Arbiter physical unclonable function modeling method and system based on extremely few excitation response pairs and application of arbiter physical unclonable function modeling method and system

Abstract

The invention relates to the technical field of Internet of Things safety, and discloses an arbiter physical unclonable function modeling method with few excitation response pairs, which comprises the following steps: step a, a group of excitation-response pairs (CRPs) are acquired, and the excitation-response pairs comprise excitation signals, corresponding response outputs and reliability data of each excitation-response pair; b, on the basis of the reliability data, the environmental noise level is calculated through probability statistical analysis, and the environmental noise level represents the noise intensity influencing the APUF response stability; and c, utilizing the ambient noise level and the reliability of a single excitation-response pair. According to the method, physical modeling of the APUF can be completed by optimizing the modeling process and only needing n + 1 CRPs, and compared with a traditional machine learning method which usually needs thousands of CRPs, the method has the advantages that the data demand quantity is reduced by 90% or above, the resource overhead of data collection, storage and processing is remarkably reduced, and convenience is provided for rapid deployment of large-scale Internet of Things equipment.

Description

Technical Field

[0001] The present invention relates to the technical field of Internet of Things security, and particularly to a modeling method, system and application of an arbiter physical unclonable function with extremely few excitation response pairs. Background Art

[0002] With the rapid development of the Internet of Things (IoT) technology, the number of networked devices has shown an explosive growth globally. According to statistics, by 2025, the total number of IoT devices is expected to exceed 50 billion, and these devices are widely used in fields such as smart homes, industrial automation, healthcare, and transportation. The IoT brings revolutionary convenience to social production and daily life by enabling interconnection between devices. Since the hardware resources of IoT devices are usually limited, such as low computing power, small storage space, and strict power consumption requirements, the application of traditional encryption algorithms (such as the Advanced Encryption Standard AES, Hash function, etc.) on these devices faces great challenges.

[0003] To meet the requirements of IoT devices in security authentication, Physical Unclonable Functions (PUFs), as an emerging hardware security primitive, have received extensive attention from academia and industry in recent years. PUFs utilize random process variations in the semiconductor manufacturing process to generate a unique input-output mapping relationship, namely Challenge-Response Pairs (CRPs). The unpredictability and non-replicability of this mapping relationship stem from the physical property differences of devices at the microscopic level. Even chips manufactured on the same production line have different PUF characteristics. Therefore, PUFs can achieve device authentication or generate encryption keys without storing keys, and have the advantages of being lightweight, low-power, and highly secure.

[0004] Arbiter Physical Unclonable Functions (APUFs) have become the focus of research and application due to their simple structure and easy implementation. The basic principle of APUF is to transmit signals through a set of symmetric delay paths and use an arbiter to judge the order of signal arrival in the two paths, and output a binary response (0 or 1). Due to the randomness of the manufacturing process, there are slight differences in the delay time of each path, and this difference determines the uniqueness of the APUF response.

[0005] The prior art has deficiencies in the modeling efficiency of APUF. Traditional machine learning modeling methods usually require collecting a large number of CRPs as training data. For example, machine learning-based modeling may require thousands or even tens of thousands of CRP samples. For Internet of Things devices in large-scale production, collecting and storing such a large amount of data one by one not only increases the production cost but also may introduce new security risks due to data leakage during the data transmission process. At the same time, existing methods often lack systematic analysis tools for dealing with environmental noise, unable to effectively quantify the specific impact of noise on the reliability of APUF, and it is also difficult to dynamically adjust the authentication strategy according to the noise level. Summary of the Invention

[0006] To solve the technical problems proposed in the background art, the present invention provides a method, system and application for modeling an arbiter physical unclonable function with extremely few excitation-response pairs.

