A Quantum-Secure Vector Dominance Judgment Method Based on Privacy Protection

Abstract

The present invention provides a quantum security vector dominance determination method based on privacy protection, including: the first participant and the second participant respectively disguise their own private vectors, and perform encoding and permutation operations to obtain the permuted and random number encrypted results; a semi-honest third party generates an entangled state sequence; the semi-honest third party retains the first particle sequence in the entangled state sequence, and distributes the second and third particles to the first participant and the second participant in order, respectively, and performs corresponding processing to obtain a comparison result. If the comparison results are all 2, the semi-honest third party declares that the vector of the first participant dominates the vector of the second participant, otherwise it declares that it does not dominate. The method also prevents input camouflage attacks by the participants themselves. The method can achieve efficient determination of vector dominance under polynomial complexity, and can resist various attacks and reduce privacy leakage.

Description

Technical Field

[0001] The present invention belongs to the field of privacy - protected computing in quantum cryptography, and particularly relates to a quantum - secure vector domination determination method based on privacy protection. Background Art

[0002] In the context of the rapid development of quantum information technology, Secure Computational Geometry (SCG) based on computational complexity assumptions has gradually exposed its security limitations under the powerful computing power of quantum computers. However, combining quantum mechanisms with encryption techniques can break through these limitations and achieve unconditional security for SCG. This combination has given rise to a new field called Quantum Secure Computational Geometry (QSCG). In recent years, solutions to geometric problems mostly rely on dividing geometric shapes into grids and using private - set computing to achieve approximate geometric calculations. However, these methods have problems such as difficulty in accurately identifying boundary nodes and high resource consumption. Summary of the Invention

[0003] Object of the Invention: The technical problem to be solved by the present invention is to provide a quantum - secure vector domination determination method based on privacy protection in view of the deficiencies of the prior art, aiming to solve the security limitations of traditional Secure Computational Geometry (SCG) faced with the powerful computing power of quantum computers and overcome the accuracy and resource - consumption problems in existing solutions to geometric problems. This method uses the quantum private permutation method to scramble the participant information encoded in an entangled state and perform domination determination on the scrambled information while ensuring the privacy of both parties' information. Compared with traditional methods, the quantum - secure vector domination protocol of the present invention has achieved significant improvements in computational efficiency and communication cost and extended its application in privacy - protected computing. In addition, the method of the present invention also has the advantages of high efficiency, scalability, and flexibility, and can meet the security computing requirements in various complex scenarios, especially in distributed computing, cloud - computing platforms, Internet - of - Things devices, and occasions with special requirements for privacy protection.

[0004] The method of the present invention includes the following steps:

[0005] Step 1, the first participant Alice and the second participant Bob respectively perform input camouflage on their private vectors.

[0006] Step 2, the first participant Alice encodes each element in the private vector into a quantum entangled state, and respectively selects the first and second particle sequences to send to the second participant Bob for permutation operations.

[0007] Step 3, the second participant Bob permutes the first particle sequence and sends it back to the first participant Alice. He adds the elements of the prepared random vector to the second particle sequence in terms of phase and then permutes it, and then sends it back to the first participant Alice;

[0008] Step 4, the first participant Alice measures the final result quantum state and obtains the result encrypted by permutation and random numbers;

[0009] Step 5, the second participant Bob performs random vector addition and permutation operations on his own private vector, and permutes the fixed vector to obtain the vector ;

[0010] Step 6, the first participant Alice and the second participant Bob share a random bit string, which is used to encrypt their respective private information when performing phase shift operations in Step 7. The semi - honest third - party Trent generates an entangled state sequence;

[0011] Step 7, the semi - honest third - party Trent retains the first particle sequence in the entangled state sequence, distributes the second and third particles to the first participant Alice and the second participant Bob in order respectively. The first participant Alice and the second participant Bob perform phase shift operations with private vector information on the sequences;

[0012] Step 8, the semi - honest third - party Trent measures the final quantum state with encrypted information in terms of phase and tells the first participant Alice the measured encoded result R i ;

[0013] Step 9, the first participant Alice performs exclusive - OR of the encoded result R i and the random bit string and takes the bit - wise negation to obtain the comparison result U;

[0014] Step 10, the first participant Alice and the second participant Bob send the comparison result U and the vector to the semi - honest third - party Trent respectively to repeat the private comparison. If the comparison result R i is all 2, the semi - honest third - party Trent announces that the vector of the first participant dominates the vector of the second participant, otherwise it announces non - domination.

