Key analysis method, computer device and storage medium

Abstract

The application relates to the technical field of algebra coding, and discloses a key analysis method, computer equipment and a storage medium, which comprise the following steps: acquiring key parameters; wherein the first key parameter comprises a first polynomial describing the mathematical properties of a first maximum scattering linear set; the second key parameter comprises a second polynomial describing the mathematical properties of a second maximum scattering linear set; performing geometric mapping processing based on the key parameters to obtain a geometric object set; performing equivalent determination of spatial geometric features based on a first subspace and a second subspace to obtain an equivalent determination result; and performing equivalent determination of the first maximum scattering linear set and the second maximum scattering linear set based on the equivalent determination result to obtain a linear set equivalent result and determine a key analysis result. The beneficial effect is that the determination process is universal and has low computational complexity, the maximum scattering linear set type key can be efficiently and accurately analyzed in terms of equivalence, and standardized support is provided for key screening and security evaluation of encryption.

Description

Technical Field

[0001] This application relates to the field of algebraic coding technology, and more particularly to a key analysis method, computer device, and storage medium. Background Technology

[0002] In the field of communications, the security of data transmission is a key concern in communication security systems. Taking encrypted communication as an example, the strength of the key directly affects the anti-attack capability of the encryption algorithm. Rank-based codes constructed from finite fields, due to the inherent mathematical relationship between their codeword structure and the security parameters of the key, can provide a theoretical basis for evaluating key strength through their rank distribution characteristics.

[0003] In a projective space over a finite field, a code that reaches the upper bound of the Singleton type in a rank metric code can be called a Maximum Rank Distance (MRD) code. The maximum scattering linear set is the core basic geometric structure for constructing maximum rank distance codes, cryptographic security components, and combinatorial designs.

[0004] In the study of maximally scattering linear sets, the same linear set can be defined by multiple different linearized polynomials. That is, different polynomials may have drastically different algebraic expressions, but the linear sets they define may be the same object in terms of geometric structure. This phenomenon is called linear set equivalence.

[0005] In practical applications of communication coding, if it is not possible to accurately determine whether two polynomials define equivalent linear sets, it may lead to duplicate analysis in code pattern screening or incorrect conclusions in key security assessment.

[0006] However, with the increase in parameter size and the number of polynomial terms, the number of equivalent maximum scattering linear sets generated by polynomials becomes extremely large. Current methods for determining the equivalence of maximum scattering linear sets suffer from excessively high computational complexity, making it difficult to meet the needs of efficient screening and analysis. Therefore, an analytical method is needed that can quickly determine whether linear sets are equivalent, thereby assisting in the evaluation of key validity. Summary of the Invention

[0007] This application aims to at least partially solve one of the technical problems in related technologies. To this end, this application proposes a key analysis method, a computer device, and a storage medium. The main technical solutions adopted in this application include: Firstly, this application provides a key analysis method, which includes: Obtain key parameters; wherein, the key parameters include finite field parameters, a first key parameter, and a second key parameter; the first key parameter includes a first polynomial describing the mathematical properties of the first maximally scattering linear set; the second key parameter includes a second polynomial describing the mathematical properties of the second maximally scattering linear set; Geometric mapping is performed based on key parameters to obtain a set of geometric objects; wherein the set of geometric objects includes a first subspace and a second subspace; An equivalence determination of spatial geometric features is performed based on the first subspace and the second subspace to obtain an equivalence determination result; wherein, the equivalence determination result is used to describe whether the first subspace and the second subspace satisfy the spatial coincidence condition; Based on the equivalence determination result, the equivalence determination of the first maximum scattering linear set and the second maximum scattering linear set is performed to obtain the linear set equivalence result, and the key analysis result is determined based on the linear set equivalence result; wherein, the key analysis result is used to describe whether the first key and the second key can be substituted for each other for encrypted communication.

[0008] By uniformly mapping key polynomials to geometric subspaces, and then using the general linear equivalence criterion of subspaces as the core, the projective semi-linear equivalence conclusion of linear sets is derived layer by layer, ultimately achieving automated analysis of key substitutability. The entire method eliminates the need to derive algebraic equations separately for keys with different structures; the judgment process is universal and computationally inexpensive; and it can efficiently and accurately perform equivalence analysis of maximally scattering linear set keys, providing standardized support for key screening and security assessment of encryption.

[0009] Optionally, geometric mapping is performed based on the key parameters to obtain a set of geometric objects, including: A mapping process is performed based on the finite field parameters and the first polynomial to construct the first subspace; A mapping process is performed based on finite field parameters and a second polynomial to construct a second subspace.

[0010] By using the unified geometric mapping process described above, the two key structures to be judged are placed under the same geometric framework, providing a comparable data basis for subsequent spatial feature equivalence determination, without the need to design differentiated preprocessing procedures for different polynomial structures.

[0011] Optionally, the mapping process for the first polynomial is the same as that for the second polynomial; A mapping process is performed based on the finite field parameters and the first polynomial to construct the first subspace, including: Based on the parameters of the finite field, all elements in the finite field are substituted into the first polynomial for calculation to obtain the target coordinate vector; The first subspace is constructed by performing vector culling and projective processing based on the target coordinate vector.

[0012] By substituting each element to generate a coordinate vector, and then standardizing the subspace structure through projective processing, a precise and unified conversion from polynomial to subspace can be achieved, providing standardized input for subsequent geometric determination.

[0013] Optionally, the second polynomial has an adjoint polynomial; the adjoint polynomial corresponds to an adjoint subspace. Equivalence determination of spatial geometric features is performed based on the first and second subspaces to obtain equivalence determination results, including: General linear equivalence determination is performed using the first subspace, the adjoint subspace, and the second subspace to obtain the equivalence determination result.

[0014] Introducing the adjoint subspace to participate in the general linear equivalence determination can cover the equivalence scenarios of all non-pseudo-regular linear sets, ensuring the completeness and accuracy of the determination results in general scenarios.

[0015] Optionally, a general linear equivalence determination is performed using the first subspace, the adjoint subspace, and the second subspace to obtain the equivalence determination result, including: The first subspace is linearly transformed using a second-order invertible matrix to obtain the transformed subspace. Spatial overlap judgment is performed based on the transformed subspace, the second subspace, and the adjoint subspace to obtain the equivalence judgment result; By combining linear transformation of second-order invertible matrices with bi-branch space coincidence verification, the abstract problem of linear set equivalence is transformed into an intuitive subspace geometric coincidence determination, realizing the standardized determination of general linear equivalence in general scenarios and providing a unified geometric judgment basis for subsequent linear set equivalence analysis.

[0016] Optionally, spatial overlap judgment is performed based on the transformed subspace, the second subspace, and the adjoint subspace to obtain an equivalence judgment result, including: The transformed subspace is compared with the second subspace to determine spatial overlap, and the first judgment result is obtained. The transformed subspace and the adjoint subspace are judged to determine spatial overlap, and the second judgment result is obtained. The equivalence determination result is determined based on the first and second judgment results.

[0017] By verifying the overlap between the atomic space and the adjoint subspace through dual-branch parallel verification, the general linear equivalence of all non-pseudo-regular linear sets in the general scenario can be fully covered, ensuring the comprehensiveness and accuracy of the judgment results.

[0018] Optionally, if both the first and second polynomials are of the four-term type structure, the equivalence determination based on the spatial geometric features of the first and second subspaces to obtain the equivalence determination result further includes: The first subspace is linearly transformed using a second-order invertible matrix to obtain the transformed subspace. The transformed subspace and the second subspace are compared to determine spatial overlap, and the equivalence determination result is obtained.

[0019] The decision branches are simplified for the unique structure of the four-term form, and the generation and verification steps of the accompanying subspace are omitted, which further reduces the computational amount of equivalence determination in the four-term form scenario and improves the analysis efficiency of specific scenarios.

[0020] Optionally, based on the equivalence determination result, an equivalence determination is made between the first maximum scattering linear set and the second maximum scattering linear set to obtain the linear set equivalence result, including: If the equivalence determination result shows that the spatial coincidence condition is met, then the first maximum scattering linear set and the second maximum scattering linear set are determined to satisfy the projective linear equivalence relation. The second maximum scattering linear set is transformed based on the finite field automorphism transformation to obtain the reconstructed linear set; The first maximum scattering linear set is compared with the reconstructed linear set, and the linear set equivalence result is obtained if the first maximum scattering linear set and the reconstructed linear set satisfy the projective linear equivalence relation.

