A multi-party semi-quantum secret size comparison method based on two-dimensional single-photon states

Abstract

This invention discloses a method for comparing the size of multi-party semi-quantum secrets based on two-dimensional single-photon states. First, each participant shares a key and encrypts their own secret, maintaining the numerical relationship. A semi-honest quantum third party (STP) randomly prepares a two-dimensional single-photon sequence using a Z-based or X-based approach and distributes it to each participant. Each participant performs corresponding operations on each pair of adjacent photons according to the encrypted bit values ​​and sends the results back to the STP. The STP receives the returned photons and decodes them to obtain encrypted data strings. After security checks, the STP performs a bitwise XOR operation on each data string to determine the size relationship of the multi-party secrets. This invention uses experimentally easily prepared and manipulated two-dimensional single-photon states instead of high-dimensional or entangled states, requiring only standard optical devices, significantly reducing the implementation difficulty. By combining additive encryption with XOR comparison rules, it achieves size comparison while protecting privacy, solving the problems of difficult physical implementation and reduced security in existing methods. It has economical, efficient, and practical technical advantages.

Description

Technical Field

[0001] This invention belongs to the field of multi-party semi-quantum secret size comparison technology, specifically involving a multi-party semi-quantum secret size comparison method based on two-dimensional single-photon states. Background Technology

[0002] Quantum private comparison (QPC) refers to a quantum communication scheme in which two or more parties holding secret values ​​use quantum states as information carriers to determine whether the secret values ​​of each party are equal or related in magnitude, while ensuring the security of each party's secrets. Based on their function, quantum private comparison schemes can be divided into two categories: one compares whether secrets are equal (QPCE); the other compares the magnitude of secrets (QPCS). The latter, because it can compare the magnitude of secrets, has broader application prospects in scenarios such as confidential bidding, secret ranking, and the millionaire's problem.

[0003] However, existing schemes capable of comparing the size of quantum secrets all share common limitations: they all rely on high-dimensional quantum states as information carriers. At the current level of quantum technology, high-dimensional quantum states face difficulties in preparation and precise manipulation, especially in the measurement stage. This results in high physical implementation difficulty and resource consumption for quantum secret size comparison schemes, keeping related research at the theoretical model level. It hinders economical and efficient experimental verification and promotion under existing quantum technology conditions, severely restricting the process of moving quantum secret comparison from theory to practical application. In 2025, the applicant proposed a new semi-quantum secret comparison method based on two-dimensional Bell entangled states to achieve size comparison. However, experiments revealed that ordinary experimental conditions cannot maintain the ideal entanglement correlation between two entangled particles during transmission, directly leading to a significant decrease in the security of this method in practical applications.

[0004] Compared to two-particle entangled states or other multi-particle entangled states, single-particle states are easier to prepare and manipulate experimentally. Preparation, transmission, and measurement can be accomplished using only common optical devices such as lasers and polarization controllers. Experimental setups built using these standard and mature optical devices are economical, simple, and universally applicable. Therefore, how to achieve multi-party secret size comparison using experimentally easily prepared and manipulated two-dimensional single-photon states while ensuring security, and how to design a feasible experimental scheme using standard and mature optical devices to truly realize the quantum secret comparison scheme experimentally, has become a pressing technical problem in this field. Summary of the Invention

[0005] The purpose of this invention is to overcome the shortcomings of existing technologies and provide a method for comparing the size of multi-party semi-quantum secrets based on two-dimensional single-photon states. This method uses experimentally easy-to-prepare and manipulate two-dimensional single-photon states as information carriers. Secret information is encoded by changing the polarization state of the photons, thus requiring only the simplest quantum optics equipment and standard optical devices to prepare and manipulate the quantum state, significantly reducing the difficulty of experimental implementation. Simultaneously, binary arithmetic addition is used to preprocess the original secret, preserving its numerical relationship while protecting data privacy. Using binary logical XOR operations, a bit-by-bit determination rule is designed for the size relationship of multiple strings of binary data, thereby achieving an effective comparison of the size of multi-party semi-quantum secrets based on two-dimensional single-particle states.

[0006] To achieve the above-mentioned technical objectives, the present invention adopts the following technical solution.