[0015] Step 1 includes: setting , , where n is the number of elements of the vector, N is the element dimension, and D is the dimension of the quantum state; the first participant Alice inputs a private vector , and the second participant Bob inputs a private vector , a permutation operation , , , is the i-th element in the private vector A, is the i-th element in the private vector B, where i ranges from 1 to n;

[0016] The first participant, Alice, performs input masking on her private vector A:

[0017] (1),

[0018] Obtain the masked private vector , with a length of 4n. Define the i-th element among the 4n elements in as , where i ranges from 1 to 4n, to obtain ;

[0019] The second participant, Bob, performs input masking on his private vector B:

[0020] (2),

[0021] Obtain the masked private vector , with a length of 4n. Define the i-th element among the 4n elements in as , to obtain ; and let the vector .

[0022] Step 2 includes: The first participant, Alice, encodes each element in the private vector into a quantum state, where i ranges from 1 to 4n, and the formula is:

[0023] (3),

[0024] where is the quantum state encoded by the first participant, Alice, for the i-th element in the private vector; is a complex phase factor, usually used to describe the phase change in the quantum state, , are the ground states in the quantum state , is stored in the particle , is stored in the particle ; is an index variable;

[0025] and respectively represent particles storing two different d - qubits. The qubit on particle is in , and the qubit on particle is in , , where e is the Euler number, is the imaginary unit;

[0026] Subsequently, the first participant, Alice, separately selects the first particle and the second particle from each quantum state to form two ordered particle sequences, namely the first sequence H and the second sequence T, , , where represents the first particle selected by the first participant, Alice, from the quantum state encoded from the 4n - th element in the disguised private vector , and represents the second particle selected by the first participant, Alice, from the quantum state encoded from the 4n - th element in the disguised private vector .

[0027] Step 2 further includes: The first participant, Alice, initializes a particle g and calculates the sum of the qubits of the particles in the first sequence H using the two - particle addition gate operation, and sends the first sequence H to the second participant, Bob;

[0028] The operation formula is:

[0029] (4)

[0030] (5)

[0031] where represents performing the two - particle addition gate operation on particles t and h. The colon ":" in formula 4 is meaningless, are two particles used to demonstrate the role of BSUM. In the operation, the quantum state is transformed into , and the quantum state of qubit y is added to the quantum state of qubit x through the BSUM operation to form a new quantum state ;

[0032] represents the particle and the particle Apply the BSUM operation so that the qubit on particle g becomes the particle and particle the sum of the qubits on; then sequentially perform to so that the qubit on particle g is ;

[0033] Indicates that after applying the qubit on particle g becomes , Indicates that the first participant Alice encodes the 4n-th element in the private vector into the quantum state the first particle in.

[0034] Step 3 includes: The second participant Bob permutes the particles in the first sequence H operation, and records the result as , then sends it to the first participant Alice; The first participant Alice performs a single-particle modular multiplication gate operation on the particles in the received result to calculate ;

[0035] Then apply gate to sequentially subtract the qubits of the particles in the received result from particle g;

[0036] Finally, measure particle g. If the measurement result is , it means that the second participant Bob has not performed measurements, forgeries, or other attacks that destroy the entangled state on the particles, and the total particles are verified correctly after the permutation operation, and the protocol continues; otherwise, the protocol terminates;

[0037] The formula for the single-particle modular multiplication gate operation is:

[0038] (6)

[0039] (7)

[0040] where is an operation example, indicating that the particle t is multiplied by the modular parameter m, and the particle t with the quantum state obtains the result after the operation, represents the particle Perform modular multiplication (D - 1). "mod" (taking the modulus) represents modular arithmetic, that is, calculating the remainder after dividing two numbers;

[0041] The applied gates subtract the received particle in the formula:

[0042] (8)

[0043] Subsequently, the first participant Alice sends the second sequence to the second participant Bob. The second participant Bob prepares a random vector , where , is the i-th element in the random vector , and i ranges from 1 to 4n. The second participant Bob applies a phase operation to each particle of the second sequence , and after performing a permutation operation, the result is obtained, and then it is sent back to the first participant Alice;

[0044] The operation has the formula:

[0045] (9),

[0046] (10),

[0047] where is a single - particle rotation gate operation; is an example operation with parameter m, acting on particle t, and parameter m represents the amplitude of rotation; represents performing a phase rotation with parameter m on qubit x to adjust the phase factor of the qubit, where y is the control factor of the rotation operation, is the quantum state of qubit x; represents performing a phase rotation operation with parameter on particle .