[0021] By transforming subspace equivalence to projective linear equivalence, and combining the two-layer judgment logic of finite field automorphic traversal and reconstructed linear set verification, the industry standard equivalence judgment of the maximum scattering linear set can be fully realized. The process is clear and adaptable to all non-pseudo-regular linear set structures, providing an accurate judgment basis for key substitutability analysis.

[0022] Secondly, this application also provides a computer device, including a memory and a processor, wherein the memory stores a computer program, and the processor executes the computer program to implement the steps of any of the methods described above.

[0023] Thirdly, this application also provides a computer-readable storage medium having a computer program stored thereon, wherein the computer program, when executed by a processor, implements the steps of any of the methods described above. Attached Figure Description

[0024] To more clearly illustrate the technical solutions in the specific embodiments of this application or the prior art, the drawings used in the description of the specific embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are some embodiments of this application. For those skilled in the art, other drawings can be obtained from these drawings without creative effort.

[0025] Figure 1 This is a flowchart of a key analysis method provided according to an embodiment of this application; Figure 2 This is a flowchart of a method for determining an equivalence determination result according to yet another embodiment of this application; Figure 3 Here is a flowchart of a method for determining the equivalence result of a linear set according to another embodiment of this application; Figure 4 This is an internal structural diagram of a computer device provided according to an embodiment of this application. Detailed Implementation

[0026] To make the objectives, technical solutions, and advantages of the embodiments of this application clearer, the technical solutions of the embodiments of this application will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of this application, not all embodiments. Based on the embodiments of this application, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of this application.

[0027] It should be noted that the application background of this scheme is the intersection of finite field projective geometry and cryptography, and the core research object is the maximum scattering linear set induced by linearized polynomials.

[0028] In related technologies, the study of the equivalence of the maximum scattering linear set adopts a polynomial perspective: that is, it is carried out through algebraic methods such as polynomial coefficient matching, norm conditions, solution of trace equations, and exponential combination analysis.

[0029] This method is effective and efficient in low-term cases such as monomials and binomials (e.g., Lunardon-Polverino, LP type). However, as the number of terms in a polynomial increases, the computational complexity and derivation difficulty increase exponentially. When extended to quadrinomials and more complex scattering linear sets, the computation becomes extremely complex.

[0030] Specifically, while the maximum scattering linear set equivalence analysis method based on a polynomial perspective can output concise final judgment conditions, its core shortcomings lie in the preliminary analysis and derivation stage, namely, the following issues: 1. The derivation of higher-order structures is difficult: For scattering linear sets with equal number of terms in four terms, the polynomial perspective requires dealing with complex combinations of higher-order exponents and multivariate algebraic constraints. The derivation is computationally intensive and logically complex, and is prone to redundancy and loopholes.

[0031] 2. Lack of a unified analytical framework and poor reusability: Different types of maximum scattering linear sets correspond to different polynomial structures, making it impossible to share a single set of analytical logic. Each type of linear set requires a complete set of algebraic derivations, resulting in extremely low research efficiency.

[0032] 3. Lack of geometric support and weak scalability: Pure algebraic derivation only focuses on the surface expression of polynomials and cannot combine the inherent geometric structure, spatial relations and group action properties of linear sets, making it difficult to extend to more novel higher-order scattering linear sets.

[0033] Based on this, according to the embodiments of this application, a key analysis method embodiment is provided, which can discard the cumbersome pure algebraic derivation mode while retaining the concise equivalence determination result, and construct a unified analysis method based on a geometric perspective, simplifying the pre-derivation process of the equivalence of the maximum scattering linear set, and adapting to the equivalence analysis scenarios of various high-order and multi-type linear sets.

[0034] Before describing the technical solution of this application, the relevant concepts will be explained first.

[0035] Finite field: A set of numbers with a finite number of elements and closed by the four arithmetic operations. It can be used as the basic mathematical carrier for keys and error correction codes. Representatives include A finite field of n elements, where Powers of odd prime numbers (which can be represented as) ,in It is an odd prime number. (positive integers) It is a positive integer (can be set to) ,in (where is a positive integer). Let be... Given a two-dimensional vector space over a finite field, each vector is of the form: ,in . Then it is The one-dimensional projective space above, that is, by The projective line formed by all one-dimensional subspaces in the equation.

[0036] Maximum scattering linear set: The set of points in a finite field projective space whose uniformity reaches the theoretical upper limit. It is the core structure for constructing encryption keys and designing MRD error-correcting codes.

[0037] Scattering subspace: A two-dimensional subspace induced by scattering polynomials. It is the geometric carrier corresponding to the polynomial algebraic structure. Each polynomial uniquely corresponds to a scattering subspace.

[0038] First, a global geometric mapping process is performed to unify the transformation of algebraic problems into geometric problems.

[0039] Specifically, for any definition in Maximum scattering polynomial on The polynomial algebraic structure can be uniquely mapped to a finite field two-dimensional subspace structure through the following fixed mapping relationship: in, Represents a polynomial The corresponding scattering subspace, i.e., the space of polynomials An induced subset is defined as It represents all forms like The set of vectors (usually) (Therefore, it contains a zero vector). Represents a polynomial The corresponding maximum scattering linear set, that is The set of points obtained by projecting all non-zero vectors in the set is: , ( The superscript * indicates that the element does not contain zero. Indicates by Zhang Cheng's one-dimensional subspace, i.e., the projective line One of the points.

[0040] The above formula enables a general transformation of the algebraic problem of arbitrarily scattering linear sets into a geometric problem, eliminating the need for differentiated algebraic simplifications for different polynomial structures.

[0041] For ease of subsequent description, the following concepts are defined: Adjoint polynomial: Suppose we are given a linearized polynomial .in, Represents the index of each term in the polynomial, with values ​​ranging from... arrive integers, corresponding to powers of; The linearized polynomial of the th The coefficient of the term.

[0042] Then, the linearized polynomial The accompanying polynomial It can be defined as: And it satisfies the trace identity For any It holds true. The subspace corresponding to this adjoint polynomial. It can be called the adjoint subspace.

[0043] General linear equivalence ( -Equivalence): An equivalence criterion for subspaces. Two subspaces are said to be equivalent if they can be completely superimposed by an invertible two-dimensional matrix (equivalent to rotation, stretching, and shearing). -equivalence.

[0044] Projective linear equivalence ( -Equivalence): The first level of equivalence criterion for linear sets. Two linear sets are said to be equivalent if they can be completely superimposed through a projective linear transformation. - Equivalence. Because scaling the entire point set in projective space does not change the point set, equivalence is essentially a subspace. - Equivalent to removing the redundancy of global scaling (global scaling in projective space does not change the point set).

[0045] Projective semilinear equivalence ( -Equivalence): A universal standard in finite geometry for determining whether linear sets are essentially the same. That is, if two maximally scattering linear sets can be completely superimposed through a projective semilinear transformation, then they are said to be equivalent. -equivalence.

[0046] It should be noted that projective semilinear ( -) The transformation is in the projective linear ( -) A transformation based on the transformation, plus an additional composite finite field automorphism. For linear sets and If a projective semilinear transformation exists Make Then the two are said to satisfy -equivalence. -Equivalence can be understood as - The combination of equivalence and finite field automorphism, that is, when making a decision, one can first apply a field automorphism to one of the linear sets, and then perform... - Equivalence determination.

[0047] Here, finite field automorphism refers to a bijective isomorphism from a finite field to itself, i.e. Its function is to map each element in the domain to its own image under the domain automorphism.

[0048] Based on the above geometric mappings and conceptual definitions, the following provides the basis for the transformation from linear set equivalence determination to subspace equivalence determination.

[0049] After completing the geometric mapping, the problem of determining the equivalence of linear sets can be transformed into the problem of determining the equivalence of subspaces.

[0050] Specifically, based on Pepe's conclusions and the properties of non-pseudoregulus (i.e., non-pseudoregulus) maximally scattering linear sets, we can derive the constraint relationship of obtaining subspaces from the equality of linear sets: For nonpseudoregulus type scattering polynomials and ( (for the adjoint polynomial), if Then there exist non-zero elements. , making or ( (For the adjoint subspace of the adjoint polynomial), then, combining the properties of the non-pseudoregulus type maximally scattering linear set, we can derive the following: That is, if two non-pseudo-canonical maximal scattering linear sets, such as the first maximal scattering linear set... Second maximum scattering linear set If PGL-equivalence is satisfied, then the corresponding subspace satisfies one of the following two conditions: First subspace Second subspace For GL-equivalent; first subspace Second subspace companion subspace It is GL-equivalent.

[0051] This conclusion holds true for all non-pseudoregulus type maximally scattering linear sets, without the need to reconstruct the decision logic for different polynomial structures.

