A non-interactive quantum homomorphic encryption method based on matrix decomposition

Abstract

The application provides a non-interactive quantum homomorphic encryption method based on matrix decomposition, which comprises the following steps: step 1, constructing a quantum network environment; step 2, a sender generates a private key, realizes quantum one-time pad encryption on a plaintext quantum state, and sends the ciphertext quantum state to a receiver through a quantum channel; step 3, the receiver acts a quantum circuit on the ciphertext quantum state, and sends a homomorphic calculation result and a key update function of the quantum circuit to the sender through a quantum secure channel; and step 4, the sender uses the private key to act a decryption operator on the homomorphic calculation result to obtain a decryption result. The application can allow the sender to entrust the receiver to realize quantum calculation service, and meanwhile, the correctness of the calculation and the security of the data are ensured. The application can eliminate phase gate errors generated when an encryption quantum gate T gate is used, and can avoid interaction between the sender and the receiver in the quantum homomorphic calculation stage, thereby reducing communication overhead and increasing the efficiency of the quantum homomorphic encryption scheme.

Description

Technical Field

[0001] This invention relates to a non-interactive quantum homomorphic encryption method based on matrix factorization, belonging to the fields of cyberspace security technology and quantum cryptography. Background Technology

[0002] Classical cryptographic schemes mostly rely on mathematical challenges such as the factorization of large prime numbers, the discrete logarithm problem over finite fields, and the discrete logarithm problem over elliptic curves to ensure their security. However, with the advent of quantum computers and the rapid development of quantum computing, classical cryptosystems face severe security challenges. For example, the security of the RSA algorithm, a public-key cryptosystem based on the difficult problem of factoring large prime numbers, is under significant threat. Shor's algorithm, proposed in 1994, can effectively factor large prime numbers using the parallelism of quantum computing. By transforming the RSA algorithm into a combinatorial optimization problem, Wang Chao et al. in 2018 used the quantum tunneling effect of the D-Wave dedicated quantum computer to search for the global optimal solution and completed the factorization of a 20-bit integer, verifying the theoretical feasibility of D-Wave breaking the RSA public-key cryptosystem. Therefore, in the post-quantum era, we need to redefine cryptosystems to resist attacks from quantum computers.

[0003] Quantum cryptography is a novel cryptographic system. Based on cryptography and quantum mechanics, it utilizes quantum physics methods to achieve the security goals of cryptography. Its security is guaranteed by physical principles such as the uncertainty principle and the no-cloning theorem. The first unconditionally secure quantum cryptography protocol was the BB84 protocol. In 1984, Bennett and Brassard proposed the first international quantum key distribution protocol. This protocol uses a single quantum state to transmit the key required for symmetric encryption algorithms, enabling completely secure transmission of classical keys between authenticated communicating parties. Subsequently, quantum cryptography has developed new protocols such as quantum teleportation, quantum secret sharing, quantum multi-party computation, quantum message authentication, and quantum signatures, further enriching the theoretical framework of quantum cryptography.

[0004] Quantum homomorphic encryption, an important branch of quantum cryptography, allows for direct computation on the quantum state of ciphertext without knowing the key, while ensuring both the correctness of the computation and the security of the data. This homomorphism gives quantum homomorphic encryption significant theoretical value, with applications in quantum delegated computation and quantum ciphertext retrieval.