[0007] A method for comparing the size of multi-directional semi-quantum secrets based on two-dimensional single-photon states includes the following steps: Step S1, Before starting communication, Each participant shares a key in advance. , , and , It is a key string The length is a positive integer, and each participant uses a key. Each person encrypts their secret information to obtain an encrypted data string. , ; Step S2: Semi-honest quantum third parties (STPs) are randomly prepared using Z-based or X-based methods. A single-photon state, , For data strings The length of the photon state, during the preparation process, STP records the position of each photon state and the preparation basis, and then prepares the first photon state. photon sequence Send to the Participating parties; Step S3, the Participants receive photon sequence Afterwards, according to the first The data string corresponding to each participant. The value in For photon sequences Each pair of adjacent photons and After performing the corresponding operation, it is sent back to STP; Step S4: STP receives photons sent back by each participant. and STP will photons Photons are measured by projection onto the Z-base. The measurement is projected onto the X-base, and the encrypted data strings of each participant are obtained based on the measurement results. ; Step S5: During the communication process, STP collaborates with each participant to perform security checks. If the check fails, the communication is terminated; if the check passes, step S6 is executed. Step S6: After the security check is passed, STP obtains the corresponding information from each participating party. A data string, STP for this The XOR operation is performed on each data string, and the size relationship between the data strings of each participant is obtained based on the operation result. Since the encryption operation does not change the relative size of the original secret, STP obtains the size relationship of the secret information of each participant by comparing the size relationship between the data strings of each participant.

[0008] Specifically, in step S1, each participating party uses a key. Each person encrypts their secret information to obtain an encrypted data string. ,in, The encryption method is represented as: (1); In the above formula, The addition operator that indicates a carry; As a participant Secret data at the m-th bit; The addition operation with carry, if the data string There was no carry at the highest position, corresponding to the participating party. Then in the data string Add a 0 before the highest bit, making Due to data string The number of digits increased by one, redefining The range of values ​​for , are positive integers and .

[0009] Specifically, the semi-honest quantum third party (STP) described in step S2 is randomly prepared using Z-based or X-based methods. A single-photon state, wherein the Z-based and X-based are respectively represented as: (2); In the above formula, and All are complex numbers, and satisfy the following conditions: ; and These represent the horizontal and vertical polarization states in optical polarization coding, respectively. and These represent the 45° polarization state and the 135° polarization state in optical polarization coding, respectively. Prepared A single-photon state is represented as: (3); The STP records the position and preparation basis of each photon, and ensures that the first photon in each column is... and the The preparation basis for each photon is the same. It is a positive odd number. ; The first photon sequence Represented as: (4).

[0010] Specifically, step S3 involves processing the photon sequence. Each pair of adjacent photons and Perform the corresponding operations, including: Operation 1: If If the value is 0, then the photon and It will be sent directly back to STP via the quantum reverse channel; Operation 2: If If the value is 1, then the participants use a polarization modulator to make the photon and The polarization direction is rotated by 45° from its original position and then sent back to STP through the quantum reverse channel.

[0011] Specifically, step S4 involves analyzing the measurement results to obtain the encrypted data strings of each participant. The analysis process of the measurement results is as follows: When STP prepares single-photon states using the Z-based method: Scenario 1: When STP prepares single-photon states using the Z-based matrix and the participant performs operation one, STP can accurately detect photons from the Z-based matrix. Photons cannot be accurately detected on the X-based matrix. At this time, STP records The value is 0; Scenario 2: When STP prepares single-photon states using the Z-based approach and the participants perform operation 2, STP can accurately detect photons from the X-based approach. However, photons cannot be accurately detected on the Z-based matrix. At this time, STP records The value is 1; When STP prepares single-photon states using the X-based method: Scenario 1: When STP prepares a single-photon state using the X-based matrix and the participant performs operation one, STP can accurately detect photons from the X-based matrix. However, photons cannot be accurately detected on the Z-based matrix. At this time, STP records The value is 1; Scenario 2: When STP prepares single-photon states using the X-based approach and the participants perform operation 2, STP can accurately detect photons from the Z-based approach. Photons cannot be accurately detected on the X-based matrix. At this time, STP records The value is 0.

[0012] Specifically, in step S5, the STP collaborates with each participating party to conduct security detection. The security detection methods include: using verification photons to verify operational consistency, using decoy state methods to detect channel security, and introducing a time synchronization system to detect eavesdropping behavior. The process of using verification photons to verify operational consistency for security testing is as follows: Each participant randomly selects two adjacent photons as check photons and publishes the operation on these two check photons. If the STP can detect these two photons from the same basis vector as the preparation basis, it means that the communication is normal and the detection is passed; otherwise, it means that the communication is abnormal and the communication is terminated.

[0013] Specifically, the STP described in step S6 applies to this... Perform an XOR operation on the data strings. The XOR operation process is as follows: After passing the security test, STP obtains... Data strings : (5); (6); In the above formula, For STP measurement The returned photon sequence The data string recorded later; According to the obtained Data strings STP calculation : (7); In the above formula, To compare intermediate values; Indicates the first Participating parties; For logical XOR operators, ; According to formula (7), if and If the values ​​are equal, then ,otherwise , is represented as: (8); right Time corresponding to and Further calculations : (9); In the above formula, To compare auxiliary values, used for determination and Size relationship; like ,but ;like ,but ; STP combines formulas (8) and (9) to obtain the judgment. and Expressions for size relationships: (10).