[0052] The derivation process of equation (1) will be explained in detail below: 1 If and It is PGL-equivalent, which means that an invertible matrix exists. Make This invertible matrix acts on For each element in the equation, apply M to... After checking all the points, the result is as follows: completely consistent; 2. According to , From the definition, we can obtain: If If their corresponding subspaces differ by a constant multiple, then there exist non-zero elements. , enabling or ; 3. Let the matrix ,but Therefore, we can conclude that... or ,Right now and It is GL-equivalent, or and It is GL-equivalent.

[0053] Therefore, according to the condition " and "It is PGL-equivalent", thus obtaining the condition " and It is GL-equivalent, or and "is GL-equivalent", i.e., equation (1).

[0054] Since the equivalence of the maximum scattering linear set is in Defined under the meaning of equivalence, therefore, determining whether two maximally scattering linear sets are equivalent is essentially determining whether they satisfy... - Equivalent. And according to the above definition, - The equivalence determination can be decomposed into - Two levels: equivalence determination and domain automorphism search.

[0055] Furthermore, Equation (1) has transformed the PGL-equivalence determination into the GL-equivalence determination of the subspace. Therefore, the entire equivalence determination problem is ultimately reduced to the determination of GL-equivalence of the subspace. Based on this, the specific determination method of GL-equivalence is given below.

[0056] Specifically, The methods for determining equivalence are as follows: For the first subspace Second subspace For two subspaces to be equivalent, a necessary and sufficient condition is that there exists a non-zero element. This non-zero element can be equivalently represented as an intermediate variable at the projective level, which can be contained within a second-order invertible matrix. In the degrees of freedom, that is The necessary and sufficient condition for equivalence can be derived as the existence of a second-order invertible matrix. , enabling .

[0057] Substituting and expanding, we get: ′ In the formula, g()' represents the transpose of g().

[0058] This expression involves algebraic computation at the polynomial level, requiring no further processing. By assigning values ​​and comparing the coefficients of the polynomials on both sides of the equation, the matrix elements can be derived. The conditions that need to be met.

[0059] For example, let The first polynomial The second polynomial To verify whether two subspaces are GL equivalent, the following steps can be taken: First, let's consider the second-order invertible matrix... Substituting the vector form of the first subspace - Equivalent definition Expanding this gives us two equations: In the formula, X is the x-coordinate variable of the vector in the first subspace, taken from the finite field; Y is the x-coordinate variable of the vector after matrix transformation, that is, the x-coordinate of the vector in the second subspace, which, like X, is a variable at the polynomial level.

[0060] Substituting the first expression into the left side of the second expression, we get: Expand to reveal: Comparing the polynomial coefficients on both sides, we can obtain , , Therefore, we only need to take... ,in This allows two subspaces to be made and satisfy -equivalence.

[0061] Based on this, further explanation - The complete process of equivalence determination.

[0062] Specifically, according to -Equivalence and PGL- Definition of equivalence: There exists an automorphism in a finite field. This makes the second maximum scattering linear set go through The reconstructed linear set obtained after transformation and the first maximum scattering linear set If the PGL-equivalence is satisfied, then the first maximally scattering linear set can be determined. Second maximum scattering linear set satisfy -equivalence.

[0063] In other words, the complete decision-making process can be broken down into three steps: traversing all possible automorphisms of finite fields. ,Bundle Acting on the second maximum scattering linear set The above yields the reconstructed linear set after transformation. For each reconstructed linear set, determine its relationship with the first maximum scattering linear set using the method described above. Does it satisfy PGL-equivalence? If any one of them exists... Let the corresponding reconstructed linear set and If PGL equivalence is satisfied, then the original first maximal scattering linear set is determined. Second maximum scattering linear set -equivalence.

[0064] Right now -The necessary and sufficient condition for equivalence determination is: .

[0065] Based on this geometric mapping framework, the automorphism group structure of linear sets can be further solved based on the GL-equivalence relation between the subspace and its own adjoint subspace.

[0066] In this context, the automorphism group refers to the group of all transformations that preserve the geometric structure of an object. The size of the group represents the degree of symmetry of the structure and is a key parameter for key and coding design.

[0067] Specifically, based on the geometric symmetry properties of the subspace, a fixed group structure solution paradigm can be established. That is, by simply determining whether the subspace and its adjoint subspace are GL-equivalent, the structure of the automorphism group can be directly derived. The structure of the automorphism group of the maximally scattering linear set can be divided into two general results: If subspace With its own accompanying subspace If GL-equivalence is satisfied, then the linear set... The automorphism group is: Otherwise (a subspace is not equivalent to its adjoint subspace), the linear set's The automorphism group is: in, , It is a general linear group The center of is composed of all non-zero scalar matrices; Representative subspace The GL-automorphism group, i.e., all invertible matrices that are invariant to the subspace, is represented by the horizontal line above. This represents the removal of the center scaling, which treats all proportional matrices as the same (scaling in projective space does not change the point set and is a repetitive transformation), and is used to remove redundant scaling transformations. It is a second-order subgroup containing only two elements (identity transformation and adjoint mirror transformation), representing additional mirror symmetry; Representing a semi-direct product, it is a combination of two groups, which can be understood as a superposition of basic symmetry and additional mirror symmetry.

[0068] In summary, the equivalence determination process of this invention can be summarized as follows: First, the linear set defined by the polynomial is transformed into the corresponding scattering subspace through global geometric mapping; second, according to equation (1), the PGL-equivalence determination of the linear set is transformed into the GL-equivalence determination between subspaces (or the GL-equivalence determination between the linear set and the accompanying subspace); finally, domain automorphism is superimposed on the basis of PGL-equivalence to complete the process. - Equivalence determination.

[0069] The entire process eliminates the need for tedious algebraic operations such as coefficient matching and trace equation solving for polynomials. It only requires finding a second-order invertible matrix that satisfies the spatial coincidence condition for the subspace. Furthermore, the above three steps constitute the core general technical framework of this invention, directly applicable to the equivalence determination and symmetry analysis of various maximum scattering linear sets. This framework relies on the equivalence relations of scattering subspaces and the structure of automorphism groups for unified derivation, eliminating the need to rebuild derivation systems for different structures. It fundamentally overcomes the technical shortcomings of related technologies, such as poor reusability, lengthy derivation processes, and high computational complexity. Therefore, it provides an efficient and unified analytical tool for the application of maximum scattering linear sets in key generation, code pattern selection, and security performance evaluation.

[0070] It should be noted that the steps shown in the flowchart in the accompanying drawings can be executed in a computer system such as a set of computer-executable instructions, and although a logical order is shown in the flowchart, in some cases the steps shown or described may be executed in a different order than that shown here.

[0071] This embodiment provides a key analysis method, such as Figure 1 As shown, the method includes the following steps: S110, Obtain key parameters.

[0072] Among them, key parameters can refer to the input parameters that need to determine the mathematical structure of the key, which are used to provide basic computational data for subsequent equivalence analysis.

[0073] Specifically, the key parameters may include finite field parameters, a first key parameter, and a second key parameter.

[0074] Among them, the finite field parameters can refer to the configuration parameters that determine the finite field environment in which the cryptographic operation takes place, and determine the size of the computation space of the key structure.

[0075] For example, this type of key can be applied to scenarios such as rank distance cryptography, MRD error correction codes, and rank-KEM key encapsulation. Taking the LP binomial scattering key as an example, we can take the odd prime power q=7 and the expansion order (i.e., the expansion dimension) n=6 (satisfying gcd(s,n)=1, taking s=1) as the finite field parameters, and the corresponding base field is... Expanding the domain to The corresponding security parameters can support the minimum rank d=4 of the MRD code, serving as the computing environment for the key structure.

[0076] The first key parameter can refer to the set of parameters that can characterize the mathematical structure of the first set of keys to be analyzed.

[0077] Specifically, the first key parameter includes a first polynomial describing the mathematical properties of the first maximum scattering linear set.

[0078] For example, the first maximum scattering linear set is the geometric structure constituting the first key. The first polynomial is a mathematical expression describing the algebraic properties of the maximum scattering linear set, which can be uniquely mapped to the corresponding linear set structure.

[0079] For example, the first maximum scattering linear set can be defined in The maximum scattering linear set on the first polynomial is given by the first polynomial. Unique representation and definition, have The first polynomial The algebraic structure of the linear set determines the linear set The geometric configuration.

[0080] Following the previous example, the first polynomial can be in binomial form. Where the coefficient δ is taken as Satisfying the norm condition The corresponding first maximum scattering linear set can be... The geometric configuration is uniquely determined by the algebraic structure of the first polynomial.