[0005] Based on the presence or absence of interaction between the client and server during the homomorphic computation phase, quantum homomorphic encryption schemes can be categorized into interactive and non-interactive types. In fact, interaction can reduce the difficulty of constructing quantum homomorphic encryption schemes. The challenge lies in implementing T-gate quantum homomorphic encryption, a process that generates S-error. Allowing interaction allows the application of an additional S-gate to eliminate the S-error. As early as 2005, before the concept of quantum homomorphic encryption existed, Childs, while researching secure-assisted quantum computing, presented a key update algorithm for related quantum gates and discussed its security. In this scheme, each execution of a quantum gate in the quantum circuit requires a round of quantum information interaction between the client and server. Specifically, two rounds of interaction are needed to complete the T-gate quantum homomorphic computation. In 2014, Fisher et al. presented a key update algorithm for quantum computation of encrypted data. In this scheme, when executing the T-gate, bidirectional quantum interaction is no longer required; instead, it involves unidirectional quantum interaction and bidirectional classical interaction. Furthermore, Fisher et al. conducted experiments on real quantum devices, verifying the correctness of the scheme. In 2015, Broadbent proposed a quantum homomorphic encryption scheme that requires only M interactions, leveraging the properties of Clifford gates, where M is the number of T gates in the quantum circuit. By transmitting auxiliary qubits to the server, this scheme eliminates the S-error through hidden S gates using only classical information exchange, ensuring security while avoiding quantum communication between the two parties. Also in 2015, Liang, through research on universal quantum circuits, presented a quantum fully homomorphic encryption scheme based on universal circuits. This scheme is similar to that of Fisher et al., but differs in that its key update algorithm depends only on the construction of the universal quantum circuit and is independent of the server's algorithm. It utilizes universal quantum circuits to implement arbitrary unitary operator operations; once the universal quantum circuit is determined, the entire key update algorithm is also determined. In contrast, in Fisher et al.'s scheme, the client needs to know the operations performed by the server to complete the key update.

[0006] However, both quantum and classical information interaction increase communication overhead between the server and client, reducing the efficiency of the scheme. Therefore, research on non-interactive quantum homomorphic encryption is essential. To address the low efficiency of existing non-interactive quantum homomorphic encryption schemes, this invention proposes a novel T-gate key update algorithm based on matrix factorization and implements homomorphic computation non-interactively, thereby improving the efficiency of quantum homomorphic encryption. Summary of the Invention

[0007] The purpose of this invention is to provide a non-interactive quantum homomorphic encryption method based on matrix factorization, so as to overcome the limitation of existing interactive quantum homomorphic encryption methods that require message passing and improve the efficiency of overall homomorphic encryption computation.

[0008] The technical solution adopted in this invention is: a non-interactive quantum homomorphic encryption method based on matrix factorization, which includes the following steps:

[0009] Step 1: Constructing a quantum network environment;

[0010] The quantum network environment includes Alice, the plaintext sender, and Bob, the ciphertext receiver. During the encryption phase, Alice uses her private key sk to encrypt the plaintext quantum state |ψ>. During the homomorphic computation phase, Bob, the receiver, uses the quantum circuit C applied to the ciphertext message. q To implement quantum computing, during the decryption phase, the sender, Alice, uses her private key `sk` to update the key and obtain the decryption operator. Then, she decrypts the data to get the correct computation result. Here, `sk` is the quantum one-time pad encryption key.

[0011] Step 2: Alice, the sender, generates a private key sk, performs quantum one-time pad encryption on the plaintext quantum state |ψ>, and sends the ciphertext quantum state |ρ> to Bob, the receiver, through a quantum channel;

[0012] 2.1 Key Generation Stage: The sender, Alice, randomly selects a 2n-classical bit private key sk = (a, b), where a, b ∈ {0, 1} n , where n is the number of qubits in the plaintext quantum state |ψ>.

[0013] 2.2 Encryption Phase: Alice, the sender, uses the private key sk generated in step 2.1 and the Pauli matrices X and Z to perform quantum one-time pad encryption on the plaintext quantum state |ψ>. The encrypted ciphertext quantum state |ρ> can be represented as:

[0014]

[0015] Where a = (a1, a2, ..., a...) n b = (b1, b2, ..., b) n ), a i b i ∈{0,1}, i=1,2,...n, This represents the tensor product operation, where the Pauli matrices X and Z are respectively... and

[0016] 2.3 Data transmission stage: Alice, the sender, transmits the encrypted quantum state |ρ> from step 2.3 to Bob, the receiver, through a quantum-secure channel.

