A Cross-Quantum Homomorphic Encryption Method Based on Quantum Network Coding

Abstract

The present invention discloses a cross quantum homomorphic encryption method based on quantum network coding. The senders S1 and S2 pre-share a maximally entangled state, perform quantum measurements on their respective particles using Bell bases, and perform classical bit encoding according to the measurement results. Then, perform secondary encryption according to the classical bit encoding results to obtain the encrypted ciphertext. Send the ciphertext state to the server through a quantum channel to perform set quantum calculations, and then transmit the executed result states to the receivers T1 and T2 respectively. The senders S1 and S2 send the key values to the intermediate node through a classical channel, and transmit the new key values to the receivers T1 and T2 through a classical channel. Decrypt and evaluate the ciphertext state according to the quantum calculations performed by the server and the received new key values to obtain the evaluated plaintext state. Under the combined action of the classical channel and the quantum channel, this method solves the bottleneck problem during simultaneous evaluation by both sending parties, realizes cross quantum homomorphic encryption, and improves the evaluation efficiency and speed.

Description

Technical Field

[0001] The present invention relates to the technical field of quantum network communication, and in particular to a cross quantum homomorphic encryption method based on quantum network coding. Background Art

[0002] At present, a large number of quantum network coding schemes for solving the bottleneck problem of quantum state transmission have been proposed. The idea of quantum homomorphic encryption comes from classical homomorphic encryption. Quantum homomorphic encryption is to send the encrypted quantum information to the server for quantum computing when the user does not have the ability of quantum computing; when the server performs quantum computing, it does not decrypt the ciphertext and then perform the calculation, but directly performs homomorphic evaluation calculation on the quantum state ciphertext and directly sends the homomorphic evaluation result to the user; the user decrypts according to the decryption algorithm to obtain the plaintext state.

[0003] However, the existing quantum homomorphic encryption schemes only consider that a single client satisfies the homomorphic property during the quantum evaluation process, and do not consider the problem of quantum evaluation congestion on the network bottleneck channel when multiple clients perform quantum homomorphic evaluation simultaneously. Summary of the Invention

[0004] The purpose of the present invention is to provide a cross quantum homomorphic encryption method based on quantum network coding. Under the combined action of the classical channel and the quantum channel, this method solves the bottleneck problem during the simultaneous evaluation of both sending parties, realizes cross quantum homomorphic encryption, and improves the evaluation efficiency and speed.

[0005] The purpose of the present invention is achieved through the following technical solutions:

[0006] A cross quantum homomorphic encryption method based on quantum network coding, the method comprising:

[0007] Step 1, pre-share a maximally entangled state between senders S1 and S2, perform quantum measurement on their respective particles using the Bell basis, and perform classical bit encoding according to the measurement results;

[0008] Step 2, after the senders S1 and S2 perform Bell basis measurement, obtain the encrypted plaintext states respectively, and then perform secondary encryption according to the classical bit encoding results to obtain the encrypted ciphertext;

[0009] Step 3, then the senders S1 and S2 respectively send the ciphertext states in Step 2 to the server through the quantum channel to perform the set quantum computing, and then transmit the result states after the server's execution to the receivers T1 and T2 respectively;

[0010] Step 4. The senders S1 and S2 respectively send the key values to the intermediate node through the classical channel, obtain a new key value through XOR operation, and then transmit the new key value to the receivers T1 and T2 through the classical channel;

[0011] Step 5. The receivers T1 and T2 decrypt the evaluated ciphertext state according to the type of quantum computation performed by the server and the received new key value, and obtain the evaluated plaintext state.

[0012] As can be seen from the technical solution provided by the present invention above, the above method solves the bottleneck problem during simultaneous evaluation by both senders under the combined action of the classical channel and the quantum channel, realizes cross quantum homomorphic encryption, and improves the evaluation efficiency and speed. BRIEF DESCRIPTION OF THE DRAWINGS

[0013] In order to more clearly illustrate the technical solutions of the embodiments of the present invention, the following will briefly introduce the drawings required for the description of the embodiments. Obviously, the drawings in the following description are only some embodiments of the present invention. For those of ordinary skill in the art, without creative efforts, other drawings can also be obtained based on these drawings.

