A method for determining the location privacy of a mobile node in an internet of things

Abstract

This invention discloses a method for securely determining the location of a mobile node within an elliptical restricted area in the Internet of Things (IoT), belonging to the technical field of restricted area location security determination. It addresses the issue that traditional geofencing or location determination schemes cannot effectively protect privacy. The invention designates Alice as the mobile node in the IoT to be determined, and Bob as the elliptical restricted area. Alice calculates a random number based on its location point P0(x0, y0) and encrypts x0, y0, and 1 using its public key based on homomorphic encryption to obtain parameters c1-c5, which is then sent to Bob. Bob, using homomorphic encryption, utilizes the elliptical equation coefficients A, B, C, D, and E of the elliptical restricted area to achieve secure communication with Alice, thereby determining whether Alice's location point P0 is within the corresponding ellipse P of the elliptical restricted area.

Description

Technical Field

[0001] This invention relates to a method for determining the confidentiality of restricted area locations. Background Technology

[0002] In the current era of booming Internet of Things (IoT), mobile nodes have become the core hubs for efficient collaboration in various intelligent systems. For example, Automated Guided Vehicles (AGVs) efficiently shuttle through large warehousing centers, accurately completing cargo transportation and sorting tasks; connected vehicles, leveraging V2X communication and edge computing platforms, can perceive the surrounding traffic environment in real time, make collaborative decisions, and execute autonomous driving operations, achieving safe and reliable road travel; furthermore, wearable devices integrating GPS, environmental, and vital sign sensors on livestock can monitor their location and health status in real time, aiding in precise feeding management and electronic fence control. However, the operating paths and work areas of these mobile nodes often involve core corporate secrets or sensitive assets, such as high-value asset storage areas. The leakage of node coordinates and area boundaries poses a serious threat to corporate security and business operations. Especially in special scenarios such as high-end electronic component warehouses, the confidentiality requirements for the location information and restricted areas of mobile nodes reach an even higher level.

[0003] However, traditional geofencing or location determination schemes often fail to consider privacy protection, directly exposing the precise coordinates and area boundaries of mobile nodes when performing spatial relationship calculations, leading to serious security risks. When a mobile node needs to determine its positional relationship with an elliptical bounded area, conventional calculation methods may not only leak its own location and area information but also, due to the lack of effective security mechanisms, allow malicious attackers to forge determination results or maliciously disrupt normal processes, thereby threatening the stability and security of the entire Internet of Things (IoT). Therefore, researching a scheme that can accurately determine the positional relationship between a mobile node and an elliptical bounded area while effectively protecting location privacy is of paramount practical significance for the secure and reliable development of the IoT. The introduction of secure multi-party computation (MPC) technology has opened up a new path to address this challenge. Since Yao proposed this concept in 1982, numerous scholars, such as Goldreich and Cramer, have conducted in-depth research on it, driving the continuous development of this field at both theoretical and practical levels. Currently, its research scope has broadly covered multiple key areas such as confidential data mining, computational geometry, set operations, and scientific computing, providing a solid theoretical foundation and technical support for the data privacy protection and collaborative computing needs of various industries.

[0004] In the Internet of Things (IoT), determining whether a mobile node is within an elliptical restricted area is essentially a problem of determining the positional relationship between a point and an ellipse. Traditional methods include distance comparison, vector methods, and polynomial algebra methods. Distance comparison, based on the first definition of an ellipse, compares the sum of the distances from a point to its two foci with a constant value. The vector method calculates the vector from the ellipse's center to the point, analyzes its angle and magnitude with the principal and secondary axes, and compares it with the axis lengths. The polynomial algebra method directly substitutes the point coordinates into the ellipse equation. To meet privacy requirements, this problem is constructed as a two-party scenario: one party holds the point coordinates, and the other party possesses the ellipse parameters. Both parties must securely collaborate to determine the position without disclosing the original data, and the result must not expose any original data characteristics. Li SD et al. calculated the relationship between the distance from a point to the two foci of the ellipse and a constant value based on the distance comparison method, but this protocol has low execution efficiency. Zhu H et al. constructed a point-circle position determination protocol based on a secure two-party computation framework, achieving a lightweight design through a cosine value calculation scheme, which avoids complex cryptographic operations while exposing the limitations of elliptical geometry adaptability. ZHANG W et al. innovatively established a mapping relationship between spatial distance and solutions to algebraic equations, and solved various spatial positional relationship determination problems by using an optimized inner product protocol. This scheme does not use public-key encryption algorithms and falls under the category of information-theoretic security, but it cannot resist attacks from malicious adversaries. Summary of the Invention

[0005] This invention aims to address the problem that traditional geofencing or location determination schemes cannot effectively protect privacy.