[0014] Specifically, in step S6, the STP obtains the size relationship of the secret information of each participant by comparing the size relationships between the data strings of each participant. The process is as follows: STP obtained the result of the XOR operation. and The size relationship, combined with the analysis obtained from the measurement results in step S4. This allows us to determine the size relationship between the data strings of each participant. Since the encryption operation does not change the relative size of the original secret, STP obtains the size relationship of the secret information of each participant by comparing the size relationship between the data strings of each participant.

[0015] Compared with the prior art, the present invention has the following beneficial effects: 1. The method of the present invention uses two-dimensional single-photon states, which are easy to prepare and manipulate experimentally, as information carriers and realizes secret information encoding by changing the polarization state of photons. This enables the design of a feasible quantum secret comparison experiment scheme with only the simplest quantum optical equipment and standard optical devices, significantly reducing the difficulty of experimental implementation.

[0016] 2. The method of this invention combines binary arithmetic addition to preprocess the original secret, which protects data privacy while keeping the numerical relationship unchanged; by using binary logical XOR operation, a bit-by-bit judgment rule for the size relationship of multiple strings of binary data is designed, thereby realizing an effective comparison of the size of multi-party semi-quantum secrets based on two-dimensional single-particle states, breaking through the bottleneck that the comparison of the size of quantum secrets generally relies on high-dimensional quantum states that are difficult to achieve experimentally as information carriers.

[0017] 3. The method of this invention comprehensively introduces a time synchronization system, a decoy state method, and an information verification mechanism to resist external attacks from both the physical implementation and procedural levels, thereby enhancing the robustness and security of the method. Finally, the security of the method is rigorously demonstrated theoretically, and the correctness of the method is verified through specific calculation examples. Detailed security analysis is conducted on typical physical attacks such as intercept-retransmission, measurement-retransmission, and photon number separation attacks, improving the security performance of the method of this invention in real-world scenarios. Attached Figure Description

[0018] To provide a more intuitive understanding of the technical implementation of this invention, the accompanying drawings involved in the embodiments of this invention are briefly described below. These drawings are used to assist in illustrating the implementation methods and are not intended to limit the invention. Those skilled in the art can make derivative designs based on the drawings without creative effort.

[0019] Figure 1 This is a flowchart illustrating a method for comparing the size of multi-directional semi-quantum secrets based on two-dimensional single-photon states according to the present invention. Figure 2 This is a schematic diagram illustrating the process of transmitting encrypted information from each participant in the method of the present invention to a semi-honest third party, STP. Figure 3 This is an optical path diagram of the experimental device in an embodiment of the present invention.

[0020] Figure 3 In the diagram, solid black lines represent optical paths, and dashed lines represent circuits; LS: Laser signal source; Decoy-IM: Intensity modulator; Pol-M1: First polarization modulator; BS: Beam splitter; VOA: Adjustable optical attenuator; QC: Quantum channel; Sy-T: Time synchronization system; CMP1: First computer; OPS1: First optical path selector; OP1: First optical path; OP2: Second optical path; Pol-M2: Second polarization modulator; OPS2: Second optical path selector; DZ: Single-photon Z-based detector; DX: Single-photon X-based detector; TDC: Time-to-digital converter; CMP2: Second computer. Detailed Implementation

[0021] To facilitate understanding and implementation of the present invention by those skilled in the art, the various steps of the method proposed in this invention are described in detail below. It should be understood that these embodiments are for illustrative purposes only and are not intended to limit the scope of the invention. Furthermore, it should be understood that after reading the teachings of this invention, those skilled in the art can make various modifications or alterations to the invention, and these equivalent forms also fall within the scope defined by the appended claims.

