Andon-RX structure zero correlation linear discriminator searching method based on miss-from-th-mid and MILP

Abstract

The invention discloses an Android-RX structure zero-correlation linear discriminator searching method based on miss-from-th-middle and MILP (Million-Independent Length Polymorphism), and the MILP modeling is applied to the miss-from-th-middle technology, so that the zero-correlation linear discriminator searching method taking a model solvability as a judgment condition is constructed. According to the method, the limitation that all search spaces cannot be traversed by a search model with no solution of the model as a judgment condition is solved, and meanwhile, the zero-correlation linear discriminator generated by an indirect contradiction reason is also considered, so that effective zero-correlation linear discriminators with longer rounds and more quantity can be searched under the new model. The method is high in accuracy, the zero-correlation linear discriminator can be searched out, meanwhile, the accurate indirect contradiction occurrence position can be given, the misjudgment situation generated based on the fact that a model has no solution is eliminated, and the tedious process of manually deducing the indirect contradiction position based on the zero-correlation linear discriminator is avoided.

Description

Technical Field

[0001] The invention relates to the field of information security, and in particular to a zero-correlation linear discriminator search method combining miss-from-the-middle and MILP technologies for an And-RX structure block cipher algorithm. Background Art

[0002] Zero-correlation linear analysis is an important method for analyzing block ciphers. Its basic principle is to use a linear approximation expression with zero correlation to recover the key of the cipher algorithm. The primary key to this attack is how to find an effective and high-round zero-correlation linear discriminator [1]. The most commonly used analysis method is to regard the search for the optimal discriminator as a combinatorial optimization problem and then automatically solve the model.

[0003] In 2014, Sun et al. [2] first proposed a MILP automated search method for bit-oriented optimal differential / linear features, which greatly improved the optimal probability / correlation of differential / linear distinguishers of multiple lightweight block cipher algorithms.

[0004] In 2017, Sasaki et al. [3] applied the bit-oriented MILP modeling method to the search for impossible differential discriminators, and proposed to use the model's unsolvability as the judgment condition to search for impossible differential discriminators. Similarly, this method was applied to the search for zero-correlation linear discriminators of block ciphers based on S-box and ARX structures, and a method for finding the location of contradictions based on deleting variable constraints was proposed [4].

[0005] Recently, Hadipour et al. [5] proposed a method combining miss-in-the-middle and CP byte-oriented modeling technology, and for the first time realized the search for zero-correlation linear discriminators with model solvability as the judgment condition. For block cipher algorithms with SPN structure and And-RX structure, based on the characterization of bit model and combining miss-in-the-middle and miss-from-the-middle technology, the team further proposed to use model solvability as the judgment condition and simultaneously consider the causes of direct and indirect contradictions to search for zero-correlation linear discriminators [6,7].

[0006] There are two basic methods for searching zero-correlation linear discriminators for block cipher algorithms with And-RX structure:

[0007] The first type is to search for zero-correlation linear discriminators by using the condition that the model has no solution. Under the condition of fixed input and output masks, a model is constructed according to the propagation properties of the linear mask, and then the model is solved. If the model has no solution, it is considered that the fixed mask is an effective zero-correlation linear discriminator.

[0008] The second category is to search for zero-correlation linear discriminators based on miss-in-the-middle technology. Under the condition of fixed input and output masks, a model is constructed in which the input and output masks are propagated along the encryption and decryption directions with probability 1 respectively. When the two models propagate to a certain round in the middle, the judgment condition is the occurrence of direct contradiction. If the model has a solution, then the discriminator is an effective zero-correlation linear discriminator.

[0009] The above two types of zero-correlation linear discriminator search methods have shortcomings. Although the first type of method does not need to consider the cause of the contradiction, it has two disadvantages: one is that it is impossible to traverse and search all possible discriminators; the other is that it is not accurate to judge whether a discriminator is a zero-correlation linear discriminator only by the model having no solution, because it is impossible to determine the specific location where the contradiction occurs.

[0010] Although the second method can directly obtain the location where the contradiction occurs, due to the particularity of the And-RX structure algorithm, the contradictions that occur are not limited to direct contradictions.

[0011] [1]Bogdanov A,Rijmen V.Linear hulls with correlation zero and linearcryptanalysis of block ciphers[J].In:Designs,Codes and Cryptography,2014,70(3):369–383.

