A Multi-Party Quantum Key Agreement Method under Phase Damping and Depolarizing Noise Channels

Abstract

The present invention relates to the fields of quantum communication and quantum cryptography, and particularly relates to a multi-party quantum key negotiation method under a phase damping and depolarization noise channel, including: constructing a blind matrix by a semi-honest third party TP, and sending the sum of each column element thereof as a blinding value to the corresponding participant; each participant embeds the secret value and the blinding value into a unitary operator, and uses the unitary operator to perform a unitary transformation on the particles with random numbers; the previous participant passes the transformed particles to the next participant, and after all participants have completed the operation, the particles return to the initiating participant's hand, and the initiating participant uses single-photon measurement to negotiate the final shared key; the present invention is relatively easy in terms of resource preparation, reduces the requirements for the quantum computing ability of the participants, expands the noise environment from joint rotation noise and joint dephasing noise to phase damping noise and depolarization noise, and enhances the security of the protocol by using the blinding value and random numbers.

Description

Technical Field

[0001] The present invention relates to the fields of quantum communication and quantum cryptography, and particularly relates to a multi-party quantum key negotiation method under a phase damping and depolarizing noise channel. Background Art

[0002] Quantum cryptography combines quantum mechanics with classical cryptography. Its security mainly depends on the basic principles of quantum mechanics, such as the superposition principle of quantum states, the quantum no-cloning principle, and the quantum uncertainty principle, etc., and can achieve unconditional security in theory. Quantum cryptography involves many fields and contains multiple important research branches such as quantum secret sharing, quantum homomorphic encryption, quantum signature, quantum multi-party computation, quantum key negotiation, etc. As one of the important branches of quantum cryptography, quantum key agreement (QKA) jointly negotiates a secure shared key among two or more participants, and before the negotiation is completed, no participant can predict or determine the final shared key. The shared key negotiated through the quantum key agreement protocol is only known to the entities participating in the communication, thus ensuring the security of data during the communication transmission process.

[0003] A secure quantum key negotiation algorithm must meet three requirements: one is correctness, all participants participating in the negotiation will finally obtain the same shared key, and only they themselves know this shared key; the second is security, an attacker cannot obtain any useful information about the final shared key without being detected, and thus cannot infer the shared key; the third is fairness, all participants make the same contribution during the negotiation process, and the final shared key must be calculated by all participants, and no participant can predict or control the final shared key in advance.

[0004] Most of the currently proposed quantum key negotiation algorithms are two-party negotiation algorithms, that is, they can negotiate a secure shared key between two participants. Since the secret values of the participants are transmitted to other participants after certain operations, the secret values of the participants are very likely to be inferred by the receiving participants, and the shared key is generated from the secret values of each participant. Therefore, there is also a certain threat to the security of the shared key. In current research, most algorithms are carried out in an ideal environment, and there are few protocols considering the noise environment. Even for the protocols considering execution in the noise environment, they all use entangled states through incoherent subspaces to resist joint dephasing noise and joint rotation noise, and there are even fewer protocols considering other noises. Summary of the Invention

[0005] To solve the above problems, the present invention provides a multi-party quantum key negotiation method under a phase damping and depolarizing noise channel. The method includes a semi-honest third party TP and n participants; each participant P i , i = {1, 2,..., n} has a secret integer k i , i = {1, 2,..., n}, and the method includes the following steps:

[0006] S1. Generate blinding values: The semi-honest third party TP randomly constructs an n×n blinding matrix, and generates n blinding values through the n×n blinding matrix and distributes them to the n participants; each participant P i embeds its corresponding blinding value and secret integer into a unitary operator for subsequent unitary transformation operations;

[0007] S2. Initiate negotiation: The i-th participant P i , as the negotiation initiator, generates n random numbers, calculates the Hash value of each random number, and publicly announces all Hash values except the i-th Hash value;

[0008] S3. Generate a particle sequence: The negotiation initiator P i prepares n random numbers according to mutually unbiased bases to obtain a particle sequence; the negotiation initiator P i performs a unitary transformation on the i-th particle in the particle sequence and sends the updated particle sequence to the next participant P i+1 ;

