformations of initial state ψ00=1/300+11+22 under selected unitary operations. The curly braces of selected unitary operations, Dirac symbol and normalization of obtained quantum states are omitted.

From Table 1, we can conclude the following:

(1) If the four unitary operations of U0, U1, U2, and U3 are chosen for the quantum secret sharing process, only three values of the minimum failure probability are available. In other words, Fmin=3/3 when two unitary operations are selected; Fmin=23/9 when three are selected; and Fmin=3/6 when all four are selected. In a word, in one-step three-level unitary operations, the fewer unitary operations the legitimate communicator selects, the larger Fmin is and the higher the security of this QSS protocol is verified.

(2) Based on the aforementioned analysis, the minimum failure probability formula in a three-level quantum system can be simplified as follows:

Fmin=23N,(10)

where N denotes the number of selected unitary operations. For the six situations listed in Formula 6, if we choose the other five situations and corresponding U0 and Xii=1,2,...,9, the same values of Fmin will be attained, along with the same simplified formula of minimum failure probability.

2.5 The security of two-step three-level unitary operations

The one-step unitary operation case is mentioned previously. We try the similar analysis in a two-step unitary operation instance.

First of all, according to the definition of a two-step unitary operation and the selected four unitary operations U0, U1, U2, and U3, all combinations can be divided into four categories as follows:

1. U0;U1,U23U1,U2,U31U1,U03U1,U2,U03U1,U2,U3,U0111 combinations;

2. U1,U0×3;U13U1,U23U1,U2,U31U2,U02U1,U2,U03U1,U2,U3,U0113×3=39 combinations;

3. U1,U2,U0×3;U13U1,U23U1,U2,U31U1,U3,U02U1,U2,U3,U0110×3=30combinations;

4. U1,U2,U3,U0;U13U1,U23U1,U2,U317combinations.

Here, the first possible unitary operations are listed before the semicolon, and the subsequent operations, after the semicolon; the symbol “×3” means that there are three similar unitary operation combinations in the first step; the symbol “(3)” denotes that there are three similar unitary operation combinations in the second step. For example, in the first category, “U1,U23” indicates that there are three similar combinations, that is, U1,U2, U1,U3, and U2,U3. In short, there are totally 11+39+30+7=87 combinations.

Second, we provide the states after continuous action of two unitary operations (shown in Table 2), supposing the initial state is ψ00=1/300+11+22. If the two unitary operations are executed in an opposite order, the same state is obtained. So there are totally 10 results, as shown in Table 2. In addition, αiji,j=0,1,2,3 denotes the obtained state after the two unitary operations, where the subscripts i and j represent the corresponding unitary operations Ui and Uj.

TABLE 2
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TABLE 2. The states after continuous action of two unitary operations. The two subscripts of α correspond to subscripts of the two used unitary operations.

Third, to calculate Fmin=1n1ijpipjϕi|ϕj, we first calculate the norm of the inner product of the two quantum states, that is, ϕi|ϕjij. The results are shown in Table 3.

TABLE 3
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TABLE 3. The norm of inner product of the two quantum states. The inner product values are symmetric about the diagonal.

From Table 3 we can see that the inner product values are symmetric about the diagonal and the inner product values between α01,α02 and α03 is 0. So the four combinations in the first category with “U1,U23" and “U1,U2,U31” in the second step can be omitted. Therefore, there are totally 874=83 combinations, which are divided into 18 situations, and the values of Fmin are shown in Table 4.

TABLE 4
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TABLE 4. The values of Fmin and the corresponding numbers of unitary operation sets. All the 83 non zero combinations are showed in 18 cases. The first-step possible unitary options are listed before the semicolon, and the second-step options after the semicolon.

From Table 4, it can be seen that the values of Fmin are the same separately in cases 1 and 4; in cases 2, 5, and 10; and in cases 3 and 16. So it can be merged into 14 cases. Thus, the values of Fmin (accurate to four decimal places) and the number of unitary operation sets corresponding to each value are shown in Table 5.

TABLE 5
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TABLE 5. The values of Fmin and the corresponding numbers of unitary operation sets when the values in Table 4 are accurate to four decimal places. Thus 18 cases are merged to 14 cases.

The results in Table 5 are also illustrated in Figure 3 for clarity. Then, some conclusions can be drawn based on the results.

(1) Among the two-step three-level unitary operations, the minimum failure probability of the Bell-state attack densely distributed between [0.34, 0.39], totally 48 sets, and there are 17 sets between [0.24, 0.29]. Furthermore, 12 sets have the highest value of Fmin, that is, 3/30.58, and 6 sets have the second highest value of Fmin=53/180.48.

(2) The sets that have the highest value of Fmin are U0;Ui,U0 and Ui,U0;Uii=1,2,3. This means the sets possessing the least selected operations have the highest security, except for U0;U1,U2, U0;U1,U3, and U0;U2,U3 whose Fmin=0. This is similar to the two-level unitary operation situation [24].