[0007] The present invention is implemented by the following technical solutions: A method for modeling an arbiter physical unclonable function with extremely few excitation-response pairs, comprising the following steps:

[0008] Step a. Obtain a set of excitation-response pairs (CRPs), where the excitation-response pairs include excitation signals and their corresponding response outputs, as well as the reliability data of each excitation-response pair;

[0009] Step b. Based on the reliability data, calculate the environmental noise level through probability statistical analysis, where: the environmental noise level represents the noise intensity affecting the response stability of APUF;

[0010] Step c. Use the environmental noise level and the reliability of a single excitation-response pair to quantify the delay difference between the two paths of the last stage of the APUF corresponding to the specified excitation-response pair;

[0011] Step d. Select n + 1 excitation-response pairs with known reliability and linearly independent after transformation from the set of excitation-response pairs, where n is the number of stages of the APUF;

[0012] Step e. Convert the excitation signals of the n + 1 excitation-response pairs into a feature matrix Φ, where each row of the feature matrix Φ corresponds to the feature vector of an excitation signal;

[0013] Step f. Based on the environmental noise level and the reliability of each selected excitation-response pair, calculate the corresponding delay difference Δ, where Δ represents the delay difference between the two paths of the last stage of the APUF;

[0014] Step g. Use the least squares method to solve the delay vector w, where the equation Φw = Δ is satisfied, thus completing the physical modeling of APUF.

[0015] Among them, the method only uses n + 1 excitation-response pairs to achieve modeling, and the modeling accuracy reaches more than 99% under a predetermined noise level.

[0016] Specifically, the probability statistical analysis in step b includes:

[0017] Using the formula to calculate the overall reliability of the APUF, where: α is the environmental noise intensity; Reliability is the average reliability measured based on multiple excitation-response pairs;

[0018] Determine the environmental noise level α from the reliability data by reverse-deriving the formula;

[0019] Specifically, the quantification of the delay difference Δ in step c includes:

[0020] For a specified excitation-response pair, when calculating reliability, P((Δ + noise)(sgn(Δ)>0) =

[0021] reliability, where: noise is the environmental noise, following the normal distribution N(0, α); reliability is the reliability of this excitation-response pair, and sgn is the sign function;

[0022] Convert the probability to the standard normal distribution form and solve for the value of the delay difference Δ, where Δ represents the delay difference between two paths in the last stage of the APUF.

[0023] Specifically, in step d, the selection of the n + 1 excitation-response pairs includes the following sub-steps:

[0024] Measure the reliability of the set of excitation-response pairs to obtain the reliability value of each excitation-response pair;

[0025] Select n + 1 excitation-response pairs from them to ensure that the corresponding characteristic matrix Φ is full rank;

[0026] Verify the linear independence of the characteristic matrix Φ to ensure the unique solution of the delay vector w.

[0027] In the present invention, under the condition that the noise level is 0.15, each excitation-response pair is repeatedly measured at least 100 times to determine its reliability value; based on the reliability value, verify that the modeling accuracy reaches more than 99%.

[0028] Adopting the method proposed by the present invention does not require additional hardware resources, and the quantification of the delay difference completely depends on the statistical analysis of environmental noise and reliability;

[0029] Predict unused excitation-response pairs using the delay vector w generated by the physical modeling;

[0030] Use the predicted excitation-response pairs for authentication between the device and the server.

[0031] The present invention also proposes an arbiter physical unclonable function modeling system with extremely few excitation-response pairs, including the following modules:

[0032] An input module configured to receive a set of excitation-response pairs (CRPs), where the excitation-response pairs include excitation signals and their corresponding response outputs, as well as reliability data for each excitation-response pair;

[0033] A noise calculation module configured to calculate the environmental noise level through probability statistical analysis based on the reliability data;

[0034] A delay quantization module configured to quantify the delay difference between two paths of the last stage of the APUF corresponding to a specified excitation-response pair by using the environmental noise level and the reliability of a single excitation-response pair;

[0035] A selection module configured to select n + 1 excitation-response pairs with known reliability and linearly independent after transformation from the set of excitation-response pairs, where n is the number of stages of the APUF;

[0036] A conversion module configured to convert the excitation signals of the n + 1 excitation-response pairs into a feature matrix Φ;

[0037] A calculation module configured to calculate the corresponding delay difference Δ based on the environmental noise level and the reliability of each selected excitation-response pair;

[0038] A modeling module configured to solve the delay vector w using the least squares method, where the equation Φw = Δ is satisfied, thereby completing the physical modeling of the APUF;

[0039] An output module configured to output the delay vector w as the modeling result;

[0040] Wherein, the system realizes modeling only using n + 1 excitation-response pairs, and the modeling accuracy reaches more than 99% under a predetermined noise level.

