CN110445609B - Quantum secret sharing method and quantum secret sharing system based on quantum walking - Google Patents
Quantum secret sharing method and quantum secret sharing system based on quantum walking Download PDFInfo
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Abstract
The invention belongs to the technical field of quantum information sharing, and discloses a quantum secret sharing method and a quantum secret sharing system based on quantum walking.A secret distributor encodes secret information to be shared into a quantum state according to rules and divides encoded particles into two parts; the two pieces of encrypted information are respectively and invisibly transmitted to a first participant and a second participant through a quantum walking system; the first participant and the second participant decrypt the secret information of the secret distributor through cooperation. The invention applies the characteristics of the quantum walking system to the invisible state of the unknown particles, and the particles are spontaneously entangled, thereby avoiding the preparation of entangled particles in the preparation stage of the particles to finish the invisible state of the particles. The invention combines the invisible state transfer method based on quantum walking with the encoding rule to complete a quantum secret sharing method based on quantum walking.
Description
Technical Field
The invention belongs to the technical field of quantum information sharing, and particularly relates to a quantum secret sharing method and a quantum secret sharing system based on quantum walking.
Background
Currently, the closest prior art: since the BB84 protocol proposed by Bennett et al, Quantum Key Distribution (QKD) and Quantum Communication (QC) have become important topics for quantum cryptography, and various QKD protocols and QC protocols have been proposed. The development of QKD and QC has prompted the study of Quantum Secret Sharing (QSS), making QSS an important branch of quantum cryptography. Assuming that the secret distributor Alice wishes to share a secret information to the participants Bob and Charlie, when only Bob and Charlie can obtain the complete secret information of Alice through mutual honest cooperation, anyone cannot obtain the complete secret information of Alice through sub-information or dishonest operation alone. QSS is a combination of classical secret sharing and quantum theory that enables secret information (either classical information or quantum encoded information) to be distributed, transmitted and recovered through quantum operations. The security of QSS is based on the basic principles of quantum mechanics, which makes QSS more secure than traditional secret sharing.
The earliest QSS schemes were proposed by Hillery et al in 1999, who used the Greenberger-Horne-Zeilinger (GHZ) entangled state to accomplish secret sharing. Since then, more and more QSS schemes based on either the Bell entangled state or the multi-particle entangled state are proposed. However, the hidden state of the particles is completed by utilizing the entanglement characteristic of the Bell state or the multi-particle entangled state, so that a secret sharing scheme is realized, the preparation of the entangled state needs to consume more resources, and the preparation and the measurement are not easy under the prior art, and the practicability of the QSS scheme is greatly reduced due to the technical obstacles. In this regard, guo-hei and guo-brilliant propose an entanglement-free QSS scheme in 2003, which accomplishes secret sharing of classical information through secret sharing based on a secret key, and thereafter, yan phoenix et al propose an entanglement-free multiparty and QSS-between-multiparty in 2005, but then documents indicate that the scheme has a potential safety hazard on particle transmission, causes secret information leakage, and provides corresponding improvement measures. Although the QSS scheme does not adopt the entanglement characteristic of entangled particles to complete secret sharing, the safety of particle transmission is difficult to guarantee.
In summary, the problems of the prior art are as follows:
(1) the existing quantum secret sharing scheme has the problem of preparation technology obstacle in the particle state preparation stage, the existing protocol mostly utilizes the entanglement characteristics of Bell state, GHZ state or multi-particle entanglement state to realize the quantum secret sharing scheme, however, the preparation of the entanglement state needs to consume more resources, and the preparation and the measurement are not easy under the prior art, and the commercialization and the popularization of the quantum secret sharing are seriously hindered by the limitation of the technical cost and the quantum resources.
(2) The existing quantum secret sharing scheme based on the single particle abandons the use of entangled particles, but the invisible transmission state of the particles is difficult to achieve, and the protocol of the type is difficult to ensure the safety of particle transmission.
The difficulty of solving the technical problems is as follows:
(1) at present, the preparation and storage of Bell-state particles, GHZ-state particles or multi-particle entangled states are very challenging, expensive equipment and more resources are needed compared with the preparation of single particle states, and the commercialization and popularization of quantum invisible transfer states realized by utilizing Bell-state particles, GHZ-state particles or multi-particle entangled states are difficult to realize under the existing conditions.
(2) The quantum secret sharing method based on Bell-state particles, GHZ-state particles or multi-particle entangled states utilizes the measurement collapsibility of entangled particles on the whole design to complete quantum secret sharing, and Bell-state particles, GHZ-state particles and multi-particle entangled states are not easy to prepare and measure in the prior art, and the technical barriers make the quantum secret scheme depending on Bell-state particles, GHZ-state particles or multi-particle entangled states impractical in commercialization and popularization.
(3) In the face of internal attack and external entanglement attack, attack is resisted through the overall design of the system, and the resistance of the system to various attacks is analyzed and evaluated through theory.
The significance of solving the technical problems is as follows:
(1) the preparation and storage of the single particle state are easier and more stable than those of the GHZ state or the multi-particle entangled state, so that the quantum secret sharing scheme adopting the single particle is easier to commercialize and popularize.
(2) Through the quantum walking system, spontaneous entanglement of single particle states in the quantum walking process can be realized, the entangled characteristic is used for invisible state transfer of particles, invisible state transfer based on GHZ states or multi-particle entangled states is replaced, communication safety is guaranteed while long-distance communication is completed, and the quantum walking system is more practical.
