Multistage security information integrity authentication method based on two-dimensional nonlinear quantum walking

Abstract

The invention provides a multilevel security information integrity authentication method based on two-dimensional nonlinear quantum walking. The multilevel security information integrity authentication method comprises the following steps: step 1, initializing parameters of a nonlinear quantum hash function with a secret key; 2, executing two-dimensional nonlinear quantum walking; 3, measuring to obtain the probability distribution of final quantum state vertexes; 4, executing a post-processing method to calculate a hash value; and 5, verifying the information integrity. According to the method, non-linear intensity is introduced as a new parameter to participate in quantum hash function construction, and the multilevel security information integrity authentication method based on two-dimensional non-linear quantum walking is constructed. In the aspect of safety, a nonlinear mechanism is introduced in underlying quantum walking to directly realize diffusion and confusion of quantum versions, so that the quantum analysis resistance of a quantum hash function is improved; in the aspect of practicability, the output flexibility and expansibility of the quantum hash function are improved, and a feasible scheme is provided for practicability of the quantum hash function.

Description

Technical Field

[0001] This invention specifically relates to a multi-level security information integrity authentication method based on two-dimensional nonlinear quantum walks, belonging to the field of cyberspace security technology. Background Technology

[0002] Hash functions, as a core component of information security systems, are widely used in various security fields such as data integrity verification, digital signatures, identity authentication, and blockchain technology. They are also the foundation of many advanced cryptographic protocols and are closely related to protocol building modules such as pseudo-random number generators and key derivation functions. A traditional hash is a one-way function that compresses an input message of arbitrary length into a fixed-length output digest using a specific algorithm. Its security is ensured by its one-wayness, collision resistance, anti-aliasing, and avalanche effect cryptographic properties. Message authentication codes (MACs) are keyed hash functions that can simultaneously verify data integrity and authenticate the data source. A MAC accepts two inputs: a message and a key, and returns a hash value as the output. A MAC requires that it is difficult to find two different messages with the same output without knowing the key. It can be constructed based on hash functions or block ciphers. Because hash function-based MACs are much faster than block cipher-based MACs, existing schemes typically construct MACs based on hash functions.

[0003] Classical hash functions typically rely on the Merkle–Damgård structure or sponge structure, processing message blocks through iterative compression functions to ultimately generate hash values. The rapid development of quantum computing theory and technology has accelerated the search for hash collisions, posing a potential threat to the security boundaries of existing hash functions. This compels us to re-examine and construct new hash mechanisms capable of resisting quantum attacks. Quantum hash functions constructed based on quantum mechanics principles have become an important research direction at the intersection of post-quantum cryptography and quantum information science.

[0004] Quantum hash functions introduce quantum mechanisms into traditional hashing schemes, utilizing properties such as quantum superposition, entanglement, and interference to generate digests of a specific length, enhancing the hash function's collision resistance and one-wayness. They rely on the uncertainty principle, the no-cloning principle for unknown quantum states, and the irreversible collapse principle of measurement quantum states from quantum mechanics to ensure dual security in both classical and quantum computing environments. Existing quantum hash functions can be mainly categorized into those based on Boson sampling, quantum walk models, and quantum chaotic systems. Among these, quantum walks, due to their inherent chaotic and diffusion properties, high-dimensional state space expansion capabilities, and parallel processing capabilities, have become an ideal candidate for constructing efficient quantum hash functions.

[0005] The construction of quantum hash function schemes based on quantum walks mainly relies on designing different underlying quantum walk algorithms so that their evolution process can be controlled by classical input messages. Then, through post-processing algorithms, a classical output hash value can be obtained. The main problems with existing schemes of this kind are as follows: (1) It cannot simultaneously improve the system space and achieve the direct realization of nonlinear quantum confusion. Existing quantum hash functions based on discrete quantum walks are mostly linear quantum walks on one-dimensional or simple graph structures. These walk systems have limited state space expansion, making it difficult to support high-intensity diffusion and confusion requirements. Furthermore, they cannot break the reversibility limit of unitary operations, posing a risk of being reverse-modeled or predicted.