[0017] Step 3: Receiver Bob interacts with the quantum circuit C of the ciphertext quantum state |ρ>. q And transmit the homomorphic computation result C through a quantum secure channel q|ρ> and the key update function of quantum circuits Send to the sender, Alice;

[0018] Receiver Bob interacts with the quantum circuit C on the ciphertext quantum state |ρ>. q During the process, it is necessary to rely on quantum circuit C q The corresponding key update function is constructed. After the quantum circuit completes its operation, the receiver Bob will input the calculation result C. q |ρ> and key update function The message is sent to the sender, Alice. Throughout the homomorphic computation phase, there is no classical or quantum information exchange between the receiver, Bob, and the sender, Alice; therefore, this invention satisfies non-interaction.

[0019] The universal set of quantum gates for quantum circuits consists of Clifford gates {H, CONT, S, X, Z} and non-Clifford gates T, where H is a Hadamard gate, S is a phase gate, T is a π / 8 gate, and CNOT is a controlled NOT gate, with matrix forms of... The following are the key update steps for Clifford gates, T gates, and general quantum circuits, respectively.

[0020] 3.1 Key Update for Clifford Gate

[0021] For any Clifford circuit C and any Pauli gate Q, there exists another Pauli gate Q′ such that CQ = Q′C. This invention uses quantum one-time pad encryption, where Q = X. a Z b The key update function for the Clifford gate {H, CONT, S, X, Z} is shown below:

[0022] f X (a, b) = (a, b)

[0023] f Z (a, b) = (a, b)

[0024] f H (a, b) = (b, a)

[0025]

[0026]

[0027] Where a and b are the private keys of the sender, Alice. This represents the classic bit XOR operation.

[0028] 3.2T gate key update

[0029] The T-gate quantum homomorphic encryption process based on matrix factorization is as follows: Figure 1 As shown. T-gate action X a Z b The result is: When encryption X exists (i.e., a = 1), the homomorphic computation process generates a redundant phase gate S. If this phase gate error S-error is not eliminated, decryption cannot be performed correctly. Using matrix factorization, S can be rewritten as... in At this point:

[0030]

[0031] Since a, b ∈ {0, 1}, the above expression can be further written as:

[0032] TX a Z b =(α1I+α2X+α2Z+α4XZ)T

[0033] Therefore, the key update function based on matrix factorization T-gate is:

[0034]

[0035] in:

[0036] 3.3 General Quantum Circuits C q Key update

[0037] The quantum homomorphic encryption process based on matrix factorization in general quantum circuits is as follows: Figure 2 As shown. Single-qubit circuit C q The universal set of quantum gates is {H, T}, and the circuit can be written as:

[0038]

[0039] Where g i ∈{H, T}, i = 1, ..., |C q |,|C q | represents the number of quantum gates in a quantum circuit.

[0040] The quantum gate g is acted upon in sequence i Using the key update rules in steps 3.1 and 3.2, the single-qubit circuit C can be obtained. q Key update function:

[0041]

[0042] in j = 1, 2, 3, 4.

[0043] Multi-qubit circuit C q The universal set of quantum gates is {H, T, CNOT}. Using the key update rules in steps 3.1 and 3.2, the key update function can be obtained:

[0044]

[0045] in

[0046] Step 4: The sender, Alice, uses her private key sk to perform homomorphic computation on the result C. q |ρ> Function decryption operator The decryption result C is obtained. q |ψ>.

[0047] Based on the key update function sent by the recipient Bob in step 3.3 Alice, the sender, inputs her key sk to obtain the decryption key dk. She then constructs a decryption operator based on the decryption key dk. The decryption result C is obtained by performing homomorphic computation. q |ψ>.

[0048] The advantages of this invention compared to the prior art are:

[0049] (1) This invention introduces matrix factorization theory into quantum homomorphic encryption method, constructs a new T-gate key update function, and solves the phase gate error S-error that occurs in the homomorphic computation process of non-Clifford gate T gate.