[0041] Specifically, use the formula Calculate the overall reliability of the APUF, where α is the environmental noise intensity; determine the environmental noise level α from the reliability data through reverse derivation.

[0042] As a further improvement of the above solution, the delay quantization module is further configured to:

[0043] For a specified excitation-response pair, calculate the probability P((Δ + noise)(sgn(Δ)>0) = reliability, where noise follows a normal distribution N(0,α);

[0044] Convert the probability to the form of a standard normal distribution and solve for the delay difference Δ.

[0045] As a further improvement to the above solution, the selection module is further configured to:

[0046] Rank the reliability of the set of excitation-response pairs;

[0047] Select n + 1 excitation-response pairs to ensure that the corresponding feature matrix Φ is full rank.

[0048] The system does not require an additional hardware measurement unit and only realizes the quantification and modeling of the delay difference through a software module.

[0049] As a further improvement to the above solution, the system further includes an authentication module configured to:

[0050] Use the delay vector w generated by the modeling module to predict new excitation-response pairs;

[0051] Apply the predicted excitation-response pairs to the security authentication between the device and the server

[0052] The application of an arbiter physically unclonable function modeling system with extremely few excitation-response pairs proposed in this solution can generate a set of responses of the excitation pairs to the APUF for prediction based on the physical modeling results.

[0053] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0054] First, high-efficiency modeling: By optimizing the modeling process, the present invention only requires n + 1 CRPs to complete the physical modeling of the APUF. Compared with traditional machine learning methods that usually require thousands of CRPs, the present invention reduces the data requirement by more than 90%. This improvement significantly reduces the resource overhead of data collection, storage, and processing, providing convenience for the rapid deployment of large-scale Internet of Things devices.

[0055] Second, high accuracy: The modeling method of the present invention can achieve a modeling accuracy of more than 99% in the working environment of typical Internet of Things devices (such as a noise level of 0.15). This high accuracy benefits from the precise quantification of the relationship between reliability and noise and the effective calculation of the delay difference, ensuring the reliability of the model in various actual application scenarios.

[0056] Third, there is no need for additional hardware overhead: Different from traditional methods that rely on hardware devices such as time-to-digital converters (TDCs), the present invention quantifies the delay difference through statistical analysis methods and does not require additional hardware support. This feature not only reduces the cost of system implementation but also simplifies the complexity of hardware design, making this method easier to implement in resource-constrained Internet of Things devices. Description of the Drawings

[0057] Figure 1 Flowchart of the method for modeling the arbiter physical unclonable function proposed by the present invention;

[0058] Figure 2 Schematic diagram of the basic APUF circuit disclosed in the embodiments of the present invention;

[0059] Figure 3 Flowchart from CRP selection to solving the delay vector in the present invention. Detailed Embodiments

[0060] Next, in combination with the drawings and specific embodiments, the present invention will be further described. It should be noted that, on the premise of no conflict, the following-described embodiments or technical features can be combined arbitrarily to form new embodiments.

[0061] Embodiment:

[0062] Referring to Figure 1 , a method for modeling the arbiter physical unclonable function with extremely few excitation responses proposed by this solution includes the following steps:

[0063] Step a. Obtain a set of excitation-response pairs (CRPs), where the excitation-response pairs include excitation signals and their corresponding response outputs, as well as the reliability data of each excitation-response pair;

[0064] Step b. Based on the reliability data, calculate the environmental noise level through probability statistical analysis, where: the environmental noise level represents the noise intensity affecting the response stability of the APUF;

[0065] Step c. Utilize the environmental noise level and the reliability of a single excitation-response pair to quantify the delay difference between the two paths of the last stage of the APUF corresponding to the specified excitation-response pair;

[0066] Step d. Select n + 1 excitation-response pairs with known reliability and linearly independent after transformation from the set of excitation-response pairs, where n is the number of stages of the APUF;

[0067] Step e. Convert the excitation signals of the n + 1 excitation-response pairs into a feature matrix Φ, where each row of the feature matrix Φ corresponds to the feature vector of an excitation signal;

[0068] Step f. Calculate the corresponding delay difference Δ based on the environmental noise level and the reliability of each selected excitation-response pair, where Δ represents the delay difference between two paths of the last stage of the APUF;

[0069] Step g. Solve the delay vector w using the least squares method, where the equation Φw = Δ is satisfied, thereby completing the physical modeling of the APUF.