[0006] (2) The output length of the quantum hash function is limited. Most existing quantum hash functions based on quantum walks can only generate a fixed-length output and cannot customize the length of the output hash value, such as those specified by NIST. Bit, Bit, Bit, Hash values ​​with bit length lack flexibility.

[0007] This invention employs a two-dimensional nonlinear quantum walk, directly breaking the reversibility of quantum walks and increasing the spatial dimension of the system. Different nonlinear intensity parameters, the size of the two-dimensional lattice, and the initial quantum state can all affect the properties of the two-dimensional nonlinear quantum walk. Compared to ordinary two-dimensional quantum walks, this method adds a nonlinear phase shift in each step of the evolution, making this invention more secure and flexible. Summary of the Invention

[0008] The purpose of this invention is to address the shortcomings of existing quantum hash functions based on quantum walks by providing a structural construction method for a quantum hash function based on two-dimensional nonlinear quantum walks. Based on this method, a message verification code is constructed to achieve a multi-level secure information integrity authentication method. This method employs a parameterized approach to control the model, solving the problems of lack of flexibility and scalability in previous solutions. It realizes classical input-classical output quantum hash functions with different security levels, thereby achieving multi-level secure information integrity authentication.

[0009] The technical solution adopted in this invention is: a multi-level secure information integrity authentication method based on two-dimensional nonlinear quantum walk, wherein a shared key and classical information are input, and each evolution operation of the two-dimensional nonlinear quantum walk is controlled by the message bit. Multi-level secure information integrity authentication is achieved by constructing a nonlinear quantum hash function with a key.

[0010] This invention includes the following steps: Step 1: Initialize the parameters of the keyed nonlinear quantum hash function 1.1 Select Security Level Choose the security level of the keyed quantum hash function, which is the binary length of the output hash value. and according to Determine the modulus coefficient and the two-dimensional lattice size of two-dimensional nonlinear quantum walks ,in, , Even number, Let be a set of positive integers, representing the number of vertices on a two-dimensional lattice. , Also positive integers and ,Right now Divisible , yes The factors, and Satisfying Relationships .

[0011] 1.2 Preprocessing Input Messages Given a plaintext input message to be compressed Transform it into The code format is then arranged in order and converted into binary format. For a set express The set of all possible finite-length sequences composed of elements in the set, in a multi-level secure information integrity authentication method based on two-dimensional nonlinear quantum walks, It is by and A sequence of arbitrary finite length. If The length is less than Use the insufficient digits to Fill in the blanks; otherwise, do nothing, and you will get the final binary quantum hash function input. .

[0012] 1.3 Initialize input parameters The parameter set for multi-level security information integrity authentication in this invention is as follows: The specific parameter meanings and setting requirements are as follows: and The value of is determined in step 1.1.

[0013] The coin operator is a four-dimensional unitary operator selected from a set that meets the safety requirements.

[0014] and represents the nonlinear intensity parameter of a two-dimensional nonlinear quantum walk.

[0015] , , , For the initial quantum state The amplitude coefficient of each coin state, where The Dirac notation in quantum mechanics. Satisfying the normalization condition .

[0016] parameter A positive integer, representing the probability multiplication factor, and... satisfy ,Right now The demand is much greater than This is to ensure the security of the hash value obtained after post-processing.

[0017] Of all the parameters, the nonlinear intensity parameter and Coin arithmetic and initial quantum state Amplitude coefficient of each coin state , , , It serves as a shared key between the two communicating parties.

[0018] Step 2: Perform a two-dimensional nonlinear quantum walk 2.1 Selecting the initial position state Choose the center point of the two-dimensional lattice as the origin. Encode it as a quantum state in the location space. The coordinates of any point on a two-dimensional lattice are... This indicates that the quantum state encoded in the location space is used... Represented by the size of the two-dimensional lattice. Encoding each vertex on a two-dimensional lattice into a quantum state in location space. Choose an initial position state from them. , , ,in, It is a set of integers, that is All are integers.