[0006] A method for secure location determination of mobile nodes and elliptical restricted regions in the Internet of Things (IoT) is proposed. The mobile node to be determined is denoted as Alice, and the elliptical restricted region is denoted as Bob. Alice calculates the location information based on her current location P0(x0,y0). Choose a random number and encrypt it using your public key based on homomorphic encryption. x0, y0, 1 obtain parameters c1-c5 and send them to Bob; Bob uses homomorphic encryption technology and the coefficients A, B, C, D, E of the elliptic equation of the elliptic restricted region to achieve secure communication with Alice, thereby determining whether Alice's position point P0 is within the ellipse P corresponding to the elliptic restricted region.

[0007] Furthermore, the specific implementation process of the method includes the following steps:

[0008] (a) Let Alice be the mobile node in the Internet of Things that needs to be judged, and Bob be the elliptical restricted area; Alice calculates based on her own location point P0(x0,y0) Choose a random number r and encrypt it using your own public key based on homomorphic encryption. x0, y0, 1 are used to obtain parameters c1-c5 and then sent to Bob;

[0009] (b) Bob calculates Send to Alice;

[0010] (c) Alice decrypts c6 to obtain And send it to Bob; if d < 0, then point P0 is inside ellipse P; if d = 0, then point P0 is on the boundary of ellipse P; if d > 0, then point P0 is outside ellipse P.

[0011] Furthermore, in step (a), Alice selects a random number r and encrypts it using her own public key based on homomorphic encryption. The process of x0, y0, and 1 is encrypted using the Paillier encryption algorithm, and the resulting parameters c1-c5 are as follows:

[0012] c5 = g 1 r N modN 2 N is the public key modulus in the Paillier encryption algorithm.

[0013] Alternatively, the specific implementation process of the method includes the following steps:

[0014] (1) Let Alice be the mobile node in the Internet of Things that needs to be judged, and Bob be the elliptical restricted area; Alice calculates based on her own location point P0(x0,y0) Choose a random number r1 and encrypt it using your public key based on homomorphic encryption. x0, y0, 1 yield parameters c1-c5, which are then sent to Bob. Bob selects a random number r2 and, based on homomorphic encryption, encrypts the coefficients A, B, C, D, E of the elliptic equation within the elliptic restricted region using his own public key to obtain parameters c6-c. 10 Alice and Bob use the same homomorphic encryption method.

[0015] (2) Alice calculation Send to Bob, Bob calculates. Send to Alice;

[0016] (3) Alice decrypts c 12 Bob obtains d1 and decrypts c. 11 We get d2;

[0017] (4) Alice and Bob each select n random numbers a i ,b i Let i = 1, ..., n, and calculate using the public key of each homomorphic encryption algorithm. And make it public;

[0018] (5) Using the split-selection method, Alice and Bob each select from n groups and Choose n / 2 groups from each and Bob and Alice announced b i d2 and a i After d1, both parties verify the agreement. If both verifications pass, the agreement continues to be executed; if one party's verification fails, the agreement is terminated.

[0019] (6) Alice and Bob each took one of the remaining n / 2 groups. and Randomly select one and And select a and b;

[0020] (7) Alice uses the homomorphic encryption algorithm and ad1 to obtain c. b Bob sends it to the other party; using the homomorphic encryption algorithm bd2, Bob obtains c. a Send it to the other party;

[0021] (8) Alice obtains m through the decryption process based on the homomorphic encryption algorithm. a And it was announced that Bob obtained m through the decryption process based on the homomorphic encryption algorithm. b And publish;

[0022] (9) Alice and Bob respectively used zero-knowledge proofs to verify and If one party cannot provide valid proof of the correctness of the agreement, that party will be deemed a malicious participant and the agreement will be terminated. λ a λ is the private key in Alice's Paillier encryption algorithm. b This is the private key in Bob's Paillier encryption algorithm;

[0023] (10) If the proof is successful, Alice and Bob obtain the calculated values ​​d1 and d2; if d1 = d2 < 0, then point P0 is inside ellipse P; if d1 = d2 = 0, then point P0 is on the boundary of ellipse P; if d1 = d2 > 0, then point P0 is outside ellipse P.

