Single-state multi-party semi-quantum secret sharing method based on d-dimensional Bell states

Abstract

This paper proposes a single-state multi-party semi-quantum secret sharing method based on d-dimensional Bell states, which achieves the goal of "the sender's key can only be recovered when all receivers work together." This method has been proven to be resistant to both external and participant attacks. The method is applicable to d-dimensional quantum systems, uses only a single d-dimensional Bell state as the initial quantum resource, and requires neither quantum entanglement exchange nor unitary operations.

Description

Technical Field

[0001] The present invention relates to the field of quantum cryptography and designs a single-state multi-party semi-quantum secret sharing method based on d-dimensional Bell states, which achieves the goal of "the sender's key can only be recovered when all receivers work together." Background Art

[0002] With the continuous expansion of practical applications of quantum information science, quantum cryptography has developed rapidly. In the past few decades, scholars have continuously studied different branches of quantum cryptography, such as quantum key distribution (QKD) [1-2], quantum secure direct communication (QSDC) [3-4], quantum dialogue (QD) [5-10], quantum key agreement (QKA) [11-12], and quantum secret sharing (QSS) [13-20]. Among them, QSS, first proposed by Hillery et al.

[13] in 1999, is a generalization of classical secret sharing to quantum scenarios, with the goal of recovering the sender's key only when all receivers cooperate.

[0003] In addition to quantum cryptography, semi-quantum cryptography has also emerged in recent years, which allows some users to have only limited quantum capabilities. In 2007, Boyer et al.

[21] first proposed the concept of "semi-quantum" in a novel single-photon-based semi-quantum key distribution (SQKD) method. Subsequently, single photons with two degrees of freedom were also used to design SQKD methods [22,23]. Restricted classical users are only allowed to perform the following four operations

[21] : (a) send particles through quantum channels; (b) measure particles using the Z basis (i.e., {|0>, |1>}); (c) prepare particles in the Z basis; (d) disrupt the order of particles.

[0004] Semiquantum secret sharing (SQSS), a combination of the "semi-quantum" concept and QSS, has seen significant development in recent years. Many SQSS methods [24-32] have been proposed from different perspectives. Unfortunately, the vast majority of previous SQSS methods [24-31] are not applicable to d-dimensional quantum systems, where d > 2. In other words, the SQSS methods in

[32] are the only two feasible SQSS methods for d-dimensional quantum systems.

[0005] Based on the above analysis, the present invention constructs a single-state multiparty semiquantum secret sharing (MSQSS) method using d-dimensional Bell states. This method is applicable to d-dimensional quantum systems, uses only one d-dimensional Bell state as the initial quantum resource, and does not utilize quantum entanglement exchange or unitary operations.

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[0042] The purpose of the present invention is to design a single-state multi-party semi-quantum secret sharing method based on d-dimensional Bell states to achieve the goal of "the sender's key can only be recovered when all receivers work together".

[0043] The single-state multi-party semi-quantum secret sharing method based on d-dimensional Bell states includes the following five steps:

[0044] S1) P0 prepares N sets of d-dimensional Bell state sequences of length 4n, where each Bell state is in make and T i j denote the first and second particles of the jth d-dimensional Bell state of the i-th group, where i = 1, 2, ..., N and j = 1, 2, ..., 4n. The first and second particles of each d-dimensional Bell state are in the same state. In addition, let and T i ={T i 1 ,T i 2 ,...,T i 4n}, where i=1,2,...,N. Then, P0 transmits S i The particles are sent to P one by one i , and T i Keep it in her own hands. i In addition to the first particle, P0 sends the next particle only when she receives the previous particle.

[0045] S2) After receiving S i After each particle, P i Randomly choose to measure it using the Z basis, prepare a new particle in the discovered state and send it to P0 (called MEASURE), or directly return it to P0 without interference (called REFLECT). Here, the Z basis is Z = {0>, 1>, ..., |d-1>}, i = 1, 2, ..., N.

[0046] S3) P0 temporarily stores data from P i All particles of , where i = 1, 2, ..., N. Then, P i Announcement S i For which particles in the equation does she choose the MEASURE operation? Two different situations need to be explained.

[0047] Case (1): For S i Medium P i Select the particle of MEASURE, P0 uses Z basis to measure S i The particles in the hand and T i The number of MEASURE particles is 2n. For safety testing, P0 randomly selects half of S i MEASURE particles in the and tell P i Their location. Afterwards, P i Then, P0 will tell P0 her measurement results of these selected MEASURE particles. i The measurement results of these selected MEASURE particles, her T i The measurement results of the corresponding particles in P i To S i The security of the quantum channel is determined by comparing the measurement results of these selected MEASURE particles. When there is no eavesdropper in the quantum channel, these measurement results should always correspond to the same.