[0070] Among them, the method only uses n + 1 excitation-response pairs to achieve modeling, and the modeling accuracy reaches more than 99% under the predetermined noise level.

[0071] Specifically, the probability statistical analysis in step b includes:

[0072] Use the formula Calculate the overall reliability of the APUF, where: α is the environmental noise intensity; Reliability is the average reliability measured based on multiple excitation-response pairs;

[0073] Determine the environmental noise level α from the reliability data by reverse derivation of the formula;

[0074] Specifically, the quantization of the delay difference Δ in step c includes:

[0075] For a specified excitation-response pair, when calculating reliability, the probability P((Δ + noise)(sgn(Δ)>0) = reliability, where: noise is the environmental noise, following the normal distribution N(0, α); reliability is the reliability of this excitation-response pair;

[0076] Convert the probability to the standard normal distribution form and solve the value of the delay difference Δ, where Δ represents the delay difference between two paths of the last stage of the APUF.

[0077] Specifically, in step d, the selection of the n + 1 excitation-response pairs includes the following sub-steps:

[0078] Measure the reliability of the set of excitation-response pairs to obtain the reliability value of each excitation-response pair;

[0079] Select n + 1 excitation-response pairs from them to ensure that the corresponding characteristic matrix Φ is full rank; n is generally 0 - 200

[0080] Verify the linear independence of the characteristic matrix Φ to ensure the unique solution of the delay vector w.

[0081] In the present invention, under the condition that the noise level is 0.15, each excitation-response pair is repeatedly measured at least 100 times to determine its reliability value; based on the reliability value, it is verified that the modeling accuracy reaches more than 99%.

[0082] The calculation method of reliability is to take the probability of 0 or 1 in the response, and the larger one. Specifically, the reliability value is calculated by the formula Reliability = max{P(R = 0), P{R = 1}}.

[0083] The method proposed by the present invention does not require additional hardware resources. The quantization of the delay difference completely depends on the statistical analysis of environmental noise and reliability;

[0084] Using the delay vector w generated by the physical modeling to predict unused excitation-response pairs;

[0085] The predicted excitation-response pairs are used for authentication between the device and the server.

[0086] The present invention also proposes an arbiter physical unclonable function modeling system with extremely few excitation-response pairs, including the following modules:

[0087] An input module configured to receive a set of excitation-response pairs (CRPs), where the excitation-response pairs include excitation signals and their corresponding response outputs, as well as the reliability data of each excitation-response pair;

[0088] A noise calculation module configured to calculate the environmental noise level through probability statistical analysis based on the reliability data;

[0089] A delay quantization module configured to quantify the delay difference between two paths of the last stage of the APUF corresponding to a specified excitation-response pair by using the environmental noise level and the reliability of a single excitation-response pair;

[0090] A selection module configured to select n + 1 excitation-response pairs with known reliability and linearly independent after transformation from the set of excitation-response pairs, where n is the number of stages of the APUF;

[0091] A conversion module configured to convert the excitation signals of the n + 1 excitation-response pairs into a feature matrix Φ;

[0092] A calculation module configured to calculate the corresponding delay difference Δ based on the environmental noise level and the reliability of each selected excitation-response pair;

[0093] A modeling module configured to solve the delay vector w using the least squares method, where it satisfies the equation Φw = Δ, thereby completing the physical modeling of the APUF;

[0094] An output module, configured to output the delay vector w as a modeling result;

[0095] Wherein, the system only uses n + 1 excitation-response pairs to implement modeling, and the modeling accuracy reaches more than 99% under a predetermined noise level.

[0096] Specifically, use the formula Calculate the overall reliability of the APUF, where α is the environmental noise intensity; determine the environmental noise level α from the reliability data through reverse derivation.