[0009] S4. The first round of verification loop: The participant P i+1 measures the (i + 1)-th particle in the particle sequence using a measurement basis and calculates the Hash value of the measurement result for verification; if the verification is successful, execute step S5, otherwise return to step S2;

[0010] S5. Update the particle sequence: The participant P i+1 performs a unitary transformation on the i-th particle in the particle sequence and sends the updated particle sequence to the next participant P i+2 to perform the next round of verification loop in the same way as step S4 until the (n - 1)-th round of verification loop is completed. The last participant updates the particle sequence and sends it to the negotiation initiator P i ;

[0011] S6. Generate a shared key: The negotiation initiator P i measures the i-th particle in the particle sequence sent by the last participant using a measurement basis and obtains the shared key according to the measurement result.

[0012] Furthermore, the transmission channels between participants, between the semi-honest third party TP and each participant are all phase damping noise channels or all depolarizing noise channels.

[0013] Furthermore, in step S1, the semi-honest third party TP selects random numbers to construct an n×n blind matrix B, expressed as:

[0014]

[0015] where b ti represents the element in the t-th row and i-th column of the blind matrix B, and the n random numbers b t1 , b t2 , …, b tn in the t-th row satisfy The semi-honest third party TP takes the sum of the elements in each column of the blind matrix as a blinding value and transmits it to the corresponding participant through a trusted classical channel.

[0016] Furthermore, the expression of a set of mutually unbiased bases is:

[0017]

[0018] where ω = e 2πi / d , d represents the dimension number of the quantum system, j = 0, 1, …, d represents the label of the unbiased basis, i = 0, 1, …, d - 1 represents the label of the vector in the given unbiased basis, and |x> represents a vector.

[0019] Furthermore, the negotiation initiator P i performs a unitary transformation on the i-th particle in the particle sequence, and its expression is:

[0020]

[0021]

[0022]

[0023] where represents the quantum state with the secret integer of participant P i after performing the unitary transformation on the i-th particle, i <U > represents the unitary operator with the secret integer of participant P i , k i represents the secret integer of participant P i , a i represents the blinding value sent by the semi-honest third party TP to participant P i , <|x > represents the initial quantum state of the i-th random number, i <X d > represents performing k times of X i gate operations, dGate operation, ω = e 2πi / d , |x> represents a vector, and <x| represents the dual vector of |x>.

[0024] Furthermore, the specific process of performing the first round of verification loop in step S4 includes:[[]]

[0025] S41. Participant P i+1 Measures the (i + 1)-th particle in the particle sequence using a measurement basis to obtain a measurement result, and calculates the Hash value of this measurement result;

[0026] S42. Participant P i+1 Queries the Hash value of the (i + 1)-th random number published by the negotiation initiator P i and compares it with the Hash value of this measurement result;

[0027] S43. If the two are equal, the verification passes; if the two are not equal, return to step S2 to restart the negotiation.

[0028] Furthermore, in step S5, the negotiation initiator P i Measures the i-th particle in the particle sequence sent by the last participant using a measurement basis. If this measurement result meets the generation condition, a shared key is obtained, and the generation condition is:[[]]

[0029]

[0030] The shared key is:[[]]

[0031]

[0032] where,[[]] represents the i-th random number generated by the negotiation initiator P i , k i represents the secret integer of participant P i , a i represents the blinding value sent by the semi-honest third party TP to participant P i , m represents the measurement result, d represents the dimension number of the quantum system, and mod represents the modulo operation.

[0033] Advantages of the present invention:[[]]

[0034] The present invention proposes a multi-party quantum key negotiation method under a phase damping and depolarizing noise channel. This method uses a semi-honest third party TP to construct a blind matrix to generate a blinding value, and participants use the blinding value and random numbers to blind their own secret values, further ensuring the confidentiality of the participants' secret values and the security of the shared key.

[0035] The present invention uses single particles as information transmission carriers, is relatively simple to prepare, and can complete key negotiation only through unitary transformation and single-photon measurement, effectively reducing the requirements for the quantum computing ability of participants and having better practicability.