[0019] 2.2 Performing a quantum walk The evolution operator controls each step of a two-dimensional nonlinear quantum walk using message bits. ,in, Indicates the number of steps taken. It is the identity matrix. The shift operator is defined as follows: , in, This represents the position quantum state during a quantum walk. for The conjugate transpose of the shift operator During the execution process, It is necessary to traverse all positions reached by the current particle; , or , or , representing one of the four ground states of the coin's quantum state during the quantum walk. Also The conjugate transpose of the shift operator During the execution process, It is necessary to traverse all four ground states of the coin quantum state; The symbol for tensor product indicates that the current state of a particle during a quantum walk is a combination of a coin state and a position state. For mathematical operations, simply put, This indicates that when the coin's ground state is At that time, the particle's current position is determined by Move to After the movement is completed, the coin's ground state remains unchanged. .

[0020] The nonlinear phase shift operator is defined as follows: .

[0021] in, Indicates the first Nonlinear phase shift operator during walking Indicates that the particle is in After walking, in position The coin's ground state is The probability density of a single component at time , Indicates that the particle is in After walking, the location is Total probability density , Represents the nonlinear intensity parameter, nonlinear phase factor It acts on the corresponding wave function component, thereby changing its phase, but not its amplitude (i.e. probability density). for The conjugate transpose of . The calculation of nonlinear phase offset does not change the current state of the walk.

[0022] Specifically, it means using the filled version The nonlinear intensity parameter is controlled bit by bit. In the first step of the walk... Step, assuming at this time The input bits are The nonlinear phase shift operator used for walking The nonlinear intensity parameters used are: , recorded as Transform the walking evolution at this time Record ,Right now , Assuming at this time The input bits are The nonlinear phase shift operator used for walking The nonlinear intensity parameters used are: , recorded as Transform the walking evolution at this time Record ,Right now , .

[0023] From the initial quantum state Begin executing a two-dimensional nonlinear quantum walk, where, Representing the initial state of the coin, the walking evolution process can be represented as follows: , express The Bits Indicates input Length, This represents the final quantum state after the evolution ends.

[0024] Step 3: Measure the probability distribution of the final quantum state vertex. The final quantum state obtained in step 2 each position state implement (Positive operator value measure) Measurement operation, to obtain each vertex on the two-dimensional lattice. The probability distribution of walking: , in, , For each vertex on a two-dimensional lattice The probability value, The vertex at the end time In coin state The amplitude coefficient at that point.

[0025] Step 4: Execute the post-processing method to calculate the hash value. The probability distribution obtained in step 3 Each of them Post-processing method That is, multiply the probability value of each vertex by After rounding down, the modulus is... The modulo operation is used to obtain the string corresponding to each vertex. Then calculate each vertex Corresponding string , Concatenate them sequentially and convert them into a new binary string. This value is the hash value of the plaintext. .

[0026] Step 5: Verify the integrity of the information After completing steps 1-4, the message sender will send the plaintext message. The obtained hash value and non-key parameters The message is sent together to the recipient. After receiving the message, the recipient uses the shared key and non-key parameters. Repeat steps 1-4 to obtain the received message. hash value , and the received hash value Perform a comparison, if If the information integrity is verified, the message... It has not been tampered with. Conversely, this indicates the message. It was altered.

[0027] The advantages of this invention compared to the prior art are: (1) This invention proposes a novel two-dimensional nonlinear quantum walk model. The keyed nonlinear quantum hash function constructed based on this model is controlled by a variable nonlinear strength parameter rather than a coin operator, which breaks the quantum walk reversibility limitation, enhances the system’s chaos and resistance to quantum analysis. By adjusting the parameters, hash values ​​that meet different security levels can be generated, which has higher flexibility and practicality.

[0028] (2) The present invention enhances the structure of quantum hash function based on quantum walk. By introducing more control parameters, it improves the collision resistance, sensitivity to plaintext messages, diffusion and confusion, and uniformity of plaintext message image of quantum hash function. In addition, the introduction of nonlinear strength parameter also enables quantum hash function to resist forgery attack. Attached Figure Description

[0029] Figure 1 This is a flowchart of the method.