[0048] Case (2): For S i Medium P i Select REFLECT particles, P0 by these particles and T i The corresponding particles in the quantum channel are measured by d-dimensional Bell basis to determine whether the quantum channel is secure. When there is no eavesdropper in the quantum channel, the Bell basis measurement result of P0 should always be |ψ 00 >.

[0049] S4) P0 checks the error rates of situations (1) and (2). If the error rate of either situation is abnormally high, the communication will be terminated immediately; otherwise, the communication will continue.

[0050] S5)S i The remaining n MEASURE particles in are used to share the key, where i = 1, 2, ..., N. Let K i Indicates P0 to S i The classical value corresponding to the measurement results of the remaining n MEASURE particles in , where K i∈{0,1,...,,d-1} and i=1,2,...,N. Here, |0>,|1>,...,|d-1> are encoded as the classical values ​​0,1,...,d-1 respectively. P0 will As her key. Obviously, P i You can automatically know K i Therefore, only when P1, P2, ..., P N Only when they work together can they recover the key K. BRIEF DESCRIPTION OF THE DRAWINGS

[0051] Figure 1 It's Eve's with a U E and U F Entanglement-measurement attack on two unitary operations. DETAILED DESCRIPTION

[0052] The technical solution of the present invention is further described below in conjunction with embodiments.

[0053] 1 Method Description

[0054] The d-dimensional Bell state can be expressed as follows:

[0055]

[0056] where u,v∈{0,1,...,d-1}, symbol represents the addition modulo d. Obviously, we can get

[0057]

[0058] In a d-dimensional quantum system, the Z basis and the X basis can be described as:

[0059] Z={0>,1>,...,|d-1>},(3)

[0060] X={F|0>,F1>,...,F|d-1>}. (4)

[0061] Here, F is the discrete quantum Fourier transform, where

[0062]

[0063] j=0,1,...,d-1. Obviously, the Z group and the X group are two groups of conjugated groups.

[0064] Assume that P0 is the party with full quantum capability, and P1, P2, ..., P N is an N-party with limited quantum power. The goal of the single-state MSQSS method based on d-dimensional Bell states is to obtain the quantum state only when P1, P2, ..., P NOnly when they work together can they reveal the key of P0. The single-state MSQSS method proposed by the present invention can be described as follows.

[0065] S1)P0 prepares N sets of d-dimensional Bell state sequences of length 4n, where each Bell state is in |ψ 00 >. Order and T i j denote the first and second particles of the jth d-dimensional Bell state of the i-th group, respectively, where i = 1, 2, ..., N and j = 1, 2, ..., 4n. According to Equation (2), the first and second particles of each d-dimensional Bell state are in the same state. In addition, let and T i ={T i 1 ,T i 2 ,...,T i 4n}, where i=1,2,...,N. Then, P0 transmits S i The particles are sent to P one by one i , and T i Keep it in her own hands. i In addition to the first particle, P0 sends the next particle only when she receives the previous particle.

[0066] S2) After receiving S i After each particle, P i Randomly choose to measure it with the Z basis, prepare a new particle in the discovered state and send it to P0 (called MEASURE), or directly return it to P0 without interference (called REFLECT). Here, i = 1, 2, ..., N.

[0067] S3) P0 temporarily stores data from P i All particles of , where i = 1, 2, ..., N. Then, P i Announcement S i For which particles in the equation does she choose the MEASURE operation? Two different situations need to be explained.

[0068] Case (1): For S i Medium P i Select the particle of MEASURE, P0 uses Z basis to measure S i The particles in the hand and T i The number of MEASURE particles is 2n. For safety testing, P0 randomly selects half of S i MEASURE particles in the and tell Pi Their location. Afterwards, P i Then, P0 will tell P0 her measurement results of these selected MEASURE particles. i The measurement results of these selected MEASURE particles, her T i The measurement results of the corresponding particles in P i To S i The security of the quantum channel is determined by comparing the measurement results of these selected MEASURE particles. When there is no eavesdropper in the quantum channel, these measurement results should always correspond to the same.