[0036] The present invention expands the noise environment from the normal joint rotation noise and joint dephasing noise to phase damping noise and depolarization noise. The proposed method is applicable to phase damping noise channels and depolarization noise channels, and analyzes the error probability of executing the negotiation process in phase damping noise channels and depolarization noise channels. Moreover, the present invention is a multi-party protocol rather than a two-party protocol, having better generality. Description of the Drawings

[0037] Figure 1 It is a flowchart of a multi-party quantum key negotiation method based on a noise environment provided by an embodiment of the present invention;

[0038] Figure 2 It is a framework diagram of a quantum key negotiation process provided by an embodiment of the present invention. Detailed Embodiments

[0039] Next, the technical solutions in the embodiments of the present invention will be clearly and completely described in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all the embodiments. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts shall fall within the protection scope of the present invention.

[0040] An embodiment of the present invention provides a framework diagram of a quantum key negotiation process, as Figure 2 shown, including a semi-honest third party TP and n participants P1, P2,..., P n (n > 2), and each participant P i (i = 1, 2,..., n) has a secret integer k i . The semi-honest third party TP will honestly execute the steps of the protocol, but also wants to steal the secret integers of some participants to deduce the shared key.

[0041] In one embodiment, based on the above block diagram, the present invention provides a multi-party quantum key negotiation method under phase damping and depolarization noise channels. The flow of the method is as Figure 1 shown, including:

[0042] S1. The semi-honest third party TP randomly constructs an n×n blind matrix and generates n blinded values through the blind matrix to distribute to n participants.

[0043] Specifically, in step S1, the semi - honest third party TP selects random numbers to construct an n×n blind matrix B, expressed as:

[0044]

[0045] where b ti represents the element in the t - th row and i - th column of the blind matrix B, and the n random numbers b t1 , b t2 , …, b tn in the t - th row satisfy Set a i to represent the sum of the elements in the i - th column of the n×n blind matrix B, that is Obviously, there exists

[0046] The semi - honest third party TP takes the sum of the elements in each column of the blind matrix as a blinding value, obtaining n blinding values {a1, a2, …, a n}}. The semi - honest third party TP transmits the blinding values {a1, a2, …, a n}} to the corresponding participants P1, P2, …, P n (n > 2) through the classical channel.

[0047] Specifically, each participant embeds its corresponding blinding value and the secret integer into the unitary operator, so that each participant has a corresponding updated unitary operator for subsequent unitary transformation operations, and the unitary operators for subsequent unitary transformation operations among the participants are different.

[0048] S2. The i - th participant P i acts as the negotiation initiator to generate n random numbers, calculate the Hash value of each random number, and publicly announce all Hash values except the i - th Hash value.

[0049] Specifically, the specific process of step S2 includes:

[0050] S21. The negotiation initiator P i generates a set including n random numbers where represents the i - th random number generated by the negotiation initiator P i ;

[0051] S22. The negotiation initiator P i calculates the Hash value of each random number in the set R, obtaining a set of Hash values where represents the Hash value of the i - th random number generated by the negotiation initiator P i ;

[0052] S23. The negotiation initiator Pi Hash value of the i-th random number Keep it confidential and only disclose the Hash values of the remaining n - 1 random numbers.

[0053] S3. Negotiation initiator P i Prepare all n random numbers as particles according to mutually unbiased bases to obtain a particle sequence.

[0054] Specifically, in this embodiment, a set of mutually unbiased bases is represented as:

[0055] |ψ x > = |x>

[0056]

[0057] where ω = e 2πi / d , d represents the dimension number of the quantum system, j = 0, 1,..., d represents the label of the unbiased basis, i = 0, 1,..., d - 1 represents the label of the vector in the given unbiased basis, and |x> represents the vector.

[0058] This embodiment includes multiple sets of mutually unbiased bases. The negotiation initiator P i Prepare all the random numbers in set R as particles according to multiple sets of mutually unbiased bases to obtain a particle sequence where represents the particle prepared from the i-th random number by a set of mutually unbiased bases, or it can be said that the initial quantum state of the i-th random number after being prepared by a set of mutually unbiased bases is

[0059] S4. Negotiation initiator P i Perform a unitary transformation on the i-th particle in the particle sequence and send the updated particle sequence to participant P i+1 .

