A threshold dynamic quantum secret sharing method based on quantum walk and lagrange interpolation

Abstract

The application relates to a threshold dynamic quantum secret sharing method based on quantum walk and Lagrange interpolation, which comprises the following steps: setting a non-negative integer secret to establish a first classical binary sequence with a target length and taking the first classical binary sequence as a secret share; preparing a decoy state photon, mixing the decoy state photon into a coin particle sequence to form a mixed sequence, judging whether a participant passes verification or not through the mixed sequence, randomly selecting a target operator for determining a second classical binary sequence if the participant passes the verification, performing bitwise XOR operation on the first classical binary sequence and the second classical binary sequence to obtain a decryption value of a private secret share; inputting the decryption value into a Lagrange interpolation model to obtain an initial hidden secret; and when personnel change or secret updating occurs in the participants during the process of obtaining the initial hidden secret, quantum resource multiplexing characteristics of quantum walk are utilized to re-distribute quantum resources. The application can efficiently add or remove participants and update secrets, and can effectively resist various attacks.

Description

Technical Field

[0001] This invention relates to the fields of quantum secret sharing and quantum walk technology, and in particular to a threshold dynamic quantum secret sharing method based on quantum walk and Lagrange interpolation. Background Technology

[0002] With the development of Internet technology, secret sharing is one of the important cornerstones of modern cryptography. Its core idea is to divide absolutely confidential information into multiple shares and have them stored separately by different participants.

[0003] However, with the rapid development of quantum computing technology, especially the introduction of Shor's algorithm, classical cryptosystems based on mathematical problems such as discrete logarithms and maximal integer factorization are facing an unprecedented threat to computing power.

[0004] To address this security crisis, quantum cryptography emerged. For the first time, existing technologies have introduced quantum mechanics principles into the field of secret sharing, proposing Quantum Secret Sharing (QSS). QSS utilizes the no-cloning theorem of quantum states and the Heisenberg uncertainty principle to guarantee unconditional security of information transmission at the level of physical laws. Early QSS protocols mostly relied on complex multi-particle entangled states. While these schemes theoretically possess perfect security, in actual physical experiments, the preparation, distribution, and joint measurement of high-dimensional entangled states are extremely difficult and highly susceptible to decoherence caused by environmental noise. This significantly limits the large-scale deployment of QSS protocols in real-world networks. Summary of the Invention

[0005] To address the problems existing in the prior art, the present invention aims to provide a threshold dynamic quantum secret sharing method based on quantum walk and Lagrange interpolation. The quantum walk-based protocol only requires a single particle and a simple Pauli operator to complete the state evolution, which has significant advantages in terms of preparation difficulty and resource consumption.

[0006] To achieve the above objectives, the present invention provides the following solution: A threshold dynamic quantum secret sharing method based on quantum walk and Lagrange interpolation includes: Set a non-negative integer secret, and use the non-negative integer secret to establish a first classical binary sequence of the target length, which is then used as a secret share; A decoy photon is prepared and mixed into a coin particle sequence to form a mixed sequence. The mixed sequence is used to determine whether the participant passes the verification. If the verification is passed, a target operator is randomly selected to determine the second classical binary sequence and is XORed with the first classical binary sequence to obtain the decryption value of the private secret share. The decrypted value is input into the Lagrange interpolation model to obtain the initial hidden secret. During the process of obtaining the initial hidden secret, if there are personnel changes or secret updates during the process of obtaining the initial hidden secret, the quantum resources are reallocated by utilizing the resource reuse property of quantum walk.