[0048] Step 4 includes: The quantum state in the hands of the first participant Alice is:

[0049] (11)

[0050] where is when the second participant Bob adds the random vector element to the phase of the particles in the second sequence and for the first sequence , the second sequence The result obtained after applying the same permutation operation; Is the i-th element of the first participant Alice's private vector after input masking, and Is the i-th quantum state In which the entangled particles are;

[0051] Subsequently, the first participant Alice applies a two-particle XOR gate Operation, inverse quantum Fourier transform Operation:

[0052] (12),

[0053] (13),

[0054] (14),

[0055] (15),

[0056] Where Is the conventional symbolic representation of the inverse Fourier transform, The symbol in the upper right corner Indicates the inverse transform, Indicates performing the inverse quantum Fourier transform operation on particle t, Indicates performing a two-particle XOR gate operation on particle t and particle h. After the operation, particle h will be XORed with the quantum state value in particle t. Among them, particle t and particle h and the quantum bits x and y on their particles are examples, which can more clearly describe the process and function of the quantum gate operation; Similarly, for After performing the two-particle XOR gate, will obtain , that is ; Is the XOR operation, 0 1 = 1, 1 1 = 0, 0 0 = 0, 1 0 = 1, Is to perform the inverse quantum Fourier transform Operation on particle To obtain the information stored in the phase of particle ;

[0057] Finally, the first participant Alice performs the inverse quantum Fourier transform on particle To extract the data in the phase, and measures to obtain , to obtain the result vector , where Is The i-th element of

[0058] Step 5 includes: The second participant Bob adds the vector to the vector , and then and are permuted by the operation:

[0059] (16)

[0060] where refers to after the i-th element undergoes a private permutation, it is permuted from the i-th element to the -th element, and the mapping of i in the permutation operation is ;

[0061] Then the second participant Bob respectively obtains , , ; is plus the random vector and the result vector obtained after the overall permutation; is the i-th element in the vector , is after the permutation operation.

[0062] Step 6 includes: The first participant Alice privately shares a set of random bit strings with the second participant Bob, where for , ; The semi-honest third party Trent generates sequences of three-particle entangled states, where the i-th entangled state is in the form of:

[0063] (17)

[0064] where is the i-th entangled state among the 4n entangled states generated by the semi-honest third party Trent, where the three entangled particles are respectively the particle retained by the semi-honest third party Trent, the particle sent to the first participant Alice, the particle sent to the second participant Bob, represents the quantum state on the particle , represents the quantum state on the particle , represents the particle quantum states on indicating the second particle in the \(i\)-th three-particle entangled state generated by the semi-honest third party Trent and indicating the third particle in the \(i\)-th three-particle entangled state generated by the semi-honest third party Trent in

[0065] Step 7 includes: The semi-honest third party Trent selects the second and third particles from each entangled state to form an ordered sequence and ; then the semi-honest third party Trent sends and to the first participant Alice and the second participant Bob respectively; indicating the second particle in the \(4n\)-th three-particle entangled state generated by the semi-honest third party Trent and the calculation formula is: , indicating the third particle in the \(4n\)-th three-particle entangled state generated by the semi-honest third party Trent;

[0066] After the first participant Alice receives , she applies the phase shift operator to and adds her private information : , where represents the phase shift operation on the particle , represents that the operation of the operator is to multiply the phase by , the same as , but shows the operation performed on the particle; represents the phase factor used to control the phase change of the quantum state, ;

[0067] The second participant Bob applies the phase shift operator to : , where represents the phase shift operation on the particle , represents that the operation of the operator is to multiply the phase by ;

[0068] Then the first participant Alice and the second participant Bob send and Send it to the semi - honest third party Trent.