(3) By combining the invisible single-particle state of transmission based on the quantum walking system, the invention makes an encoding rule and a secret sharing scheme, and provides a more practical, safe and feasible quantum secret sharing method based on quantum walking.
Disclosure of Invention
Aiming at the problems in the prior art, the invention provides a quantum secret sharing method and a quantum secret sharing system based on quantum walking.
The invention is realized in such a way that a quantum secret sharing method based on quantum walking comprises the following steps:
step one, the secret distributor encodes the shared secret information into a quantum state according to the secret information encoding and splitting rule. And (3) splitting the encoding particles, wherein in the stage of splitting the secret information encoding, the secret distributor generates a string of random binary sequences L with n bits. Then, the original information M of n bits is coded into a two-particle bit string { | b according to L1c1>，|b2c2>，|b3c3>...|bncn>}。
And step two, the linear quantum walking system transmits the two pieces of encrypted information to the first participant and the second participant in a hidden mode respectively.
Step three, after eavesdropping detection is carried out, the first participant and the second participant measure the particle strings in the hands according to the result published by the secret distributor, and the invisible transmission state is completed;
and step four, the first participant and the second participant decrypt the secret information of the secret distributor through cooperation.
Further, the step two in which the linear quantum walking system respectively and invisibly transmits the two pieces of encrypted information to the first participant and the second participant specifically includes: the linear quantum walking system occurs in a complex hilbert space, consisting of two quantum spaces, respectively a position space and a coin space, expressed as:
the method comprises the following steps that Hp represents position span { n, n belongs to Z }, Hc represents coin direction { |0>, |1> } of a linear quantum walking system, and a total subsystem in the linear quantum walking system is described as follows:
wherein
S is called the shift operator and is denoted as S ═ Σn|n+1><n |, and C is a coin operator. When the coin operator is |0>In state, coin is driven from state | n>Move to state | n +1>. When the coin operator is |1>In the state, the coin moves backwards to state | n-1>。
Further, after eavesdropping detection is carried out, the two-step quantum walking invisible state transmission system selects a two-coin-based quantum walking system, and selects a proper initial state and a matched measuring base to perform invisible transmission of any unknown qubits between the first participant and the secret distributor and between the second participant and the secret distributor.
Further, the secret distributor of the two-step quantum walking hidden state system is the unknown qubit of the hidden state of the first participant
Wherein | α |2+|β|2Completing stealth pass-through state, secret distributor prepares a1And ApTwo particles, A1Is an unknown qubit of a secret distributor wanting an invisible transition, also denoted coin1, ApIs a state of a position space. The first participant prepares particle B, also denoted coin 2. A. thepAnd B are both |0 in the initial state>. Through two-step quantum walking, particle A1Can complete the invisible state transmission with the particles B.
The quantum walking of the two-step quantum walking is described as follows:
wherein,
in the formula, C1Representing the operator of coin1-A1, the protocol selects the I operation as the operator.
The quantum walking of the two-step quantum walking is expressed as follows:
wherein:
where H denotes performing a Hadamard operation on the coin2-B particle when the initial state of the B particle is in the |0> state, the operator can also be replaced by an I operation when the initial state of the B particle is in the | + > state.
Further, the two-step quantum walking invisible state transfer system specifically comprises:
first, the first participant measures A with the X basis1The measurement result of the particle is marked as lambda 1(| +)>And | ->Respectively noted as 1 and-1). First participant measures A with Q-basepParticles of wherein | Q>＝{|-2'>,|-1>,|0>,|1>,|2'>}，
The measurement result was recorded as λ 2(| -2'>，|0>，|2'>Respectively-1, 0, 1).
Then, the first participant informs the first participant of the measurement results λ 1 and λ 2, and the first participant performs corresponding Pauli recovery operations on the B particles according to λ 1 and λ 2, and completes the operation on A1Invisible transport states of the particles.
Further, the quantum secret sharing method based on quantum walking specifically includes:
the first step, the particle preparation phase, the secret issuer, the first participant, the second participant prepares some particles for the stealth states of the quantum walking system. Secret issuers prepare a particle string Ap for stealth propagation of a quantum walking system, where Ap ═ 010203...0n>. The first participant prepares the initial state to be |0>The particle string Bp is used for completing an invisible state of transferring the sub-secret information of the secret issuer, wherein Bp is |010203...0n>. The second participant, like the first participant, prepares an initial state |0> particle string Cp for completing an invisible state of the sub-secret information with the secret issuer, where Cp | -010203...0n>。
Secondly, in the secret information encoding stage, a secret publisher generates a string of n-bit binary random sequence L, and then the secret publisher generates two particle bit strings { | b of n-bit original information M by using L according to an encoding rule1c1〉,|b2c2>,|b3c3>...|bncn>}. When a secret issuer generates a two-particle bit string | bc>Then, the secret issuer extracts | B > particles in order to generate B ═ B1b2b3...bn>Sequentially extracting | C > particles to generate C ═ C1c2c3...cnSuch as (c). The encoding of the secret information is completed.
And thirdly, in a secret information distribution stage, the secret publisher invisibly transmits the sub-secret information B of the M to the first participant and the sub-secret information C to the second participant through a quantum walking system.
Fourthly, the secret information is recovered.