[0081] The second key parameter can refer to a set of parameters that characterize the mathematical structure of the first set of keys to be analyzed, and may include a second polynomial describing the mathematical properties of the second maximum scattering linear set.

[0082] Similarly, the second maximum scattering linear set is the geometric structure that constitutes the second key. The second polynomial is a mathematical expression describing the algebraic properties of this maximum scattering linear set, which can be uniquely mapped to the corresponding linear set structure.

[0083] For example, the second maximum scattering linear set can be defined in The maximum scattering linear set on, given by the second polynomial Unique representation and definition, have The second polynomial The algebraic structure of the linear set determines the linear set The geometric configuration.

[0084] Specifically, the preset structure parameters of two sets of keys to be evaluated can be read from the key configuration library of the existing key management system, and the corresponding finite field configuration items, first polynomial expression and second polynomial expression can be extracted and organized into a standardized set of key parameters; or the two sets of key polynomial parameters and finite field parameters input by the user can be received through an external interface to complete the key parameter collection.

[0085] It should be noted that the key configuration library is a conventional storage component in existing key management systems, used to centrally store the structural parameters of various scattering keys, and is a mature technology in this field. Furthermore, the finite field configuration terms and polynomial expressions are standard representations of this type of maximum scattering linear set key, which can be directly read and used by those skilled in the art for subsequent geometric mapping and equivalence determination operations, making the implementation clear and explicit.

[0086] S120. Perform geometric mapping processing based on key parameters to obtain a set of geometric objects.

[0087] Among them, the set of geometric objects can refer to the set of geometric structures generated by the key polynomial through geometric mapping, that is, the unified computation carrier obtained by transforming the algebraic key structure into a geometric spatial structure.

[0088] Specifically, the set of geometric objects may include a first subspace and a second subspace.

[0089] The first subspace can refer to the two-dimensional vector subspace generated by the mapping of the first polynomial, corresponding to the geometric shape of the first key, and the second subspace can refer to the two-dimensional vector subspace generated by the mapping of the second polynomial, corresponding to the geometric shape of the second key.

[0090] For example, the first subspace may refer to the space defined by the first polynomial. The uniquely induced scattering subspace is specifically defined as follows: That is, all forms such as The second subspace is a subspace structure within a two-dimensional vector space formed by vectors. The uniquely induced scattering subspace is specifically defined as follows: That is, all forms such as The example is a subspace structure in a two-dimensional vector space composed of vectors. This example is only used to help understand the specific forms of the first and second subspaces. In actual applications, it is not limited to the above form.

[0091] It should be noted that the "induction" here refers to taking each element in the finite field as the x-coordinate input, substituting it into the first polynomial to calculate the corresponding y-coordinate output. Thus, the two-dimensional vectors with all input and output pairs together constitute a subspace set.

[0092] Optionally, geometric mapping processing is performed based on key parameters to obtain a set of geometric objects, including: mapping processing based on finite field parameters and a first polynomial to construct a first subspace; and mapping processing based on finite field parameters and a second polynomial to construct a second subspace.

[0093] It should be noted that since the mapping rules from all scattering polynomials to subspaces are completely uniform and independent of the number of terms or structure of the polynomial itself, the mapping processing of the first polynomial is the same as that of the second polynomial. The following explanation uses the construction of the first subspace as an example.

[0094] Specifically, the mapping process based on the finite field parameters and the first polynomial is used to construct the first subspace, including: firstly, based on the finite field parameters, all elements in the finite field are substituted into the first polynomial for calculation to obtain the target coordinate vector; then, vector culling and projection processing are performed based on the target coordinate vector to construct the first subspace.

[0095] The target coordinate vector can refer to the set of two-dimensional coordinate vectors generated by substituting the elements of the finite field into the scattering polynomial.

[0096] For example, first, all elements are determined based on the finite field parameters. Then, all elements are iterated through, and each element is used as an input value and substituted into the first polynomial to calculate the corresponding output value. Next, the input value of each element is paired with its corresponding output value to form a two-dimensional coordinate vector. All coordinate vectors together form the target coordinate vector.

[0097] Next, duplicate redundant vectors are removed from the target coordinate vector, and all distinct coordinate vectors are retained, thus completing the process of removing redundant terms.

[0098] Finally, projective processing is performed and a subspace is constructed. Projective normalization is performed on the non-zero vectors in the target coordinate vector. Based on the rule that all proportional vectors correspond to the same projective point, each non-zero vector is labeled with its corresponding projective point number, establishing a mapping relationship between the subspace vectors and the subsequent linear set. Finally, all retained coordinate vectors are combined into a two-dimensional vector subspace to construct the first subspace. For example, a quartic scattering polynomial is used as the first polynomial for the mapping process. Here, it is assumed that the first polynomial has a quartic structure. Its expression is as follows: Where h is the coefficient control parameter; t is the dimension control parameter, used to determine the dimension size of the finite field extension; s is the exponential step size parameter; and q is the odd prime power corresponding to the finite field.

[0099] It should be noted that the constraint condition of the first polynomial is... The two constraints guarantee that the polynomial can generate the maximum scattering linear set that meets the requirements.

[0100] Next, substitute the values ​​into the calculation to generate the target coordinate vector. First, the extended field dimension can be determined based on the finite field parameters. , to obtain Finite field of elements Then, iterate through all elements in the finite field, substituting each element as the input value x into the above four-term equation to calculate the corresponding output value. Each input value x is paired with its corresponding output value f(x) to form a two-dimensional coordinate vector. All coordinate vectors together form the target coordinate vector set.

[0101] Then, vector culling is performed. Redundant vectors are removed from the target coordinate vectors, retaining only the distinct coordinate vectors. Since the four-term polynomial is a scattering polynomial, it satisfies the injective property, and all coordinate vectors are distinct. It is only necessary to confirm the existence of the zero vector to satisfy the subspace closure requirement.

[0102] Finally, projective processing is performed, and the preserved coordinate vectors are combined into a scattering subspace. This serves as the first subspace that has been constructed, and also provides the corresponding first maximally scattering linear set. .

[0103] By substituting each element to generate a coordinate vector, and then standardizing the subspace structure through projective processing, a precise and unified conversion from polynomial to subspace can be achieved, providing standardized input for subsequent geometric determination.

[0104] It should be noted that the detailed steps for mapping based on finite field parameters and the second polynomial to construct the second subspace can be found in the relevant description of mapping based on finite field parameters and the first polynomial to construct the first subspace, and will not be repeated here.

[0105] By using the unified geometric mapping process described above, the two key structures to be judged are placed under the same geometric framework, providing a comparable data basis for subsequent spatial feature equivalence determination, without the need to design differentiated preprocessing procedures for different polynomial structures.

[0106] S130. Based on the first subspace and the second subspace, perform equivalence determination of spatial geometric features to obtain equivalence determination results.

[0107] The equivalence determination result can refer to the Boolean determination result used to describe whether the first subspace and the second subspace satisfy the space coincidence condition.

[0108] The spatial coincidence condition can refer to the criterion that two subspaces can be completely overlapped through a second-order invertible linear transformation, corresponding to the structural equivalence standard at the geometric level. For example, if rotating, stretching, and shearing the first subspace makes all points of it completely correspond to those of the second subspace, then the two subspaces satisfy the spatial coincidence condition.

[0109] Understandably, in general scenarios, the projective linear equivalence of linear sets not only corresponds to the general linear equivalence of atomic spaces, but may also correspond to the general linear equivalence of adjoint subspaces. Therefore, it is necessary to introduce the adjoint structure of the second polynomial to participate in the determination.

[0110] In other words, the second polynomial Having an adjoint polynomial The adjoint polynomial can be regarded as the dual form of the second polynomial under linear trace operation. The two satisfy the trace symmetric identity, and the structure type of the adjoint polynomial is completely consistent with that of the original polynomial.

[0111] Furthermore, accompanied by polynomials There is a corresponding adjoint subspace The accompanying subspace can serve as a mirror geometry of the second subspace, and its construction method is the same as that of the first and second subspaces.

[0112] Optionally, equivalence determination of spatial geometric features is performed based on the first subspace and the second subspace to obtain equivalence determination results, including: performing general linear equivalence determination using the first subspace, the adjoint subspace, and the second subspace to obtain equivalence determination results.