[0024] Furthermore, in step (1), Alice selects a random number r1 and encrypts it using her own public key based on homomorphic encryption. The process of x0, y0, and 1 is encrypted using the Paillier encryption algorithm, and the resulting parameters c1-c5 are as follows:

[0025] g a The modulo term in the public key of the Paillier encryption algorithm corresponding to Alice The elements in the multiplicative group N a Let g be the modulus of the public key in Alice's Paillier encryption algorithm. Alice's public key is (g a N a ), λ a This is the private key in Alice's Paillier encryption algorithm.

[0026] Furthermore, in step (1), Bob selects a random number r2 and encrypts the coefficients A, B, C, D, and E of the elliptic equation in the elliptic restricted region using his own public key based on homomorphic encryption. This process is then performed using the Paillier encryption algorithm, resulting in the parameter c6-c. 10 as follows:

[0027] g b The modulo of the public key in Bob's corresponding Paillier encryption algorithm The elements in the multiplicative group N b Let g be the modulus of the public key in Bob's Paillier encryption algorithm. Bob's public key is (g b N b ), λ b This is the private key in Bob's Paillier encryption algorithm.

[0028] Furthermore, in step (4), Alice and Bob each select n random numbers a. i ,b i And calculated using the public key of their respective homomorphic encryption algorithms.

[0029] Furthermore, in step (5), after Bob and Alice publish b i d2 and a i After d1, both parties verify. and

[0030] Furthermore, in step (7), Alice uses the c obtained from ad1 b And Bob obtained c using bd2 a as follows:

[0031]

[0032] Furthermore, in step (8), Alice's decryption process based on the homomorphic encryption algorithm yields m.a Bob obtained m through the decryption process based on the homomorphic encryption algorithm. b The process is as follows:

[0033] Alice and Bob each used λ a and λ b calculate and λ a λ is the private key in Alice's Paillier encryption algorithm. b This is the private key in Bob's Paillier encryption algorithm.

[0034] Beneficial effects:

[0035] This invention proposes a confidentiality determination scheme for the positional relationship between a point and an arbitrary ellipse. This scheme can efficiently solve the problem of confidentiality determination of the positional relationship between a mobile node and an arbitrary elliptical restricted area in the Internet of Things (IoT). Furthermore, since a circle is a special case of an ellipse, this scheme is also applicable to the confidentiality determination of circular restricted areas.

[0036] This invention designs a highly efficient confidential determination protocol for the positional relationship between points and ellipses in a semi-honest model. This protocol is based on a polynomial substitution method, which improves the determination efficiency. Simultaneously, a confidential determination protocol for the positional relationship between points and ellipses in a malicious model is also designed, which can effectively resist attacks from malicious adversaries, avoid privacy leaks and incorrect judgments, and provides correctness analysis and security proof for the protocol. Attached Figure Description

[0037] Figure 1 This is a schematic diagram of P0 inside the ellipse P.

[0038] Figure 2 This is a schematic diagram of P0 on the boundary of ellipse P.

[0039] Figure 3 This is a schematic diagram of P0 outside the ellipse P.

[0040] Figure 4 A logic diagram for a confidentiality determination protocol for elliptical positional relationships. Detailed Implementation

[0041] To address the problems in the background technology, this invention proposes an efficient and concise confidentiality determination protocol for the positional relationship between points and ellipses under a semi-honest model based on the polynomial substitution method, and designs a confidentiality determination protocol resistant to malicious adversary attacks. Before proceeding with the specific details, some basic knowledge will be explained.

[0042] Paillier encryption algorithm:

[0043] The Paillier encryption algorithm is constructed as follows: Choose two large prime numbers p and q. Calculate N = pq, λ = lcm(p-1, q-1), and choose the values ​​that satisfy gcd(L(g λ modN 2 g, where N) = 1 The public key is (g, N), and the private key is λ.

[0044] When encrypting messages, m∈Z N Calculate c = g using random numbers. m r N modN 2 .

[0045] Decrypting ciphertext calculate

[0046] The Paillier encryption algorithm is a probabilistic cryptographic algorithm with semantic security, meaning that each plaintext m can be encrypted into different ciphertexts, and there is no multinomial-time algorithm that can determine whether these ciphertexts correspond to the same plaintext.