[0030] Figure 2 The diagram shows the structure of a quantum hash function based on a two-dimensional nonlinear quantum walk.

[0031] The symbols are explained as follows: For input message The Bit, ; For the first quantum walk Evolutionary operators during walking , See step 2.2 for details. For input message The Bit, Because every step of walking is determined by the input message. It is determined by one bit, so and One-to-one correspondence, both are of length the input message. length ; For the first quantum walk The quantum state after walking. , This is the initial state; Let be the evolution function for each step of the quantum hash function; For quantum measurement symbols; For the final quantum state (shown in the figure) Each vertex measured Probability distribution; This is the final message hash value. Detailed Implementation

[0032] Please see Figure 1-2As shown, the multi-level secure information integrity authentication method based on two-dimensional nonlinear quantum walks proposed in this invention mainly aims to construct a keyed nonlinear quantum hash function based on two-dimensional nonlinear quantum walks, thereby realizing a multi-level secure information integrity authentication method. By studying the properties of two-dimensional nonlinear quantum walks, secure parameters are selected to ensure that the keyed nonlinear quantum hash function possesses the required security properties.

[0033] This invention proposes a multi-level secure information integrity authentication method based on two-dimensional nonlinear quantum walks, where the entire system's evolution occurs on a two-dimensional lattice. In two-dimensional quantum walks, a two-dimensional lattice refers to a two-dimensional grid structure composed of discrete lattice points, representing the space of positions that the quantum walking particle can occupy. The proposed multi-level secure information integrity authentication method based on two-dimensional nonlinear quantum walks utilizes a Hilbert space for its system. ,in, Represents a location space, whose orthogonal basis consists of sizes. The labels of each vertex on a two-dimensional lattice, arranged in a dictionary order, are given, defined as... , It is a set of integers. Representing the four-dimensional coin space, defined as The particle starts from its initial quantum state. Begin walking, in an area of ​​size A two-dimensional nonlinear quantum walk is performed on a two-dimensional lattice, and the final quantum state of the walk is... ,in, For input message The product of a series of evolutionary transformations controlled bit by bit. The vertex at the end time In coin state The amplitude coefficient at that point.

[0034] The multi-level security information integrity authentication method proposed in this invention can be computed simultaneously on both classical and quantum computers in polynomial time complexity. When computed on a classical computer, it essentially simulates the evolution of a two-dimensional nonlinear quantum walk, a process involving basic arithmetic and logical operations, requiring only... The time complexity is low. When computed using a quantum computer, this invention requires only logarithmic-level resources. The quantum hash function proposed in this invention has unidirectionality. In the quantum walk, measurement, and measurement result post-processing stages of this invention, the quantum walk evolution process, quantum measurement operation, modulo operation, and truncation operation are irreversible, making the output and input of the hash function a one-to-many mapping.

[0035] One embodiment of the present invention is a multi-level secure information integrity authentication method based on two-dimensional nonlinear quantum walks, which is applied to the data integrity authentication process. It is assumed that the communicating parties are Alice and Bob, and they share a key. ,in For coin operators, for Step Operator, and The nonlinear intensity parameter of the two-dimensional nonlinear quantum walk. , , , For the initial quantum state The amplitude coefficient of each coin state. Alice, as the message sender, needs to send a plaintext message. Bob receives the message and needs to perform integrity authentication to prevent it from being tampered with. The authentication method includes the following five steps: Step 1: Initialize the parameters of the keyed nonlinear quantum hash function 1.1 Select Security Level Alice chooses the security level of the keyed quantum hash function, which is the binary length of the output hash value. and according to Determine the modulus coefficient and the two-dimensional lattice size of two-dimensional nonlinear quantum walks ,in, ,Right now It is a positive integer, and An even number indicates that the number of vertices on a two-dimensional lattice is 1. , and ,Right now Divisible ,yes The factors, and Satisfying Relationships .