[0113] Specifically, the corresponding adjoint polynomial can first be derived from the second polynomial, and the adjoint subspace can be generated using the same mapping rule. Then, it is determined whether a second-order invertible matrix exists. This makes it possible to utilize For the first subspace Transformed subspace obtained after performing linear transformation With the second subspace Satisfies general linear equivalence; or, the transformed subspace With the second subspace companion subspace The conditions must be met to satisfy general linear equivalence. If either of the above two conditions is met, the equivalence determination result is valid; if neither is met, the equivalence determination result is invalid.

[0114] Introducing the adjoint subspace to participate in the general linear equivalence determination can cover the equivalence scenarios of all non-pseudo-regular linear sets, ensuring the completeness and accuracy of the determination results in general scenarios.

[0115] S140. Based on the equivalence determination result, perform equivalence determination on the first maximum scattering linear set and the second maximum scattering linear set to obtain the linear set equivalence result, and determine the key analysis result based on the linear set equivalence result.

[0116] Among them, the linear set equivalence result can refer to the judgment result characterizing whether two maximum scattering linear sets satisfy the industry-standard equivalence criteria.

[0117] The key analysis result can refer to the conclusion that reflects whether the core structure of two sets of keys is essentially the same, that is, the final analysis result that can be used to describe whether the first key and the second key can be substituted for each other for encrypted communication.

[0118] Specifically, firstly, based on the subspace equivalence determination, it is determined whether two linear sets satisfy projective linear equivalence. Then, all Frobenius automorphisms in the finite field are traversed, and an automorphism transformation is performed on the second maximally scattering linear set to obtain a reconstructed linear set. Each reconstructed linear set is then evaluated to determine whether the first maximally scattering linear set and each reconstructed linear set satisfy projective linear equivalence. If at least one reconstructed linear set satisfies the condition, then the two maximally scattering linear sets are determined to satisfy projective semi-linear equivalence, yielding the final linear set equivalence result.

[0119] After obtaining the linear set equivalence result, the key analysis result can be determined based on the linear set equivalence result.

[0120] Specifically, the determination conclusion of the linear set equivalence result can be read. If two maximally scattering linear sets satisfy projective semi-linear equivalence, then the two key structures are determined to be essentially the same, and the analysis conclusion that they can be substituted for each other is output. If they do not satisfy the condition, then the two key structures are determined to be different, and the analysis conclusion that they cannot be substituted for each other is output.

[0121] For example, if the maximum scattering linear sets corresponding to the first and second polynomials are determined to be projectively semi-linearly equivalent, it indicates that the core scattering structures of the two keys are essentially the same and can be used interchangeably in encrypted communication scenarios. If they are determined not to be equivalent, then the security performance and encoding characteristics of the two keys are fundamentally different and they cannot be used interchangeably.

[0122] In the above implementation, by uniformly mapping the key polynomial to a geometric subspace, and then using the general linear equivalence determination of the subspace as the core, the projective semi-linear equivalence conclusion of the linear set is derived layer by layer, ultimately achieving automated analysis of key substitutability. The entire method does not require deriving algebraic equations separately for keys with different structures. The determination process is universal and has low computational complexity, enabling efficient and accurate equivalence analysis of maximally scattering linear set keys, providing standardized support for key screening and security assessment of encryption.

[0123] In some implementation methods, please refer to the appendix. Figure 2The first subspace, the adjoint subspace, and the second subspace are used to perform general linear equivalence determination to obtain equivalence determination results, including: S210. Use a second-order invertible matrix to perform a linear transformation on the first subspace to obtain the transformed subspace.

[0124] Among them, a second-order invertible matrix can refer to an invertible second-order square matrix that can perform linear transformations such as rotation, stretching, and shearing on a two-dimensional vector space, and is the computational carrier for realizing linear transformations of subspaces.

[0125] For example, a second-order invertible matrix can be represented as In the form of a diagonal matrix. The overall scaling and transformation of the corresponding subspace, and the exchange matrix. The transformation of the two coordinate dimensions of the corresponding subspace.

[0126] The transformed subspace can refer to the new subspace obtained by linear transformation of the first subspace using a second-order invertible matrix, that is, the new geometric shape after linear deformation of the original geometric structure.

[0127] Specifically, firstly, based on verification experience, a second-order invertible matrix containing unknowns can be constructed. Then, each two-dimensional coordinate vector in the first subspace is multiplied by this invertible matrix to obtain the transformed new coordinate vectors. Subsequently, all the transformed new coordinate vectors are integrated to obtain the transformed subspace.

[0128] For example, suppose the first subspace is Therefore, for the verification scenario based on the classical scattering structure of the first polynomial, we can choose a structure of the form... The target matrix (where b ≠ 0 is guaranteed to be invertible). Then, each vector in the first subspace... Multiplying by M yields the transformed vector. Ultimately, all the transformed vectors together form the transformed subspace.

[0129] S220. Based on the transformed subspace, the second subspace, and the adjoint subspace, perform spatial overlap judgment to obtain the equivalence judgment result.

[0130] Specifically, the spatial overlap judgment is performed based on the transformed subspace, the second subspace, and the adjoining subspace to obtain an equivalence judgment result, including: performing a spatial overlap judgment between the transformed subspace and the second subspace to obtain a first judgment result; performing a spatial overlap judgment between the transformed subspace and the adjoining subspace to obtain a second judgment result; and determining an equivalence judgment result based on the first judgment result and the second judgment result.

[0131] The first judgment result can refer to a Boolean judgment result used to characterize whether the transformed subspace and the second subspace coincide, that is, intermediate judgment data reflecting whether the two satisfy general linear equivalence (GL-equivalence). The second judgment result can refer to a Boolean judgment result used to characterize whether the accompanied subspace of the transformed subspace and the second subspace coincide, that is, intermediate judgment data reflecting whether the two satisfy general linear equivalence (GL-equivalence).

[0132] Space overlap judgment can refer to the judgment operation that verifies whether all vectors in two subspaces are completely corresponding.

[0133] Specifically, first, the complete vector sets of the two subspaces to be compared are extracted. Then, taking any one of the subspaces as the target subspace, each vector in the other subspace is checked to see if it exists in the target subspace. Simultaneously, the reverse process is performed to check if each vector in the target subspace exists in the other subspace. This completes the spatial overlap judgment operation.

[0134] It should be noted that the above-described vector-wise bidirectional verification method is a general spatial coincidence determination method, which can be adapted to the subspace corresponding to any scattering polynomial structure. The specific implementation is a common technique in this field and will not be elaborated here. If both polynomials to be determined are of the four-term type structure, spatial coincidence determination can be completed directly through parameter matching, without the need to perform vector-wise traversal verification, resulting in better determination efficiency.

[0135] For example, taking the case where both polynomials to be judged are of the tetranomial type, let's assume the first polynomial is... f For a four-term The second polynomial g For a four-term .in h These are the coefficient control parameters of the first polynomial; t For common dimension control parameters, is the exponential step size parameter of the first polynomial; k These are the coefficient control parameters for the second polynomial. Let be the exponential step size parameter of the second polynomial. Then parameter matching can be performed as follows: Specifically, four judgment conditions can be set to correspond to the overlapping cases under typical geometric deformations: (a) ,and .

[0136] The above expression means: when the first polynomial mold The result and the second polynomial mold The results are equal, and the parameters satisfy... When the above condition is met, the two corresponding subspaces can coincide only through basic linear transformations such as scaling and rotation. That is, if the above condition is met, the transformed subspace can be considered to coincide with the second subspace, and the first judgment result is true.

[0137] (b) ,and .

[0138] The above expression means: when the first polynomial mold The result and the second polynomial mold The results are equal, and the parameters satisfy... When the above equation is satisfied, the corresponding two subspaces can coincide after a mirror-type linear transformation. That is, if the above equation is satisfied, it can be considered that the subspace after the transformation coincides with the adjoint subspace of the second subspace, and the second judgment result is true.

[0139] (c) ( (even number), and .

[0140] The above expression means: when t is even, if the first polynomial mold The result and the second polynomial mold The results are equal, and the parameters satisfy... Then, the atomic spaces coincide under the corresponding half-cycle flip transformation. That is, if the above equation is satisfied, the transformed subspace can also be considered to coincide with the second subspace, and the first judgment result is true.

[0141] (d) ( (even number), and .

[0142] The above expression means: when t is even, if the first polynomial mold The result and the second polynomial mold The results are equal, and the parameters satisfy... Then, the corresponding half-cycle flip plus mirror transformation coincides, that is, under the condition of satisfying the above formula, it can be considered that the corresponding transformed subspace coincides with the companion subspace of the second subspace, and the second judgment result is true.