[0014] Figure 1 Schematic diagram of the cross quantum homomorphic encryption method based on quantum network coding provided by the embodiment of the present invention;

[0015] Figure 2 Schematic diagram of cross quantum homomorphic evaluation of the butterfly network provided by the embodiment of the present invention;

[0016] Figure 3 Implementation process diagram of cross quantum homomorphic evaluation when the type of quantum computation in the embodiment of the present invention is single-particle Clifford gate;

[0017] Figure 4 Implementation process diagram of cross quantum homomorphic evaluation when the type of quantum computation in the embodiment of the present invention is controlled NOT gate CNOT;

[0018] Figure 5 Implementation process diagram of cross quantum homomorphic evaluation when the type of quantum computation in the embodiment of the present invention is non-Clifford T gate. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0019] The following will clearly and completely describe the technical solutions in the embodiments of the present invention with reference to the drawings in the embodiments of the present invention. Obviously, the described embodiments are only some, rather than all, embodiments of the present invention. This does not constitute a limitation to the present invention. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts fall within the protection scope of the present invention.

[0020] As Figure 1 shown in the schematic flowchart of the cross-quantum homomorphic encryption method based on quantum network coding provided by an embodiment of the present invention, the method includes:

[0021] Step 1: The senders S1 and S2 pre-share a maximally entangled state, perform quantum measurements on their respective particles using the Bell basis, and perform classical bit encoding according to the measurement results;

[0022] In this step, the senders S1 and S2 respectively have the plaintexts to be evaluated, perform quantum measurements on their respective particles using the Bell basis, and perform classical bit encoding according to the measurement results; the process of Bell basis measurement is the process of encrypting the plaintext state and also the process of encoding;

[0023] Among them, it is set that S1 and T1, S2 and T2 are two sender-receiver pairs, S0 is an intermediate node, C(S0,T0) is a bottleneck channel, the quantum states |ψ> and |φ> are plaintext states; then the corresponding relationships between the Bell basis measurement results |Ψ + >, |Ψ - >, |Φ + >, |Φ - > encoded as classical bits are 00, 10, 01, 11, and the classical bits are also encryption keys. The encryption operation is represented by the matrix multiplied by the matrix ; and a i b i ∈{00,10,01,11}, where a i and b i belong to the encryption key and are randomly selected from the integers 0 and 1. [[ID=�7]]

[0024] Step 2: After the senders S1 and S2 perform Bell basis measurements, they respectively obtain the encrypted plaintext states, and then perform secondary encryption according to the classical bit encoding results to obtain the encrypted ciphertext;

[0025] Step 3: Then the senders S1 and S2 respectively send the ciphertext states in Step 2 to the server through quantum channels to perform the set quantum calculations, and then transmit the result states after the server's execution to the receivers T1 and T2 respectively;

[0026] In this step, the set types of quantum calculations include: single-particle Clifford gates, controlled-NOT gates CNOT, and non-Clifford T gates.

[0027] For example, as Figure 2The figure shows a schematic diagram of cross - quantum homomorphic evaluation of the butterfly network according to an embodiment of the present invention. The dashed line represents the classical channel for communicating classical information, and the solid line represents the quantum channel for communicating quantum information. The arrow direction is the transmission direction. A server is arranged on the quantum channel to perform set quantum calculations on the encrypted ciphertext. The quantum calculation U performed by the server includes single - particle Clifford gates, controlled - NOT gates (CNOT), and non - Clifford T gates.

[0028] Step 4: The senders S1 and S2 respectively send the key values to the intermediate node through the classical channel, obtain a new key value through an exclusive - OR operation, and then transmit the new key value to the receivers T1 and T2 through the classical channel.

[0029] Step 5: The receivers T1 and T2 decrypt the evaluated ciphertext state according to the type of quantum calculation performed by the server and the received new key value to obtain the evaluated plaintext state.