[0007] Optionally, establishing the first classical binary sequence includes: The non-negative integer secret is set by a secret distributor, and a large prime number is determined to set a finite threshold for the non-negative integer secret; Multiple coefficients are randomly generated, and a k-1 order stochastic polynomial model is established by combining them with the aforementioned non-negative integer secret: ; in, for polynomials of order 1 The secret is a non-negative integer. For large prime numbers, Randomly generated One coefficient, for The independent variable of the order polynomial, This is the modulo operation; Assign non-zero identity identifiers to the participants, input each non-zero identity identifier into the random multinomial model, and obtain the ordinate corresponding to each non-zero identity identifier: ; in, For k, which will be passed to each other via quantum walk The ordinate, Assign k to each The x-coordinate; The ordinate is converted into a first classical binary sequence of the target length.

[0008] Optionally, forming the mixed sequence includes: The decoy photons are prepared by constructing a quantum walking system on the complete graph using a secret distributor; the quantum walking system includes: position space particles and coin space particles; The decoy photons are mixed into the coin particle sequence according to the classical identity key pre-shared between the secret distributor and the participant, forming the mixed sequence.

[0009] Optionally, determining the second classical binary sequence includes: After the secret distributor sends the mixed sequence to the participant via a quantum channel, the position and measurement basis of the decoy state are announced. The participant measures the decoy state, and the secret distributor uses the measurement results to calculate the error rate. The error rate is used to determine whether the participant has passed the verification. If the error rate is higher than the security threshold, the protocol is terminated. If the error rate is lower than the security threshold, the participant's identity is confirmed to be legitimate and the channel is not being eavesdropped on. The participant randomly selects the Pauli operator as the coin operator and applies the coin operator to the coin particle sequence to obtain the classical operator sequence as the private secret share. At the same time, the position particle state of the secret distributor undergoes a conditional translation based on the participant's coin state to obtain the position state. The second classical binary sequence is obtained by measuring the positional particle sequence locally using the positional state through the secret distributor.

[0010] Optionally, obtaining the decrypted value of the private secret share includes: The secret distributor performs a bitwise XOR operation between the second classic binary sequence as a one-time cipher and the first classic binary sequence to generate an auxiliary sequence that is published to the participants. The auxiliary sequence is decrypted based on the participant's private secret share, and the polynomial ordinate is obtained as the decryption value of the private secret share.

[0011] Optionally, obtaining the initially hidden secret includes: When a participant needs to recover a secret, the coordinates of each participant are shared through a secure classic authentication channel; the coordinates include: a non-zero identity identifier and a decryption value; Input the coordinates into the Lagrange interpolation model to obtain the initial hidden secret.

[0012] Optionally, the Lagrange interpolation model includes: ; in, To initially conceal the secret, The ordinate of the polynomial is... Indicates the participants currently involved in the calculation. Represents the set except Other participants, For the collection of participants, , For participants , The assigned x-coordinate.

[0013] Optionally, reallocating quantum resources using the resource reuse property of the quantum walk includes: As the number of participants increases, the new ordinate is calculated locally by the secret distributor, and a new coin particle is prepared separately for the new participant. A random Pauli operator is then applied until a new auxiliary sequence is published. When the number of participants decreases, the original hidden secret remains unchanged. A new random polynomial model is generated by the secret distributor, invalidating the old coordinates held by the reduced participants. The remaining participants then choose a new coin operator until a new auxiliary sequence is announced, and the new coordinate points are regained.

[0014] Optionally, reallocating quantum resources using the resource reuse property of the quantum walk includes: When the secret is updated, the secret distributor uses the new initial hidden secret to generate a new random polynomial model, substitutes the non-zero identity of all participants into the new random polynomial model, calculates a new set of ordinates, and randomly selects a coin operator based on the participants to act on the coin particle sequence until a new auxiliary sequence is announced and the new coordinate points are regained.

[0015] The beneficial effects of this invention are as follows: When performing all the above dynamic updates (adding, subtracting, and changing passwords), this method perfectly utilizes the "resource reuse" property of quantum walks: there is no need to re-prepare and emit expensive photon sequences; participants only need to reapply the Pauli operator to the existing quantum coin locally.