[0012] [2] Sun S, Hu L, Wang M, et al. Towards finding the best characteristics of some bit-oriented block ciphers and automatic enumeration of (related-key) differential and linear characteristics with predefined properties [J]. Cryptology ePrint Archive, 2014.

[0013] [3]Sasaki Y,Todo Y.New impossibledifferential search tool from designand cryptanalysis aspects:Revealing structural properties of several ciphers[C] / / Advances in Cryptology–EUROCRYPT 2017:36th Annual InternationalConference on the Theory and Applications of CryptographicTechniques.Springer International Publishing,2017:185-215。

[0014] [4]Cui T,Chen S,Fu K,et al.New automatic tool for finding impossibledifferentials and zero-correlation linear approximations[J].Science ChinaInformation Sciences,2021,64:1-3。

[0015] [5]Hadipour H,Sadeghi S,Eichlseder M.Finding the Impossible:Automatedsearch for full impossible-differential,zero-correlation,and integral attacks[C] / / Annual International Conference on the Theory and Applications ofCryptographic Techniques.Cham:Springer Nature Switzerland,2023:128-157。

[0016] [6]Hadipour H,Gerhalter S,Sadeghi S,et al.Improved search forintegral,impossibledifferential and zero-correlation Attacks[J]。

[0017] [7]Chakraborty D,Hadipour H,Nguyen PH,et al.Finding completeimpossible differential attacks on AndRX ciphers and efficient distinguishers for ARX designs[J].IACR Transactions on Symmetric Cryptology,2024,2024(3). Summary of the invention

[0018] In view of the problems existing in the two types of methods in the background technology, the present invention proposes a zero-correlation linear discriminator search method combining miss-from-the-middle and MILP technology, which is used to search for zero-correlation linear discriminators of block cipher algorithms with And-RX structure. This method combines miss-from-the-middle with MILP technology for the first time to construct a perfect zero-correlation linear discriminator search method, which can not only solve the limitation of not being able to traverse all discriminator search spaces and not being able to derive the specific location where the contradiction occurs when the model has no solution as the judgment condition, but also consider the causes of indirect contradictions.

[0019] The technical solution for achieving the purpose of the present invention is:

[0020] A zero-correlation linear discriminator search method of And-RX structure based on miss-from-the-middle and MILP, comprising the following steps:

[0021] (1) Establish a linear mask propagation model based on the block cipher algorithm structure:

[0022] Based on the algorithm round function structure and the linear mask propagation properties, the nonlinear component And operation, linear component branch operation, etc. are constrained and characterized in bits to construct the input mask Output Mask An r-round MILP model is constructed with probability 1, which propagates along the encryption direction and the decryption direction respectively, and the judgment constraints of the conflict points are given;

[0023] (2) Merge the linear masks encountered in the mth round:

[0024] When the input mask in step (1) Output Mask When the propagation reaches the middle of the mth round, 0≤m≤r, under the premise that no contradiction occurs, the two linear masks that meet Merge into a new linear mask according to the given mask merging rule

[0025] (3) Based on merged linear mask Build the back-propagation model:

[0026] The linear mask after merging in step (2) With probability 1, reverse propagation is performed in the encryption and decryption directions respectively. At a certain bit position of the mask in a certain round, Compare with the propagation path to determine whether there will be a contradiction with a mask value of 0 and 1 with a probability of 1;

[0027] (4) Solve the constructed model:

[0028] The back propagation model and the mask propagation model are connected based on the contradiction points to form a unified model, and then the solution is performed to select and determine the value of the initial parameter m (0≤m≤r), set the initial constraints to ensure that the searched discriminator is effective, and set the objective function.

[0029] The present invention proposes a discriminator search method combining miss-from-the-middle and MILP technology, which is used to search for zero-correlation linear discriminators of the And-RX structure algorithm. It avoids two problems that may occur when searching for zero-correlation linear discriminators with the condition that the model is unsolvable, and also considers the causes of indirect contradictions. At the same time, the discriminator can be expanded with the condition that the MILP model is solvable, laying a foundation for unifying the search for zero-correlation linear discriminators and key recovery into an overall problem for attack.