[0047] Its important property is additive homomorphism, given any two ciphertexts: ,Right now ,

[0048] Zero-knowledge proof:

[0049] Zero-knowledge proof (ZKP) is an interactive verification mechanism in which the prover exchanges data with the verifier to convince the verifier of the correctness of the conclusion, but does not reveal any other useful information to the verifier in the process. Only when the verifier can only know the result of "whether the conclusion is correct or not" is the interaction called a zero-knowledge proof.

[0050] Segmentation-selection method:

[0051] The cut-and-choose method is a cryptographic tool widely used in protocols to counter malicious actors. Its basic process is as follows: one party generates and sends multiple circuits to another; the receiver randomly selects half for inspection, requesting the other party to publish these circuits to verify their correctness; the unselected circuits are used in subsequent secure computations. For example:

[0052] Input phase:

[0053] (1) Alice inputs a vector consisting of s pairs. Right now There are 1 in total, and input 's' to check X1, X2, ..., X s Is the value within {0,1}? n middle;

[0054] (2) Bob inputs σ1, σ2, ..., σ∈{0,1} and parameters.

[0055] Output phase: When the sender has no output, the receiver receives:

[0056] (1) Vector The j-th pair, i.e.

[0057] (2) From vector Obtain σ in each pair i ,Right now Where i = 1, 2, ..., l, j ∈ ξ, Final output X k .

[0058] Security definition:

[0059] In the field of secure multi-party computation, participants are categorized into two types: semi-honest and malicious. Semi-honest participants honestly input data and strictly follow the protocol's steps to process intermediate results during computation, but may attempt to infer information about the other party from the computation process. Malicious participants, on the other hand, not only attempt to obtain information but may also forge or transmit false information within the protocol. The protocol design of this invention is based on the scenario of two participants.

[0060] The ideal-practice paradigm is an effective way to prove the security of protocols under a malicious model, in which at least one party is honest, otherwise the security of the protocol cannot be guaranteed.

[0061] Ideal Protocol: Alice possesses information x, and Bob possesses information y. They jointly compute the function f(x,y) = (f1(x,y), f2(x,y)). After the protocol ends, Alice only obtains f1(x,y), and Bob only obtains f2(x,y), but cannot obtain any other information.

[0062] (1) Data sent by participants to TTP: Alice and Bob each possess information x and y, which they send to TTP (trusted third party). Honest participants input real data into TTP, while malicious participants may not execute the protocol or input false data x′ or y′ into TTP.

[0063] (2) The output of TTP to Alice: TTP calculates the function f(x,y). If Alice is an honest participant, it will send f1(x,y) to Alice; otherwise, it will send the special symbol ⊥.

[0064] (3) The output of TTP to Bob: If Alice is a malicious participant, she will not execute the protocol after receiving f1(x,y). TTP will send ⊥ to Bob after detecting Alice's malicious behavior; otherwise, it will send f2(x,y) to Bob.

[0065] If the execution result under the malicious model is that both parties can only obtain their own results and cannot obtain any other information, then it is consistent with the ideal model, which proves that the protocol under the malicious model is secure.

[0066] Definition 1. Existence strategy pairs in an ideal protocol In actual protocols, there exist corresponding strategy pairs. Make The protocol is then considered secure. Let F represent the probability polynomials under the actual and ideal protocols, respectively, where x and y are the information possessed by each party, and F is {0,1}. * ×{0,1} * →{0,1} * ×{0,1} * The time function is denoted by z, where z is auxiliary input information. This indicates that, under the ideal model, participants, with the help of auxiliary input information z, use a strategy Calculate F(x,y), and This represents the output pair generated after the interaction between A1=(x,z) and A2=(y,z).

[0067] Based on the above description, this invention will be described in detail in conjunction with the confidentiality determination protocol for the relationship between points and ellipses under the semi-honest model and the malicious model. Specific implementation method one:

[0069] This embodiment is a method for confidentiality determination of the location of mobile nodes and elliptical restricted areas in the Internet of Things, which is implemented through a confidentiality determination protocol for the relationship between the location of points and the ellipse under a semi-honest model.

[0070] In the Internet of Things (IoT), determining the location relationship between mobile nodes and restricted areas is a critical security issue. When a mobile node is operating, the system needs to determine whether it is within a restricted area based on its location and restricted area data. If the mobile node is within an elliptical restricted area, the system must issue a warning and take appropriate measures; otherwise, the mobile node is allowed to continue operating. However, malicious attackers may tamper with the mobile node's location information or forge restricted data, interfering with the determination results and threatening warehouse security.