[0030] In a specific implementation, as Figure 3 shown is a diagram of the implementation process of cross - quantum homomorphic evaluation when the type of quantum calculation in the embodiment of the present invention is a single - particle Clifford gate. Referring to Figure 3 , if the type of quantum calculation is a single - particle Clifford gate, then:

[0031] In step 1, two pairs of maximally entangled states are pre - shared between the senders S1 and S2, that is where the particle s 11 , s 12 At the sender S1, the particle s 21 , s 22 At the sender S2; the plaintext to be evaluated by the sender S1 is The plaintext to be evaluated by the sender S2 is The senders S1 and S2 respectively apply the Bell basis to perform quantum measurements on the particle pairs (s 11 , s1), (s 22 , s2). The Bell basis is According to the measurement results, classical bit encoding is performed on them. The corresponding relationship between the Bell basis measurement results |Ψ + >, |Ψ - >, |Φ + >, |Φ - > encoded as classical bits is 00, 10, 01, 11. Their classical bits are also encryption keys. The encryption operation is And a1b1, a2b2 ∈ {00, 10, 01, 11};

[0032] In step 2, after the senders S1 and S2 perform the Bell basis measurement, the encrypted plaintext states obtained are respectively The senders S1 and S2 then perform secondary encryption according to the encoding results. After the secondary encryption is completed, the senders S1 and S2 respectively obtain the encrypted ciphertext states as

[0033] In step 3, the senders S1 and S2 respectively send the ciphertext states through the quantum channels Q(S1,T2) and Q(S2,T1) to the server to perform the set quantum computation, and at the same time send the result states after the execution to the receivers T1 and T2 respectively;

[0034] In step 4, the senders S1 and S2 respectively send the key values a1b1 and a2b2 to the intermediate node S0 through the classical channels C(S1,S0) and C(S2,S0), and perform an exclusive OR operation at the intermediate node S0 to obtain a new key value and transmit the new key value to the receivers T1 and T2 through the classical channels C(T0,T1) and C(T0,T2);

[0035] In step 5, the receivers T1 and T2 decrypt the evaluated ciphertext states according to the type of quantum computation performed by the server and the received new key value, where:

[0036] 1) When the quantum homomorphic evaluation operator U performed by the server is X, Y, Z, both receiving parties apply the unitary operator to the ciphertext state That is, the homomorphic evaluation on the plaintext state is obtained through decryption and where X, Y, Z refer to the quantum Pauli operators, and the specific matrix representation forms are as follows:

[0037]

[0038] 2) When the quantum homomorphic evaluation operator U performed by the server is a single-particle H gate, both receiving parties apply the unitary operator to the ciphertext state That is, the homomorphic evaluation on the plaintext state is obtained through decryption and where the H gate is the Hadamard gate, and the matrix representation form is Its actions on the single qubit and are as follows respectively:

[0039]

[0040]

[0041] 3) When the quantum homomorphic evaluation operator U executed by the server is a single-particle P gate, both receiving parties apply the unitary operator to the ciphertext state That is, the homomorphic evaluation on the plaintext state is obtained through decryption and where the P gate is a special quantum gate in quantum mechanics, also known as the phase gate, and its matrix representation is:

[0042] Its action on a single qubit and is shown as follows respectively:

[0043] P|0> = |0>; P|1> = i|1>.

[0044] As Figure 4 shown is the implementation process diagram of the cross quantum homomorphic evaluation where the quantum computing type in the embodiment of the present invention is the controlled NOT gate CNOT. Referring to Figure 4 , if the quantum computing type is the controlled NOT gate CNOT, then:

[0045] In step 1, four pairs of maximally entangled states are pre-shared between senders S1 and S2, that is

[0046] where the particles s 11 , s 12 , s 13 , s 14 are at sender S1, and the particles s 21 , s 22 , s 23 , s 24 are at sender S2; the plaintext to be evaluated by sender S1 is The plaintext to be evaluated by sender S2 is Senders S1 and S2 respectively apply the Bell basis to perform quantum measurements on the particle pairs (s 11 , s1), (s 13 , s3) and (s 22 , s2), (s 24 , s4), where the Bell basis is Classical bit encoding is performed according to the measurement results. The encoding correspondence of the Bell basis |Ψ + >, |Ψ - >, |Φ + >, |Φ - > is 00, 10, 01, 11, and its classical bits are also the encryption keys. The encryption operation is And a1b1, c1d1, a2b2, c2d2 ∈ {00, 10, 01, 11}; where, represents a matrix multiplied by the matrix to obtain represents a matrix multiplied by the matrix to obtain represents a matrix multiplied by the matrix to obtain represents a matrix multiplied by the matrix to obtain; where a1, b1, c1, d1 and a2, b2, c2, d2 are classical bits 0 or 1. When it is 0, it means applying the identity matrix I to the target state, and when it is 1, it means applying the corresponding unitary matrix to the target state;