[0030] The method of the present invention applies the miss-from-the-middle technology to MILP modeling, and constructs a zero-correlation linear discriminator search method with the model being solvable as the judgment condition for the block cipher algorithm of the And-RX structure. The limitation that the search model cannot traverse all search spaces when the model is unsolvable as the judgment condition is solved, and the zero-correlation linear discriminator generated by indirect contradictions is also considered, so that the new model can search for more effective zero-correlation linear discriminators with longer rounds. BRIEF DESCRIPTION OF THE DRAWINGS

[0031] Figure 1 This is a schematic diagram of the model construction principle in the And-RX structure zero-correlation linear discriminator search method of the present invention. DETAILED DESCRIPTION

[0032] The present invention will be described in detail below in conjunction with the embodiments and drawings, but the present invention is not limited thereto.

[0033] Example

[0034] A zero-correlation linear discriminator search method based on miss-from-the-middle and MILP in And-RX structure, referring to Figure 1 , including the following steps:

[0035] (1) Establish a linear mask propagation model based on the block cipher algorithm structure:

[0036] Based on the algorithm round function structure and the linear mask propagation properties, the nonlinear component And operation, linear component branch operation, etc. are constrained and characterized in bits to construct the input mask Output Mask An r-round MILP model is constructed with probability 1, which propagates along the encryption direction and the decryption direction respectively, and the judgment constraints of the conflict points are given;

[0037] (2) Merge the linear masks encountered in the mth round:

[0038] When the input mask in step (1) Output Mask When the propagation reaches the middle of the mth round, 0≤m≤r, under the premise that no contradiction occurs, the two linear masks that meet Merge into a new linear mask according to the given mask merging rule

[0039] (3) Based on merged linear mask Build the back-propagation model:

[0040] The linear mask after merging in step (2) With probability 1, reverse propagation is performed in the encryption and decryption directions respectively. At a certain bit position of the mask in a certain round, Compare with the propagation path to determine whether there will be a contradiction with a mask value of 0 and 1 with a probability of 1;

[0041] (4) Solve the constructed model:

[0042] The back propagation model and the mask propagation model are connected based on the contradiction points to form a unified model, and then the solution is performed to select and determine the value of the initial parameter m (0≤m≤r), set the initial constraints to ensure that the searched discriminator is effective, and set the objective function.

[0043] Furthermore, in step (1), based on the algorithm round function structure and the linear mask propagation property, a MILP model of linear mask propagation with probability 1 along the encryption and decryption directions and a constraint characterization of contradiction point determination are constructed respectively. It is assumed that the encryption algorithm block length is 2n bits, and a two-dimensional variable (x1, x2) is used to represent the value state of a 1-bit mask, where (0,0) represents the mask value of 0, (0,1) represents the mask value of 1, (1,1) represents the mask value is unknown, and "*" represents an unknown state. The constraint characterization of the linear mask through each algorithm component and the constraint characterization of contradiction point determination are as follows:

[0044] (1.1) Branch operation linear mask propagation constraints:

[0045] The linear mask is propagated with a probability of 1 through the branch operation. When the values ​​of the two output masks are determined, the value of the input mask is the XOR value of the two output masks; when the value of any output mask is unknown, the value of the input mask is also unknown.

[0046] Since each bit mask has three value states and the value of a 1-bit mask is represented by a two-dimensional variable, there are a total of 9 possible value states, including (0,0,0,0,0,0), (0,0,0,1,0,1), (0,0,1,1,1,1), (0,1,0,0,0,1), (0,1,0,1,0,0), (0,1,1,1,1,1), (1,1,0,0,1,1), (1,1,0,1,1,1), and (1,1,1,1,1,1);

[0047] Therefore, let Indicates the input mask of the i-th round of branch operation The value state of the jth bit, and Respectively represent the output mask of the i-th round of branch operation The value state of the j-th bit, 0≤i≤r-1, 0≤j≤n-1;

[0048] Then the linear mask propagation constraint of the branch operation can be expressed by Equation 1:

[0049]

[0050] (1.2) “AND” operation linear mask propagation constraint:

[0051] Think of the AND operation as a 2×1 S-box: when the output mask value is 0, the two input mask values ​​must also be 0; when the output mask value is 1, the two input mask values ​​can be any value;

[0052] Similar to step (1.1), there are three possible input and output masks for the AND operation, including (0,0,0,0,0,0,), (1,1,1,1,0,1), and (1,1,1,1,1,1);