Further, the secret information distribution stage of the third step specifically includes:
(1) the secret issuer randomly inserts k bits of decoy particles in B for eavesdropping detection. Through the quantum walking system introduced earlier, the secret issuer invisibly transmits information BK inserted with decoy particles to the first participant, wherein BK ═ b1,b2,b3...bn+k}. With biInvisible states of particles are examples. The secret publisher has prepared an initial state of |0>The particle Api of (a) is used for the invisible transmission state of a quantum walking system, and a secret issuer (b) usesiAs particles in the quantum walking system which need to be stealthy transferred, wherein bi＝α|0>+β|1>，|α|2+|β|21. The first participant has prepared the qubit string Bp in the particle preparation phase, the first participant will particle BpiAs receiving particles in quantum walking systems, BpiAt |0>State. The initial state of the whole quantum walking system is written as:
after quantum walking W1, the overall system state becomes:
|Φ>(1)＝(α|100>+β|-110>)p12。
after quantum walking W2, the overall system state becomes:
|Φ>(2)＝(α|200>+α|001>+β|010>+β|-211>)p12。
(2) secret publishers measure b with X baseiMeasurement result | +>And>are denoted as 1 and-1, respectively. Secret publishers continue to use Q-base | Q>＝{|-2′>，|-1>，|0>，|1>，|2′>Measure Api. Wherein,
measurement results | -2'>、|0>、|2′>Are noted as-1, 0, 1, respectively. The measurement result sequences of the X group and the Q group are respectively expressed as a lambda 1 sequence and a lambda 2 sequence.
(3) The secret publisher publishes the lambda 1 sequence and the lambda 2 sequence to the first participant, and the first participant performs Pauli recovery operation on the particles Bp by combining with the encoding rule to obtain the target state. The first participant completes the stealth of the unknown particle, Bp, from the secret publisheriThe state of the particles is converted to BKiThe state of the particles.
(4) After the first participant claims receipt of all particles, the secret issuer initiates eavesdropping detection. The secret publisher announces the locations of the bait particles and the measurement bases, the first participant selects the appropriate measurement base to measure each bait particle, and the secret publisher evaluates the error rate in the particle transmission process based on the first participant's measurements. If the error rate exceeds a specified threshold ε, the communication is terminated and the scheme is repeated from the beginning until the error rate is accepted. Otherwise, the secret issuer continues secret information distribution.
(5) The first participant discards the k-bit decoy particles in the particle string Bp to obtain n-bit quantum secret information Mb.
(6) The secret publisher also uses a quantum walking system to invisibly transmit the sub-secret information CK added with the k-bit decoy particles to a second participant, and the secret publisher uses gamma respectively1And gamma2The sequence of measurements of the measurement bases X and Q is indicated.
(7) Secret hairCloth will be gamma1Sequence and gamma2The sequence is published to a second participant, which incorporates the encoding rules. Pauli recovery operations are performed on the particles Cp to obtain the target state. The second participant completes the stealth of the unknown particle from the secret issuer, CpiConversion of the state of the particle to CKiThe state of the particles.
(8) After the second participant claims receipt of all particles, the secret issuer initiates eavesdropping detection. The secret publisher announces the locations of the bait particles and the measurement bases, the second participant selects the appropriate measurement base to measure each bait particle, and the secret publisher evaluates the error rate during the particle transmission based on the second participant's measurements. If the error rate exceeds a specified threshold epsilon, the second participant's communication with the secret publisher is terminated and the secret publisher repeats from scratch.
(9) The second participant discards the k-bit decoy particles in the particle string Cp to obtain n-bit quantum secret information Mc. At the moment, the secret publisher completes the invisible transfer of the sub-secret information of the secret publisher and the first participant and the second participant through the quantum walking system.
Further, the fourth secret information recovery stage specifically includes:
(1) the first participant has an unknown qubit string Mb in hand and the second participant has an unknown qubit string Mc in hand. The secret issuer publishes to the first participant and the second participant a character string L that determines the measurement basis of both particle strings.
(2) The first participant and the second participant select each qubit in the measurement basis Z or X measurement hand according to the string L, and adopt the Z basis measurement if the bit value of L is 0 and adopt the X basis measurement if the bit value of L is 1. The measurement results of Mb and Mc are represented as Rb and Rc, respectively.
(3) The first participant and the second participant recover the original secret information M of the secret issuer according to Mb XOR Mc by cooperation.
Another object of the present invention is to provide a quantum communication control system implementing the quantum secret sharing method based on quantum walking.
Another object of the present invention is to provide a quantum secret sharing system implementing the quantum walking-based quantum secret sharing method.
In summary, the advantages and positive effects of the invention are: the invention provides a quantum secret sharing scheme based on quantum walking, which completes the invisible state transfer of a single particle state through quantum walking.
Compared with the prior quantum secret sharing scheme, the invention has the following advantages:
firstly, the scheme of the invention does not need to prepare entangled-state particles in the preparation of initial particles, but adopts single particles, and the difficulty in preparing the entangled-state particles can indicate that the quantum secret sharing scheme based on the entangled state is not worth in some cases, and after all, the practicability is an important pursuit of quantum information theory.
Second, the original information of the secret distributor is skillfully split into sub-secret information and encoded into quantum states, and nobody can deduce the original information from the subsets.
Thirdly, the quantum communication process of the scheme is completed through quantum walking, and single particles spontaneously generate entanglement in the two-step quantum walking process, so that the invisible state transfer of the single particle state is realized.