[0060] Specifically, the negotiation initiator P i Selects the i-th particle from the particle sequence E Performs a unitary transformation Expressed as:

[0061]

[0062]

[0063]

[0064] where represents the quantum state of the i-th particle after the participant P i performs a unitary transformation, carrying the secret integer of the participant P i , represents carrying the participant Pi Unitary operator of the secret integer, k i Denote the participant P i Secret integer of, a i Denote the blinded value sent by the semi - honest third party TP to the participant P i Denote the initial quantum state after the preparation of the i - th random number Denote the execution of k i times of X d gate operations Denote the execution of a i times of Y d gate operations, ω = e 2πi / d , |x> represents a vector, <x| represents the dual vector of |x>

[0065] Update the particle sequence E to obtain the particle sequence Negotiation initiator P i Send the particle sequence S i to the next participant P i+1 .

[0066] S5. Participant P i+1 Measure the (i + 1)-th particle in the particle sequence using the measurement basis and calculate the Hash value of the measurement result for verification

[0067] Specifically, participant P i+1 Receives the particle sequence S i and uses the measurement basis {|e0>, |e1>, …, |e d-1 >} to measure the (i + 1)-th particle in this particle sequence to obtain the measurement result Participant P i+1 Calculates the Hash value of the measurement result Compares the Hash value of the (i + 1)-th random number published by the negotiation initiator P i with the Hash value of the measurement result If the two are equal, the verification passes; if the two are not equal, terminate the step and start the negotiation again

[0068] S6. If the verification is successful, then participant P i+1 Performs a unitary transformation on the i - th particle in the particle sequence and sends the updated particle sequence to the next participant P i+2 .

[0069] Specifically, participant P i+1 Performs a unitary transformation on the i - th particle in the particle sequence S i ​​​​​ Particle sequence S i The i-th particle in will be converted to:

[0070]

[0071] Update the particle sequence S i Obtain the particle sequence Participant P i+1 Send the particle sequence S i+1 To the next participant P i+2 .

[0072] S7. Participant P i+2 Measure the (i + 2)-th particle in the particle sequence using the measurement basis and calculate the Hash value of the measurement result for verification; if the verification is successful, then participant P i+2 Perform a unitary transformation on the i-th particle in the particle sequence and send the updated particle sequence to the next participant P i+3 .

[0073] Specifically, participant P i+2 Receives the particle sequence S i+1 And uses the measurement basis {|e0>, |e1>, …, |e d-1 >} to measure the (i + 2)-th particle in this particle sequence Obtain the measurement result Participant P i+2 Calculates the Hash value of the measurement result Of Compare the Hash value of the (i + 2)-th random number published by the negotiation initiator P i With the Hash value of the measurement result Of If the two are equal, the verification passes, and participant P Performs a unitary transformation on the i-th particle in the particle sequence S i+2 Update the particle sequence S i+1 Obtain the particle sequence S And send it to the next participant P i+1 Obtain the particle sequence S i+2 And send it to the next participant P i+3 ; if the two are not equal, terminate the step and start the negotiation again.

[0074] S8. Participant P i+3 , P i+4 , …, P n , P1, …, P i-1 Execute the above similar steps until participant P i-1 After performing the operations, return the updated particle sequence to the negotiation initiator P i .

[0075] Negotiation Initiator P i Measure the i-th particle in the received particle sequence using the measurement basis to obtain the final shared key.

[0076] Specifically, all participants perform the unitary transformation on the i-th particle in the particle sequence, that is, the i-th particle in the updated particle sequence returned by participant P in S8 is transformed into: i-1

[0077]

[0078] Negotiation Initiator P i Use the measurement basis {|e0>, |e1>, …, |e d -1>} to measure the i-th particle in particle sequence S i-1 The measurement result is m. Since The following equation is satisfied:

[0079]

[0080] Since Negotiation Initiator P i knows the value of the random number r i i So he can deduce That is, the final shared key

[0081] In practical applications, multi-party quantum negotiation is carried out in a noisy quantum channel. The noise will affect the quantum states transmitted during the negotiation process, thereby resulting in an error-prone shared key. The multi-party quantum negotiation method proposed in the embodiments of the present invention can be carried out under a phase-damping noise channel or a depolarizing noise channel, and finally obtain the correct shared key.