[0047] In step 2, after the senders S1 and S2 perform the Bell - basis measurement, they respectively obtain the encrypted plaintext states as The senders S1 and S2 then perform secondary encryption according to the encoding results. After the secondary encryption, the senders S1 and S2 respectively obtain the encrypted ciphertext states as

[0048] In step 3, the sender S1 sends the ciphertext states to the server through the quantum channels Q1 and Q2 respectively to perform the set quantum computation; at the same time, the sender S2 sends the ciphertext states to the server through the quantum channels Q4 and Q3 respectively to perform the set quantum computation; the server transmits the resultant states to the receivers T1 and T2 respectively;

[0049] In step 4, the senders S1 and S2 send the key values a1b1c1d1 and a2b2c2d2 to the intermediate node S0 through the classical channels C(S1,S0) and C(S2,S0) respectively; at the intermediate node S0, an XOR operation is performed to obtain a new key value and the new key value is transmitted to the receivers T1 and T2 through the classical channels C(T0,T1) and C(T0,T2);

[0050] In step 5, the receivers T1 and T2 apply the unitary operation to decrypt the evaluated ciphertext states to obtain the homomorphic evaluation on the plaintext state through decryption and

[0051] As Figure 5 shown, this is the implementation process diagram of the cross - quantum homomorphic evaluation when the quantum computation type in the embodiment of the present invention is a non - Clifford T gate. Refer toFigure 5 If the type of quantum computing is non-Clifford T gate, then:

[0052] In step 1, two pairs of maximally entangled states are pre-shared between senders S1 and S2, that is where particle s 11 , s 12 At sender S1, particle s 21 , s 22 At sender S2; the plaintext to be evaluated by sender S1 is The plaintext to be evaluated by sender S2 is Senders S1 and S2 respectively apply the Bell basis to perform quantum measurements on the particle pairs (s 11 , s1), and (s 22 , s2), where the Bell basis is Perform classical bit encoding according to the measurement results. The encoding correspondence of the Bell basis |Ψ + >, |Ψ - >, |Φ + >, |Φ - > is 00, 10, 01, 11, and its classical bits are also the encryption keys. The encryption operation is And a1b1, a2b2 ∈ {00, 10, 01, 11};

[0053] In step 2, after senders S1 and S2 perform the Bell basis measurement, they respectively obtain the encrypted plaintext states as Senders S1 and S2 then perform secondary encryption according to the encoding results. After performing the secondary encryption, senders S1 and S2 respectively obtain the encrypted ciphertext states as

[0054] In step 3, when the server performs the non-Clifford T gate evaluation, according to the formula P c TX a Z b |ψ> = X a′ Z b′ T|ψ>, where P, T, X, and Z are all unitary matrices, the superscripts c, a, b are the encryption keys 0 or 1, and the superscripts a′, b′ are the decryption keys 0 or 1;

[0055] When c = 1, a P error will occur. To eliminate the P error, two auxiliary particles 1 and 2 are added to the two senders and initialized as |0>1 and |0>2. At the same time, to eliminate the P error, a classical secure channel is established between senders S1 and S2 for transmitting the bit information a1 and a2 of the auxiliary particles 1 and 2. After senders S1 and S2 obtain the corresponding a1 and a2, they respectively perform the H gate on the auxiliary particles and then perform encryption operations to obtain the states of the auxiliary particles as Senders S1 and S2 respectively send the ciphertext states and the auxiliary particles to the server through quantum channels (Q1, Q2) and (Q4, Q3) to perform specific quantum computations; the server first performs the T gate and then the controlled-NOT gate on the ciphertext states, and performs Z-basis measurements on the auxiliary particles, and transmits the measurement results m, n and the measured ciphertext states to receivers T1 and T2 through classical channels C1, C2 and quantum channels Q1, Q4 respectively;

[0056] In step 4, senders S1 and S2 respectively send the key values a1b1 and a2b2 to the intermediate node S0 through classical channels C(S1, S0) and C(S2, S0); at the intermediate node S0, an exclusive-OR operation is performed to obtain a new key value and the new key value is transmitted to receivers T1 and T2 through classical channels C(T0, T1) and C(T0, T2);

[0057] In step 5, receiver T1, according to the received new key value and the measurement result m, let decrypt the ciphertext state and obtain the homomorphic evaluation on the plaintext state through decryption Receiver T2, according to the received new key value and the measurement result n, let decrypt the ciphertext state and obtain the homomorphic evaluation on the plaintext state through decryption

[0058] It should be noted that the content not described in detail in the embodiments of the present invention belongs to the prior art well known to those skilled in the art.