[0050] (2) In the process of homomorphic computation, the present invention avoids the transmission of classical or quantum information between the sender Alice and the receiver Bob, thereby improving the efficiency of the quantum homomorphic encryption scheme. Attached Figure Description

[0051] Figure 1 This is a T-gate quantum homomorphic encryption process based on matrix factorization.

[0052] Figure 2 This is a quantum homomorphic encryption process for general quantum circuits based on matrix factorization.

[0053] Figure 1 and Figure 2 The symbols used in Chinese are explained below:

[0054] Alice is the sender of the plaintext quantum state |ψ>;

[0055] Bob is the encrypted quantum state X a Z b The receiver of |ψ>;

[0056] (a, b) is the private key used by the sender Alice to implement quantum one-time pad encryption;

[0057] X and Z are Pauli matrices.

[0058] Figure 1 The following is an explanation of the symbols unique to Chinese:

[0059] T is a quantum π / 8 gate;

[0060] unitary matrix U T =α1I+α2X+α2Z+α4XZ is the decryption operator corresponding to the key update function of the T gate.

[0061] Figure 2 The following is an explanation of the symbols unique to Chinese:

[0062] C q It is a typical quantum circuit;

[0063] unitary matrix It is the decryption operator corresponding to the key update function of a general quantum circuit. Detailed Implementation

[0064] The non-interactive quantum homomorphic encryption method based on matrix factorization proposed in this invention needs to solve the following three problems: (1) How to use matrix factorization to solve the phase gate error S-error that occurs in the homomorphic computation process of non-Clifford gate T gate; (2) Prove that the method has homomorphism, that is, the computation result of the quantum circuit applied to the ciphertext quantum state after decryption is the same as the computation result of the quantum circuit applied to the plaintext quantum state directly; (3) Prove that the method satisfies non-interaction, that is, there is no classical or quantum information transmission between the sender Alice and the receiver Bob in the homomorphic computation process.

[0065] Using quantum one-time pad, quantum homomorphic encryption of Clifford gates can be achieved simply by updating the key. For non-Clifford gate T gates, the action X a Z b The result is: When X encryption exists (i.e., a=1), the homomorphic computation process will generate an extra phase gate S. If this phase gate error S-error is not eliminated, then decryption cannot be performed correctly.

[0066] The main idea behind this invention is to utilize matrix factorization to decompose the phase gate S into a linear combination of Pauli matrices I and Z, and then eliminate the S-error by constructing a new T-gate key update function. From simple single-qubit circuits to complex multi-qubit circuits, the impact of phase gate decomposition on the subsequent quantum gate key update process in the quantum circuit is analyzed. Finally, a key update function for general quantum circuits is given.

[0067] This invention proposes a non-interactive quantum homomorphic encryption method based on matrix factorization. The specific implementation steps of this invention are described in four parts below:

[0068] Step 1: Constructing a quantum network environment;

[0069] The quantum network environment includes Alice, the plaintext sender, and Bob, the ciphertext receiver. During the encryption phase, Alice uses her private key sk to perform a quantum one-time pad encryption on the plaintext quantum state |ψ>. During the homomorphic computation phase, Bob, the receiver, applies the quantum circuit C to the ciphertext message. q Implement quantum computing and provide the key update function corresponding to the quantum circuit. During the decryption phase, the sender Alice updates the function based on the private key sk and the key. The key is updated to obtain the decryption key dk, and then the decryption operator is constructed using the decryption key. Decryption is achieved by applying the homomorphic computation results.

[0070] Step 2: Alice, the sender, generates a private key sk, performs quantum one-time pad encryption on the plaintext quantum state |ψ>, and sends the ciphertext quantum state |ρ> to Bob, the receiver, through a quantum channel;

[0071] 2.1 Key Generation Stage: The sender, Alice, randomly selects a 2n-classical bit private key sk = (a, b), where a, b ∈ {0, 1} n , where n is the number of qubits in the plaintext quantum state |ψ>.