[0016] This method uses a (k,n) threshold, which allows for the perfect and lossless reconstruction of the core secret S by arbitrarily gathering k valid nodes, thus achieving high fault tolerance.

[0017] This method utilizes the no-cloning theorem and decoy state detection in quantum mechanics at the bottom layer to create an absolutely eavesdroppable one-time codebook generation channel. At the top layer, it employs the information-theoretical security of the Shamir threshold scheme to establish an algebraic barrier against internal collusion. Even if the quantum device experiences some side-channel leakage, as long as the number of leaked coordinates is less than k, the core secret S remains mathematically absolutely secure.

[0018] This method does not introduce any high-dimensional complex entangled states (such as GHZ states) and uses single photons as quantum resources entirely, which is extremely easy to achieve with current physical experimental techniques. Attached Figure Description

[0019] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0020] Figure 1 This is a flowchart of a threshold dynamic quantum secret sharing method based on quantum walk and Lagrange interpolation according to an embodiment of the present invention; Figure 2 This is a quantum circuit diagram of quantum walk according to an embodiment of the present invention; Figure 3 A flowchart illustrating the addition of participants in an embodiment of the present invention; Figure 4 A flowchart illustrating the reduction of participants in an embodiment of the present invention; Figure 5 This is a flowchart illustrating the password change process according to an embodiment of the present invention. Detailed Implementation

[0021] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0022] To make the above-mentioned objects, features and advantages of the present invention more apparent and understandable, the present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0023] Key terms and constraints are as follows: Lagrange Interpolation: This is the core mathematical tool secretly shared by Shamir. In geometry, "two points determine a straight line, three points determine a parabola." Generalizing, k points can uniquely determine a (k-1)th order polynomial curve. Lagrange interpolation is an algebraic formula that, using k known coordinate points on a curve, derives the entire polynomial equation (especially finding the constant term, our core secret S) in reverse. One-Time Codebook: The only encryption method in information theory mathematically proven to be "absolutely secure." It requires the key to be completely random, at least equal in length to the plaintext, and never reused. Quantum Walk: The quantum mechanical counterpart of classical random walks (such as flipping a coin to decide whether to go left or right).

[0024] This invention perfectly achieves extremely high fault tolerance and physical-level anti-eavesdropping mechanisms. The method described in this invention includes the following three types of participating entities: Secret Distributor: Usually named Alice. She is the initiator of the entire secret-sharing protocol and the sole knowledgeer of the global polynomial, responsible for constructing the classical mathematical layer, preparing the quantum physical layer, protecting the communication channel with other participants, and initiating dynamic updates. Legitimate Participant: In this invention, she is named... After receiving the mixed quantum sequence from Alice, the participant verifies its identity by comparing the result of the decoy state, performs a quantum coin toss, and then calculates the coordinates, working with any k other legitimate participants to solve for the secret S. Potential attackers include external entities lurking in the quantum channel and classical public channel, attempting to steal the quantum state through "intercept-retransmission attacks" or "entanglement-measurement attacks," which the decoy state mechanism specifically uses to keep them out; participants colluding privately within the system, whose conspiracy is thwarted by the mathematical dimensionality reduction properties of Lagrange polynomials; and participants kicked out by Alice, who possess the old coordinates and attempt to intercept the public broadcast during system updates to derive the new coordinates. The scheme mandates that legitimate members "reapply the Pauli operator" during updates to defend against "double XOR" plaintext difference attacks launched by this entity.

[0025] like Figure 1 As shown, this embodiment discloses a threshold dynamic quantum secret sharing method based on quantum walk and Lagrange interpolation, including: setting a non-negative integer secret; using the non-negative integer secret to establish a first classical binary sequence of a target length, which is used as a secret share; preparing decoy photons; mixing the decoy photons into a coin particle sequence to form a mixed sequence; using the mixed sequence to determine whether a participant passes verification; if verification is successful, randomly selecting a target operator to determine a second classical binary sequence; and performing a bitwise XOR operation with the first classical binary sequence to obtain the decryption value of the private secret share; inputting the decryption value into a Lagrange interpolation model to obtain the initial hidden secret; and during the process of obtaining the initial hidden secret, when there are personnel changes or secret updates among the participants, the quantum resources are reallocated using the resource reuse characteristics of quantum walk.