[0069] Case (2): For S i Medium P i Select REFLECT particles, P0 by these particles and T i The corresponding particles in the quantum channel are measured by d-dimensional Bell basis to determine whether the quantum channel is secure. When there is no eavesdropper in the quantum channel, the Bell basis measurement result of P0 should always be |ψ 00 >.

[0070] S4) P0 checks the error rates of situations (1) and (2). If the error rate of either situation is abnormally high, the communication will be terminated immediately; otherwise, the communication will continue.

[0071] S5)S i The remaining n MEASURE particles in are used to share the key, where i = 1, 2, ..., N. Let K i Indicates P0 to S i The classical value corresponding to the measurement results of the remaining n MEASURE particles in , where K i ∈{0,1,...,d-1} and i=1,2,...,N. Here, |0>,|1>,...,|d-1> are encoded as the classical values ​​0,1,...,d-1 respectively. P0 will As her key. Obviously, P i You can automatically know K i Therefore, only when P1, P2, ...., P N Only when they work together can they recover the key K.

[0072] 2 Security Analysis

[0073] 2.1 External Attacks

[0074] (1) Entanglement-Measurement Attack

[0075] In order to obtain K, the external eavesdropper Eve should first know K i, where i = 1, 2, ..., N. Eve may try her best to Figure 1 Shown includes U E and U F Entanglement-measurement attack of two unitary operations to obtain K i Here, Eve sends a i The particles apply U E and from P i The particle returning to P0 applies U F In addition, U E and U F Share a common detection space with an initial state of |ε>. As described in

[21] , the shared detection state allows Eve to E With the help of information obtained from P i The particle returned to P0 initiates an attack; U F Depends on executing U E Any attack on the subsequent measurements can be performed by U E and U F This is achieved through controlled doors.

[0076] Theorem 1: Assume that Eve sends a i (i=1,2,...,N) and from P i The particle returning to P0 attacks (U E ,U F ). In order to avoid introducing errors in step S3, Eve’s final detection state should not only be consistent with P i The operation has nothing to do with P0 and P i Thus, if Eve wants to remain undetected, she will have no effect on K by launching this attack. i Still clueless.

[0077] Proof: U E The effect on particle |k> can be expressed as

[0078]

[0079] Here, |β kt > is dependent on U E Eve's detection state, and k,t=0,1,...,d-1. In addition, for k=0,1,...,d-1,

[0080] Before Eve attacks, the global state of the composite system consisting of P0 and Eve’s particles can be expressed as |ψ 00 >|ε>. Execute U in Eve E After that, the composite system was evolved into

[0081]

[0082] When P i Receive S i When measuring particles, she chooses MEASURE or REFLECT.

[0083] ① Consider P i Select MEASURE. According to formula (7), when P i To S i The measurement result of the particle is |t>, the state of the composite system is collapsed to Where k, t = 0, 1, ..., d-1.

[0084] Eve is a little bit confused about the P i The particle that returns to P0 executes U F In order to prevent Eve from being detected in the case (1) of step S3, U F Should satisfy

[0085]

[0086] This means that U F Cannot change P i After operation, i S to P0 i The particle and P0 in the hand T i The state of the corresponding particle.

[0087] ② Consider P i Select REFLECT. In this case, according to formula (7), the state of the composite system is Where k, t = 0, 1, ..., d-1.

[0088] Eve is a little bit confused about the P i The particle that returns to P0 executes U F The state of the composite system is transformed into

[0089]

[0090] Inserting equation (8) into equation (9) yields

[0091]

[0092] In order to prevent Eve from being detected in case (2) of step S3, the measurement result of P0 is |ψ 00 The probability of > should be 1. Therefore, based on formula (2) and formula (10), we can conclude that

[0093] |F0>=|F1>=...=F d-1>=|F>. (11)

[0094] ③ Inserting equation (11) into equation (8) yields

[0095]

[0096] According to formula (2), substituting formula (11) into formula (10) yields

[0097]

[0098] According to equations (13) and (14), it can be concluded that in order to avoid errors in step S3, Eve’s final detection state should not only be consistent with P i The operation has nothing to do with P0 and P i Thus, if Eve wants to remain undetected, she will have no effect on K by launching this attack. i Still clueless.

[0099] (2) Trojan Horse Attack

[0100] S i The particles in (i=1,2,...,N) are in P0 and P i Therefore, P i A wavelength filter and a photon number splitter (PNS) can be placed in front of her device to avoid invisible photon eavesdropping attacks and delayed photon Trojan horse attacks, respectively [33,34].