[0072] 2.2 Encryption Phase: Alice, the sender, uses the private key sk generated in step 2.1 to perform quantum one-time pad encryption on the plaintext quantum state |ψ> using the X and Z basic quantum gates. The encryption result |ρ> can be expressed as:

[0073]

[0074] Where |ρ> represents the ciphertext quantum state, and a=(a1, a2, …, a n b = (b1, b2, ..., b) n ), a i b i ∈{0,1}, i=1,2,...n, This represents the tensor product operation, where the Pauli matrices X and Z are respectively... and

[0075] 2.3 Data transmission stage: Alice, the sender, transmits the ciphertext quantum state |ρ> to Bob, the receiver, through a quantum-secure channel.

[0076] Step 3: Receiver Bob interacts with the quantum circuit C of the ciphertext quantum state |ρ>. q And transmit the homomorphic computation result C through a quantum secure channel q |ρ> and the key update function of quantum circuits Send to the sender, Alice;

[0077] Receiver Bob interacts with the quantum circuit C on the ciphertext quantum state |ρ>. q During the process, it is necessary to rely on quantum circuit C q Construct the corresponding key update function After the quantum circuit completes its operation, the receiver Bob will transmit the calculation result C. q |ρ> and key update function The message is sent to the sender, Alice. Throughout the homomorphic computation phase, there is no classical or quantum information exchange between the receiver, Bob, and the sender, Alice; therefore, this invention satisfies non-interaction.

[0078] The universal set of quantum gates for quantum circuits consists of Clifford gates {H, CONT, S, X, Z} and non-Clifford gates T, where H is a Hadamard gate, S is a phase gate, T is a π / 8 gate, and CNOT is a controlled NOT gate, with matrix forms of... The following are the key update steps for Clifford gates, T gates, and general quantum circuits, respectively.

[0079] 3.1 Key Update for Clifford Gate

[0080] For any Clifford circuit C and any Pauli gate Q, there exists another Pauli gate Q′ such that CQ = Q′C. Since this invention uses quantum one-time pad encryption, Q = X. a Z b The Clifford gate {H, CONT, S, X, Z} acts on X. a Z b The results are shown below:

[0081] X(X a Z b )=(X a Z b )X

[0082] Z(X a Z b )=(X a Z b )X

[0083] H(X a Z b )=(X b Z a )H

[0084]

[0085]

[0086] The key update function for the Clifford gate {H, CONT, S, X, Z} is as follows:

[0087] f X (a, b) = (a, b)

[0088] f Z (a, b) = (a, b)

[0089] f H (a, b) = (b, a)

[0090]

[0091]

[0092] Where a and b are the private keys of the sender, Alice. This represents the classic bit XOR operation.

[0093] 3.2T gate key update

[0094] T-gate function X a Z b The result is: When encryption X exists (i.e., a = 1), the homomorphic computation process generates a redundant phase gate S. If this phase gate error S-error is not eliminated, decryption cannot be performed correctly. Using matrix factorization, S can be rewritten as... in At this point:

[0095]

[0096] Since a, b ∈ {0, 1}, the above expression can be further written as:

[0097] TX a Z b =(α1I+α2X+α2Z+α4XZ)T

[0098] in:

[0099] Using matrix factorization, the key update function for the T-gate is:

[0100]

[0101] in:

[0102] 3.3 General Quantum Circuits C q Key update

[0103] For a single-qubit circuit C q Since the universal set of quantum gates is {H, T}, the circuit can be written as:

[0104]

[0105] Where g i ∈{H, T}, i = 1, ..., |C q |,|C q | represents the number of quantum gates in the quantum circuit. The index i, from smallest to largest, indicates the order in which the quantum gates are applied, i.e., quantum gate g1 is applied to the ciphertext quantum state first, and quantum gate g2 is applied to the ciphertext quantum state last.