[0082] In one embodiment, to verify the performance of the present invention, the state and operation vectorization form is used to analyze the error probability generated when the above negotiation process is executed in phase-damping noise or depolarizing noise.

[0083] Specifically, the phase-damping noise is expressed as:

[0084]

[0085] where p represents the noise parameter, are all the qubit density matrices of the quantum states. Since any qubit density matrix can be converted into a vector form Therefore The vector form of The vector form of E1(ρ) is ​There is a relational equation between |ρ> and |ρ'> as follows:

[0086] ε1|ρ> = |ρ'>

[0087]

[0088] Unitary transformation is a diagonal matrix, defined as:

[0089]

[0090] Assume that the dimension of the quantum Hilbert space is 2. During the key negotiation process, the initiator P i The i-th particle in the particle sequence E prepared is expressed as:

[0091]

[0092] The i-th particle is represented by the density matrix as:

[0093]

[0094] Convert it to vector representation:

[0095]

[0096] Based on the vector representation |ρ0> of the i-th initial quantum state (the i-th particle), when each participant performs a unitary transformation on the i-th initial quantum state and transmits the transformed quantum state in a phase-damping noise channel, after one participant after another performs the transformation and transmission, the initial quantum state finally evolves into:

[0097]

[0098] Its corresponding density matrix representation is:

[0099]

[0100] Since The initial state is When is even, When is odd, Therefore, when the initiator P i measures the final state ρ with {|+>, |->} as the measurement basis f1 the conditional probabilities of obtaining |+> and |-> are respectively:

[0101]

[0102]

[0103] It can be found that in the absence of noise, that is, when p = 0, there is a perfect correlation between the result of the last measurement and the phase sum. For example, when is even, P(|+>) = 1 and P(|->) = 0, and when is odd, P(|+>) = 0 and P(|->) = 1.

[0104] The quantum states sent by each participant are affected by noise during transmission, which may introduce errors, reducing this correlation and leading to mistakes. For example, when is even and the measurement basis is |->, is odd and the measurement basis is |+>, then errors will occur, resulting in incorrect measurement results. Assuming that the probability of each case is 1 / 2, therefore, the error probability can be obtained as:

[0105]

[0106] Assuming the requirement where δ is a parameter less than 1, we can obtain:

[0107]

[0108] It can be found that the noise parameter is inversely proportional to the number of participants.

[0109] Similarly, the depolarizing noise is expressed as:

[0110]

[0111] where, tr(ρ) = a + d, and the quantum operator of the depolarizing noise channel is:

[0112]

[0113] After each participant successively performs a unitary operation on the initial quantum state |ρ0> and transmits it through the depolarizing noise channel, the final quantum state evolves into:

[0114]

[0115] It can be found that the above final quantum state is the same as the final quantum state after transmission through the phase damping noise. Therefore, similarly, the depolarizing noise level is inversely proportional to the number of participants.

[0116] The above-described embodiments illustrate the advantages of the present invention. The semi-honest third party TP added in the present invention only constructs the blind matrix and generates the blinded values, and does not participate in the subsequent negotiation process. Moreover, the initial quantum state embeds the random numbers of the participants. Therefore, the semi-honest third party cannot obtain any useful information about the secret values of the participants and the final shared key. All participants in the present invention only need to complete the preparation of single particles, perform unitary transformations, and measure single particles, which is simple in operation and effectively reduces the requirements for the quantum computing capabilities of the participants.

[0117] Although the embodiments of the present invention have been shown and described, those of ordinary skill in the art can understand that various changes, modifications, substitutions, and variations can be made to these embodiments without departing from the principles and spirit of the present invention. The scope of the present invention is defined by the appended claims and their equivalents.