[0022] Example 1 like Figure 1 As shown, this embodiment discloses a method for comparing the size of multi-party semi-quantum secrets based on two-dimensional single-photon states. This method uses two-dimensional single-photon states, which are experimentally easy to prepare and manipulate, as information carriers. Secret information is encoded by changing the photon polarization state, thus requiring only the simplest quantum optical equipment and standard optical devices to prepare and manipulate the quantum state, significantly reducing the difficulty of experimental implementation. Simultaneously, binary arithmetic addition is used to preprocess the original secret, preserving its numerical relationship while protecting data privacy. Using binary logical XOR operations, a bit-by-bit determination rule for the size relationship of multiple strings of binary data is designed, thereby achieving an effective comparison of the size of multi-party semi-quantum secrets based on two-dimensional single-particle states. The method includes the following steps: Step S1, Before starting communication, Each participant shares a key in advance. , , and , It is a key string The length is a positive integer, and the participants use a key. Each person encrypts their secret information to obtain an encrypted data string. , ; Step S2: Semi-honest quantum third parties (STPs) are randomly prepared using Z-based or X-based methods. A single-photon state, , For data strings The length of the photon state, during the preparation process, STP records the position of each photon state and the preparation basis, and then prepares the first photon state. photon sequence Send to the Participating parties; Step S3, the Participants receive photon sequence Afterwards, according to the first The data string corresponding to each participant. The value in For photon sequences Each pair of adjacent photons and After performing the corresponding operation, it is sent back to STP; Step S4: STP receives photons sent back by each participant. and STP will photons Photons are measured by projection onto the Z-base. The measurement is projected onto the X-base, and the encrypted data strings of each participant are obtained based on the measurement results. ; Step S5: During the communication process, STP collaborates with each participant to perform security checks. If the check fails, the communication is terminated; if the check passes, step S6 is executed. Step S6: After the security check is passed, STP obtains the corresponding information from each participating party. A data string, STP for this The XOR operation is performed on each data string, and the size relationship between the data strings of each participant is obtained based on the operation result. Since the encryption operation does not change the relative size of the original secret, STP obtains the size relationship of the secret information of each participant by comparing the size relationship between the data strings of each participant.

[0023] Specifically, in step S1, each participating party uses a key. Each person encrypts their secret information to obtain an encrypted data string. The encryption method is represented as: (1); In the above formula, The addition operator that indicates a carry; As a participant Secret data at the m-th bit; The addition operation with carry, if the data string There was no carry at the highest position, corresponding to the participating party. Then in the data string Add a 0 before the highest bit, making Due to data string The number of digits increased by one, redefining The range of values ​​for , are positive integers and .

[0024] Specifically, the semi-honest quantum third party (STP) described in step S2 is randomly prepared using Z-based or X-based methods. A single-photon state, the process is as follows: In optical polarization coding, the four polarization states of a photon—0°, 90°, 45°, and 135°—can be used for coding. The horizontal polarization state is denoted as... The vertical polarization state is denoted as The 45° polarization state is denoted as The 135° polarization state is denoted as In two-dimensional Hilbert space, let , , , Then the polarization quantum state of any single photon can be represented by the Z-basis or X-basis as follows: (2); In the above formula, and All are complex numbers, and satisfy the following conditions: ; and These represent the horizontal and vertical polarization states in optical polarization coding, respectively. and It is a set of orthogonal and normalized complete bases; and These represent the 45° and 135° polarization states in optical polarization coding, respectively. and It is another set of orthogonal and unified complete bases; Prepared A single-photon state is represented as: (3); The STP records the position and preparation basis of each photon, and ensures that the first photon in each column is... and the The preparation basis for each photon is the same. It is a positive odd number. ; The first photon sequence Represented as: (4); It should be noted that in this embodiment, STP is a semi-honest quantum third party, which has the following characteristics: (1) They do not collude with eavesdroppers to leak information, nor do they actively attack equipment and channels to obtain information, but they will use the information they have to deduce the secrets of each participant; (2) Possesses quantum capabilities and can prepare, measure, and transmit two-dimensional single-photon states using Z-based or X-based methods; (3) It also has the ability to perform simple binary addition operations.

[0025] The participants in the quantum field do not possess quantum capabilities, as shown in Table 1 below, and can only perform the following two operations: operate( ): The particle is returned directly through the quantum reverse channel without any other operation; operate( ): Using a polarization modulator, the polarization direction of the photon is rotated by 45° from its original position, and then returned to the photon through a quantum reverse channel.

[0026] Specifically, step S3 involves processing the photon sequence. Each pair of adjacent photons and The corresponding operations are performed, as shown in Table 1 below. The operations include: Table 1. Participants' Views on Photons and Operation ; Operation (i): If If the value is 0, then the photon and It will be sent directly back to STP via the quantum reverse channel; Operation (ii): If If the value is 1, then the participants use a polarization modulator to make the photon and The polarization direction is rotated by 45° from its original position and then sent back to STP through the quantum reverse channel.

[0027] Specifically, such as Figure 2 As shown, in step S4, the encrypted data strings of each participant are obtained by analyzing the measurement results. The analysis process of the measurement results is as follows: like Figure 2 As shown in (a), when STP prepares single-photon states using the Z-based matrix and the participant performs operation (i), STP is able to detect photons from the Z-based matrix. Photons cannot be detected on the X-based matrix. At this point, the STP records the measurement results. The value is 0; when STP prepares a single-photon state using the Z-based approach and the participant performs operation (ii), STP can detect photons from the X-based approach. Photons cannot be detected on the Z-based substrate. At this point, the STP records the measurement results. The value is 1.