[0059] In summary, the method described in the embodiments of the present invention aims at the quantum evaluation bottleneck problem in the process of double quantum homomorphic encryption, applies the maximally entangled state, studies the realization of cross quantum homomorphic encryption based on the butterfly network, combines the advantages of quantum network coding, and through the design of coding and decoding methods, under the joint action of classical channels and quantum channels, realizes the cross perfect evaluation of the plaintext state on the butterfly network, improving the evaluation efficiency and speed.

[0060] Meanwhile, different solutions are designed on the butterfly network according to different types of quantum computing, and the quantum circuits of each solution are designed. The design of the quantum circuit based on the cross quantum homomorphic encryption scheme of quantum network coding has important theoretical significance for quantum secure network communication.

[0061] In addition, those of ordinary skill in the art can understand that all or part of the steps in implementing the above-described method embodiments can be completed by instructing relevant hardware through a program, and the corresponding program can be stored in a computer-readable storage medium. The above-mentioned storage medium can be a read-only memory, a magnetic disk, an optical disk, or the like.

[0062] The above is only a preferred specific embodiment of the present invention, but the protection scope of the present invention is not limited thereto. Any changes or substitutions that can be easily thought of by those skilled in the art within the technical scope disclosed by the present invention should be covered by the protection scope of the present invention. Therefore, the protection scope of the present invention should be subject to the protection scope of the claims. The information disclosed in the background art section of this article is only intended to deepen the understanding of the overall background art of the present invention, and should not be regarded as an admission or any form of implication that this information constitutes the prior art known to those skilled in the art.

Claims

1. A cross quantum homomorphic encryption method based on quantum network coding, characterized in that, The method includes the following steps: Step 1: The senders S1 and S2 pre-share a maximally entangled state. Apply the Bell basis to perform quantum measurements on their respective particles, and perform classical bit encoding according to the measurement results; Step 2: After the senders S1 and S2 perform Bell basis measurements, they respectively obtain the encrypted plaintext states, and then perform secondary encryption according to the classical bit encoding results to obtain the encrypted ciphertext; Step 3: Then the senders S1 and S2 respectively send the ciphertext states of Step 2 to the server through a quantum channel to perform set quantum calculations, and then transmit the result states after the server's execution to the receivers T1 and T2 respectively; Step 4: The senders S1 and S2 respectively send the key values to the intermediate node through a classical channel, and obtain a new key value through an exclusive OR operation, and then transmit the new key value to the receivers T1 and T2 through a classical channel; Step 5: The receivers T1 and T2 decrypt the evaluated ciphertext states according to the type of quantum calculation performed by the server and the received new key value to obtain the evaluated plaintext states.

2. The cross quantum homomorphic encryption method based on quantum network coding according to claim 1, characterized in that In Step 1, the senders S1 and S2 respectively have the plaintexts to be evaluated, apply the Bell basis to perform quantum measurements on their respective particles, and perform classical bit encoding according to the measurement results; Among them, let S1 and T1, S2 and T2 be two sending-receiving pairs, S0 be an intermediate node, C(S0,T0) be a bottleneck channel, and the quantum states |ψ> and |φ> be plaintext states. Then the corresponding relationships between the Bell basis measurement results |Ψ + >, |Ψ - >, |Φ + >, |Φ - > encoded as classical bits are 00, 10, 01, 11 respectively. These classical bits are also encryption keys, and the encryption operation is denote the matrix multiplied by the matrix ; and a i b i ∈{00, 10, 01, 11}, where a i and b i belong to the encryption key and are randomly selected from the integers 0 and 1.

3. The cross quantum homomorphic encryption method based on quantum network coding according to claim 1, characterized in that In Step 3, the set quantum calculation types include: single-particle Clifford gates, controlled-NOT gates CNOT, and non-Clifford T gates.