[0106] C q Acting on the encrypted quantum state X a Z b At |ψ>, the state of the entire quantum system is:

[0107]

[0108] Applying the first quantum gate g1, if g1 is a quantum gate H, then according to the key update function of the Clifford gate in step 3.1, the result is calculated as follows:

[0109]

[0110] If g1 is a quantum gate T, then according to the key update function of gate T in 3.2, the result is calculated as follows:

[0111]

[0112] in

[0113] Since a, b ∈ {0, 1}, then X b Z a There are only four possibilities: I, X, Z, and XZ. Therefore, g1 being an H or T gate can be represented by a unified calculation result:

[0114]

[0115] in

[0116] Applying the second quantum gate g2, if g2 is an H-gate, then the following holds true:

[0117] g2(α1I+α2X+α2Z+α4XZ)g1|ψ>

[0118] =H(α1I+α2X+α2Z+α4XZ)g1|ψ>

[0119] =(α1HI+α2HX+α2HZ+α4HXZ)g1|ψ>

[0120] =(α1IH+α2ZH+α2XH+α4XZH)g1|ψ>

[0121] =(α1I+α2X+α2Z+α4XZ)Hg1|ψ>

[0122] If g2 is a quantum gate T, then the following holds true:

[0123] T(α1I+α2X+α2Z+α4XZ)g1|ψ>

[0124] =(α1TI+α2TX+α2TZ+α4TXZ)g1|ψ>

[0125] =(α1IT+α2(α′1I+α′2X+α′2Z+α′4XZ)T+α2ZT+α4(α ″ 1I+α ″ 2X+α ″ 2Z+α ″ 4XZ)T)g1|ψ>

[0126] =α1I+α2(a′1I+a′2X+a′3Z+a′4XZ)+α2Z+α4(a″1I+a″2X+a″3Z+a″4XZ))Tg1|ψ>

[0127] =(α″′1I+α″′2X+α″′3Z+α″′4XZ)Tg1|ψ>

[0128] in It is a set Combination of multiplication and addition.

[0129] Similarly, g2 can be written as an H or T gate to achieve a unified calculation result:

[0130]

[0131] in It is a set Combination of multiplication and addition.

[0132] The remaining quantum gates After all the effects are applied, the final calculation result is:

[0133]

[0134] in It is a set Combination of multiplication and addition.

[0135] Therefore, the single-qubit circuit C q The key update function is:

[0136]

[0137] in

[0138] For multi-qubit circuits C q At this point, the universal set of quantum gates for the quantum circuit is {H, T, CONT}. When the CNOT gate appears alone, the key can be updated according to the Clifford update function in step 3.1. When both the CNOT gate and the T gate exist in the circuit, let the CNOT gate act on the i-th and j-th lines of the quantum circuit, and the T gate act on the i-th line (C... q (This is an n-qubit circuit, where each line corresponds to one qubit). Let's assume the i-th line is the control bit and the j-th line is the target bit. According to the key update functions for the T-gate and CNOT gate in step 3.1, the following holds true:

[0139]

[0140] Where (a) i b i ) and (a j b j ) are the quantum one-time pad encryption keys for the i-th and j-th qubits, respectively.

[0141] Therefore, the multi-qubit circuit C q The key update function is:

[0142]

[0143] in k is the number of T gates in the quantum circuit, satisfying:

[0144]

[0145] in for:

[0146]

[0147] Step 4: The sender, Alice, uses her private key sk to perform homomorphic computation on the result C. q |ρ> Function decryption operator The decryption result C is obtained. q |ψ>.

[0148] Based on the key update function sent by the recipient Bob in step 3.3 Alice, the sender, inputs her key sk to obtain the decryption key dk. She then constructs a decryption operator based on the decryption key dk. The decryption result C is obtained by performing homomorphic computation. q |ψ>.

[0149] 4.1 Calculate the decryption key: Based on the key update function sent by the receiver Bob in step 3.3. Alice, the sender, inputs her key sk to obtain the decryption key dk:

[0150]

[0151] in

[0152] 4.2 Decryption Phase: The sender, Alice, constructs the decryption operator based on the decryption key dk. And apply the homomorphic calculation result C q |ρ> Obtain the decryption result C q |ψ>.