[0036] Alice selects the binary length of the output hash value. Bits, select modulus coefficients Therefore, the corresponding two-dimensional lattice size of the two-dimensional nonlinear quantum walk .

[0037] 1.2 Preprocessing Input Messages The plaintext message Alice needs to enter is Transform it into The code format is then arranged in order and converted into binary format. .if The length is less than Use the insufficient digits to The input to the final binary quantum hash function is either padded or left unprocessed. The purpose of padding is to prevent the hash values ​​generated by short messages from exhibiting a regularity.

[0038] Assume Alice's input is Convert it to binary:

[0039] because The length is greater than Therefore, no filling is required.

[0040] 1.3 Initialize input parameters The parameter set for multi-level security information integrity authentication in this invention is as follows: The specific parameter meanings and setting requirements are as follows: and The value of is determined in step 1.1.

[0041] The coin operator is a four-dimensional unitary operator selected from a set that meets security requirements, used to control the directional evolution of a two-dimensional quantum walk. Its selection should satisfy security requirements to resist spoofing attacks on hash functions. A spoofing attack on a hash function can be defined as: for a hash function... Given an input message A polynomial-time adversary can construct another input message. Make This means that a collision occurs in the hash function.

[0042] and The nonlinear intensity parameter for the two-dimensional nonlinear quantum walk must be selected from a set that meets safety requirements to prevent changes in the nonlinear intensity from causing the quantum walk dynamics to change from a diffuse state to a local state or to produce a stable arc.

[0043] , , , For the initial quantum state The amplitude coefficient of each coin state, where The Dirac notation in quantum mechanics. Satisfying the normalization condition .

[0044] Coin Counter Nonlinear strength parameters and initial quantum state Amplitude coefficient of each coin state , , , As shared keys between the communicating parties, they are determined before the instance steps begin, and they determine the specific evolution and properties of the two-dimensional nonlinear quantum walk.

[0045] parameter , is the probability multiplication factor, and satisfy .

[0046] The final parameter set is: , Step 2: Perform a two-dimensional nonlinear quantum walk 2.1 Selecting the initial position state Choosing the center point of the two-dimensional lattice as the origin, the encoding is as follows: Encoding each vertex on a two-dimensional lattice into a quantum state in location space. Choose an initial position state from them. , , Assume the initial position state is .

[0047] 2.2 Performing a quantum walk The evolution operator controls each step of a two-dimensional nonlinear quantum walk using message bits. ,in Indicates the number of steps taken. For location space The identity matrix on, The shift operator is defined as follows: , The nonlinear phase shift operator is defined as follows: .

[0048] Specifically, it means using the filled version The nonlinear intensity parameter is controlled bit by bit. In the first step of the walk... Step, assuming at this time The input bits are The nonlinear phase shift operator used for walking The nonlinear intensity parameters used are: , recorded as Transform the walking evolution at this time Record ,Right now , Assuming at this time The input bits are The nonlinear phase shift operator used for walking The nonlinear intensity parameters used are: , recorded as Transform the walking evolution at this time Record ,Right now , .

[0049] From the initial quantum state Begin a two-dimensional nonlinear quantum walk, in which Given the initial state of the coin, the walking evolution process can be represented as follows: , express The Bits Indicates after filling Length, This represents the final quantum state after the evolution ends.

[0050] Step 3: Measure the probability distribution of the final quantum state vertex. After performing step 2, the quantum walk system is finally in a quantum state. ,in, The vertex at the end time In coin state The amplitude coefficient at each vertex of the two-dimensional lattice. conduct The corresponding probability distribution is obtained by measurement. When simulated on a classical computer, the simulation program can be used to calculate the probability distribution of walking at each location. ,in, , For each vertex of the particle on the two-dimensional lattice The probability value.