4. The cross quantum homomorphic encryption method based on quantum network coding according to claim 3, wherein If the quantum calculation type is a single-particle Clifford gate, then: In Step 1, two pairs of maximally entangled states are pre-shared between the senders S1 and S2, that is Among them, particle s 11 , s 12 At sender S1, particle s 21 , s 22 At sender S2; the plaintext to be evaluated by sender S1 is The plaintext to be evaluated by sender S2 is Senders S1 and S2 respectively apply the Bell basis to perform quantum measurements on the particle pairs (s 11 , s1), (s 22 , s2), and the Bell basis is According to the measurement results, perform classical bit encoding on them. The corresponding relationships between the Bell basis measurement results |Ψ + >>, |Ψ - >>, |Φ + >>, |Φ - >> encoded as classical bits are 00, 10, 01, 11, and their classical bits are also the encryption keys. The encryption operation is And a1b1, a2b2 ∈ {00, 10, 01, 11}; In step 2, after the sender S1 and S2 perform the Bell - basis measurement, they respectively obtain the encrypted plaintext states as Then, the sender S1 and S2 perform secondary encryption according to the encoding results. After the secondary encryption, the sender S1 and S2 respectively obtain the encrypted ciphertext states as In step 3, the senders S1 and S2 respectively send the ciphertext states to the server through the quantum channels Q(S1,T2) and Q(S2,T1) to perform the set quantum computation, and at the same time, transmit the resultant states to the receivers T1 and T2 respectively; In step 4, senders S1 and S2 respectively send the key values a1b1 and a2b2 to the intermediate node S0 through the classical channels C(S1,S0) and C(S2,S0), and perform an XOR operation at the intermediate node S0 to obtain a new key value and transmit the new key value to the receivers T1 and T2 through the classical channels C(T0,T1) and C(T0,T2); In Step 5, the receivers T1 and T2 decrypt the evaluated ciphertext states according to the type of quantum calculation performed by the server and the received new key value, where: 1) When the quantum homomorphic evaluation operator U executed by the server is X, Y, or Z, both receiving parties apply the unitary operator to the ciphertext state That is, the homomorphic evaluation on the plaintext state is obtained by decryption and where X, Y, and Z refer to the quantum Pauli operators, and the specific matrix representation forms are as follows: 2) When the quantum homomorphic evaluation operator U executed by the server is a single-particle H gate, both receiving parties apply the unitary operator to the ciphertext state That is, the homomorphic evaluation on the plaintext state is obtained through decryption and where the H gate is a Hadamard gate, and its matrix representation is 3) When the quantum homomorphic evaluation operator U executed by the server is a single-particle P gate, both receiving parties apply the unitary operator to the ciphertext state That is, the homomorphic evaluation on the plaintext state is obtained through decryption and where the P gate is a phase gate, and its matrix representation is:

5. The cross quantum homomorphic encryption method based on quantum network coding according to claim 3, characterized in that, If the quantum calculation type is a controlled-NOT gate CNOT, then: In Step 1, four pairs of maximally entangled states are pre-shared between the senders S1 and S2, that is Among them, particle s 11 , s 12 , s 13 , s 14 At the sender S1, particle s 21 , s 22 , s 23 , s 24 At the sender S2; the plaintext to be evaluated by the sender S1 is The plaintext to be evaluated by the sender S2 is The senders S1 and S2 respectively apply the Bell basis to perform quantum measurements on the particle pairs (s 11 , s1), (s 13 , s3) and (s 22 , s2), (s 24 , s4), where the Bell basis is Perform classical bit encoding according to the measurement results. The encoding correspondence of the Bell basis |Ψ + >, |Ψ - >, |Φ + >, |Φ - > is 00, 10, 01, 11, and its classical bit is also the encryption key. The encryption operation is And a1b1, c1d1, a2b2, c2d2 ∈ {00, 10, 01, 11}; among them, represents the multiplication of the matrix and the matrix , represents the multiplication of the matrix and the matrix , represents the multiplication of the matrix and the matrix , represents the multiplication of the matrix and the matrix ; where a1, b1, c1, d1 and a2, b2, c2, d2 are classical bits 0 or 1; In step 2, after the sender S1 and S2 perform the Bell basis measurement, they respectively obtain the encrypted plaintext states as The sender S1 and S2 then perform secondary encryption according to the encoding results. After the secondary encryption is completed, the sender S1 and S2 respectively obtain the encrypted ciphertext states as In step 3, the sender S1 sends the ciphertext states to the server through the quantum channels Q1 and Q2 respectively to perform the set quantum computation; at the same time, the sender S2 sends the ciphertext states to the server through the quantum channels Q4 and Q3 respectively to perform the set quantum computation; the server transmits the resulting states to the receivers T1 and T2 respectively; In step 4, the senders S1 and S2 respectively send the key values a1b1c1d1 and a2b2c2d2 to the intermediate node S0 through the classical channels C(S1,S0) and C(S2,S0); at the intermediate node S0, an exclusive-or operation is performed to obtain a new key value and the new key value is transmitted to the receivers T1 and T2 through the classical channels C(T0,T1) and C(T0,T2); In step 5, the receivers T1, T2 apply unitary operations according to the received new key values Decrypt the evaluated ciphertext state Obtain the homomorphic evaluation on the plaintext state through decryption And 6. The cross quantum homomorphic encryption method based on quantum network coding according to claim 3, characterized in that, If the quantum calculation type is a non-Clifford T gate, then: In Step 1, two pairs of maximally entangled states are pre-shared between the senders S1 and S2, that is where particle s 11 , s 12 At sender S1, particle s 21 , s 22 At sender S2; the plaintext to be evaluated by sender S1 is The plaintext to be evaluated by sender S2 is Senders S1 and S2 respectively apply the Bell basis to perform quantum measurements on the particle pairs (s 11 , s1) and (s 22 , s2), where the Bell basis is According to the measurement results, perform classical bit encoding on them. The encoding correspondence of the Bell basis |Ψ + >>, |Ψ - >>, |Φ + >>, |Φ - >> is 00, 10, 01, 11, and its classical bits are also the encryption keys. The encryption operation is And a1b1, a2b2 ∈ {00, 10, 01, 11}; In step 2, after the sender S1 and S2 perform the Bell basis measurement, they respectively obtain the encrypted plaintext states as The sender S1 and S2 then perform secondary encryption according to the coding results. After the secondary encryption is completed, the sender S1 and S2 respectively obtain the encrypted ciphertext states as In step 3, when the server performs a non-Clifford T gate evaluation, according to the formula P c TX a Z b |ψ> = X a′ Z b′ T|ψ>, where P, T, X, and Z are all unitary matrices, the superscripts c, a, b are encryption keys 0 or 1, and the superscripts a′, b′ are decryption keys 0 or 1; When c = 1, a P error will occur. To eliminate the P error, two auxiliary particles 1 and 2 are added to the two senders and initialized as |0>1 and |0>2. At the same time, to eliminate the P error, a classical secure channel is established between senders S1 and S2 for transmitting the bit information a1 and a2 of the auxiliary particles 1 and 2. After senders S1 and S2 obtain the corresponding a1 and a2, they respectively perform the H gate on the auxiliary particles and then perform encryption operations to obtain the states of the auxiliary particles as Senders S1 and S2 respectively send the ciphertext states and auxiliary particles to the server through quantum channels (Q1, Q2) and (Q4, Q3) to perform specific quantum computations; the server first performs the T gate on the ciphertext state and then performs the controlled-NOT gate, and performs the Z-basis measurement on the auxiliary particles, and transmits the measurement results m, n and the measured ciphertext state to receivers T1 and T2 through classical channels C1, C2 and quantum channels Q1, Q4 respectively; In step 4, the senders S1 and S2 respectively send the key values a1b1 and a2b2 to the intermediate node S0 through the classical channels C(S1,S0) and C(S2,S0); at the intermediate node S0, an exclusive-or operation is performed to obtain a new key value and the new key value is transmitted to the receivers T1 and T2 through the classical channels C(T0,T1) and C(T0,T2); In step 5, the recipient T1, according to the received new key value and the measurement result m, makes decrypt the ciphertext state and obtain, through decryption, the homomorphic evaluation on the plaintext state The recipient T2, according to the received new key value and the measurement result n, makes decrypt the ciphertext state and obtain, through decryption, the homomorphic evaluation on the plaintext state

Images