[0028] like Figure 2 As shown in (b), when the STP prepares a single-photon state using the X-based matrix and the participant performs operation (i), the STP is able to detect photons from the X-based matrix. Photons cannot be detected on the Z-based substrate. At this point, the STP records the measurement results. The value is 1; when STP prepares a single-photon state using the X basis and the participant performs operation (ii), STP can detect photons from the Z basis. Photons cannot be detected on the X-based matrix. At this point, the STP records the measurement results. The value is 0.

[0029] Specifically, in step S5, the STP collaborates with each participating party to conduct security detection. The security detection methods include: using verification photons to verify operational consistency, using decoy state methods to detect channel security, and introducing a time synchronization system to detect eavesdropping behavior. The process of using verification photons to verify operational consistency for security testing is as follows: Each participant randomly selects two adjacent photons as check photons and publishes the operation on these two check photons. If the STP can detect these two photons from the same basis vector as the preparation basis, it means that the communication is normal and the detection is passed; otherwise, it means that the communication is abnormal and the communication is terminated.

[0030] Specifically, the STP described in step S6 applies to this... Perform an XOR operation on the data strings. The XOR operation process is as follows: After passing the security test, STP obtains... Data strings : (5); (6); In the above formula, For STP measurement The returned photon sequence The data string recorded later; According to the obtained Data strings STP calculation : (7); In the above formula, To compare intermediate values; Indicates the first Participating parties; For logical XOR operators, ; According to formula (7), if and If the values ​​are equal, then ,otherwise , is represented as: (8); right Time corresponding to and Further calculations : (9); In the above formula, To compare auxiliary values, used for determination and Size relationship; like ,but ;like ,but ; STP combines formulas (8) and (9) to obtain the judgment. and Expressions for size relationships: (10).

[0031] Specifically, in step S6, the STP obtains the size relationship of the secret information of each participant by comparing the size relationships between the data strings of each participant. The process is as follows: STP obtained the result of the XOR operation. and The size relationship, combined with the analysis obtained from the measurement results in step S4. This allows us to determine the size relationship between the data strings of each participant. Since the encryption operation does not change the relative size of the original secret, STP obtains the size relationship of the secret information of each participant by comparing the size relationship between the data strings of each participant.

[0032] The feasibility and safety of the method of the present invention will be further illustrated below through a practical case.

[0033] like Figure 3 The image shown is the optical path diagram of the experimental device used in this case. Figure 3 In the diagram: solid black lines represent optical paths, and dashed lines represent circuits; LS: laser signal source; Decoy-IM: intensity modulator, used to prepare photon states of different intensities; Pol-M1: first polarization modulator, used to prepare the required polarization quantum states; BS: beam splitter, which splits some photons for sampling by the time synchronization system; VOA: tunable optical attenuator, which attenuates light of different polarizations to the single-photon level; QC: quantum channel, a free-space channel is used in this experiment; Sy-T: time synchronization system, used to achieve optical synchronization to ensure STP and The same photon is manipulated; CMP1: First computer, receives time synchronization system signals and sends instructions to OPS1; OPS1: First optical path selector, allowing the photon to pass through the first or second optical path. In the experiment, a direction-adjustable total reflection mirror can be used to achieve optical path selection; OP1: First optical path, when the photon passes through this optical path, it does not receive any operation and passes directly; OP2: Second optical path, when the photon passes through this optical path, it will pass through the second polarization modulator; Pol-M2: Second polarization modulator, rotates the polarization direction of the photon by 45°. For example, the original 45° polarized optical path becomes 90° or 0° after passing through this polarization modulator; OPS2: Second optical path selector; DZ: Single-photon Z-based detector; DX: Single-photon X-based detector; TDC: Time-to-digital converter, converts optical signals into electrical signals; CMP2: Second computer, equipped with the special software of this scheme, for STP to complete data processing.

[0034] The following are the specific experimental steps: Step 1*: Each participant encrypts its own secret information according to formula (1) and inputs the encrypted information into CMP1; Step 2*: STP first generates weakly coherent light using a laser source LS, and then uses an intensity modulator Decoy-IM to randomly modulate the light source into three different intensities. Next, the modulated light pulses are passed through a polarization modulator Pol-M1 to randomly modulate light pulses with Z-based or X-based polarization directions. Finally, these light pulses are passed through an adjustable optical attenuator VOA to attenuate them to the single-photon level. This prepares the single-photon states required for the experiment, which include decoy states and signal states. However, to ensure that all participants manipulate the same photon as STP, the light pulses exiting the first polarization modulator Pol-M1 need to pass through a beam splitter for sampling by the time synchronization system. Step 3*: When each participant receives the signal from the synchronization system, computer CMP1 synchronously sends instructions to optical path selector OPS1 according to the encrypted information. When the photon sent by STP is... or Then the participants, based on the encrypted information The value sends a command to the photon; Specific circumstances: When When the photon is in the first optical path selector OPS1, it causes the photon to pass through the first optical path, meaning no operation is performed on the photon, and it returns directly; when... At that time, the first optical path selector OPS1 causes the photon to pass through the second optical path, that is, the second polarization modulator Pol-M2 is used to rotate the polarization direction of the photon by 45° from the original, and then it returns to it through the quantum reverse channel; Step 4*: When the STP receives a photon He will use the second optical path selector OPS2 to project it onto the Z-base for measurement; when the STP receives the photon Similarly, using the second optical path selector OPS2, the light is projected onto the X-base for measurement. The measurement result is input to the time-to-digital converter TDC, which converts the optical signal into an electrical signal and inputs it to the computer CMP2. The CMP2 has a pre-programmed sequence that automatically records the measurement results. The value of .