(3) Among the six situations shown in Formula 6, if another situation is selected, that is, another Bell-state combination and corresponding U0 and Xii=1,2,,9 are selected, the similar calculation process and conclusions will be obtained because their mutual relationship is consistent with the first situation.

FIGURE 3
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FIGURE 3. The minimum failure probability of two-step three-level unitary operations. The values of Fmin are accurate to four decimal places and three are totally 84 unitary operation sets.

2.6 The selection of three-level unitary operations

Based on the quantitative computing results and analysis on one-step and two-step unitary operations in a three-level system, the selection rules of unitary operations are summarized as follows:

(1) The unitary operations in QSS protocols based on qutrits should be chosen carefully. First, the unitary operations whose Fmin is 0 should not be chosen since the protocol is obviously insecure in such a situation. Second, the unitary operation to transform the basis is a necessary but not a sufficient condition for the security. In other words, the unitary operation like U0 can change the basis; so if U0 is not selected, the QSS protocol is totally insecure. However, although U0 is the possible choice, the security of the protocol still cannot be guaranteed. For example, Fmin of U0;U1,U2, U0;U1,U3, U0;U2,U3, and U0;U1,U2,U3 is 0.

(2) More complex unitary operations cannot be counted on to generate higher security. In a one-step unitary operation, Max [Fmin (U1,U2,,Ui)] > Max [Fmin (U1,U2,,Ui,Ui+1)], that is, fewer unitary operations lead to larger Fmin and higher security of the QSS protocol. The two-step unitary operation has the similar situation. This is because in one-step scene, the more the unitary operations, the more the pairwise non-orthogonal quantum states obtained.

(3) When the same unitary operations are selected, the security of two-step unitary operations is not necessarily higher than that of one-step unitary operations. There exist possibilities. For example, Fmin ({U0,U1,U2}) = Fmin ({U0;U0,U1,U2}) 0.38; Fmin ({U0,U1,U2}) 0.38 < Fmin ({U0,U1;U0,U2}) 0.48; Fmin ({U0,U1,U2}) 0.38 > Fmin ({U0,U1,U2;U1,U2}) 0.26. So we should choose the two-step unitary operations with the highest value of Fmin.

(4) The maximum of Fmin of the three-level unitary operations (0.58) is less than the maximum of Fmin of the two-level unitary operations (0.71) [23, 24]. This does not mean that two-level unitary operations are safer than three-level unitary operations because the value of Fmin is influenced by specified unitary operation types. We have restricted the types of two-level and three-level unitary operations, so this is just a comparison between two preset conditions.

In brief, we should choose the unitary operation sets that have a higher value of Fmin to ensure the security of QSS protocols based on qutrits.

3 Conclusion

In this paper, we first present a general QSS protocol based on single qutrits, and then propose the Bell-state attack and the definition of minimum failure probability for the attack. In this way, QSS protocols based on single qutrits and three-level unitary operations are considered, and the quantitative security analysis is performed corresponding to different sets of four three-level unitary operations. The results show that the selection of unitary operations will significantly affect the security of such QSS protocols. As a result, some crucial rules for choosing unitary operations are given to ensure the security or achieve a higher security. This work can serve as an important guidance in designing and implementing QSS protocols based on single qutrits and three-level unitary operations. The method and results may also contribute to analyze the security of other high-level quantum cryptography protocols based on unitary operations, such as secure computation [25], quantum secure direct communication [26], quantum key agreement [27], quantum private query [28], and quantum oblivious transfer [29]. Furthermore, unitary operations are also used in other quantum algorithms, for example, the quantum blockchain algorithm [30] and quantum artificial intelligence algorithm [31], and it will be interesting in attempting to analyze their security using our method.

Data availability statement

The original contributions presented in the study are included in the article/Supplementary Material; further inquiries can be directed to the corresponding author.

Author contributions

JX, XL, and ZL conceived the presented idea. YH and YZ verified the methods and developed the analyses. ZZ and YS completed the calculation. All authors contributed to the article and approved the submitted version.

Funding

This work is supported by the National Natural Science Foundation of China (Grant nos 62201252 and 62071240), the Innovation Program for Quantum Science and Technology (Grant no. 2021ZD0302900), and the Natural Science Foundation of Jiangsu Province, China (Grant no. BK20220804).

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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Keywords: quantum cryptography, quantum secret sharing, three-level unitary operation, quantitative security analysis, qutrit

Citation: Xu J, Li X, Han Y, Zhou Y, Liu Z, Zhang Z and Song Y (2023) Quantitative security analysis of three-level unitary operations in quantum secret sharing without entanglement. Front. Phys. 11:1213153. doi: 10.3389/fphy.2023.1213153

Received: 27 April 2023; Accepted: 23 May 2023;
Published: 19 June 2023.

Edited by:

Weibin Li, University of Nottingham, United Kingdom

Reviewed by:

Lihua Gong, Nanchang University, China
Su-Juan Qin, Beijing University of Posts and Telecommunications (BUPT), China

Copyright © 2023 Xu, Li, Han, Zhou, Liu, Zhang and Song. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Juan Xu, juanxu@nuaa.edu.cn

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