[0051] The probability distribution of the final quantum state obtained is as follows:

[0052] Step 4: Execute the post-processing method to calculate the hash value. The probability distribution obtained in step 3 Each vertex in probability distribution Post-processing method This involves multiplying the probability values ​​of each vertex by... Amplify, round down, and then perform modulo operation. The modulo operation yields the string. Next, the calculated values ​​for each vertex will be... Corresponding string Concatenate them sequentially and convert them into a new binary string. This value is the hash value of the plaintext. .

[0053] when For each of the results obtained in step 3 enlarge Retain The hash value of each vertex is obtained by dividing the number of decimal places: .

[0054] By arranging and concatenating all the characters in order, we obtain a... A binary string of bits as a message hash value For ease of description, it will be converted to hexadecimal. .

[0055] Step 5: Verify the integrity of the information Alice will send plaintext messages The obtained hash value and non-key parameters They were sent together to Bob, and Bob received the message. Then, use the shared key. Non-key parameters Repeat steps 1-4 to obtain the received message. hash value , and the received hash value Perform a comparison, if If the information integrity is verified, the message... It has not been tampered with. Conversely, it indicates the message It was tampered with during transmission.

[0056] The above description is merely an example of a multi-level security information integrity authentication method based on two-dimensional nonlinear quantum walks according to the present invention. It should be noted that these examples are only used to illustrate the implementation process of the present invention and do not limit the scope of the present invention. For those skilled in the art, several improvements, refinements, or equivalent modifications can be made without departing from the present invention. These improvements, refinements, or equivalent modifications should also be considered as the scope defined by the appended claims of the present invention.

Claims

1. A multi-level secure information integrity authentication method based on two-dimensional nonlinear quantum walks, characterized in that, Includes the following steps: Step 1: Initialize the parameters of the keyed nonlinear quantum hash function Step 1.1: Select the security level; select the security level of the keyed quantum hash function, i.e., the binary length of the output hash value. and according to Determine the modulus coefficient and the two-dimensional lattice size of two-dimensional nonlinear quantum walks , Step 1.2: Preprocess the input message; given an input plaintext message to be compressed. Transform it into The code format is then arranged in order and converted into binary format. For a set express The set of all possible sequences of finite length composed of the elements in the set. It is by and A sequence of arbitrary finite length; Step 1.3: Initialize input parameters; the parameter set for multi-level security information integrity authentication is as follows: ,in, For coin operators, and The nonlinear intensity parameter of the two-dimensional nonlinear quantum walk; , , , For the initial quantum state The amplitude coefficient of each coin state, where The Dirac notation in quantum mechanics. Satisfying the normalization condition ;parameter A positive integer, representing the probability multiplication factor, and... satisfy ,Right now The demand is much greater than ; Step 2: Perform a two-dimensional nonlinear quantum walk Step 2.1: Select the initial position state Choose the center point of the two-dimensional lattice as the origin. Encode it as a quantum state in the location space. The coordinates of any point on a two-dimensional lattice are... This indicates that the quantum state encoded in the location space is used... express; Step 2.2: Perform a quantum walk The evolution operator controls each step of a two-dimensional nonlinear quantum walk using message bits. ,in, Indicates the number of steps taken. It is the identity matrix. For shift operators; Step 3: Measure the probability distribution of the final quantum state vertex. For each position state of the final quantum state obtained Perform positive operator value measurement Measurement operations are performed to obtain the vertices of the two-dimensional lattice. The probability distribution of walking; Step 4: Execute the post-processing method to calculate the hash value. Each of the obtained probability distributions Post-processing method That is, multiply the probability value of each vertex by After rounding down, the modulus is... The modulo operation is used to obtain the string corresponding to each vertex. Then calculate each vertex Corresponding string , Concatenate them sequentially and convert them into a new binary string. That is, the hash value of the plaintext. ; Step 5: Verify the integrity of the information Plain text message The obtained hash value and non-key parameters The message is sent together to the recipient. After receiving the message, the recipient uses the shared key and non-key parameters. Repeat steps 1-4 to obtain the received message. hash value , and the received hash value Perform a comparison, if Then the integrity of the information is authenticated, and the message... It has not been tampered with. Conversely, this indicates the message. It was altered.