[0035] Step 5*: Based on the measurement results, CMP2 completes security checks such as information verification and determines whether the current environment is suitable for communication. If it is not suitable for communication or there is eavesdropping, CMP2 will send a command to the laser LS to stop preparing the light source, and the entire process will be interrupted; otherwise, proceed to the next step. Step 6*: This step is entirely completed by the computer CMP2, which will organize the data according to formulas (5) and (6) into... The data strings are then sorted out using formulas (8) to (10). and The size relationship is determined by the size relationship of the secret information of each participant.

[0036] I. Feasibility Analysis: There were three participants They each possess binary secret data. , , These participants share a key. ;because The length of the shared key The length is small. In Add a zero before it to make its length 3; therefore, With the help of STP, the three participants implemented the plan as follows: Step S1: After the three participants complete the secret information encryption task using formula (1), they can obtain the following results: ; Because the data string in the above formula Only 3 people, exist Add a zero before it to make ; Step S2: Since there are a total of 3 participants, And the length of the shared key The value is 3, therefore Therefore, STP is randomly prepared. Single-photon state: ; STP prepares the The single photon state is organized into 3 photon sequences: ; Finally, STP will sequence the photons. Send to , Send to , Send to ; Step S3: After each participant receives a photon, they will perform corresponding operations on these photons according to the rules in Table 1. For example, due to ,so The operations on the received 8 photons are as follows: Operation ( ),operate( ),operate( ),operate( ),operate( ),operate( ),operate( ),operate( ); Step S4: After receiving the returned photons, the STP analyzes them. When the participant directly returns photons, the STP can measure and record the photons from the same substrate as the preparation substrate. Otherwise, record ; Step S5: Since only the feasibility of the solution is analyzed, it is assumed that the safety test is passed, and proceed directly to the next step; Step S6: After passing the security check and measuring all returned photons, STP will obtain three data strings, namely... , , By comparing these data with the original data string, we can obtain... , , Next, STP will compare using formula (10) to... and Relationship: ; Furthermore, we can obtain: ; Right now, ; Similarly, we can conclude that: ; And thus obtain ; Finally, we can obtain .

[0037] Based on the above results, it can be seen that the size relationship compiled according to the method of this invention is consistent with the actual secret values ​​of each participating party. , , The size relationships are consistent; therefore, STP compared the size relationships of each participant's secrets without knowing their secrets, making the method of this invention practically feasible.

[0038] II. Security Analysis The decoy state method used in this invention can successfully resist photon number separation attacks; however, there are many other common attack methods, such as intercept-retransmission attacks and intercept-measurement-retransmission attacks.

[0039] Next, this example will analyze in detail the impact of these attacks on the method of the present invention from both internal and external perspectives.

[0040] 1) External attacks The external attacker, Eve, possesses full quantum capabilities. He can use various means to attack channels, devices, and other systems, all in an attempt to obtain the secret information of the participants.

[0041] Suppose Eve launches an intercept-retransmission attack, intercepting a light pulse in the channel and then randomly transmitting another light pulse. It's important to note that in actual experiments, there are no ideal single-photon sources; light pulses are prepared to achieve single-photon levels. Because STP uses a three-intensity decoy state method, Eve doesn't know the intensity of the intercepted light pulse, so the intensity of her retransmitted pulse is random. STP can determine whether eavesdropping is occurring in the channel through measurement, statistics, and comparison.

[0042] If Eve launches an intercept-measure-retransmit attack, by intercepting and measuring the light pulse, she knows its intensity and can send a pulse of the same intensity. However, Eve doesn't know which polarizer (Z-based or X-based) modulated the pulse. Choosing the wrong measurement basis prevents Eve from correctly measuring the photon. Even if she measures correctly... The probability of a photon is only... When there are only 3 participants and the secret message is only 3 bits long, Eve's probability of correctly measuring the photon is only... Even if Eve correctly measures the photons, she cannot obtain information secretly related to the participants. This is because Eve does not know how the participants use the positive coding of light to load information. The scheme also includes a crucial anti-eavesdropping system: the "time synchronization system." Eve's interception, measurement, and retransmission all require time. If the STP does not receive a light pulse within the specified time sequence, the system will report an error. When the error rate exceeds a certain threshold, communication is determined to be abnormal, and communication is terminated.