[0097] As a further improvement of the above solution, the delay quantization module is further configured to:

[0098] For a specified excitation-response pair, calculate the probability P((Δ + noise)(sgn(Δ)>0) = reliability, where noise follows a normal distribution N(0,α);

[0099] Convert the probability to the standard normal distribution form and solve for the delay difference Δ.

[0100] As a further improvement of the above solution, the selection module is further configured to:

[0101] Perform a reliability ranking on the set of excitation-response pairs;

[0102] Select n + 1 excitation-response pairs to ensure that the corresponding characteristic matrix Φ is full rank.

[0103] The full rank determination criterion for Φ is that when rank(Φ) = n + 1, it is regarded as full rank. And the rank of the transformed characteristic matrix rank(Φ) = n + 1, where When k = 0, 1,..., n - 1 and Φ n = 1, c is the excitation.

[0104] The system does not require an additional hardware measurement unit and only realizes the quantization and modeling of the delay difference through software modules.

[0105] As a further improvement of the above solution, the system further includes an authentication module, configured to:

[0106] Use the delay vector w generated by the modeling module to predict a new excitation-response pair;

[0107] Apply the predicted excitation-response pair to the security authentication between the device and the server

[0108] The application of an arbiter physically unclonable function modeling system with extremely few excitation-response pairs proposed in this solution can generate a set of excitations to predict the responses of the APUF based on the physical modeling results.

[0109] More specifically:

[0110] Referring to Figures 1 - 3 , this solution proposes an efficient and reliable modeling method for an arbitration PUF (APUF). This method quantifies the relationship between the reliability of the APUF response and environmental noise, and then quantifies the delay difference inside the APUF, and constructs a PUF model based on a linear model.

[0111] Referring to Figure 2 shown in the figure, the embodiments of the method of the present invention include the following main steps:

[0112] 1. Quantification of APUF reliability and environmental noise.

[0113] First, quantify the relationship between the reliability of the APUF and environmental noise, and calculate the environmental noise. Starting from the perspective of probability statistics, its mathematical model is deduced.

[0114] As Figure 1 shown, for the n-stage APUF, the delay difference Δ(n - 1) of the last stage follows a normal distribution N(0, nσ 2 ), where n is the number of APUF stages, and σ 2 is the single-stage delay variance.

[0115] The environmental noise noise follows a normal distribution N(0, nασ 2 ), where α represents the noise intensity coefficient.

[0116] Response flip probability: The flip probability P(flip) of the APUF response is defined as the probability that the sign changes after adding the delay difference and the noise, that is:

[0117] P(flip) = P(Δ(n - 1) + noise < 0 ∣ Δ(n - 1) > 0) + P(Δ(n - 1) + noise > 0 ∣ Δ(n - 1) < 0)

[0118] Due to the symmetry of Δ(n - 1) and noise, the above conditional probabilities can be calculated together.

[0119] Derivation process: Let X = Δ(n - 1), then X ~ N(0, nσ 2 ), and Y = Δ(n - 1) + noise ~ N(0, (1 + α)nσ 2 ). The flip probability is:

[0120] P(flip) = 2 * P(Y < 0 ∣ X > 0)

[0121] Normalize the two variables X and Y,

[0122] Using the probability formula for joint distribution:

[0123] where ρ is the correlation coefficient between X and Y, which is calculated by:

[0124] Calculated.

[0125] where Cov(X,Y) represents the covariance of X and Y:

[0126] Cov(X,Y) = Var(X) + Cov(X,noise) = nσ 2 .

[0127] The reliability of the APUF, Reliability = 1 - P(flip), can be written in the following form:

[0128]

[0129] This formula shows that the reliability decreases as the noise intensity α increases, laying a theoretical foundation for subsequent delay quantization.

[0130] 2. Quantification of delay difference.

[0131] Based on the above relationship between reliability and noise, further quantify the delay difference Δ between the two paths of the APUF.