[0026] Specifically, this embodiment discloses a threshold dynamic quantum secret sharing method based on quantum walk and Lagrange interpolation, including: S1: Initialization and Construction of Classical Polynomials. Alice defines the finite threshold. (where p is a large prime number), define a non-negative integer secret S ( Alice then constructs a k-1 order random polynomial, assigns a public and unique non-zero identity to each of the n participants, substitutes each identity into the polynomial, calculates the corresponding ordinate, and converts the ordinate into a binary sequence of fixed length.

[0027] S1-1: Alice determines a large prime number p such that the secret S to be shared satisfies All arithmetic operations will be performed over a finite threshold.

[0028] S1-2: Alice randomly generates k-1 coefficients and constructs a k-1 order random polynomial: .

[0029] S1-3: Alice assigns a public and unique non-zero identity to each of the n participants. Alice substituted each identifier into a polynomial to calculate the corresponding ordinate. .

[0030] S1-4: Alice will process each decimal integer... Convert to a fixed-length classical binary sequence This sequence is the secret share that will later be sent to each participant via a quantum channel.

[0031] S2: Quantum Resource Preparation and Identity Authentication. Alice prepares a quantum walking system on a complete graph with two vertices, prepares decoy photons, mixes the decoy photons into a sequence of coin particles, and sends them to... Alice then announces the location and measurement basis of the decoy state, calculates the error rate of the measurement results, and continues if it is below the safety threshold; otherwise, the protocol is terminated.

[0032] S2-1: Alice constructs a quantum walking system on a complete graph with two vertices. The system consists of position-space particles. and coin space particles composition, , ,..., For the first A legitimate participant A quantum operator is applied sequentially to the sequence of coin particles, and the initial state of each particle is prepared as follows: ,like Figure 2 As shown.

[0033] S2-2: Alice Preparation A decoy photon, according to Alice and Pre-shared classic identity key Alice mixed the decoy photon sequence into the coin particle sequence. In, a mixed sequence is formed. .

[0034] S2-3: Alice will Send via quantum channel . Upon receiving the information, Alice announced the location and measurement base of the decoy state. Extract and measure the decoy state, and publish the results. Alice calculates the error rate; if it is below the safety threshold, the test is confirmed. If the identity is valid and the channel is not being eavesdropped on, proceed to the next step; otherwise, terminate the agreement.

[0035] S3: Share distribution based on quantum walk. Randomly select either the Pauli I or Pauli X operator to flip the positional particle state of Alice. Alice records the operator sequence, then XORs the position particle with the coordinate binary sequence, and publishes the result. , This result allows us to determine the polynomial coordinates belonging to the program.

[0036] S3-1: Bob randomly selects either the Pauli I operator or the Pauli X operator as his coin operator. Bob applies the operator to the sequence of coin particles in his hand. He also recorded the sequence of classical operators he randomly selected as his private secret share. .

[0037] S3-2: At this point, the position particle The status will be based on The coin's state undergoes a conditional translation if the coin is flipped to... The position state is flipped; if it is The position state remains unchanged.

[0038] S3-3: Alice retrieves or directly sets the particle sequence at the location. use The base is measured to obtain the classical binary sequence. .at this time, Numerically equal to Private share .

[0039] S3-4: Alice's use As a one-time codebook, it contains the binary sequence of coordinates he actually intended to send. Perform a bitwise XOR operation. Alice generates and sends... Release of auxiliary sequences: .