[0043] In summary, during the communication process, an external attacker cannot obtain information related to the participants' secrets through various attack methods.

[0044] 2) Internal attack In quantum communication schemes, insiders are familiar with the communication rules, have access to secret information, and are closer to the secret information.

[0045] In the method of this invention, STP acts as a semi-honest third party, neither actively attacking the channel nor colluding with Eve to leak secret information, but deducing the secrets of each participant based on the information it already possesses. However, the information sent by each participant to STP through polarization operations has been encrypted using formula (1), and since STP cannot obtain the key... Therefore, it is impossible to obtain what is in hand. Information deduces the secret content.

[0046] In the method of this invention, each participant is aware of the rules of the encryption information and shares the same key. It is assumed that there is one or more external attackers with super quantum capabilities who disguise themselves as classical participants and carry irrelevant data to try to extract secret information from other participants.

[0047] The attacker first launches an intercept-retransmission attack. Since the intensity of the intercepted light pulse is unknown, the attacker can only randomly select the intensity of the light pulse for retransmission. At this time, the attacker will be identified as Eve, an external attacker. STP will be able to determine whether there is eavesdropping in the channel by measuring, statistically analyzing and comparing.

[0048] Suppose the attacker launches an intercept-measure-retransmission attack. Like the external attacker Eve, the attacker cannot determine whether the light pulse is modulated by a Z-based or X-based polarizer. If the wrong measurement basis is selected, the attacker will be unable to measure the photons correctly, just like Eve, and thus will be unable to obtain information that is secretly related to each participant.

[0049] Another scenario is that if a participant unintentionally obtains secret information, it could be considered deliberate sabotage. For example, they could secretly place another polarizer in the channel, arbitrarily rotating the polarization angle of photons, causing the STP to be unable to receive the correct encrypted information, thus leading to communication chaos. Step S5 of the method in this invention adds an information verification step. If the information verification fails, it indicates a problem in the communication process, making it unsuitable to continue communication, and the communication will be terminated.

[0050] In summary, the decoy state method, information verification mechanism, shared key, and time synchronization system enable the technical solution of this invention to effectively resist various attack methods such as photon number splitting attacks, intercept-retransmission attacks, and intercept-measurement-retransmission attacks. Therefore, neither internal nor external attacks can obtain any information related to the participants' secrets from the communication process of this solution; moreover, the information verification method can also determine whether the current environment is suitable for communication.

[0051] The above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention in any other way. Any person skilled in the art may make changes or modifications to the above-disclosed technical content to create equivalent embodiments. However, any simple modifications, equivalent changes, and modifications made to the above embodiments based on the technical essence of the present invention without departing from the scope of the present invention shall still fall within the protection scope of the present invention.

Claims

1. A method for multi-party semi-quantum secret size comparison based on two-dimensional single-photon states, characterized in that, Includes the following steps: Step S1, Before starting communication, Each participant shares a key in advance. , , , , It is the length of the key, a positive integer, which is used by each participant through the key. Each person encrypts their secret information to obtain an encrypted data string. , ; Step S2: Semi-honest quantum third parties (STPs) are randomly prepared using Z-based or X-based methods. A single-photon state, , For data strings The length of the photon state, during the preparation process, STP records the position of each photon state and the preparation basis, and then prepares the first photon state. photon sequence Send to the Participating parties; Step S3, the Participants receive photon sequence Afterwards, according to the first The data string corresponding to each participant. The value in For photon sequences Each pair of adjacent photons and After performing the corresponding operation, it is sent back to STP; Step S4: STP receives photons sent back by each participant. and STP will photons Photons are measured by projection onto the Z-base. The measurement is projected onto the X-base, and the encrypted data strings of each participant are obtained based on the measurement results. ; Step S5: During the communication process, STP collaborates with each participant to perform security checks. If the check fails, the communication is terminated; if the check passes, step S6 is executed. Step S6: After the security check is passed, STP obtains the corresponding information from each participating party. A data string, STP for this Perform an XOR operation on each data string and determine the size relationship between the data strings of each participant based on the result.

2. The method for comparing the size of multi-directional semi-quantum secrets based on two-dimensional single-photon states according to claim 1, characterized in that, In step S1, each participant uses a key. Each person encrypts their secret information to obtain an encrypted data string. The encryption method is represented as: （1）； In the above formula, The addition operator that indicates a carry. As a participant Secret data at the m-th position; The addition operation with carry, if the data string There was no carry at the highest position, corresponding to the participating party. Then in the data string Add a 0 before the highest bit, making Due to data string The number of digits increased by one, redefining The range of values ​​for , are positive integers and .