In addition, security analysis shows that the encrypted quantum invisible state-transfer system based on quantum walking can resist internal attack and external entanglement attack according to the inherent unpredictable chaotic nonlinear dynamic behavior, and the technology is safe and feasible.
Drawings
Fig. 1 is a flowchart of a quantum secret sharing method based on quantum walking according to an embodiment of the present invention.
Fig. 2 shows the number of correctly guessed particles k and the probability P of successfully inferring the entire message when N is 100,500, and 1000, as provided by an embodiment of the present inventioniThe relationship between them.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is further described in detail with reference to the following embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention.
The prior quantum secret sharing scheme has the problem of preparation technical obstacle in the particle state preparation stage, the prior protocol mostly utilizes the entanglement characteristics of the Beel state, the GHZ state or the multi-particle entangled state to realize the quantum secret sharing scheme, however, the preparation of the entangled state needs to consume more resources, and the preparation and the measurement are not easy under the prior art, and the technical obstacle restricts the practicability of the quantum secret sharing scheme. While the quantum secret sharing scheme based on the single particle abandons the use of entangled particles, the invisible transmission state of the particles is difficult to achieve, and the protocol of the type is difficult to ensure the safety of particle transmission.
Aiming at the problems in the prior art, the invention provides a quantum secret sharing method based on quantum walking,
the present invention will be described in detail with reference to the following embodiments.
The concept of quantum walking was first proposed in 1993 by Aharonov et al, which was originally used to study the phenomenon of quantum diffusion, a quantum simulation of classical random walking. Subsequently, Ambainis et al proposed a linear quantum walking model and was promoted to the quantum walking model on the graph by Aharonov et al in 2001. Since then, many studies on quantum walking have been proposed. Recently, quantum walking has proven to be a promising resource in quantum information processing tasks, as it can be used to implement a set of gates and has been fabricated in many physical systems. Since the conditional walking operation in the quantum walking is a non-local operation, the quantum walking can be used to cause an entanglement relationship between particles. Inspired by the advantage of quantum walking, the invention provides a QSS scheme which is independent of entangled-state particle preparation and is more effective. In the scheme of the invention, the quantum walking system is adopted to cause the single particles to be spontaneously entangled in the quantum walking process, so that the scheme of the invention only needs to prepare the single particles without preparing entangled-state particles, and the single particles are easier to generate and measure in the prior art. In addition, the secret information is coded into the quantum state by combining with the coding rule of the non-entangled QSS scheme, the invisible state transmission of the particles is completed by adopting a two-step quantum walking system, and the safety of the particles in the communication process is ensured.
The quantum secret sharing method based on quantum walking provided by the embodiment of the invention comprises the following steps:
the method comprises a particle preparation stage, a secret information encoding stage, a secret information invisible transmission stage and a secret information recovery stage.
The secret distributor encodes the secret information to be shared into quantum states according to rules and divides the encoded particles into two parts.
And then, the two pieces of encrypted information are respectively and invisibly transmitted to the first participant and the second participant through the quantum walking system.
After eavesdropping detection is carried out, the first participant and the second participant measure the particle strings in the hands according to the result published by the secret distributor, and the invisible transmission state is completed.
Thereafter, the first participant and the second participant are able to decrypt the secret information of the secret distributor through collaboration.
In the embodiment of the invention, the invention relates to a secret information coding and splitting technology and a two-step linear quantum walking technology, which comprises the following steps:
(1) the secret information encoding splitting rule is as follows:
in the secret information encoding and splitting stage, a secret distributor generates a string of n-bit random binary sequences L, and then encodes n-bit original information M into two particle bit strings { | b according to L1c1〉,|b2c2〉,|b3c3>...|bncn>And the coding rules are shown in table 1.
TABLE 1 corresponding bit values for strings L, M and | bc >.
The meanings of Table 1 areThe measurement basis of the two-particle bit string | bc > is determined by L, and their eigenvectors are determined by M. This means that when L is 0, | biciThe measurement base of (g) is a Z base. When L is 1, | bici>The measured radical of (2) is the X radical. | bi>XOR|ci>Result of (2) is equal to Mi. Here | +>＝1/2(|0>+|1>)，|->＝1/2(|0>-|1>). Then, the secret distributor extracts | b in order>Particle generation B ═ B1b2b3...bn>In order extract | c>Particle generation C ═ C1c2c3...cn>. For example, if L is 0110, M is 1011, the two-particle bit string | bc>Can be generated into { |01>,|-->,|+->,|10>Thus B ═ 0- +1, C ═ 1-0, only the secret distributor knows exactly which two particle bits | B she producediciThe string "B" or "C" alone cannot infer the information of M. At this time, the encoding of the secret information is completed.
(2) The linear quantum walking technique is as follows:
quantum walking is quantum simulation of classical random walking, and a quantum transmission system can be encrypted according to the inherent unpredictable chaotic nonlinear dynamic behavior of the quantum transmission system.
Quantum walking occurs in a complex Hilbert space, consisting of two main quantum spaces, position space and coin space, respectively, denoted as
Where Hp represents the position span { n, n ∈ Z }, Hc represents the coin direction of quantum walking { |0>, |1> }, and the evolution of the total subsystem at each step of quantum walking can be described as an equation
Wherein
S is called the shift operator and is denoted as S ═ Σn|n+1><n |, and C is a coin operator, when the coin operator is |0>In state, it takes the coin from state | n>Move to state | n +1>When the coin operator is |1>In state, it causes the coin to move back to state | n-1 >.