[0132] Definition and assumption: For a single stimulus-response pair (CRP), assume the delay difference Δ > 0 (when it is less than 0, it can be calculated by the same method), the environmental noise noise ~ (0, α), and the reliability is:

[0133] reliability = P(Δ + noise > 0)

[0134] noise follows the probability distribution function:

[0135]

[0136] Calculated using the normalization function:

[0137]

[0138] By inverting the cumulative distribution function of the standard normal distribution:

[0139]

[0140] This formula can calculate Δ only with the known reliability of the CRPs, without directly measuring the hardware delay, reducing the implementation cost.

[0141] 3. Physical modeling method

[0142] An efficient modeling method is proposed by using the linear accumulation model of APUF and combining the quantized delay difference.

[0143] Linear model: The delay vector w of APUF is related to the feature vector Φ after excitation conversion and the delay vector Δ:

[0144] Δ = w T Φ.

[0145] Modeling steps:

[0146] Selection of CRPs: Select n + 1 CRPs with known reliability and full rank.

[0147] Delay calculation: According to the formula in step 2, calculate the average reliability of this group of CRPs and the delay difference Δ of each CRP respectively.

[0148] Solve the delay vector w: Calculate the delay vector using the least squares method to obtain the delay vector Δ.

[0149] Advantages: Only n + 1 CRPs are required to complete the modeling, which reduces the large amount of data requirements compared with the traditional method. And the value of n in this scheme is generally between 0 - 256. Compared with the traditional method that requires thousands of data, the data operation amount is greatly reduced, and the efficiency is improved.

[0150] Compared with the traditional physical modeling method, the present invention only needs at least n + 1 CRPs to complete the construction of the model, significantly reducing the required data volume. To verify the effectiveness and accuracy of the method of the present invention, modeling experiments were carried out on APUFs of different levels. Table 1 below shows the average prediction accuracy achieved by using only n + 1 CRPs for modeling with the method of the present invention for different APUF levels (n).

[0151] Number of APUF stages 16 32 64 128 256 Number of CRPs 17 33 65 129 257 Modeling accuracy (%) 99.55 98.93 99.24 99.42 99.21

[0152] As shown in Table 1, although only far fewer CRP numbers than those required by the traditional method are used, the modeling method of the present invention can achieve a high prediction accuracy (all higher than 98.9%) on APUFs of different scales, verifying the efficiency and high-precision characteristics of the method.

[0153] Through the above embodiments, it can be seen that the present invention proposes an innovative APUF modeling method, which solves the problems of reliability, efficiency and security in the prior art. This method realizes the hardware-free measurement of delay difference by quantifying the relationship between reliability and noise, and only needs a small number of CRPs to complete high-precision modeling. In addition, the CRP screening strategy further optimizes the authentication and modeling performance, and is applicable to resource-constrained Internet of Things devices.

[0154] The above embodiments are only the preferred embodiments of the present invention, and the scope of protection of the present invention cannot be limited thereby. Any non-substantial changes and substitutions made by those skilled in the art based on the present invention fall within the scope of protection required by the present invention.

Claims

1. A method for modeling an arbiter physically unclonable function with extremely low excitation response, characterized in that It includes the following steps: Step a: Obtain a set of stimulus-response pairs, where each stimulus-response pair includes a stimulus signal and its corresponding response output, as well as reliability data for each stimulus-response pair; Step b: Based on the reliability data, calculate the environmental noise level through probability statistical analysis, where: the environmental noise level represents the noise intensity affecting the response stability of the APUF; Step c: Utilize the environmental noise level and the reliability of a single stimulus-response pair to quantify the delay difference between two paths of the last stage of the APUF corresponding to a specified stimulus-response pair; Step d: Select n + 1 stimulus-response pairs with known reliability and linearly independent after transformation from the set of stimulus-response pairs, where n is the number of stages of the APUF; Step e: Convert the stimulus signals of the n + 1 stimulus-response pairs into a feature matrix Φ, where each row of the feature matrix Φ corresponds to the feature vector of a stimulus signal; Step f: Based on the environmental noise level and the reliability of each selected stimulus-response pair, calculate the corresponding delay difference Δ, where Δ represents the delay difference between two paths of the last stage of the APUF; Step g: Use the least squares method to solve the delay vector w that satisfies the equation Φw = Δ, thereby completing the physical modeling of the APUF.