[0071] To address the aforementioned problem, this invention transforms it into a confidentiality determination problem concerning the positional relationship between a point and an ellipse. Specifically, the positional information of the moving node is abstracted as a point on a two-dimensional plane, and the restricted region is represented as an ellipse on that plane.

[0072] Theorem 1. The general form equation is Ax 2 +By 2 The equation +Cx + Dy + E = 0 (A > 0, B > 0 and A ≠ B) can represent any ellipse whose center is not at the origin and whose principal axes are not parallel to the coordinate axes. When A = B = 1, it is a circle, which is a special case of an ellipse.

[0073] Let Alice be the mobile node in the Internet of Things that needs to be judged, and Bob be the elliptical restricted area. Alice has a point P0(x0,y0), and Bob has an ellipse P. Without disclosing the privacy of either party, Alice and Bob can determine the positional relationship between P0 and P based on the general equation of the ellipse.

[0074] To illustrate the positional relationship between the point and the ellipse, we provide three diagrams for clarity. Figure 1 This indicates that point P0 is inside the ellipse P. Figure 2 This indicates that point P0 lies on the boundary of ellipse P. Figure 3 This indicates that point P0 is outside the ellipse P.

[0075] Substitute the coordinates of point P0 into the general equation of ellipse P, and determine the positional relationship between the point and any ellipse by the substitution result d. If d < 0, the point is inside the ellipse; if d = 0, the point is on the boundary of the ellipse; if d > 0, the point is outside the ellipse.

[0076] The plaintext space of the Paillier encryption algorithm is Z. N (Z) N The addition group (+) forms an additive group, which does not distinguish between positive and negative numbers. The plaintext m ∈ {0,...,N / 2-1} can be restricted to be processed; similarly, N is the public key modulus in the Paillier encryption algorithm; for Z... NFor elements x and y in it, if x + y = 0 mod N and 0 < x < N / 2, then it must be that y > N / 2. At this time, y is the additive inverse of x. If we consider x ≥ 0, then y can be regarded as a negative number. Further assume that x, y < N / 2. If (x - y) mod N < N / 2, then x > y; conversely, if (x - y) mod N > N / 2, it means that x < y.

[0077] The elliptical position relationship confidentiality determination protocol is as Figure 4 shown.

[0078] Table 1

[0079]

[0080]

[0081] Analysis of correctness and security:

[0082] (1) Protocol 1 aims to substitute the point P0(x0, y0) into the general equation of the ellipse Ax 2 + By 2 + Cx + Dy + E = 0 (A > 0, B > 0 and A ≠ V). The specific operation is that Alice encrypts x0, y0, 1 and sends them to Bob. After receiving the ciphertext, Bob uses homomorphic operations to multiply the ciphertext with the coefficients A, B, C, D, E of the general equation of the ellipse respectively, and returns the result to Alice. After Alice decrypts, she gets the result d, so as to judge the position relationship between the point P0 and the ellipse. If d > 0, the point is outside the ellipse; if d < 0, the point is inside the ellipse; if d = 0, the point is on the ellipse boundary. This protocol can correctly judge the position relationship between the point and the ellipse.

[0083] (2) In the first step of Protocol 1, Alice calculates the value, then encrypts x0, y0, 1, and sends the ciphertext to Bob. In the second step, Bob performs homomorphic operations on the ciphertext sent by Alice, and then sends the calculation result to Alice. In the third step, Alice decrypts the calculation result and makes it public. During the execution of Protocol 1, both Alice and Bob can obtain the correct result, and since both sides perform interactive operations with 5 unknowns and 1 equation, it is impossible to deduce the data of the other party based on the information they each have. The protocol is secure. Specific implementation method two:

[0085] This implementation method is a method for determining the confidentiality of the position of a mobile node and an elliptical restricted area in the Internet of Things. This implementation method is realized through the point-elliptical position relationship confidentiality determination protocol under the malicious model.

[0086] When designing protocols under a malicious model, a semi-honest model is typically constructed first. Then, potential malicious attacks are analyzed, and preventative measures are designed to ensure that malicious adversaries cannot execute or are detectable. Finally, participants engage in the protocol in a semi-honest manner. It's important to note that malicious behaviors such as refusing to participate, providing false input, or prematurely ending the protocol cannot be prevented in an ideal protocol, and practical protocols do not consider these situations. Furthermore, while it's impossible to completely prevent other malicious behaviors from either party during protocol execution, it's crucial to ensure that these malicious behaviors can be detected.