[0101] (3) Intercept-retransmit attack

[0102] In order to know K i (i=1,2,...,N), Eve prepares fake sequences in advance In Z base, intercept S from P0 i , and Send to P i ; In P i right After applying her manipulation, Eve intercepts the i The particle sequence sent out and S i Send to P0. Consider P i When REFLECT is selected, P i Returns the jth (j = 1, 2, ..., 4n) pseudoparticle she received to P0; then P0 uses the d-dimensional Bell basis to measure S sent from Eve i The jth particle in its hand T iTherefore, in this case, Eve cannot be detected. Consider P i When MEASURE is selected, P i Use the Z basis to measure the jth pseudoparticle received, prepare a new particle in the discovered state and send it back to P0; P0 uses the Z basis to measure the S sent from Eve i The jth particle in the hand and T i Therefore, in this case, Eve will It can be concluded that when Eve launches an intercept-retransmit attack, she will probability of being detected.

[0103] (4) Measurement-Retransmission Attack

[0104] In order to know K i (i=1,2,...,N), Eve intercepts S from P0 i , using the Z basis to measure its particles and transmit the corresponding state to P i Consider P i When REFLECT is selected, P i Returns the jth (j=1,2,...,4n) particle she received to P0; P0 uses the d-dimensional Bell basis to measure the particle from P i The jth particle and the T in its hand i Therefore, in this case, Eve will The probability of being detected. Consider P i When MEASURE is selected, P i Measure the jth particle received using the Z basis, prepare a new particle in the discovered state and transmit it to P0. Obviously, in this case, Eve's attack cannot be detected. Now we can conclude that when Eve launches this measurement-retransmission attack, the probability that she will be detected is

[0105] 2.2 Participant Attack

[0106] In the method of the present invention, different P i They play the same role and are independent of each other, where i = 1, 2, ..., N. Here, we need to discuss two different actor attack cases.

[0107] First, assume that P1, P2, ..., P N Only one of them is untrustworthy. Without loss of generality, assume that P1 is untrustworthy. P1 may m and from P mThe transmission particle returning to P0 initiates the attack, where m = 2, 3, ..., N. Unfortunately, as analyzed above, since P1 is independent of P0 and P m , she is essentially an outside eavesdropper and will inevitably be discovered.

[0108] Secondly, suppose P1, P2, ..., P N More than one of them is untrustworthy. The worst case is that P1, P2, ..., P N There are n-1 parties in the network that are untrustworthy. Without loss of generality, assume that P1 is the only trustworthy party. P2, P3, ..., P N It is possible to attack the transmission particles from P0 to P1 and from P1 back to P0. Unfortunately, due to the N Independent of P0 and P1, they are essentially external eavesdroppers, which means they will undoubtedly be detected.

[0109] Example:

[0110] 1 Application examples of the method of the present invention

[0111] Ignoring security checks, here's an example to further explain the principle of secret sharing.

[0112] P0 prepares N sets of d-dimensional Bell state sequences of length 4n, where each Bell state is in |ψ 00 >. Order and T i j denote the first and second particles of the jth d-dimensional Bell state of the i-th group, respectively, where i = 1, 2, ..., N and j = 1, 2, ..., 4n. In addition, let and T i ={T i 1 ,T i 2 ,...,T i 4n}, where i=1,2,...,N. Then, P0 transmits S i The particles are sent to P one by one i , and T i Keep it in her own hands. i In addition to the first particle of S, P0 sends the next particle only when it receives the previous particle. i After each particle, P i Randomly select MEASURE or REFLECT operation; here, i = 1, 2, ..., N. P0 temporarily stores the data from P iAll particles of , where i = 1, 2, ..., N. Then, P i Announcement S i Which particles in the MEASURE operation she selected. After eavesdropping detection, S i The remaining n MEASURE particles in are used to share the key, where i = 1, 2, ..., N. Let K i Indicates P0 to S i The classical value corresponding to the measurement results of the remaining n MEASURE particles in , where K i ∈{0,1,...,d-1} and i=1,2,...,N. Here, |0>,|1>,...,|d-1> are encoded as the classical values ​​0,1,...,d-1 respectively. Therefore, P0 will As her key. Obviously, P i You can automatically know K i Therefore, only when P1, P2, ..., P N Only when they work together can they recover the key K.

[0113] 2 Discussion and Conclusion

[0114] In quantum communication methods for a d-dimensional quantum system, the quantum bottom efficiency is usually used to evaluate the communication efficiency, which is defined as

[35]

[0115]

[0116] Here, b, q, and c are the length of the shared key, the number of consumed quantum bottom numbers, and the amount of consumed classical information, respectively. The classical resources required for eavesdropping detection are not considered here.