[0069] Step 8 includes: After integrating all the received particles, the semi - honest third party Trent obtains the following quantum state:

[0070] ,

[0071] where is the initial i - th quantum state is the overall quantum state form after the first participant Alice and the second participant Bob perform phase - shift operations on the received particles respectively;

[0072] Subsequently, the semi - honest third party Trent applies gates and operations, and then measures the particles:

[0073] ,

[0074] where is the two - particle XOR gate operation on particles and , which processes the quantum state into , is the two - particle XOR gate operation on particles and , which processes the quantum state into , is the inverse quantum Fourier transform operation on particle because the following two particles and need to be processed by the two - particle XOR gate into in order to perform the inverse quantum Fourier transform to obtain the phase information, otherwise the result obtained is meaningless random information;

[0075] The result obtained after measurement is ;

[0076] Step 9 includes: The semi - honest third party Trent statistically analyzes the result. If , the semi - honest third party Trent records as 0; if , Trent records as 2, otherwise records as 1, and then notifies the first participant Alice of the result . The first participant Alice, through and Perform an exclusive - OR operation and bit - wise inversion to obtain and comparison result U of;

[0077] Step 10 includes: The first participant Alice and the second participant Bob repeatedly execute Steps 7 - 9, but the first participant Alice inputs U and the second participant Bob inputs ; After the protocol execution, the semi - honest third - party Trent outputs the result ; If all of the are equal to 2, then , the semi - honest third - party Trent announces the result , that is, vector A dominates vector B, and the symbol means that for i = 1, 2, …, n, the i - th element in vector A is greater than the i - th element in vector B ; Otherwise, the semi - honest third - party Trent announces the result , that is, vector A does not dominate vector B, and the symbol means that for i = 1, 2, …, n, there is at least one .

[0078] Beneficial effects: The present invention proposes a quantum - secure vector domination determination method based on privacy protection. Using the quantum private permutation protocol, it permutes the elements of the private input vectors of the participants on the premise of ensuring that the domination determination is not affected, thereby preventing other parties from stealing information. Compared with other perturbation methods, this method also prevents input - camouflage attacks carried out by the participants themselves. This method can efficiently determine vector domination with polynomial complexity and can resist various attacks, reducing privacy leakage. BRIEF DESCRIPTION OF THE DRAWINGS

[0079] Figure 1 is the flowchart of the method of the present invention.

[0080] Figure 2 is a schematic diagram of the quantum - secure vector domination process.

[0081] Figure 3 is a schematic diagram of quantum private comparison. DETAILED DESCRIPTION OF THE INVENTION

[0082] The following further describes the present invention in detail with reference to the drawings and specific embodiments, and the above and / or other advantages of the present invention will become clearer.

[0083] The embodiment of the present invention provides a quantum - secure vector domination determination method based on privacy protection, including:

[0084] Step 1, as shown in Figure 1 (where U in Figure 1 represents the result obtained after the first participant Alice and the second participant Bob perform the same addition of a random vector to the private vectors after disguise, followed by a permutation operation and then a private comparison), the first participant Alice and the second participant Bob disguise their respective private vectors, and set the vector length to , the dimension of the quantum system . The first participant Alice inputs a private vector , and the second participant Bob inputs a private vector and a permutation operation . As shown in Figure 2 , both parties perform the input disguise operation simultaneously. Alice performs input disguise on her private vector, denoted as , where according to the input disguise rule, is obtained. Bob performs the above-mentioned input disguise operation on his private vector and obtains , and prepares a .

[0085] Step 2, Alice encodes each element in the private vector into a quantum state: , for example, the first element is encoded as:

[0086] (1)

[0087] The 8th element is encoded as:

[0088] (2)

[0089] The same applies to other elements, where . Subsequently, Alice selects the first particle and the second particle from each state respectively, to form two ordered particle sequences and .

[0090] Alice initializes a particle g, i.e., , and uses the operation to calculate the sum of the particles in the first sequence H, and sends H to Bob:

[0091] (3)

[0092] Step 3, Bob performs a permutation Perform the operation and record the result as , where , and then send it back to Alice. Alice performs on the particles in to calculate :

[0093] (4)

[0094] (5)

[0095] Then apply gates to subtract these particles in turn:

[0096] (6)

[0097] Finally, measure particle g. If the measurement result is , it means that Bob has not performed any attacks such as measuring or forging the particles to destroy the entangled state, and the total particles are verified correctly after the permutation operation, and the protocol continues; otherwise, the protocol terminates.

[0098] Alice sends the sequence to Bob. Bob prepares a random vector . Bob applies the operation to each particle in the sequence , where . After the permutation operation, the change of the first particle is: , the change of the second particle is: , and the same is true for other particles. Then perform a permutation operation on the particles after the phase rotation operation to obtain: , and send it back to Alice.

[0099] Step 4, the quantum state in Alice's hand is as follows:

[0100] (7)

[0101] That is: ;

[0102] Subsequently, Alice applies to the result, :

[0103] (8)

[0104] (9)

[0105] Finally, Alice measures particle and obtains the result , where , so as to obtain .