[0087] In Protocol 1, step 1 involves Alice encrypting data and sending it to Bob; step 2 involves Bob performing a homomorphic operation on the ciphertext and returning the result; and step 3 involves Alice decrypting the data and publishing the symbol value of the result. Steps 1-3 are susceptible to malicious attacks, and Alice and Bob may provide false information during data exchange.

[0088] To resist these malicious attacks, Protocol 1 is improved using cryptographic tools such as zero-knowledge proofs and split-choice methods.

[0089] Table 2

[0090]

[0091]

[0092] Correctness analysis:

[0093] (1) In the first three steps of Protocol 2, since Alice and Bob may provide false information when interacting, the Socialist Millionaires' Problem (SMP) is invoked to determine whether the two parties are malicious participants.

[0094] (2) When applying the Paillier encryption algorithm, to ensure correctness, the parameters must satisfy a. i |d1-d2| <N a / 2 and b i |d1-d2| <N b / 2. Theoretically, d1, d2, a i ,b i None of them exceed τ bits, only N is needed a N b The number of bits exceeds 2τ+1 bits. Steps 4 and 5 of the protocol are designed to ensure that the above conditions are met. If the protocol execution is not aborted, it indicates that Alice and Bob are executing the protocol in a semi-honest manner, thus guaranteeing the correctness of the protocol.

[0095] (3) Alice's deception succeeds when she selects option a, which does not meet the requirements. i It was not selected in step 5 but was selected in step 6, leading Bob to a wrong conclusion. Its maximum probability of successful deception is: in n groups... In this scenario, if one set of data does not meet the requirements, the maximum probability is 1 / n; if half of the data does not meet the requirements, the probability of successfully deceiving the user is... If more than half of the data does not meet the requirements, the probability of successful deception will drop to 0. Bob's malicious behavior has the same probability of success.

[0096] Security Analysis:

[0097] Under the malicious model, the security of Protocol 2 is demonstrated using a widely accepted ideal-practice paradigm.

[0098] Theorem 2 Protocol 2 (denoted as Protocol Π) is secure in the presence of malicious actors.

[0099] Proof. According to Definition 1, when executing protocol Π, calculating the time function F requires the strategy pair in the actual protocol. Strategies in the Ideal Protocol They exist and are computationally indistinguishable. To ensure the security of the protocol, at least one party must be honest; otherwise, the protocol cannot be implemented.

[0100] Scenario 1: A1 is honest, A2 is dishonest.

[0101] If A1 honestly executes protocol Π, then: Where S is the sequence message received by A2 through zero-knowledge proof.

[0102] Now that B1 is determined, it is only necessary to prove that A2 and B2 are computationally indistinguishable. For this, a strategy pair needs to be found. Make its output the same as They are computationally indistinguishable. Since A2 is the actual executor of the protocol, the correctness of the protocol must be verified based on A2(d2).

[0103] (1) If A1 is honest, B1 will send the real input information d1 to TTP.

[0104] (2) If A1 is dishonest, B2 will send input information A2(d2) to TTP.

[0105] (3) TTP obtains input information (d1, A2(d2)) and calculates F(d1, A2(d2)).

[0106] (4) B2 receives F(d1,A2(d2)) sent by TTP, uses it to generate, and compares it with the result obtained by A2. Indistinguishable in computation And send it to A2 to get A2's output. B2 selects the d′1 simulation protocol, such that F(d′1,A2(d2))=F(d1,A2(d2)).

[0107] During the execution of the B2 protocol, the following situations may occur:

[0108]

[0109] In steps 4-9 of the protocol, if the protocol uses the same encryption algorithm, then... And zero-knowledge proofs guarantee... therefore:

[0110] Scenario 2: A1 is dishonest, A2 is honest. In this case, there are two possibilities:

[0111] If Alice ignores TTP after receiving the message, TTP will send a ⊥ to Bob. In this case, the actual protocol output will be:

[0112] Conversely, if TTP sends Bob the correct information F2(A1(d1),d2), then the actual protocol output will be:

[0113]

[0114] Here, S is the sequence message received by A1 through zero-knowledge proof.