[0053] Therefore, let and Respectively represent the input mask of the i-th round of the “AND” operation The value state of the jth bit, Indicates the output mask of the i-th round of the "AND" operation The value state of the j-th bit, 0≤i≤r-1, 0≤j≤n-1;

[0054] Then the linear mask propagation constraint of the “AND” operation can be expressed by Equation 2:

[0055]

[0056] (1.3) Determination constraints of contradiction points:

[0057] When the input and output masks are propagated in the encryption and decryption directions respectively, when the masks meet in a certain round in the middle, only when the probability of generating a contradiction between the mask values ​​0 and 1 is 1, it is considered a contradiction. An extra bit of flag variable d is required for each bit position to indicate whether a contradiction occurs. d=0 means that no contradiction occurs, and d=1 means that a contradiction occurs. There are 8 possible situations, including (0,0,0,0,0), (0,0,0,1,1), (0,1,0,0,1), (0,0,1,1,0), (1,1,0,0,0), (0,1,1,1,0), (1,1,0,1,0), (1,1,0,1,0), and (1,1,1,1,0);

[0058] Therefore, let and Represents two masks respectively and The value state encountered at the j-th bit position in the i-th round, d i,j As a flag variable, it indicates whether a contradiction occurs at this position;

[0059] In order to characterize the generation of indirect contradictions, the decision function of the contradiction point is defined as The specific constraints are expressed as formula 3:

[0060]

[0061] Furthermore, in step (2), the two linear masks that meet are merged into a new linear mask according to the given mask merging rule. The merging rules are:

[0062] When input and output masks When the two masks propagate along the encryption and decryption directions with probability 1 and meet in the middle of the mth round, if and only if the masks meet at the same bit position, they are merged according to the rules in Table 1 without any contradiction;

[0063] Therefore, the mask that meets in the middle No contradiction can occur, that is, the value of each bit flag variable that has a contradiction is 0. The constraint condition of formula 4 needs to be satisfied, and then the mask is merged through formula 7 to obtain a new mask.

[0064] Table 1 Mask merging rules

[0065]

[0066] (2.1) Characterization of the prerequisite constraints of mask merging:

[0067] and Respectively represent the encounter mask and The value state of the jth bit, Represents the merge mask at round m The value status of the j-th bit;

[0068] When merging by encountering mask, make sure and There can be no contradiction between them, that is, the contradiction flag variable d in the mth round m,j (0≤j≤n-1) all values ​​are 0, and and The value status cannot all be unknown, ensuring the merge mask When propagating along the encryption and decryption directions with probability 1, direct contradictions can occur. The constraints are as follows:

[0069]

[0070]

[0071] (2.2) Constraint characterization of mask merging:

[0072] As can be seen from Table 1, there are 7 possible cases of mask merging, including (0,0,0,0,0,0,0,), (0,1,0,1,0,1), (0,1,1,1,0,1), (0,0,1,1,0,0,), (1,1,0,0,0,0,), (1,1,0,1,0,1), and (1,1,1,1,1,1);

[0073] Then the mask merging constraint can be expressed by Equation 7:

[0074]

[0075] Furthermore, step (3) is based on the mask merging obtained in step (2) Then, construct the propagation model along the encryption and decryption directions respectively. The establishment of the model is the same as step (1), and the judgment constraints of indirect contradictions are set:

[0076] Connect the mask Γ based on the contradiction point d U With (Γ W ,Γ V ) propagation path, its constraints are shown in Equation 8:

[0077]

[0078] And make sure to merge the mask There is at least one bit position in the back propagation path where a contradiction occurs, and its constraint is shown in Equation 9:

[0079]

[0080] Furthermore, in step (4), the constructed propagation model is solved by first setting the initial parameters and initial condition constraints to ensure the solvability and effectiveness of the model, and taking searching for as many zero-correlation linear discriminators as possible as the objective function. The solution process is as follows:

[0081] (4.1) Initial condition constraints:

[0082] The initial constraints are to restrict the input and output masks The value of cannot be all zero, and its conditional constraints are as follows

[0083] As shown in formula 10:

[0084]

[0085] (4.2)Objective function:

[0086] In order to search for as many zero-correlation linear discriminators as possible while satisfying all conditional constraints, that is, to include as many unknown "*" bits as possible in the input and output masks, each bit (x1, x2) in the input and output masks needs to use an additional variable t i (0≤i≤2n-1) to indicate whether the value is unknown, and the variable t i The relationship between and the mask can be expressed by Equation 11, and the objective function is set to Equation 12:

[0087]

[0088] Finally, the model is solved by Gurobi solver to obtain as many zero-correlation linear discriminators as possible and the corresponding specific locations of indirect contradictions.