[0153] Based on the decryption key obtained in step 4.1 Constructing the decryption operator

[0154]

[0155] In step 3, the receiver Bob interacts with the quantum circuit C on the ciphertext quantum state |ρ>. q The homomorphic computation results are as follows:

[0156]

[0157] The sender, Alice, computes the homomorphic result C. q |ρ> Function decryption operator Established:

[0158]

[0159] According to step 4.2, the sender Alice can finally obtain the decryption result C. q|ψ>, which is related to the direct action of the plaintext quantum state |ψ> on the quantum circuit C. q The result is the same, therefore the non-interactive quantum homomorphic encryption method based on matrix factorization in this invention satisfies homomorphism.

[0160] The contents not described in detail in this specification are existing technologies known to those skilled in the art.

[0161] The above description is merely a preferred embodiment of a non-interactive quantum homomorphic encryption method based on matrix factorization according to the present invention. It should be noted that, for those skilled in the art, several improvements and modifications can be made without departing from the principle of the non-interactive quantum homomorphic encryption method based on matrix factorization according to the present invention, and these improvements and modifications should also be considered within the protection scope of the non-interactive quantum homomorphic encryption method based on matrix factorization according to the present invention.

Claims

1. A non-interactive quantum homomorphic encryption method based on matrix factorization, characterized in that, It includes the following steps: Step 1: Constructing a quantum network environment; The quantum network environment includes Alice, the plaintext sender, and Bob, the ciphertext receiver; Alice uses a private key. For plaintext quantum states Encryption, then the receiver Bob interacts with the quantum circuit of the ciphertext quantum state. Homomorphic computation is performed, and finally, the sender Alice decrypts the homomorphic computation result. Step 2: Alice, the sender, generates a private key. For plaintext quantum states Achieve quantum one-time pad encryption and transmit the ciphertext quantum state through a quantum channel. Send to recipient Bob; Step 3: Receiver Bob checks the ciphertext quantum state. Quantum circuits And transmit the homomorphic computation results through a quantum secure channel Key update function for quantum circuits Send to sender Alice; in step three, the universal set of quantum gates for the quantum circuit is composed of Clifford gates. Non-Clifford Gate The door consists of, among which For the Hadamard door, For phase gate, for Door, It is a controlled NOT gate; non-Clifford gates are eliminated based on matrix factorization. Phase gate error in homomorphic computation -error, the specific process is as follows: Homomorphic computation The result of the gate is Using matrix decomposition, the phase gate S is decomposed into a linear combination of Pauli matrices I and Z. At this point, we have: ； therefore, The key update function for the gate is: ； in: ; Step 4: Alice, the sender, uses her private key. Homomorphic computation results Functional decryption operator Decryption result obtained ; The sender, Alice, first updates the key according to the function. and private key Calculate the decryption key Then use the decryption key Constructing the decryption operator Finally, the homomorphic calculation results effect Decryption result obtained .

2. The non-interactive quantum homomorphic encryption method based on matrix factorization according to claim 1, characterized in that: In step two, the generation of the encrypted quantum state includes: ; in, Representing the quantum state of the ciphertext, , Represents tensor product operation, encryption operator and They are respectively and .

3. The non-interactive quantum homomorphic encryption method based on matrix factorization according to claim 1, characterized in that: For single-qubit circuits The universal set of quantum gates is Single-qubit circuit The key update function is: ； in, ; For multi-qubit circuits The universal set of quantum gates for quantum circuits is Multi-qubit circuit The key update function is: ； in, , For quantum circuits The number of doors.

4. The non-interactive quantum homomorphic encryption method based on matrix factorization according to claim 1, characterized in that: The specific process of step four is as follows: The sender Alice updates the key based on the key sent by the receiver Bob. and your private key Obtain the decryption key : ； in, ; Then based on the decryption key Constructing the decryption operator And the homomorphic calculation results Functional decryption operator get Among them, the decryption operator for: 。

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