[0115] At this point, B2 is determined. It needs to be proven that A1 and B1 are computationally indistinguishable; for this, a strategy pair needs to be found. Make its output the same as They are computationally indistinguishable. Since A1 is the actual executor of the protocol, the correctness of the protocol must be verified based on A1(d1).

[0116] (1) If A1 is dishonest, B1 will send input information A1(d1) to TTP.

[0117] (2) If A2 is honest, B2 will send the real input information d2 to TTP.

[0118] (3) TTP receives input information (A1(d1),d2) and calculates F(A1(d1),d2).

[0119] (4) B1 receives F(A1(d1),d2) sent by TTP and generates a value obtained by A1 executing the actual protocol. Indistinguishable in computation And send it to a1 to get the output of a1. B1 selects the d′2 simulation protocol, such that F(A1(d1),d′2)=F(A1(d1),d2).

[0120] During the execution of the B1 protocol, the following two situations exist:

[0121] (1) When A1 receives the message and no longer responds to TTP, we get:

[0122] (2) Conversely, we get

[0123] If the same encryption algorithm is used in steps 4-9 of the protocol, then... Zero-knowledge proof guarantees therefore:

[0124] In conclusion, under the malicious model, Protocol 2 (Protocol Π) is secure.

[0125] Protocols 1 and 2 of this invention are implemented based on the Paillier encryption algorithm. The computational complexity analysis only considers the main overhead of the protocol—modulo exponentiation (M). m Communication complexity is compared using the number of protocol rounds as a benchmark.

[0126] A. Computational complexity:

[0127] Protocol 3 of "Secure two-party computational geometry" performs two OT protocol executions and two encrypted random number generation operations during execution, with a total computational cost of (32m+28)M. m .

[0128] Protocol 1 involved encryption five times and decryption once during execution, with a total computational overhead of 12MB. m .

[0129] Protocol 2 involved 10 encryptions and 2 decryptions during execution, requiring 24 modular exponentiation operations. It generated 2n sets of modular exponents, verified n of them, requiring a total of 5n modular exponentiation operations; and performed two zero-knowledge proofs of discrete logarithms, requiring a total of 12 modular exponentiation operations. The total computational cost was (5n+36)M. m .

[0130] B. Communication complexity:

[0131] During the execution of Protocol 3 of "Secure two-party computational geometry", the OT protocol exchanged 16 rounds and exchanged information 10 rounds, with a total communication overhead of 26 rounds.

[0132] During the execution of Protocol 1, the total communication overhead is 3 rounds.

[0133] During the execution of Protocol 2, the first three steps involved exchanging information for 4 rounds, while the Socialist Millionaire Protocol was used for 3 rounds, resulting in a total communication overhead of 7 rounds.

[0134] Table 3 Comparison of Protocol Efficiency

[0135]

[0136] As shown in Table 3, Protocol 3 of "Secure two-party computational geometry" is inferior to Protocols 1 and 2 of this invention in terms of communication complexity and computational complexity, and it cannot resist attacks from malicious adversaries. Preprocessing or service outsourcing can be used to improve the protocol efficiency under malicious models.

[0137] This invention addresses the problem of confidential determination of the positional relationship between mobile nodes and elliptical restricted regions in IoT environments. It proposes a method based on secure multi-party computation, utilizing the Paillier encryption algorithm to achieve confidential positional relationship determination. A confidential determination protocol for point-ellipse positional relationships under a semi-honest model is designed, and its potential malicious attack behaviors are analyzed in depth. Furthermore, a protocol resistant to malicious adversaries is designed, and the correctness, security, probability of successful malicious attacks, and efficiency of the two protocols are analyzed. Analysis shows that the proposed solution significantly improves efficiency compared to existing solutions, effectively balancing privacy protection and computational efficiency. It provides an efficient and secure solution to the privacy protection problem of mobile node positions within elliptical restricted regions in IoT, possessing significant theoretical and practical value.

[0138] This invention may have other embodiments. Without departing from the spirit and essence of this invention, those skilled in the art can make various corresponding changes and modifications according to this invention, but these corresponding changes and modifications should all fall within the protection scope of the appended claims.