[0106] Step 5, Bob adds the vector to the vector , and then performs a permutation operation on their sum , where , , that is, to obtain , .

[0107] Step 6, as Figure 3 shown, at this time the semi - honest third party starts to participate in assisting the two participants in private comparison. Alice privately shares a set of random bit strings = 1010111 with Bob. Trent generates tri - particle entangled state sequences:

[0108] (10)

[0109] The subscript corresponds to that the particle is to be sent to Trent, Alice, and Bob respectively. Trent selects the second and third particles from each quantum state to form ordered sequences and . Then Trent sends them to Alice and Bob respectively.

[0110] Step 7, after Alice receives , she applies the phase - shift operator to the particles , as Figure 3 shown, and adds her own private information : , for example, applying the phase - shift operator to the particle , she can obtain . Similarly, Bob also applies his own phase - shift operator : , for example, applying the phase - shift operator to the particle , and can obtain: . Then she sends the and sequences after the operation back to Trent.

[0111] Step 8, as Figure 3 shown, Trent obtains the quantum state Among them, , , ……, .

[0112] Subsequently, Trent applies gates and operations to each entangled state, and then measures the particles to obtain:

[0113] ,

[0114] ,

[0115] That is, for , ;

[0116] For , ;

[0117] After measuring these, the obtained results are . , , , , , , , .

[0118] Step 9, Trent statistically analyzes the results. If , Trent records as 0; if , Trent records as 2, otherwise records as 1, calculates the result as , and then notifies Alice of the result. Alice performs an exclusive OR operation on and the pre-shared key: , obtains , and takes the bitwise complement to obtain , denoted as U, as shown in Figure 1 .

[0119] Step 10, Alice and Bob repeat Steps 7 to 9, but Alice inputs U and Bob inputs . That is, Alice and Bob call the above steps to let Trent compare U and . After the protocol is executed, Trent calculates . Since U and are both , all are equal to 2. According to the protocol steps, Trent discovers , so the announced result: . It solves the problem of information leakage during the comparison of single vector element pairs in reality. Compared with the prior art, when dealing with similar problems, the present invention uses input masking technology to enhance the confidentiality of vector elements, and at the same time can resist internal and external attacks, ensuring the security of the secret information of both parties during the comparison. Even in the case of the participation of a third party, the privacy of the information can be maintained. Using a semi-quantum model reduces the user's requirements for quantum resources, reduces the demand for expensive quantum devices, and saves hardware costs.

[0120] The present invention provides a quantum-secure vector domination determination method based on privacy protection. There are many methods and ways to specifically implement this technical solution. The above is only the preferred embodiment of the present invention. It should be noted that for those of ordinary skill in the art, without departing from the principle of the present invention, several improvements and refinements can be made, and these improvements and refinements should also be regarded as the protection scope of the present invention. Each component not clearly defined in this embodiment can be implemented by the prior art.

Claims

1. A privacy-preserving quantum security vector dominance determination method, characterized in that: The following steps are involved: Step 1: The first participant Alice and the second participant Bob disguise their own private vectors respectively; Step 2: Alice, the first participant, encodes each element in the private vector into a quantum entangled state, selects the first and second particle sequences and sends them to Bob, the second participant, for permutation operation. Step 3, the second participant Bob performs a permutation operation on the first particle sequence and sends it back to the first participant Alice, adds the prepared random vector elements to the phase of the second particle sequence and performs a permutation operation, and then sends it back to the first participant Alice; Step 4: Alice, the first participant, measures the final quantum state and obtains the result after being permuted and encrypted with random numbers. Step 5: The second participant Bob performs random vector addition and permutation operations on his own private vector, and performs random vector permutation operations on the fixed vector Permutate and get the vector ; Step 6: The first participant Alice and the second participant Bob share a random bit string, and the semi-honest third party Trent generates an entangled state sequence; Step 7, the semi-honest third party Trent retains the first particle sequence in the entangled state sequence, and distributes the second and third particles to the first participant Alice and the second participant Bob in order, respectively. The first participant Alice and the second participant Bob apply a phase shift operation with private vector information to the sequence; Step 8: The semi-honest third party Trent measures the final quantum state with encrypted information on the phase and sends the measured encoding result R i Inform the first participant Alice; Step 9: The first participant Alice encodes the result R i XOR with the random bit string and invert it bit by bit to get the comparison result U; Step 10: The first participant Alice and the second participant Bob compare the result U and the vector Send it to the semi-honest third party Trent for repeated private comparison. If the comparison result R i All are 2, and the semi-honest third party Trent declares that the first participant's vector dominates the second participant's vector, otherwise it declares that it does not dominate.