(3) The two-step quantum walking invisible transmission system comprises the following steps:
through quantum walking, the shift operator can bring entanglement between a position space and a coin space, so that the invention can apply the entanglement resource to the invisible state of the particle. Specifically, the invention selects a quantum walking system based on two coins, and by selecting a proper initial state and a matched measuring base, the invention can successfully conceal any unknown quantum bit in the state of transmission between the two parties, so that the concealed state of transmission of the particles is not dependent on the preparation of initial entangled-state particles.
Assuming that the secret distributor wants to conceal the unknown qubits in the propagation state to the first participant
Wherein | α |2+|β|21. To complete the stealth pass, the secret distributor needs to prepare two particles, A1And Ap，A1Is an unknown qubit of a secret distributor wanting an invisible transition, also denoted coin1, ApIs a state of a position space. Likewise, the first participant prepares particle B, which is also denoted as coin 2. A. thepAnd the initial state of B is all |0 >. Through two-step quantum walking, particle A1Can complete the invisible state transmission with the particles B.
The step 1 quantum walking can be described as:
wherein,
in the formulae (3) and (4), C1Representing the operator of coin1-A1, the protocol selects the I operation as the operator.
The 2 nd quantum walking can be expressed as:
wherein,
in equation (5), H indicates that Hadamard operation is performed on the coin2-B particle when the initial state of the B particle is in the |0> state, this operator can also be replaced by an I operation when the initial state of the B particle is in the | + > state.
At this point, the first participant measured A with the X basis1The measurement result of the particle is marked as lambda 1(| +)>And | ->Respectively noted as 1 and-1). Thereafter, the first participant measured A with Q-basepParticles of wherein | Q>＝{|-2'>,|-1>,|0>,|1>,|2'>}，
The measurement result was recorded as λ 2(| -2'>，|0>，|2'>Respectively-1, 0, 1). Then, the first participant informs the first participant of the measurement results λ 1 and λ 2, and the first participant performs corresponding Pauli recovery operations on the B particles according to λ 1 and λ 2, and completes the operation on A1Invisible transport states of the particles.
In the embodiment of the invention, the following two-step quantum walking scheme is derived as follows:
through analytical calculation, it can be found that through W1, an entanglement relationship is generated between Ap particles and A1 particles, and the complex state of the Ap particles and the A1 particles is changed from the complex state
Become (a | 10)>+b|-11>)p1This is why quantum walking can be used for the stealth transport of particles without relying on the initial stage for the preparation of entangled particles.
Now, A1And B are also in an entangled state when the secret distributor measures particle A1According to the quantum mechanics theory, particles Ap and B will collapse to the corresponding states.
The secret distributor then measures Ap particles with the Q basis and particle B also collapses to the corresponding state.
Finally, the secret distributor informs the first participant of the measurement result. From these measurements, the first participant performs a corresponding Pauli operation on particle B to restore the unknown quantum state. The relationship between the measurements and Pauli operation is shown in Table 2.
TABLE 2 relationship between measurement results and Pauli operation
The above shows that the quantum walking theoretically has the capability of generating random space to resist violent attack due to the unpredictable chaotic nonlinear dynamic behavior inherent in the coin state, and further the application of the invisible state system based on the two-step quantum walking in quantum secret sharing can be researched. Therefore, a quantum secret sharing scheme based on quantum walking is proposed.
The invention is further described with reference to the following figures and examples.
Examples
As shown in fig. 1, the quantum secret sharing method based on quantum walking provided by the embodiment of the present invention includes a secret distributor Alice, a participant Bob, Charlie. The protocol is divided into a particle preparation phase, a secret information encoding phase, a secret information distribution phase and a secret information recovery phase. And the Alcie encodes the secret information to be shared into a quantum state according to a rule, divides the encoded particles into two parts, and then transmits the two pieces of encrypted information to Bob, Charlie in a hidden mode through a quantum walking system. After eavesdropping detection, Bob and Charlie measure the particle strings in the hand according to the result published by Alice, and invisible transmission is completed. Thereafter, Bob and Charlie are able to decrypt Alice's secret information by cooperation.
The method specifically comprises the following steps:
the first step, the particle preparation phase, Alice, Bob, Charlie, prepares some particles for the stealth states of the quantum walking system. Alice prepares a particle string Ap for the stealth pass state of the quantum walking system, where Ap ═ 010203...0n>. Bob prepares the initial state to be |0>The particle string Bp is used for completing the invisible transmission state of the sub-secret information with Alice, wherein Bp is |010203...0n>. Charlie is the same as Bob, who prepares the initial state as |0>The particle string Cp is used for completing an invisible state of transfer of the sub-secret information with Alice, where Cp ═ 010203...0n>。
Secondly, in the secret information encoding stage, Alice generates a string of n-bit binary random sequence L, and then Alice uses L to generate two particle bit strings { | b of n-bit original information M according to the rule in Table 11c1>，|b2c2>，|b3c3>...|bncn>}. When Alice generates a two-particle bit string | bc>Then, Alice extracts | b in order>Particle generation B ═ B1b2b3...bn>In order extract | c>Particle generation C ═ C1c2c3...cn>. At this time, the encoding of the secret information is completed.
And thirdly, in a secret information distribution stage, Alice invisibly transmits the sub-secret information B of the M to Bob and the sub-secret information C to Charlie through a quantum walking system.