Claims

1. A multi-party quantum key negotiation method under a phase damping and depolarizing noise channel, characterized in that including a semi - honest third - party TP and n participants; each participant P i , i = {1, 2,..., n} has a secret integer k i , i = {1, 2,..., n}, and the method includes the following steps: S1. Generate blinding values: The semi - honest third - party TP randomly constructs an \(n\times n\) blind matrix, and generates \(n\) blinding values through the \(n\times n\) blind matrix and distributes them to \(n\) participants; each participant P i embeds its corresponding blinding value and the secret integer into the unitary operator for subsequent unitary transformation operations; S2. Initiate negotiation: The i-th participant P i As the negotiation initiator, generate n random numbers, calculate the Hash value of each random number, and publicly announce all Hash values except the i-th Hash value; S3. Generate a particle sequence: The negotiation initiator P i Prepares n random numbers according to mutually unbiased bases to obtain a particle sequence; the negotiation initiator P i Performs a unitary transformation on the i-th particle in the particle sequence and sends the updated particle sequence to the next participant P i+1 ; Negotiation initiator P i Perform a unitary transformation on the i-th particle in the particle sequence, and its expression is: Among them, represents the participant P i After performing the unitary transformation on the i-th particle, the quantum state with the secret integer of the participant P i of represents the unitary operator with the secret integer of the participant P i k i represents the secret integer of the participant P i a i represents the blinding value sent by the semi-honest third party TP to the participant P i ; represents the initial quantum state of the i-th random number, represents the execution of k i times of X d gate operations, represents the execution of a i times of Y d gate operations, ω = e 2πi / d , |x> represents a vector, <x| represents the dual vector of |x>; S4. First round of verification loop: Participant P i+1 Measure the (i + 1)-th particle in the particle sequence using the measurement basis, and calculate the Hash value of the measurement result for verification; if the verification is successful, execute step S5, otherwise return to step S2; S5. Update the particle sequence: Participant P i+1 Perform a unitary transformation on the i-th particle in the particle sequence and send the updated particle sequence to the next participant P i+2 Execute the next round of verification loop in the same way as in step S4 until the (n - 1)-th round of verification loop is completed. The last participant updates the particle sequence and sends it to the negotiation initiator P i ; S6. Generate a shared key: The negotiation initiator P i Measures the i-th particle in the particle sequence sent by the last participant using a measurement basis, and obtains a shared key based on the measurement result; Negotiation Initiator P i Measure the i-th particle in the particle sequence sent by the last participant using the measurement basis. If the measurement result meets the generation condition, a shared key is obtained. The generation condition is as follows: Among them, represents the i-th random number generated by the negotiation initiator P, m represents the measurement result, d represents the dimension number of the quantum system, and mod represents the modulo operation. i ​ 2. The multi-party quantum key negotiation method under a phase damping and depolarizing noise channel according to claim 1, wherein The transmission channels between participants and between the semi - honest third - party TP and each participant are all phase - damping noise channels or all depolarizing noise channels.

3. A multi-party quantum key negotiation method under a phase damping and depolarizing noise channel according to claim 1, characterized in that In step S1, the semi - honest third - party TP selects random numbers to construct an n×n blind matrix B, which is expressed as: Among them, b ti represents the element in the t-th row and i-th column of the blind matrix B. The n random numbers b t1 , b t2 , …, b tn satisfy The semi-honest third party TP takes the sum of the elements in each column of the blind matrix as a blinding value and transmits it to the corresponding participant through a trusted classical channel.

4. A multi-party quantum key negotiation method under a phase damping and depolarizing noise channel according to claim 1, wherein The expression of a set of mutually unbiased bases is: where ω = e 2πi / d , d represents the dimension number of the quantum system, j = 0, 1, …, d represents the label of the unbiased basis, i = 0, 1, …, d - 1 represents the label of the vectors in a given unbiased basis, and |x> represents a vector.

5. A multi-party quantum key negotiation method under a phase damping and depolarizing noise channel according to claim 1, characterized in that, The specific process of the first - round verification loop in step S4 includes: S41. Participant P i+1 Measure the (i + 1)-th particle in the particle sequence using a measurement basis to obtain a measurement result, and calculate the Hash value of the measurement result; S42. Participant P i+1 Query negotiation initiator P i The Hash value of the (i + 1)-th random number published, and compare it with the Hash value of this measurement result; S43. If the two are equal, the verification passes; if the two are not equal, return to step S2 to start the negotiation again.

6. A multi-party quantum key negotiation method under a phase damping and depolarizing noise channel according to claim 1, characterized in that The shared key is:

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