2. The method according to claim 1, characterized in that Step 1 includes: Setting , , where n is the number of elements in the vector, N is the element dimension, and D is the dimension of the quantum state; the first participant Alice enters a private vector , the second participant Bob enters a private vector , a permutation operation , , , Is a private vector The i-th element in Is a private vector The i-th element in , i ranges from 1 to n; The first participant Alice disguises the input of her private vector A: (1)， Get the disguised private vector , with a length of 4n, The i-th element of the 4n elements in is defined as , i ranges from 1 to 4n, and we get ; The second participant Bob performs input disguise on his private vector B: (2)， Get the disguised private vector , with a length of 4n, The i-th element of the 4n elements in is defined as ,get ; and let the vector .

3. The method according to claim 2, characterized in that Step 2 includes: the first participant Alice sends the private vector Each element in Encoded as quantum state, i takes the value of 1~4n, the formula is: (3)， in The first participant, Alice, adds the i-th element in the private vector Encoded quantum state; is a complex phase factor, , It is a quantum state The ground state in Stored in particles middle, Stored in particles middle; is an index variable; and Respectively represent particles storing two different d-qubits, particles The quantum bits on ,particle The quantum bits on , , e is the Euler number, is an imaginary unit; Then, the first participant Alice selects the first particle and the second particle from each quantum state, forming two ordered particle sequences, the first sequence H and the second sequence T. , ,in, It means that the first participant Alice obtains the disguised private vector The 4nth element in The first particle selected from the quantum state encoded, It means that the first participant Alice obtains the disguised private vector The 4nth element in The second particle selected from the quantum state encoded.

4. The method according to claim 3, characterized in that: Step 2 also includes: the first participant Alice initializes a particle g, using a two-particle addition gate Operation on the first sequence The sum of the quantum bits of the particles in the first sequence is calculated and Sent to the second participant Bob; Said The formula for the operation is: (4)， (5)， in represents the double particle addition gate operation performed on particles t and h, are two particles used to demonstrate the effect of BSUM. In operation, the quantum state is transformed into , the quantum state of quantum bit y is added to quantum bit x through BSUM operation to form a new quantum state ; Represents the particle and particles Apply the BSUM operation to turn the quantum bit on particle g into particle and particles The sum of the quantum bits; then arrive , so that the quantum bit on particle g is ; Indicates application After that, the quantum bit on particle g becomes , It means that the first participant Alice adds the 4nth element in the private vector Encoded quantum state The first particle in .

5. The method according to claim 4, characterized in that Step 3 includes: the second participant Bob takes the first sequence Replace the particles in Operation, and record the result as , and then sent to the first participant Alice; the first participant Alice receives the result The particles in the single particle modular multiplication gate Operation to calculate ; Then apply The gates sequentially subtract the received result from particle g The quantum bits of particles in the Finally, the particle g is measured. If the measurement result is , it means that the second participant Bob did not attack the particle to destroy the entanglement state, and the total particle verification is correct after the replacement operation, and the protocol continues; otherwise, the protocol terminates; The single particle modular multiplication gate The formula for the operation is: (6)， (7)， in It means to multiply the particle t by parameter m, and the quantum state is The particle t passes through Operation results , Represents the particle Perform modular multiplication D-1, mod represents modular operation; The application The gates subtract the received The formula for the particle in is: (8)， Subsequently, the first participant Alice sends the second sequence T to the second participant Bob, who prepares a random vector ,in, , is a random vector The second participant Bob applies a phase operation to each particle of the second sequence T. , after the permutation operation, the result is , and then send it back to the first participant Alice; The operation The formula is: (9)， (10)， in This is an example operation with parameter m, acting on particle t, where parameter m represents the magnitude of the rotation; Indicates that the phase factor of the quantum bit x is adjusted by performing a phase rotation with parameter m on the quantum bit x. is the quantum state with qubit x; Represents the particle The parameters are Phase rotation operation.