Claims

1. A method for confidentiality determination of the location of mobile nodes and elliptical restricted areas in the Internet of Things, characterized in that: In the Internet of Things (IoT), the mobile node requiring judgment is denoted as Alice, and the elliptical restricted area is denoted as Bob; Alice determines her location based on her position... Calculated , Choose a random number and encrypt it using your own public key based on homomorphic encryption. Get parameters And send it to Bob; Bob uses homomorphic encryption technology, utilizing the coefficients of the elliptic equation within the elliptic confinement region. To achieve secure communication with Alice, and thus to determine Alice's location. Is it within the ellipse corresponding to the elliptical restricted region? Internal judgment; The method described under the semi-honest model includes the following steps: (a) Let Alice be the mobile node in the Internet of Things that needs to be judged, and Bob be the elliptical restricted area; Alice determines its location based on its position. Calculated , The parameters under the semi-honest model Let it be denoted as parameter Choose a random number And encrypt using your own public key based on homomorphic encryption. Get parameters And send it to Bob; Alice selects a random number. And encrypt using your own public key based on homomorphic encryption. The process uses the Paillier encryption algorithm for encryption, and the resulting parameters... as follows: , , , , N is the modulus of the public key in the Paillier encryption algorithm; The modulo term in the public key of the Paillier encryption algorithm corresponding to Alice Elements in the multiplication group; (b) Bob computes Send to Alice; (c) Alice decryption get And send it to Bob; if Then point In an ellipse within; if Then point In an ellipse On the boundary; if Then point In an ellipse outside.

2. The method for confidentiality determination of the location of mobile nodes and elliptical restricted areas in the Internet of Things according to claim 1, characterized in that: The method described under the malicious model includes the following steps: (1) Let Alice be the mobile node in the Internet of Things that needs to be judged, and Bob be the elliptical restricted area; Alice determines its location based on its position. Calculated , Select random number And encrypt using its own public key based on homomorphic encryption. Get parameters And send it to Bob; Bob selects a random number. And based on homomorphic encryption, the coefficients of the elliptic equation in the elliptic restricted region are encrypted using the user's own public key. Get parameters Alice and Bob use the same homomorphic encryption method. Parameters As follows: , , , , ; is an element in the multiplicative group of modulo under the public key of the Paillier encryption algorithm corresponding to Alice, is the modulus in the public key of the Paillier encryption algorithm corresponding to Alice, and the public key of Alice is , is the private key of the Paillier encryption algorithm corresponding to Alice; Parameters As follows: , , , , ; The modulo of the public key in Bob's corresponding Paillier encryption algorithm Elements in the multiplication group below, This is the modulus of the public key in Bob's Paillier encryption algorithm. Bob's public key is... , This is the private key in Bob's Paillier encryption algorithm; (2) Alice computes and sends to Bob, who computes and sends to Alice; (3) Alice decrypts gets , Bob decrypts gets ; (4) Alice and Bob each select a random number , and compute , and publish them using the public key of their respective homomorphic encryption algorithm; , ;​ (5) Using the split-select method, Alice and Bob each select from group and group and After Bob and Alice publish and , both verify. If both pass, the protocol continues. If one fails, the protocol aborts.​ (6) Alice and Bob each took from the remaining... Group and Randomly select one and , and select ; (7) Alice uses homomorphic encryption algorithm to... get Send it to the other party; Bob uses homomorphic encryption algorithm to... ,get Send it to the other party; , ; (8) Alice's decryption process based on the homomorphic encryption algorithm yields... And it was announced that Bob's decryption process based on the homomorphic encryption algorithm was obtained. And publish; (9) Alice and Bob respectively used zero-knowledge proofs to verify and If one party cannot provide valid proof of the correctness of the agreement, that party will be deemed a malicious participant and the agreement will be terminated. , , This is the private key in Alice's Paillier encryption algorithm. This is the private key in Bob's Paillier encryption algorithm; (10) If the proof is successful, Alice and Bob obtain the calculated value. and ;like Then point In an ellipse within; if Then point In an ellipse On the boundary; if Then point In an ellipse outside.

3. The method for confidentiality determination of the location of mobile nodes and elliptical restricted areas in the Internet of Things according to claim 2, characterized in that: In step (5), Bob and Alice publish... and Afterwards, both parties verified and .

4. The method for confidentiality determination of the location of mobile nodes and elliptical restricted areas in the Internet of Things according to claim 3, characterized in that: In step (8), Alice obtains the decryption process based on the homomorphic encryption algorithm. Bob obtained the decryption process based on the homomorphic encryption algorithm. The process is as follows: Alice and Bob used respectively and calculate and ; This is the private key in Alice's Paillier encryption algorithm. This is the private key in Bob's Paillier encryption algorithm.

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