2. The multi-level secure information integrity authentication method based on two-dimensional nonlinear quantum walks according to claim 1, characterized in that: In step 1.1, , Even number, Let be a set of positive integers, representing the number of vertices on a two-dimensional lattice. , Also positive integers and ,Right now Divisible , yes The factors, and Satisfying Relationships .

3. A multi-level secure information integrity authentication method based on two-dimensional nonlinear quantum walks according to claim 1 or 2, characterized in that: In step 1.2, if The length is less than Use the insufficient digits to Fill in the blanks; otherwise, do nothing, and you will get the final binary quantum hash function input. .

4. The multi-level secure information integrity authentication method based on two-dimensional nonlinear quantum walks according to claim 3, characterized in that: In step 1.3, among all parameters, the nonlinear intensity parameter and Coin arithmetic and initial quantum state Amplitude coefficient of each coin state , , , It serves as a shared key between the two communicating parties.

5. The multi-level secure information integrity authentication method based on two-dimensional nonlinear quantum walks according to claim 1, characterized in that: In step 2.1, based on the size of the two-dimensional lattice Encoding each vertex on a two-dimensional lattice into a quantum state in location space. Choose an initial position state from them. , , ,in, It is a set of integers, that is All are integers.

6. A multi-level secure information integrity authentication method based on two-dimensional nonlinear quantum walks according to claim 1 or 5, characterized in that: In step 2.2, the shift operator is defined as follows: ， in, This represents the position quantum state during a quantum walk. for The conjugate transpose of the shift operator During the execution process, It is necessary to traverse all positions reached by the current particle; , or , or , representing one of the four ground states of the coin's quantum state during the quantum walk. Also The conjugate transpose of the shift operator During the execution process, It is necessary to traverse all four ground states of the coin quantum state; The symbol for tensor product indicates that the current state of a particle during a quantum walk is a combination of a coin state and a position state; For mathematical operations.

7. The multi-level secure information integrity authentication method based on two-dimensional nonlinear quantum walks according to claim 6, characterized in that: In step 2.2, message bits are used to control each step of the two-dimensional nonlinear quantum walk, i.e., the evolution operator. ,in, Indicates the number of steps taken. It is the identity matrix. The nonlinear phase shift operator is defined as follows: ； in, Indicates the first Nonlinear phase shift operator during walking Indicates that the particle is in After walking, in position And the coin's ground state is The probability density of a single component at time , Indicates that the particle is in After walking, the location is Total probability density , Represents the nonlinear intensity parameter, nonlinear phase factor It acts on the corresponding wave function component, thereby changing its phase, but not its amplitude. for The conjugate transpose of . The calculation of nonlinear phase offset does not change the current state of the walk.

8. The multi-level secure information integrity authentication method based on two-dimensional nonlinear quantum walk according to claim 7, characterized in that: In step 2.2, use the filled... The nonlinear intensity parameter is controlled bit by bit; in the first step of the walk... Step, let's assume at this time The input bits are The nonlinear phase shift operator used for walking The nonlinear intensity parameters used are: , recorded as Transform the walking evolution at this time Record ,Right now Suppose at this time The input bits are The nonlinear phase shift operator used for walking The nonlinear intensity parameters used are: , recorded as Transform the walking evolution at this time Record ,Right now ; From the initial quantum state Begin executing a two-dimensional nonlinear quantum walk, where, The initial state of the coin is represented as follows: [Insert coin description here] , express The bit Indicates input Length, This represents the final quantum state after the evolution ends.

9. The multi-level secure information integrity authentication method based on two-dimensional nonlinear quantum walks according to claim 1, characterized in that: In step 3, the quantum state is: ； The probability distribution of walking is: ， in, , For each vertex on a two-dimensional lattice The probability value, The vertex at the end time In coin state The amplitude coefficient at that point.

10. The multi-level secure information integrity authentication method based on two-dimensional nonlinear quantum walk according to claim 1, characterized in that: In step 4, the probability distribution is: .

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