6. The method according to claim 5, characterized in that Step 4 includes: the quantum state in the hand of the first participant Alice is: (11)， in When the second participant Bob adds a random vector element to the phase of the second sequence of particles, and , second sequence The result obtained after applying the same permutation operation; is the i-th element of the private vector of the first participant Alice after input disguise; Then the first participant Alice applies a two-particle XOR gate to the result Operation, quantum inverse Fourier transform operate: (12)， (13)， (14)， (15)， in The symbol in the upper right corner represents the inverse transformation, represents the inverse quantum Fourier transform operation on particle t, It means that a double-particle XOR gate operation is performed on particles t and h. After the operation, particle h will XOR the value of the quantum state in particle t. After performing a double particle XOR gate, we get , that is ; is an XOR operation, It is a particle Perform inverse quantum Fourier transform operations to obtain particles Information stored on the phase; Finally, the first participant Alice Perform inverse quantum Fourier transform to extract phase data and measure , and the resulting vector ,in yes The ith element of .

7. The method according to claim 6, characterized in that Step 5 includes: the second participant Bob converts the vector Add to Vector Then, and Replace operate: (16)， in refers to After the i-th element is privately replaced, it is replaced from the i-th element to the elements, the mapping of i in the permutation operation is ; Then the second participant Bob gets , , ; yes Add random vector And the result vector obtained after the overall permutation; For vector The i-th element in yes The result after the permutation operation.

8. The method according to claim 7, characterized in that Step 6 includes: the first participant Alice privately shares a set of random bits with the second participant Bob , among which , ; Semi-honest third party Trent generation A sequence of three-particle entangled states, where the i-th entangled state is of the form: (17)， in is the i-th entangled state among the 4n entangled states generated by the semi-honest third party Trent, in which the three entangled particles are the particles retained by the semi-honest third party Trent. , the particle sent to the first participant Alice , the particle sent to the second participant Bob , Represents particles The quantum state of Represents particles The quantum state of Represents particles The quantum state of represents the i-th three-particle entangled state generated by the semi-honest third party Trent The second particle in represents the i-th three-particle entangled state generated by the semi-honest third party Trent The third particle in .

9. The method according to claim 8, characterized in that Step 7 includes: the semi-honest third party Trent selects the second and third particles from each entangled state to form an ordered sequence and ; then the semi-honest third party Trent will and Send to the first participant Alice and the second participant Bob respectively; Represents the 4nth three-particle entangled state generated by the semi-honest third party Trent The second particle in The calculation formula is , represents the third particle in the 4nth three-particle entangled state generated by the semi-honest third party Trent; The first participant, Alice, receives Afterwards, Applying a phase shift operator , add your own private information : ,in, It means that the particle The phase shift operation is performed. Representation Operator The operation performed is to multiply the phase by ; represents the phase factor, which is used to control the phase change of the quantum state. ; The second participant, Bob, Applying a phase shift operator : ,in, It means that the particle The phase shift operation is performed. Representation Operator The operation performed is to multiply the phase by ; Then the first participant Alice and the second participant Bob will and Sent to the semi-honest third party Trent.

10. The method according to claim 9, characterized in that Step 8 includes: After integrating all received particles, the semi-honest third party Trent obtains the following quantum state: ， in is the initial i-th quantum state The overall quantum state form after the first participant Alice and the second participant Bob respectively perform phase shift operations on the received particles; Then the semi-honest third party Trent imposes Door and Operation, then Particles to measure: ， in It is a particle and The two-particle XOR gate operation converts the quantum state Processed as , It is a particle and The two-particle XOR gate operation converts the quantum state Processed as , It is a particle The inverse quantum Fourier transform operation performed; The results after measurement ; Step 9 includes: the semi-honest third party Trent counts the results. If , the semi-honest third party Trent will Recorded as 0; if Trent will Recorded as 2, otherwise Record it as 1, and then Inform the first participant Alice, who and Applying XOR operation and bitwise inversion yields and The comparison result U; Step 10 includes: the first participant Alice and the second participant Bob repeat steps 7 to 9, but the first participant Alice inputs U and the second participant Bob inputs ; After the protocol is executed, the semi-honest third party Trent outputs the result ; If all are all equal to 2, then , the semi-honest third party Trent publishes the results , that is, vector A dominates vector B, symbol It means that for i=1,2,…,n, there is the i-th element in vector A Greater than the i-th element in vector B Otherwise, the semi-honest third party Trent publishes the results , that is, vector A does not dominate vector B, the symbol It means that for i=1,2,…,n, there is at least one .

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