(1) Alice randomly inserts k bits of decoy particles in B for eavesdropping detection. Through the quantum walking system introduced earlier, Alice can invisibly transmit information BK inserted with decoy particles to Bob, where BK ═ b1，b2，b3...bn+k}. The invention uses biInvisible states of particles are examples. Previously, Alice had prepared an initial state of |0>The particle Api is used for the invisible transmission state of a quantum walking system, and Alice uses biAs particles in the quantum walking system which need to be stealthy transferred, wherein bi＝α|0>+β|1>，|α|2+|β|21. Bob has prepared the qubit string Bp in the particle preparation phase, he has prepared the particle BpiAs receiving particles in quantum walking systems, BpiAt |0>State. Now, the initial state of the whole quantum walking system can be written as:
after quantum walking W1 in step 1, the overall system state becomes:
|Φ>(1)＝(α|100>+β|-110>)p12 (13)。
after quantum walking W2 in step 2, the overall system state becomes:
|Φ>(2)＝(α|200>+α|001>+β|010>+β|-211>)p12 (14)。
(2) for AliceMeasurement of X baseiMeasurement result | +>And | ->Are denoted as 1 and-1, respectively. Alice continues to use Q-base | Q>＝{|-2′>，|-1>，|0>，|1>，|2′>Measuring the Api, wherein,
measurement results | -2'>、|0>、|2′>Are noted as-1, 0, 1, respectively. Finally, the measurement result sequences of the X group and the Q group are respectively expressed as a λ 1 sequence and a λ 2 sequence.
(3) Alice publishes the λ 1 sequence and λ 2 sequence to Bob, who performs Pauli recovery operation on the particle Bp to obtain the target state in combination with Table 1. Thereafter, Bob completes the invisible state of the unknown particle, Bp, from AliceiThe state of the particles is converted to BKiThe state of the particles.
(4) After Bob claims to receive all the particles, Alice starts eavesdropping detection. Alice announces the locations of the bait particles and the measurement bases, Bob selects the appropriate measurement base to measure each bait particle, and Alice can evaluate the error rate during particle transmission based on Bob's measurements. If the error rate exceeds a specified threshold ε, they terminate the communication and then repeat the scheme from the beginning until the error rate is acceptable. Otherwise, Alice continues to distribute the secret information.
(5) Bob discards k-bit decoy particles in the particle string Bp to obtain n-bit quantum secret information Mb.
(6) Alice also uses a quantum walking system to invisibly transmit the sub-secret information CK added with the k-bit decoy particles to Charlie, and the Alice uses gamma to respectively transmit the state to Charlie at this time1And gamma2The sequence of measurements of the measurement bases X and Q is indicated.
(7) Alice will gamma1Sequence and gamma2The sequence is published to Charlie, which, in conjunction with Table 1, performs Pauli recovery operations on the particles Cp to obtain the target state. Thereafter, Charlie completes the invisible state of the unknown particle, Cp, from AliceiThe state of the particles is converted to CKiThe state of the particles.
(8) Alice starts eavesdropping detection after Charlie asserts that all particles are received. Alice announces the location of the bait particles and the measurement basis, Charlie selects the appropriate measurement basis to measure each bait particle, and Alice, based on the Charlie measurements, can evaluate the error rate during the particle transmission. If the error rate exceeds a specified threshold ε, they terminate the communication and Alice repeats the scheme from scratch.
(9) Charlie discards k-bit decoy particles in the particle string Cp to obtain n-bit quantum secret information Mc. At the moment, Alice completes the invisible transmission of the sub-secret information of Bob and Charlie through the quantum walking system.
Fourthly, a secret information recovery stage specifically comprises the following steps:
(1) bob has an unknown qubit string Mb in hand and Charlie has an unknown qubit string Mc in hand. Alice publishes to Bob and Charlie a string L that determines the measurement basis of the two particle strings.
(2) Bob and Charlie select each qubit in the measurement basis Z or X measurement hand from the string L, i.e., take the Z basis measurement if the bit value of L is 0 and the X basis measurement if the bit value of L is 1. The measurement results of Mb and Mc are represented as Rb and Rc, respectively.
(3) Bob and Charlie can recover the original secret information M of Alice according to Mb XOR Mc through cooperation.
The invention is further described below in connection with a security analysis.
The invention can effectively resist the attack of internal participants and the attack of external participants, and ensure the safety of the secret sharing process. The method comprises the following specific steps:
(1) an internal attack.
Bob cannot infer Alice's original information M by guessing the measurement of the particle string Mc in Charlie's hand, because only Alice knows exactly the pair of particles | b that she preparedici>. The present invention assumes that Bob has a 50% probability of guessing another particle ciThe probability P that Bob successfully deduces the entire message M can be quantitatively evaluated from statistical datai。
Where k denotes the total number of correctly guessed particles and N denotes the length of the whole message M. Probability PiFitting the binomial distribution and binomial coefficients.
The probability P of successfully inferring the entire message by calculating the number k of correctly guessed particles for N100, N500, and N1000iIt can be seen that for different N, PiIts maximum value (P) exists over the interval (0, k)max(N＝256)≈0.0575，Pmax(N＝512)≈0.0407，Pmax(N1024) ≈ 0.0288) and decreases with increasing N, as in fig. 2, the number of correctly guessed particles k and the probability P of successfully inferring the entire message when N100, 500 and 1000iThe relationship between them. Thus, it can be concluded that the encoding scheme can efficiently accomplish the encoding of secret information, and Bob cannot successfully infer the original information M by guessing.