[0089] The method of the present invention has high accuracy. While searching for a zero-correlation linear discriminator, it can also give its accurate indirect contradiction location, eliminating misjudgment caused by the model having no solution, and avoiding the tedious process of manually deriving the indirect contradiction location based on the zero-correlation linear discriminator.

Claims

1. A zero-correlation linear discriminator search method based on miss-from-the-middle and MILP in And-RX structure, characterized in that: The following steps are involved: (1) Establish a linear mask propagation model based on the block cipher algorithm structure: Based on the algorithm round function structure and the linear mask propagation properties, the nonlinear component And operation, linear component branch operation, etc. are constrained and characterized in bits to construct the input mask Output Mask An r-round MILP model is constructed with probability 1, which propagates along the encryption direction and the decryption direction respectively, and the judgment constraints of the conflict points are given; (2) Merge the linear masks encountered in the mth round: When the input mask in step (1) Output Mask When the propagation reaches the middle of the mth round, 0≤m≤r, under the premise that no contradiction occurs, the two linear masks that meet Merge into a new linear mask according to the given mask merging rule (3) Based on merged linear mask Build the back-propagation model: The linear mask after merging in step (2) With probability 1, reverse propagation is performed in the encryption and decryption directions respectively. At a certain bit position of the mask in a certain round, Compare with the propagation path to determine whether there will be a contradiction with a mask value of 0 and 1 with a probability of 1; (4) Solve the constructed model: The back propagation model and the mask propagation model are connected based on the contradiction points to form a unified model, and then the solution is performed to select and determine the value of the initial parameter m, 0≤m≤r, set the initial constraints to ensure that the searched discriminator is effective, and set the objective function.

2. The And-RX structure zero-correlation linear discriminator search method according to claim 1, characterized in that: In step (1), based on the algorithm round function structure and the linear mask propagation properties, a MILP model of linear mask propagation with probability 1 along the encryption and decryption directions and a constraint characterization of conflict point determination are constructed respectively. It is assumed that the encryption algorithm block length is 2n bits, and a two-dimensional variable (x1, x2) is used to represent the value state of a 1-bit mask, where (0,0) represents the mask value of 0, (0,1) represents the mask value of 1, (1,1) represents the mask value is unknown, and "*" represents an unknown state. The constraint characterization of the linear mask through each algorithm component and the constraint characterization of conflict point determination are as follows: (1.1) Branch operation linear mask propagation constraints: The linear mask is propagated with a probability of 1 through the branch operation. When the values ​​of the two output masks are determined, the value of the input mask is the XOR value of the two output masks; when the value of any output mask is unknown, the value of the input mask is also unknown. Since each bit mask has three value states and the value of a 1-bit mask is represented by a two-dimensional variable, there are a total of 9 possible value states, including (0,0,0,0,0,0), (0,0,0,1,0,1), (0,0,1,1,1,1), (0,1,0,0,0,1), (0,1,0,1,0,0), (0,1,1,1,1,1), (1,1,0,0,1,1), (1,1,0,1,1,1), and (1,1,1,1,1,1); Therefore, let Indicates the input mask of the i-th round of branch operation The value state of the jth bit, and Respectively represent the output mask of the i-th round of branch operation The value state of the j-th bit, 0≤i≤r-1, 0≤j≤n-1; Then the linear mask propagation constraint of the branch operation can be expressed by Equation 1: (1.2) "AND" operation linear mask propagation constraint: Think of the "AND" operation as a 2×1 S-box: when the output mask value is 0, the values ​​of the two input masks must also be 0; when the output mask value is 1, the values ​​of the two input masks can be any value; Similar to step (1.1), there are three possible input and output masks for the "AND" operation, including (0,0,0,0,0,0,), (1,1,1,1,0,1) and (1,1,1,1,1,1); Therefore, let and Respectively represent the input mask of the i-th round of "AND" operation The value state of the jth bit, Indicates the output mask of the i-th round of the "AND" operation The value state of the j-th bit, 0≤i≤r-1, 0≤j≤n-1; Then the linear mask propagation constraint of the "AND" operation can be expressed by Equation 2: (1.3) Determination constraints of contradiction points: When the input and output masks are propagated in the encryption and decryption directions respectively, when the masks meet in a certain round in the middle, only when the probability of generating a contradiction between the mask values ​​0 and 1 is 1, it is considered a contradiction. An extra bit of flag variable d is required for each bit position to indicate whether a contradiction occurs. d=0 means that no contradiction occurs, and d=1 means that a contradiction occurs. There are 8 possible situations, including (0,0,0,0,0), (0,0,0,1,1), (0,1,0,0,1), (0,0,1,1,0), (1,1,0,0,0), (0,1,1,1,0), (1,1,0,1,0), (1,1,0,1,0), and (1,1,1,1,0); Therefore, let and Represents two masks respectively and The value state encountered at the j-th bit position in the i-th round, 0≤i≤r-1,0≤j≤n-1, d i,j As a flag variable, it indicates whether a contradiction occurs at this position; In order to characterize the generation of indirect contradictions, the decision function of the contradiction point is defined as The specific constraints are expressed as formula 3:

3. The And-RX structure zero-correlation linear discriminator search method according to claim 1, characterized in that: Step (2) merges the two linear masks into a new linear mask according to the given mask merging rule. The merging rules are: When input and output masks When the masks propagate along the encryption and decryption directions with probability 1 to the middle of the mth round, if and only if the masks that meet at the same bit position are merged according to the rules of Table 1 without any contradiction, therefore, the masks that meet in the middle No contradiction can occur, that is, the value of each bit flag variable that has a contradiction is 0. The constraint condition of formula 4 needs to be satisfied, and then the mask is merged through formula 7 to obtain a new mask. Table 1 Mask merging rules (2.1) Characterization of the prerequisite constraints of mask merging: and Respectively represent the encounter mask and The value state of the jth bit, Represents the merge mask at round m The value state of the j-th bit, 0≤m≤r, 0≤j≤n-1; When merging by encountering mask, make sure and There can be no contradiction between them, that is, the contradiction flag variable d in the mth round m,j All values ​​are 0, 0≤j≤n-1, and and The value status cannot be all unknown, ensuring the merge mask When propagating along the encryption and decryption directions with probability 1, direct contradictions can occur. The constraints are as follows: (2.2) Constraint characterization of mask merging: As can be seen from Table 1, there are 7 possible cases of mask merging, including (0,0,0,0,0,0,0,), (0,1,0,1,0,1), (0,1,1,1,0,1), (0,0,1,1,0,0,), (1,1,0,0,0,0,), (1,1,0,1,0,1), and (1,1,1,1,1,1); Then the mask merging constraint can be expressed by Equation 7:

4. The And-RX structure zero-correlation linear discriminator search method according to claim 1, characterized in that: Step (3) is based on the mask merging in step (2) obtained Then, the propagation model is constructed along the encryption and decryption directions respectively. The establishment of the model is the same as step (1). The judgment constraints of indirect contradiction are set: based on the contradiction point d, the connection mask Γ U With (Γ W ,Γ V ) propagation path, its constraints are shown in Equation 8: And make sure to merge the mask There is at least one bit position in the back propagation path where a contradiction occurs, and its constraint is shown in Equation 9:

5. The And-RX structure zero-correlation linear discriminator search method according to claim 1, characterized in that: In step (4), the constructed propagation model is solved by first setting the initial parameters and initial condition constraints to ensure the solvability and effectiveness of the model, and searching for as many zero-correlation linear discriminators as possible is used as the objective function. The solution process is as follows: (4.1) Initial condition constraints: The initial constraints are to restrict the input and output masks The value of cannot be all zero, and its conditional constraints are shown in formula 10: (4.2)Objective function: In order to search for as many zero-correlation linear discriminators as possible while satisfying all conditional constraints, that is, to include as many unknown "*" bits as possible in the input and output masks, each bit (x1, x2) in the input and output masks needs to use an additional variable t i To indicate whether the value is an unknown state, 0≤i≤2n-1, and the variable t i The relationship between and the mask can be expressed by Equation 11, and the objective function is set to Equation 12: Finally, the model is solved by Gurobi solver to obtain as many zero-correlation linear discriminators as possible and the corresponding specific locations of indirect contradictions.

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