2. The method for modeling the physical unclonable function of an arbiter with extremely few excitation responses as described in claim 1, wherein The probability statistical analysis in step b includes: Use the formula to calculate the overall reliability of the APUF; where: α is the environmental noise intensity; Reliability is the average reliability measured based on multiple stimulus-response pairs; Derive the above formula backward to determine the environmental noise level α from the reliability data.

3. The method for modeling a physical unclonable function of an arbiter with extremely few excitation responses as described in claim 1, characterized in that The quantification of the delay difference Δ in step c includes: For a specified stimulus-response pair, when calculating reliability, P((Δ + noise)(sgn(Δ)>0) = reliability, where: noise is the environmental noise, following the normal distribution N(0,α); reliability is the reliability of this stimulus-response pair, and sgn is the sign function; Convert the probability to the standard normal distribution form and solve for the value of the delay difference Δ, where Δ represents the delay difference between two paths of the last stage of the APUF.

4. The method for modeling a physical unclonable function of an arbiter with extremely few excitation response pairs according to claim 1, characterized in that, In step d, the selection of the n + 1 stimulus-response pairs includes the following sub-steps: Perform reliability measurements on the set of stimulus-response pairs to obtain the reliability values of each stimulus-response pair; Select n + 1 stimulus-response pairs from them to ensure that the corresponding feature matrix Φ is full rank; Verify the linear independence of the feature matrix Φ to ensure a unique solution for the delay vector w.

5. An arbiter physical unclonable function modeling system with extremely few excitation response pairs, which is established by using the method described in any one of claims 1-4, and is characterized in that, It includes the following modules: Input module, configured to receive a set of stimulus-response pairs, where each stimulus-response pair includes a stimulus signal and its corresponding response output, as well as reliability data for each stimulus-response pair; Noise calculation module, configured to calculate the environmental noise level through probability statistical analysis based on the reliability data; Delay quantification module, configured to utilize the environmental noise level and the reliability of a single stimulus-response pair to quantify the delay difference between two paths of the last stage of the APUF corresponding to a specified stimulus-response pair; A selection module, configured to select n + 1 excitation-response pairs with known reliability and linearly independent after transformation from the set of excitation-response pairs, where n is the number of stages of the APUF; A transformation module, configured to transform the excitation signals of the n + 1 excitation-response pairs into a feature matrix Φ; A calculation module, configured to calculate the corresponding delay difference Δ based on the environmental noise level and the reliability of each selected excitation-response pair; A modeling module, configured to solve for the delay vector w using the least squares method, where the equation Φw = Δ is satisfied, thereby completing the physical modeling of the APUF; An output module, configured to output the delay vector w as the modeling result.

6. The physical unclonable function modeling system for an arbiter with extremely few excitation responses according to claim 1, characterized in that The noise calculation module uses the formula to calculate the overall reliability of the APUF, where α is the environmental noise intensity; and determines the environmental noise level α from the reliability data through reverse derivation using the above formula.

7. The physical unclonable function modeling system of an arbiter with extremely few excitation response pairs as claimed in claim 1, wherein The delay quantization module is further configured to: For a specified excitation-response pair, calculate the probability P(Δ + noise > 0) = reliability, where noise follows a normal distribution N(0, α); Convert the probability to the standard normal distribution form and solve for the delay difference Δ.

8. The physical unclonable function modeling system for an arbiter with extremely few excitation responses as described in claim 1, characterized in that, The selection module is further configured to: Screen out n + 1 excitation-response pairs to ensure that the corresponding feature matrix Φ is full rank.

9. The physical unclonable function modeling system for an arbiter with extremely few excitation responses as described in claim 1, wherein The system further includes an authentication module, configured to: Use the delay vector w generated by the modeling module to predict new excitation-response pairs; Apply the predicted excitation-response pairs to the security authentication between the device and the server.

10. Application of an arbiter physically unclonable function modeling system with extremely few excitation responses as described in claim 1, characterized in that, Based on the physical modeling result, generate a set of excitation pairs to predict the response of the APUF.

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