(2) And (4) external attack.
The first method comprises the following steps: interception/retransmission attacks. An external attacker Eve or any participant cannot effectively steal secret information through interception/retransmission attack because the invention adds decoy particles in the quantum walking process of the particles to avoid the attack. Suppose that Eve intercepts the receiving particle string Bp of Bob in the quantum walking system and then retransmits another string of particle sequence to the receiving party. Since Eve has no knowledge of the location of the spoofed particles and the measurement basis, the recipient will obtain irrelevant measurements, which increases the error rate during the particle transmission, resulting in the termination of the session. Therefore, the intercept/retransmit attack of Eve is not effective for this scheme.
And the second method comprises the following steps: entanglement attack. Nor can an external attacker Eve, or any participant, obtain information through an entanglement attack because the information is encoded into a sequence of particles and encrypted with a discrete-time quantum walking system as it is transmitted in a quantum channel. Assuming that Eve is used for acquiring information of receiving particles in the invisible transmission state, capturing a receiving particle string Cp of Charlie in a quantum walking system, and usingNovel particles e and CpiTwisted together to form a larger Hilbert space, in which Cpi＝{|0>、|1>、|+>、|->}。
Where E is the single operation matrix of Eve, denoted as
Four { E } determined by the E operator00,e01,e10,e11The pure state satisfies the normalization condition
Because EE ═ 1, a, b, a ', b' satisfy the following relationships
|a|2+|b|2＝1,|a'|2+|b'|2＝1,ab*＝(a')*b' (23)。
The invention can obtain the following results:
|a|2＝|a'|2,|b|2＝|b'|2 (24)。
if Eve's attack particles are in an entangled state, such eavesdropper interference will not occurInevitably introducing errors, the invention can use PEDetects the presence of an eavesdropper.
PE＝|b|2＝1-|a|2＝|b'|2＝1-|a'|2 (25)。
If Eve does not want to introduce errors, the total particles must be related to the auxiliary particles of Eve in a direct-product state. However, in the direct-integration state, the auxiliary particles e and CpiThere is no correlation between the particles, so Eve does not get any useful information, proving that the entanglement attack is fruitless.
The above description is only for the purpose of illustrating the preferred embodiments of the present invention and is not to be construed as limiting the invention, and any modifications, equivalents and improvements made within the spirit and principle of the present invention are intended to be included within the scope of the present invention.
Claims (5)
1. A quantum secret sharing method based on quantum walking is characterized by comprising the following steps:
firstly, a secret publisher encodes shared secret information into a quantum state according to a secret information encoding and splitting rule; splitting the encoded particles, wherein in the secret information encoding stage, a secret publisher generates a string of n-bit random binary sequence L; then, the original information M of n bits is coded into a two-particle bit string { | b according to L1c1>,|b2c2>,|b3c3>...|bncn>}；
Step two, the linear quantum walking system transmits the two pieces of encrypted information to a first participant and a second participant in a hidden mode respectively;
step three, after eavesdropping detection is carried out, the first participant and the second participant measure the particle strings in the hands according to the result published by the secret publisher, and the invisible transmission state is completed;
step four, the first participant and the second participant decrypt the secret information of the secret publisher through cooperation;
the quantum secret sharing method based on quantum walking further comprises the following steps:
a particle preparation stage, namely, a secret publisher, a first participant and a second participant prepare the invisible-state particles for the quantum walking system; secret issuers prepare a particle string Ap for stealth propagation of a quantum walking system, where Ap ═ 010203...0n>(ii) a The first participant prepares the initial state to be |0>Is used for completing the invisible transmission state of the sub-secret information of the secret issuer, wherein Bp is |010203...0n>(ii) a The second participant prepares the initial state as |0 like the first participant>Is used to complete an invisible state of transfer of sub-secret information with a secret issuer, where Cp ═ 010203...0n>；
In the secret information encoding stage, when the secret publisher generates two-particle bit string | bc>Thereafter, the secret issuer extracts | b in order>Particle generation B ═ B1b2b3...bn>In order extract | c>Particle generation C ═ C1c2c3...cn>(ii) a The encoding of the secret information is completed;
in the secret information issuing stage, a secret issuer invisibly transmits the sub-secret information B of the M to a first participant and invisibly transmits the sub-secret information C to a second participant through a quantum walking system;
recovering the secret information;
the secret information issuing stage specifically includes:
(1) a secret issuer randomly inserts k bits of decoy particles into B for eavesdropping detection; through a quantum walking system, a secret publisher invisibly transmits information BK inserted with decoy particles to a first participant, wherein BK ═ b1,b2,b3...bn+k}; for biInvisible transport states of the particles; the secret publisher has prepared an initial state of |0>The particle Api of (a) is used for the invisible transmission state of a quantum walking system, and a secret issuer (b) usesiAs particles in the quantum walking system which need to be stealthy transferred, wherein bi＝α|0>+β|1>，|α|2+|β|21 is ═ 1; first ginsengThe AND party has prepared the qubit string Bp in the particle preparation phase, and the first participant will prepare the particle BpiAs receiving particles in quantum walking systems, BpiAt |0>State; the initial state of the whole quantum walking system is written as:
after quantum walking W1, the overall system state becomes:
|Φ>(1)＝(α|100>+β|-110>)p12；
after quantum walking W2, the overall system state becomes:
|Φ>(2)＝(α|200>+α|001>+β|010>+β|-211>)p12；
(2) secret publishers measure b with X baseiMeasurement result | +>And | ->Are noted as 1 and-1, respectively; secret publishers continue to use Q-base | Q>＝{|-2'>,|-1>,|0>,|1>,|2'>Measuring Api; wherein,