[0040] S3-5: Received public Then, using their locally retained private share Decryption: At this point, His polynomial ordinates were obtained safely and accurately. The binary representation of the coordinates. Alice repeats this process for the remaining participants until everyone has their own coordinates.

[0041] S4: Threshold Cooperation and Secret Reconstruction. k Bobs use the coordinates they possess, combined with Lagrange interpolation, to collaboratively access the secret.

[0042] S4-1: When k legitimate participants in the system (denoted as set K) need to recover a secret, they share the coordinates they possess through a secure classical authentication channel. .

[0043] S4-2: The participants substituted the numerical values ​​into the Lagrange interpolation formula to calculate the constant term in the finite field: The result of the calculation is Alice's original hidden secret, S.

[0044] S5: Dynamic operation mechanism. For example... Figure 3-5 As shown, there is no need to reallocate quantum resources from scratch, which provides extremely high flexibility in the event of personnel changes or secret updates.

[0045] S5-1: Adding a participant: Alice only needs to calculate a new coordinate point locally. and convert to binary. Then for new participants individually Prepare new coin particles and send them (carrying a decoy state). Applying a random Pauli operator yields a new key. Alice measured The auxiliary sequence is published, and the existing n participants do not need to perform any operations, such as... Figure 3 As shown.

[0046] S5-2: Reduce the number of participants: Keep S in the polynomial unchanged, Alice generates a new polynomial. The participants who were kicked out The old coordinates are discarded. The remaining participants then choose a new set of coin operators and perform another quantum walk with Alice to obtain the result. Alice calculates new auxiliary values ​​and distributes the new coordinates to them, such as Figure 4 As shown.

[0047] S5-3: Changing the Secret: Alice uses a new secret Generate a new (k-1)th order polynomial Identify all participants Substituting the new polynomial, a new set of coordinates is calculated. Participant Bob randomly selects a coin operator to act on the existing coin particles. Alice calculates new auxiliary values ​​and distributes the new coordinates to the participants, such as... Figure 5 As shown.

[0048] The embodiments described above are merely preferred embodiments of the present invention and are not intended to limit the scope of the present invention. Various modifications and improvements made to the technical solutions of the present invention by those skilled in the art without departing from the spirit of the present invention should fall within the protection scope defined by the claims of the present invention.

Claims

1. A threshold dynamic quantum secret sharing method based on quantum walk and Lagrange interpolation, characterized in that, include: Set a non-negative integer secret, and use the non-negative integer secret to establish a first classical binary sequence of the target length, which is then used as a secret share; A decoy photon is prepared and mixed into a coin particle sequence to form a mixed sequence. The mixed sequence is used to determine whether the participant passes the verification. If the verification is passed, a target operator is randomly selected to determine the second classical binary sequence and is XORed with the first classical binary sequence to obtain the decryption value of the private secret share. The decrypted value is input into the Lagrange interpolation model to obtain the initial hidden secret. During the process of obtaining the initial hidden secret, if there are personnel changes or secret updates during the process of obtaining the initial hidden secret, the quantum resources are reallocated by utilizing the resource reuse property of quantum walk.

2. The threshold dynamic quantum secret sharing method based on quantum walk and Lagrange interpolation according to claim 1, characterized in that, Establishing the first classical binary sequence includes: The non-negative integer secret is set by a secret distributor, and a large prime number is determined to set a finite threshold for the non-negative integer secret; Multiple coefficients are randomly generated, and a k-1 order stochastic polynomial model is established by combining them with the aforementioned non-negative integer secret: ； in, for polynomials of order 1 The secret is a non-negative integer. For large prime numbers, Randomly generated One coefficient, for The independent variable of the order polynomial, This is the modulo operation; Assign non-zero identity identifiers to the participants, input each non-zero identity identifier into the random multinomial model, and obtain the ordinate corresponding to each non-zero identity identifier: ； in, For k, which will be passed to each other via quantum walk The ordinate, Assign k to each The x-coordinate; The ordinate is converted into a first classical binary sequence of the target length.