[0143] If any one of the above four conditions is true, it means that there exists a suitable second-order invertible matrix such that the transformed subspace coincides with the second subspace or the adjoint subspace, thus obtaining the equivalence determination result that the general linear equivalence of the first and second subspaces holds. If none of the four conditions are true, then the equivalence determination result is false.

[0144] By verifying the overlap between the atomic space and the adjoint subspace through dual-branch parallel verification, the general linear equivalence of all non-pseudo-regular linear sets in the general scenario can be fully covered, ensuring the comprehensiveness and accuracy of the judgment results.

[0145] Optionally, if both the first and second polynomials are of the four-term type structure, the equivalence determination based on the spatial geometric features of the first and second subspaces to obtain the equivalence determination result further includes: performing a linear transformation on the first subspace using a second-order invertible matrix to obtain the transformed subspace; and performing a spatial overlap determination between the transformed subspace and the second subspace to obtain the equivalence determination result.

[0146] It should be noted that if both the first and second polynomials are of the four-term type structure, then the second polynomial... The accompanying polynomial It also has a fixed simplified structure and is still a tetranomial of the same type, that is, the adjoint polynomial can be expressed as .

[0147] Combining the aforementioned criteria for general linear equivalence of subspaces (b), this parameter transformation naturally satisfies the matching requirement of criterion (b). Therefore, it can be deduced that the scattering subspace of this type of four-term scattering linear set is... Its accompanying subspace It always satisfies the general linear equivalence relation.

[0148] Furthermore, this characteristic is unique to this four-term structure, which eliminates the need to distinguish between the atomic space and the adjoint space in its equivalence determination, greatly simplifying the analysis process and ensuring the stability of the quadratic symmetric expansion structure when solving for the automorphism group.

[0149] Specifically, the steps of using a second-order invertible matrix to perform a linear transformation on the first subspace to obtain the transformed subspace are exactly the same as the transformation steps in S210 mentioned above, and will not be repeated here.

[0150] Furthermore, the spatial overlap determination in the quaternary polynomial scenario is as follows: for two polynomials that are both quaternary polynomials, the parameters of the first polynomial are directly extracted. h and With the parameters of the second polynomial k and Combined with common dimension parameters tThe verification can be performed to check whether the conditions in case (a) or case (c) (the case where only atomic spaces overlap) of the above four conditions are met.

[0151] In other words, since the companion subspace and the atomic space are always equivalent ( Without needing to verify the accompanying branches, as long as the atomic space coincidence condition is met, it can be directly determined that the two subspaces are generally linearly equivalent, and the equivalence determination result can be obtained.

[0152] The decision branches are simplified for the unique structure of the four-term form, and the generation and verification steps of the accompanying subspace are omitted, which further reduces the computational amount of equivalence determination in the four-term form scenario and improves the analysis efficiency of specific scenarios.

[0153] In the above implementation, by combining linear transformation of second-order invertible matrices with bi-branch space coincidence verification, the abstract problem of linear set equivalence is transformed into an intuitive subspace geometric coincidence determination, realizing the standardized determination of general linear equivalence in general scenarios, and providing a unified geometric judgment basis for subsequent linear set equivalence analysis.

[0154] In some implementation methods, please refer to the appendix. Figure 3 Based on the equivalence determination results, the equivalence determination of the first maximum scattering linear set and the second maximum scattering linear set is performed to obtain the linear set equivalence results, including: S310. If the equivalence determination result shows that the spatial coincidence condition is met, then the first maximum scattering linear set and the second maximum scattering linear set are determined to satisfy the projective linear equivalence relation.

[0155] Among them, the projective linear equivalence relation can refer to the equivalence relation that two linear sets in the projective space can completely overlap through the projective linear transformation, that is, the linear set level equivalence standard (PGL-equivalence) after removing the overall scaling redundancy.

[0156] Specifically, since the general linear equivalence of a subspace and the projective linear equivalence of a linear set are a necessary and sufficient correspondence, as long as the equivalence determination result of the subspace satisfies the space coincidence condition, it can be directly deduced that the two linear sets satisfy the projective linear equivalence relation. The derivation logic of this correspondence has been explained above and will not be repeated here.

[0157] For example, for a non-pseudo-regular maximally scattering linear set, the general linear equivalence of the subspace and the projective linear equivalence of the linear set satisfy a bidirectional reciprocal relationship: .

[0158] That is, as long as the first subspace and the second subspace satisfy general linear equivalence (space coincidence), it can be directly determined that the first maximum scattering linear set and the second maximum scattering linear set corresponding to them satisfy projective linear equivalence.

[0159] S320. The second maximum scattering linear set is transformed based on the finite field automorphism transformation to obtain the reconstructed linear set.

[0160] Among them, the finite field automorphism transformation can refer to the bijective isomorphism mapping from a finite field to itself, that is, the transformation operation that renames each element in the finite field without changing the operation rules.

[0161] The reconstructed linear set can refer to the new linear set obtained by the second maximum scattering linear set after the finite field automorphism transformation, that is, the new linear set form after the original linear set is renamed by the field elements.

[0162] Specifically, we can first traverse all automorphic transformations in the finite field, applying each automorphism sequentially to all points of the second maximally scattering linear set, performing automorphic mapping on the coordinate elements of each point. Reintegrating all the transformed points yields the reconstructed linear set corresponding to each automorphism.

[0163] For example, let the finite field automorphism be denoted as The second largest scattering linear set is Automorphism Acting on the second maximum scattering linear set At each coordinate element, the transformed point is obtained. All the transformed points together constitute the reconstructed linear set. .

[0164] S330. Compare the first maximum scattering linear set with the reconstructed linear set, and obtain the linear set equivalence result if the first maximum scattering linear set and the reconstructed linear set satisfy the projective linear equivalence relation.

[0165] It should be noted that, as mentioned earlier, projective semi-linear equivalence is a common standard in the industry for determining whether the maximum scattering linear sets are essentially the same. It is defined as the existence of finite field automorphisms such that the two transformed linear sets satisfy projective linear equivalence. Therefore, as long as any reconstructed linear set satisfies projective linear equivalence with the first maximum scattering linear set, it can be determined that the two original linear sets satisfy the final projective semi-linear equivalence relation.

[0166] Specifically, the first maximum scattering linear set can be projectively linearly equivalent to each of the reconstructed linear sets corresponding to each automorphism. If at least one reconstructed linear set satisfies the projective linear equivalence relation with the first maximum scattering linear set, then the two original maximum scattering linear sets are determined to satisfy the projective semi-linear equivalence relation, and this conclusion is taken as the final linear set equivalence result. If none of the reconstructed linear sets satisfy the condition, then the two sets are determined not to satisfy the projective semi-linear equivalence relation.

[0167] For example, taking the linear set corresponding to two four-term scattering polynomials as an example, let the first polynomial be... The corresponding linear set is the first maximum scattering linear set. Let the second polynomial be... The corresponding linear set is the second largest scattering linear set. .

[0168] There exists an automorphism in a finite field If any of the following four conditions are met, it can be determined that the corresponding reconstructed linear set is true. With the first maximum scattering linear set It satisfies projective linear equivalence.

[0169] (a) ,and .

[0170] The above equation corresponds to the automorphic scaling equivalence under basic step size matching. If the above equation is satisfied, it means that the linear set has been reconstructed. With the first maximum scattering linear set They can overlap after a basic transformation.

[0171] (b) ,and .

[0172] The above equation corresponds to the automorphic scaling equivalence under mirror step matching. If the above equation is satisfied, it means that the linear set has been reconstructed. With the first maximum scattering linear set They overlap after mirror transformation.

[0173] (c) (t is an even number), and .

[0174] The above equation corresponds to the automorphic scaling equivalence under half-cycle flip step size matching. If the above equation is satisfied, it means that the linear set has been reconstructed. With the first maximum scattering linear set They overlap after a half-cycle flip transformation.

[0175] (d) (t is an even number), and .

[0176] The above equation corresponds to the automorphic scaling equivalence under half-cycle flipping plus mirror step size matching. If the above equation is satisfied, it means that the linear set has been reconstructed. With the first maximum scattering linear set They coincide after a half-cycle flip and mirror transformation.

[0177] If any one of the above four conditions is true, it means that there exists a corresponding finite field automorphism such that the reconstructed linear set and the first maximally scattering linear set are projectively linearly equivalent. This yields the linear set equivalence result where the first maximally scattering linear set and the second maximally scattering linear set are projectively semi-linearly equivalent. If none of the four conditions are true under any of the automorphisms, then the equivalence result is that projectively semi-linear equivalence is not satisfied.