3. The method for comparing the size of multi-directional semi-quantum secrets based on two-dimensional single-photon states according to claim 2, characterized in that, The semi-honest quantum third party (STP) described in step S2 is randomly prepared using Z-based or X-based methods. A single-photon state, wherein the Z-based and X-based are respectively represented as: （2）； In the above formula, and All are complex numbers, and satisfy the following conditions: ; and These represent the horizontal and vertical polarization states in optical polarization coding, respectively. and These represent the 45° polarization state and the 135° polarization state in optical polarization coding, respectively. Prepared A single-photon state is represented as: （3）； The STP records the position and preparation basis of each photon, and ensures that the first photon in each column is... and the The preparation basis for each photon is the same. It is a positive odd number. ; The first photon sequence Represented as: （4）。 4. The method for comparing the size of multi-directional semi-quantum secrets based on two-dimensional single-photon states according to claim 3, characterized in that, The photon sequence described in step S3 Each pair of adjacent photons and Perform the corresponding operations, including: Operation 1: If If the value is 0, then the photon and It will be sent directly back to STP via the quantum reverse channel; Operation 2: If If the value is 1, then the participants use a polarization modulator to make the photon and The polarization direction is rotated by 45° from its original position and then sent back to STP through the quantum reverse channel.

5. The method for comparing the size of multi-directional semi-quantum secrets based on two-dimensional single-photon states according to claim 4, characterized in that, Step S4 describes analyzing the measurement results to obtain the encrypted data strings of each participant. The analysis process of the measurement results is as follows: When STP prepares single-photon states using the Z-based method: Scenario 1: When STP prepares single-photon states using the Z-based matrix and the participant performs operation one, STP can accurately detect photons from the Z-based matrix. Photons cannot be accurately detected on the X-based matrix. At this time, STP records The value is 0; Scenario 2: When STP prepares single-photon states using the Z-based approach and the participants perform operation 2, STP can accurately detect photons from the X-based approach. However, photons cannot be accurately detected on the Z-based matrix. At this time, STP records The value is 1; When STP prepares single-photon states using the X-based method: Scenario 1: When STP prepares a single-photon state using the X-based matrix and the participant performs operation one, STP can accurately detect photons from the X-based matrix. However, photons cannot be accurately detected on the Z-based matrix. At this time, STP records The value is 1; Scenario 2: When STP prepares single-photon states using the X-based approach and the participants perform operation 2, STP can accurately detect photons from the Z-based approach. Photons cannot be accurately detected on the X-based matrix. At this time, STP records The value is 0.

6. The method for comparing the size of multi-directional semi-quantum secrets based on two-dimensional single-photon states according to claim 5, characterized in that, In step S5, the STP collaborates with each participant to conduct security checks. The security checks include: verifying operational consistency using verification photons, detecting channel security using decoy state methods, and introducing a time synchronization system to detect eavesdropping behavior. The process of using verification photons to verify operational consistency for security testing is as follows: Each participant randomly selects two adjacent photons as check photons and publishes the operation on these two check photons. If the STP can detect these two photons from the same basis vector as the preparation basis, it means that the communication is normal and the detection is passed; otherwise, it means that the communication is abnormal and the communication is terminated.

7. The method for comparing the size of multi-directional semi-quantum secrets based on two-dimensional single-photon states according to claim 6, characterized in that, The STP described in step S6 is for this Perform an XOR operation on the data strings. The XOR operation process is as follows: After passing the security test, STP obtains... Data strings : （5）； （6）； In the above formula, For STP measurement participants Return photon sequence The data string recorded later; According to the obtained Data strings STP calculation : （7）； In the above formula, To compare intermediate values; Indicates the first Participating parties; For logical XOR operators, ; According to formula (7), if and If the values ​​are equal, then ,otherwise , is represented as: （8）； right Time corresponding to and Further calculations : （9）； In the above formula, To compare auxiliary values, used for determination and Size relationship; like ,but ;like ,but ; STP combines formulas (8) and (9) to obtain the judgment. and Expressions for size relationships: （10）。 8. The method for comparing the size of multi-directional semi-quantum secrets based on two-dimensional single-photon states according to claim 7, characterized in that, In step S6, the STP obtains the size relationship of the secret information of each participant by comparing the size relationships between the data strings of each participant. The process is as follows: STP obtained the result of the XOR operation. and The size relationship, combined with the analysis obtained from the measurement results in step S4. This allows us to obtain the size relationship between the data strings of each participant. Since the encryption operation uses arithmetic addition, it does not change the relative size of the original secret. STP obtains the size relationship of the secret information of each participant by comparing the size relationship between the data strings of each participant.

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