measurement results | -2'>、|0>、|2'>Are noted as-1, 0, 1, respectively; the measurement result sequences of the X group and the Q group are respectively expressed as a lambda 1 sequence and a lambda 2 sequence;
(3) the secret publisher publishes the lambda 1 sequence and the lambda 2 sequence to a first participant, and the first participant performs Pauli recovery operation on the particles Bp by combining with an encoding rule to obtain a target state; the first participant completes the stealth of the unknown particle, Bp, from the secret publisheriConversion of the state of the particles to BKiThe state of the particle;
(4) after the first participant claims to receive all the particles, the secret publisher starts to perform eavesdropping detection; the secret publisher announces the positions and measurement bases of the bait particles, the first participant selects a proper measurement base to measure each bait particle, and the secret publisher evaluates the error rate in the particle transmission process according to the measurement result of the first participant; terminating the first participant's communication with the secret publisher if the error rate exceeds a specified threshold epsilon; otherwise, the secret publisher continues to publish the secret information;
(5) the first participant discards k-bit decoy particles in the particle string Bp to obtain n-bit quantum secret information Mb;
(6) the secret publisher also uses a quantum walking system to invisibly transmit the sub-secret information CK added with the k-bit decoy particles to a second participant, and the secret publisher respectively uses gamma 1 and gamma 2 to represent the measurement result sequences of the measurement bases X and Q;
(7) the secret publisher publishes the gamma 1 sequence and the gamma 2 sequence to a second participant, the second participant incorporating the encoding rule; pauli recovery operation is carried out on the particles Cp to obtain a target state; the second participant completes the stealth of the unknown particle from the secret issuer, CpiConversion of the state of the particle to CKiThe state of the particle;
(8) after the second participant claims to receive all the particles, the secret publisher starts to perform eavesdropping detection; the secret publisher announces the positions and the measurement bases of the bait particles, the second participant selects a proper measurement base to measure each bait particle, and the secret publisher evaluates the error rate in the particle transmission process according to the measurement result of the second participant; terminating the second participant's communication with the secret publisher if the error rate exceeds a specified threshold epsilon;
(9) the second participant discards k-bit decoy particles in the particle string Cp to obtain n-bit quantum secret information Mc; at the moment, the secret publisher completes the invisible transfer of the sub-secret information of the secret publisher, the first participant and the second participant through the quantum walking system.
2. The quantum secret sharing method based on quantum walking as claimed in claim 1, wherein the step two of invisibly transmitting two pieces of encrypted information to the first participant and the second participant by the linear quantum walking system comprises: the linear quantum walking system occurs in a complex hilbert space, consisting of two quantum spaces, respectively a position space and a coin space, expressed as:
the method comprises the following steps that Hp represents position span { n, n belongs to Z }, Hc represents coin direction { |0>, |1> } of a linear quantum walking system, and a total subsystem in the linear quantum walking system is described as follows:
wherein
S is called the shift operator and is denoted as S ═ Σn|n+1><n |, and C is a coin operator; when the coin operator is |0>In state, coin is driven from state | n>Move to state | n +1>(ii) a When the coin operator is |1>In the state, the coin moves backwards to state | n-1>。
3. The quantum secret sharing method based on quantum walking as claimed in claim 1, wherein said step three is that after eavesdropping detection, the two-step quantum walking stealth state system selects a two-coin-based quantum walking system, selects a suitable initial state and a matching measurement base to stealth any unknown qubit in state between the first participant and the secret issuer and between the second participant and the secret issuer.
4. The quantum secret sharing method based on quantum walking as claimed in claim 3, wherein the secret issuer of the two-step quantum walking hidden-state system is the unknown qubit of the hidden-state of the first participant
Wherein | α |2+|β|2Completing invisible transmission state, secret issuer prepares A1And ApTwo particles, A1Is an unknown qubit in which a secret issuer wants to pass invisibly, also denoted coin1, ApIs a state of a position space; the first participant prepares particle B, also denoted as coin 2; a. thepAnd B are both |0 in the initial state>(ii) a Through two-step quantum walking, particle A1The invisible state transmission between the particles B can be completed;
the 1 st step quantum walking is described as:
wherein,
in the formula, C1Representing an operator of coin1-A1, and taking a protocol selection I operation as the operator;
the 2 nd step quantum walking is described as follows:
wherein:
where H represents the Hadamard operation performed on the coin2-B particle.
5. The quantum secret sharing method based on quantum walking as claimed in claim 1, wherein the secret information recovery phase specifically comprises:
(1) the first participant has an unknown qubit string Mb in hand and the second participant has an unknown qubit string Mc in hand; the secret publisher publishes a character string L which determines two particle string measurement bases to a first participant and a second participant;
(2) the first participant and the second participant select each qubit in the measurement basis Z or X measurement hand according to the character string L, and adopt Z basis measurement if the bit value of L is 0 and adopt X basis measurement if the bit value of L is 1; the measurement results of Mb and Mc are represented as Rb and Rc, respectively;
(3) the first participant and the second participant cooperate to recover the original information M of the secret issuer according to Mb XOR Mc.
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