[0178] In the above implementation, by transforming subspace equivalence to projective linear equivalence, and combining the two-layer judgment logic of finite field automorphic traversal and reconstructed linear set verification, the industry standard equivalence judgment of the maximum scattering linear set can be fully realized. The process is clear and adaptable to all non-pseudo-regular linear set structures, providing an accurate judgment basis for key substitutability analysis.

[0179] The embodiments of this specification also provide a method for characterizing the automorphism group of a four-term linear set. When the equivalence results of the linear sets show that the first maximum scattering linear set and the second maximum scattering linear set satisfy the projective semilinear equivalence, the method further includes: determining the automorphism group of the first maximum scattering linear set using a first subspace.

[0180] In this context, an automorphism group refers to the group of all transformations that preserve the geometric structure of an object. The size of the group represents the degree of symmetry of the structure and is a key parameter for key and coding design.

[0181] Specifically, the general linear automorphism group of the first subspace is first determined. Then, a modulus centering process is performed on this general linear automorphism group to remove redundant transformations caused by global scaling. Finally, a second-order mirror-symmetric extension induced by the accompanying transformation is superimposed to obtain the projective linear automorphism group of the linear set. Then, finite-field automorphisms satisfying the parameter constraints are selected and superimposed onto the projective linear automorphism group, ultimately obtaining the projective semi-linear automorphism group of the first maximally scattering linear set. Furthermore, the order of the automorphism group can be directly calculated based on the final group structure.

[0182] For example, taking a four-term scattering polynomial as an example, let the first polynomial be... The corresponding linear set is the first maximum scattering linear set. Its corresponding scattering subspace is the first subspace. The specific solution process for its automorphism group can be described as follows: First, based on the structural symmetry of the four-term, according to the parameters... The parity of subspaces can be obtained in two cases to form the automorphism group. : (a) when When the number is even, the first subspace automorphism group Non-zero elements are generated, all in diagonal matrix form: in, For a finite field The non-zero elements in the matrix represent the differential scaling transformations of the two coordinate dimensions of the subspace for each diagonal matrix, and the overall structure of the subspace remains unchanged after the transformation.

[0183] (b) When When the trace is odd, the automorphism group of the subspace is generated by elements satisfying the trace condition, and all of them are in symmetric matrix form: in, For the equation In a finite field The roots within are used to control the coefficients of off-diagonal terms, ensuring that the structure of the subspace remains stable after the transformation.

[0184] Furthermore, based on the aforementioned As can be seen from the derivation of equation (2), since the quadratic subspace and its adjoint subspace always satisfy general linear equivalence, a second-order symmetric extension induced by the adjoint transformation can be introduced. First, the automorphism group of the subspace is processed by the modulus center, and proportionally repeated matrices are removed to obtain... Then superimpose second-order subgroups The first maximally scattering linear set can be obtained through a semi-direct product combination. The projective linear (PGL) automorphism group is given by: Among them, the second-order subgroup Similarly, according to parameters Determining the parity: (a) when When the number is even, the following formula applies: (b) When When the number is odd, the following formula applies: Next, by superimposing finite-field automorphisms that satisfy parameter constraints on the projective linear automorphism group, the final projective semilinear group can be obtained. -) Automorphism group: Furthermore, based on the above group structure, the total order of the automorphism group can be directly calculated: This order result It can directly reflect the symmetry of a linear set and can be used directly as an evaluation parameter for key security performance and coding error correction performance.

[0185] In the above implementation, relying on the existing geometric mapping and equivalence determination framework, the automorphism group characterization and order calculation of the four-term linear set can be completed directly and synchronously without the need to build an additional independent algebraic derivation system. It can quickly output the symmetry quantification parameters of the linear set, providing direct data support for the security assessment of encryption keys and the performance analysis of error correction codes.

[0186] This specification also provides a key analysis method, which includes the following steps: S402, Obtain key parameters.

[0187] The key parameters include finite field parameters, a first key parameter, and a second key parameter; the first key parameter includes a first polynomial describing the mathematical properties of the first maximally scattering linear set; and the second key parameter includes a second polynomial describing the mathematical properties of the second maximally scattering linear set.

[0188] S404. Based on the finite field parameters, substitute all elements in the finite field into the first polynomial for calculation to obtain the target coordinate vector.

[0189] S406. Based on the target coordinate vector, perform vector culling and projection processing to construct the first subspace.

[0190] S408. The mapping process for the first polynomial is the same as that for the second polynomial. The mapping process is based on the finite field parameters and the second polynomial to construct the second subspace.

[0191] The second polynomial has an adjoint polynomial, and the adjoint polynomial corresponds to an adjoint subspace.

[0192] S410. Use a second-order invertible matrix to perform a linear transformation on the first subspace to obtain the transformed subspace.

[0193] If both the first and second polynomials are of the four-term type structure, then jump to step S411; if either the first or second polynomial is not of the four-term type structure, then jump to step S412.

[0194] S411. If both the first and second polynomials are of the four-term type structure, then the transformed subspace and the second subspace are judged to be spatially coincident, and the equivalence judgment result is obtained, and then the process jumps to step S420.

[0195] S412. Perform a spatial overlap judgment between the transformed subspace and the second subspace to obtain the first judgment result; S414. Perform a spatial overlap judgment between the transformed subspace and the adjoint subspace to obtain the second judgment result; S416. Determine the equivalence determination result based on the first judgment result and the second judgment result.

[0196] The equivalence determination result is used to describe whether the first subspace and the second subspace satisfy the spatial coincidence condition.

[0197] S418. If the equivalence determination result shows that the spatial coincidence condition is met, then the first maximum scattering linear set and the second maximum scattering linear set are determined to satisfy the projective linear equivalence relation.

[0198] S420. The second maximum scattering linear set is transformed based on the finite field automorphism transformation to obtain the reconstructed linear set.

[0199] S422. Compare the first maximum scattering linear set with the reconstructed linear set, and obtain the linear set equivalence result if the first maximum scattering linear set and the reconstructed linear set satisfy the projective linear equivalence relation.

[0200] S424. Determine the key analysis result based on the linear set equivalence result.

[0201] The key analysis results are used to describe whether the first key and the second key can be substituted for each other for encrypted communication.

[0202] It should be understood that although the steps in the flowchart above are shown sequentially as indicated by the arrows, these steps are not necessarily executed in the order indicated by the arrows. Unless explicitly stated herein, there is no strict order restriction on the execution of these steps, and they can be executed in other orders. Moreover, at least some steps in the flowchart above may include multiple steps or stages, which are not necessarily completed at the same time, but can be executed at different times. The execution order of these steps or stages is not necessarily sequential, but can be performed alternately or in turn with other steps or at least some of the steps or stages in other steps.

[0203] Please see Figure 4 , Figure 4 This is a schematic diagram of the structure of a computer device provided in an embodiment of this application, such as... Figure 4As shown, the computer device includes one or more processors 10, memory 20, and interfaces for connecting the components, including high-speed interfaces and low-speed interfaces. The components communicate with each other via different buses and can be mounted on a common motherboard or otherwise installed as needed. The processor can process instructions executed within the computer device, including instructions stored in or on memory to display graphical information of a GUI on an external input / output device (such as a display device coupled to the interface). In some alternative implementations, multiple processors and / or multiple buses can be used with multiple memories and multiple memory modules, if desired. Similarly, multiple computer devices can be connected, each providing some of the necessary operations. Figure 4 Take a processor 10 as an example.

[0204] Processor 10 may be a central processing unit, a network processor, or a combination thereof. Processor 10 may further include a hardware chip. The hardware chip may be an application-specific integrated circuit (ASIC), a programmable logic device (PLD), or a combination thereof. The programmable logic device may be a complex programmable logic device (CAMP), a field-programmable gate array (FPGA), a general-purpose array logic (GDA), or any combination thereof.

[0205] The memory 20 stores instructions executable by at least one processor 10 to cause the at least one processor 10 to perform the method shown in the above embodiments.

[0206] The memory 20 may include a program storage area and a data storage area. The program storage area may store the operating system and applications required for at least one function; the data storage area may store data created based on the use of the computer device. Furthermore, the memory 20 may include high-speed random access memory and may also include non-transitory memory, such as at least one disk storage device, flash memory device, or other non-transitory solid-state storage device. In some alternative embodiments, the memory 20 may optionally include memory remotely located relative to the processor 10, and these remote memories may be connected to the computer device via a network. Examples of such networks include, but are not limited to, the Internet, intranets, local area networks, mobile communication networks, and combinations thereof.

[0207] The memory 20 may include volatile memory, such as random access memory; the memory may also include non-volatile memory, such as flash memory, hard disk or solid-state drive; the memory 20 may also include a combination of the above types of memory.