[0117] In the method of the present invention, the length of K is n, so it can be known that b = n. P0 needs to prepare N groups of d-dimensional Bell state sequences with a length of 4n, and i Send to P i When P i For the received S i When the particle selects the MEASURE operation, it needs to generate 2n new particles and transmit them to P0. This shows that q = 4n × 2 × N + 2n × N = 10nN. No classical resources are consumed in classical communication, so c = 0. From this, we can conclude that the quantum efficiency of the method of the present invention is

[0118] Here, the method of the present invention is further compared with the method of reference

[32] . The method of reference

[32] is the only two SQSS methods that are feasible in d-dimensional quantum systems to date. It is easy to see from Table 1 that the method of the present invention outperforms the method of reference

[32] in terms of the number of initial quantum states; and in terms of the use of unitary operations, the method of the present invention is superior to the method of reference

[32] .

[0119] Table 1 Comparison between different MSQSS methods

[0120]

[0121] This paper designs a tree-type single-state MSQSS method based on d-dimensional Bell states. This method has been shown to be resistant to external and participant attacks. It is applicable to d-dimensional quantum systems, employs only a single d-dimensional Bell state as the initial quantum resource, and uses neither quantum entanglement exchange nor unitary operations.

Claims

1. A single-state multi-party semi-quantum secret sharing method based on d-dimensional Bell states, which can achieve the goal of "the sender's key can only be recovered when all receivers work together" and is resistant to external attacks and participant attacks; Applicable to d-dimensional quantum systems, only one d-dimensional Bell state is used as the initial quantum resource; Neither quantum entanglement exchange nor unitary operation is required; it includes the following five processes: S1) P0 prepares N sets of d-dimensional Bell state sequences of length 4n, where each Bell state is in make and T i j denote the first and second particles of the jth d-dimensional Bell state of the i-th group, where i = 1, 2, ..., N and j = 1, 2, ..., 4n; the first and second particles of each d-dimensional Bell state are in the same state; in addition, let and Where i=1,2,...,N; Then, P0 transmits S i The particles are sent to P one by one i , and T i kept in her own hands; In addition to S i In addition to the first particle, P0 sends the next particle only when P0 receives the previous particle; S2) After receiving S i After each particle, P i Randomly choose to measure it using the Z basis, prepare a new particle in the discovered state and send it to P0 (called MEASURE), or directly return it to P0 without interference (called REFLECT); here, the Z basis is Z = {|0>,|1>,...,|d-1>}, i = 1,2,...,N; S3) P0 temporarily stores data from P i All particles of , where i = 1, 2, ..., N; then, P i Announcement S i For which particles in the equation did she choose the MEASURE operation? Two different situations need to be explained. Case (1): For S i Medium P i Select the particle of MEASURE, P0 uses Z basis to measure S i The particles in the hand and T i The corresponding particles in the MEASURE are 2n. For safety detection, P0 randomly selects half of the S i MEASURE particles in the and tell P i Their location; after that, P i P0 informs her measurement results of these selected MEASURE particles; then, P0 will pass her measurement results of S i The measurement results of these selected MEASURE particles, her T i The measurement results of the corresponding particles in P i To S i The measurement results of these selected MEASURE particles are compared to determine whether the quantum channel is secure; when there is no eavesdropper in the quantum channel, these measurement results should always correspond to the same; Case (2): For S i Medium P i Select REFLECT particles, P0 by these particles and T i The corresponding particles in the quantum channel perform d-dimensional Bell basis measurements to determine whether the quantum channel is secure; when there is no eavesdropper in the quantum channel, the Bell basis measurement result of P0 should always be |ψ 00 > S4) P0 checks the error rates of situations (1) and (2); if the error rate of any situation is abnormally high, the communication will be terminated immediately, otherwise, the communication will continue; S5)S i The remaining n MEASURE particles in are used to share the key, where i = 1, 2, ..., N; let K i Indicates P0 to S i The classical value corresponding to the measurement results of the remaining n MEASURE particles in , where K i ∈{0,1,...,d-1} and i=1,2,...,N; here, |0>,|1>,...,|d-1> are encoded as the classical values ​​0,1,...,d-1 respectively; P0 will As her key; obviously, P i You can automatically know K i ; Therefore, only when P1, P2, ..., P N Only when they work together can they recover the key K.

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