3. The threshold dynamic quantum secret sharing method based on quantum walk and Lagrange interpolation according to claim 1, characterized in that, Forming the mixed sequence includes: The decoy photons are prepared by constructing a quantum walking system on the complete graph using a secret distributor; the quantum walking system includes: position space particles and coin space particles; The decoy photons are mixed into the coin particle sequence according to the classical identity key pre-shared between the secret distributor and the participant, forming the mixed sequence.

4. The threshold dynamic quantum secret sharing method based on quantum walk and Lagrange interpolation according to claim 1, characterized in that, Determining the second classical binary sequence includes: After the secret distributor sends the mixed sequence to the participant via a quantum channel, the position and measurement basis of the decoy state are announced. The participant measures the decoy state, and the secret distributor uses the measurement results to calculate the error rate. The error rate is used to determine whether the participant has passed the verification. If the error rate is higher than the security threshold, the protocol is terminated. If the error rate is lower than the security threshold, the participant's identity is confirmed to be legitimate and the channel is not being eavesdropped on. The participant randomly selects the Pauli operator as the coin operator and applies the coin operator to the coin particle sequence to obtain the classical operator sequence as the private secret share. At the same time, the position particle state of the secret distributor undergoes a conditional translation based on the participant's coin state to obtain the position state. The second classical binary sequence is obtained by measuring the positional particle sequence locally using the positional state through the secret distributor.

5. The threshold dynamic quantum secret sharing method based on quantum walk and Lagrange interpolation according to claim 4, characterized in that, The decryption value for obtaining the aforementioned private secret share includes: The secret distributor performs a bitwise XOR operation between the second classic binary sequence as a one-time cipher and the first classic binary sequence to generate an auxiliary sequence that is published to the participants. The auxiliary sequence is decrypted based on the participant's private secret share, and the polynomial ordinate is obtained as the decryption value of the private secret share.

6. The threshold dynamic quantum secret sharing method based on quantum walk and Lagrange interpolation according to claim 1, characterized in that, Obtaining the initially hidden secret includes: When a participant needs to recover a secret, the coordinates of each participant are shared through a secure classic authentication channel; the coordinates include: a non-zero identity identifier and a decryption value; Input the coordinates into the Lagrange interpolation model to obtain the initial hidden secret.

7. The threshold dynamic quantum secret sharing method based on quantum walk and Lagrange interpolation according to claim 1, characterized in that, The Lagrange interpolation model includes: ； in, To initially conceal the secret, The ordinate of the polynomial is... Indicates the participants currently involved in the calculation. Represents the set except Other participants, For the collection of participants, , For participants , The assigned x-coordinate.

8. The threshold dynamic quantum secret sharing method based on quantum walk and Lagrange interpolation according to claim 1, characterized in that, The reallocation of quantum resources using the resource reuse properties of quantum walks includes: As the number of participants increases, the new ordinate is calculated locally by the secret distributor, and a new coin particle is prepared separately for the new participant. A random Pauli operator is then applied until a new auxiliary sequence is published. When the number of participants decreases, the original hidden secret remains unchanged. A new random polynomial model is generated by the secret distributor, invalidating the old coordinates held by the reduced participants. The remaining participants then choose a new coin operator until a new auxiliary sequence is announced, and the new coordinate points are regained.

9. The threshold dynamic quantum secret sharing method based on quantum walk and Lagrange interpolation according to claim 1, characterized in that, The reallocation of quantum resources using the resource reuse properties of quantum walks includes: When the secret is updated, the secret distributor uses the new initial hidden secret to generate a new random polynomial model, substitutes the non-zero identity of all participants into the new random polynomial model, calculates a new set of ordinates, and randomly selects a coin operator based on the participants to act on the coin particle sequence until a new auxiliary sequence is announced and the new coordinate points are regained.

Images