[0208] The computer device also includes an input device 30 and an output device 40. The processor 10, memory 20, input device 30, and output device 40 can be connected via a bus or other means. Figure 4 Taking the example of a connection between China and Israel via a bus.

[0209] Input device 30 can receive input numerical or character information, and generate key signal inputs related to user settings and function control of the computer device, such as a touchscreen, keypad, mouse, trackpad, touchpad, joystick, one or more mouse buttons, trackball, joystick, etc. Output device 40 may include display devices, auxiliary lighting devices (e.g., LEDs), and haptic feedback devices (e.g., vibration motors). The aforementioned display devices include, but are not limited to, liquid crystal displays, light-emitting diodes, displays, and plasma displays. In some alternative embodiments, the display device may be a touchscreen.

[0210] This application also provides a computer-readable storage medium. The methods described in this application can be implemented in hardware or firmware, or implemented as recordable on a storage medium, or implemented as computer code downloaded over a network and originally stored on a remote storage medium or a non-transitory machine-readable storage medium and subsequently stored on a local storage medium. Thus, the methods described herein can be processed by software stored on a storage medium using a general-purpose computer, a dedicated processor, or programmable or dedicated hardware. The storage medium can be a magnetic disk, optical disk, read-only memory, random access memory, flash memory, hard disk, or solid-state drive, etc.; further, the storage medium can also include combinations of the above types of memory. It is understood that computers, processors, microprocessor controllers, or programmable hardware include storage components capable of storing or receiving software or computer code. When the software or computer code is accessed and executed by the computer, processor, or hardware, the methods shown in the above embodiments are implemented.

[0211] The devices, modules, or units described in the above embodiments can be implemented by computer chips or entities, or by products with certain functions. A typical implementation device is a computer. Specifically, a computer can be, for example, a personal computer, laptop computer, cellular phone, camera phone, smartphone, personal digital assistant, media player, navigation device, email device, game console, tablet computer, wearable device, or any combination of these devices.

[0212] For ease of description, the above devices are described separately by function as various units. Of course, in implementing this application, the functions of each unit can be implemented in one or more software and / or hardware.

[0213] Those skilled in the art will understand that embodiments of this application can be provided as methods, apparatus, or computer program products. Therefore, this application can take the form of a completely hardware embodiment, a completely software embodiment, or an embodiment combining software and hardware aspects. Furthermore, this application can take the form of a computer program product embodied on one or more computer-usable storage media (including but not limited to disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.

[0214] This application is described with reference to flowchart illustrations and / or block diagrams of methods, apparatus, and computer program products according to embodiments of this application. It will be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, special-purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, generate instructions for implementing the flowchart... Figure 1 One or more processes and / or boxes Figure 1 A device that provides the functions specified in one or more boxes.

[0215] These computer program instructions may also be stored in a computer-readable storage medium that can direct a computer or other programmable data processing device to function in a particular manner, such that the instructions stored in the computer-readable storage medium produce an article of manufacture including instruction means, which are implemented in a process Figure 1 One or more processes and / or boxes Figure 1 The function specified in one or more boxes.

[0216] These computer program instructions may also be loaded onto a computer or other programmable data processing equipment to cause a series of operational steps to be performed on the computer or other programmable equipment to produce a computer-implemented process, thereby providing instructions that execute on the computer or other programmable equipment for implementing the process. Figure 1 One or more processes and / or boxes Figure 1 The steps of the function specified in one or more boxes.

[0217] It should also be noted that the terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such a process, method, article, or apparatus. Without further limitation, an element defined by the phrase "comprising one..." does not exclude the presence of other identical elements in the process, method, article, or apparatus that includes said element.

[0218] The various embodiments in this specification are described in a progressive manner. Similar or identical parts between embodiments can be referred to mutually. Each embodiment focuses on describing the differences from other embodiments. Because some embodiments are described relatively simply, relevant parts can be referred to the descriptions of similar embodiments.

[0219] The above description is merely an embodiment of this application and is not intended to limit the scope of this application. Various modifications and variations can be made to this application by those skilled in the art. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of this application should be included within the scope of the claims of this application.

[0220] Although embodiments of this application have been described in conjunction with the accompanying drawings, those skilled in the art can make various modifications and variations without departing from the spirit and scope of this application, and such modifications and variations all fall within the scope defined by the appended claims.

Claims

1. A key analysis method, characterized in that, The method includes: Obtain key parameters; wherein, the key parameters include finite field parameters, a first key parameter, and a second key parameter; the first key parameter includes a first polynomial describing the mathematical properties of a first maximally scattering linear set; the second key parameter includes a second polynomial describing the mathematical properties of a second maximally scattering linear set; Geometric mapping is performed based on the key parameters to obtain a set of geometric objects; wherein the set of geometric objects includes a first subspace and a second subspace; An equivalence determination of spatial geometric features is performed based on the first subspace and the second subspace to obtain an equivalence determination result; wherein, the equivalence determination result is used to describe whether the first subspace and the second subspace satisfy the spatial coincidence condition; Based on the equivalence determination result, the first maximum scattering linear set and the second maximum scattering linear set are evaluated for equivalence to obtain the linear set equivalence result, and the key analysis result is determined based on the linear set equivalence result; wherein, the key analysis result is used to describe whether the first key and the second key can be substituted for each other for encrypted communication.

2. The method according to claim 1, characterized in that, The geometric mapping process based on the key parameters to obtain a set of geometric objects includes: The first subspace is constructed by mapping based on the finite field parameters and the first polynomial. The second subspace is constructed by mapping based on the finite field parameters and the second polynomial.

3. The method according to claim 2, characterized in that, The mapping process for the first polynomial is the same as that for the mapping process for the second polynomial. The mapping process based on the finite field parameters and the first polynomial to construct the first subspace includes: Based on the finite field parameters, all elements in the finite field are substituted into the first polynomial for calculation to obtain the target coordinate vector; The first subspace is constructed by performing vector culling and projective processing based on the target coordinate vector.

4. The method according to claim 1, characterized in that, The second polynomial has an adjoint polynomial; the adjoint polynomial corresponds to an adjoint subspace; The equivalence determination based on the spatial geometric features of the first subspace and the second subspace to obtain the equivalence determination result includes: A general linear equivalence determination is performed using the first subspace, the accompanying subspace, and the second subspace to obtain the equivalence determination result.

5. The method according to claim 4, characterized in that, The step of performing a general linear equivalence determination using the first subspace, the adjoint subspace, and the second subspace to obtain the equivalence determination result includes: The first subspace is linearly transformed using a second-order invertible matrix to obtain the transformed subspace. Based on the transformed subspace, the second subspace, and the accompanying subspace, a spatial overlap judgment is performed to obtain the equivalence judgment result.

6. The method according to claim 5, characterized in that, The step of performing spatial overlap determination based on the transformed subspace, the second subspace, and the accompanying subspace to obtain the equivalence determination result includes: The transformed subspace is compared with the second subspace to determine spatial overlap, and a first judgment result is obtained. The transformed subspace and the accompanying subspace are spatially overlapped to obtain a second judgment result. The equivalence determination result is determined based on the first determination result and the second determination result.

7. The method according to claim 1, characterized in that, If both the first polynomial and the second polynomial are of the four-term type structure, the equivalence determination based on the spatial geometric features of the first subspace and the second subspace to obtain the equivalence determination result further includes: The first subspace is linearly transformed using a second-order invertible matrix to obtain the transformed subspace. The transformed subspace and the second subspace are compared to determine spatial overlap, and the equivalence determination result is obtained.

8. The method according to claim 1, characterized in that, The step of performing equivalence determination on the first maximum scattering linear set and the second maximum scattering linear set based on the equivalence determination result to obtain the linear set equivalence result includes: If the equivalence determination result indicates that the spatial coincidence condition is met, then the first maximum scattering linear set and the second maximum scattering linear set are determined to satisfy the projective linear equivalence relation. The second maximum scattering linear set is transformed based on the finite field automorphism transformation to obtain the reconstructed linear set; The first maximum scattering linear set is compared with the reconstructed linear set, and the linear set equivalence result is obtained if the first maximum scattering linear set and the reconstructed linear set satisfy the projective linear equivalence relation.

9. A computer device, characterized in that, include: A memory and a processor, the memory and the processor being communicatively connected to each other, the memory storing computer instructions, the processor executing the computer instructions to perform the method of any one of claims 1 to 8.

10. A computer-readable storage medium, characterized in that, The computer-readable storage medium stores computer instructions for causing a computer to perform the method of any one of claims 1 to 8.

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