Signature method, device, system and equipment based on GR-SIS problem

Abstract

The invention discloses a signature method, device, system and equipment based on a GR-SIS problem, and belongs to the technical field of information security, and the method comprises the following steps: step 1, based on dihedral non-exchangeable group ring construction, defining a group ring structure and a sampling rule, and generating a key pair containing a public key matrix and a private key vector; 2, mapping a target message into a group ring element through a hash function, and generating a signature vector in combination with a private key and lattice sampling; and 3, verifying the legality of the message by utilizing the linear operation of the public key matrix and the signature vector. According to the method, the signature scheme is expanded to the dihedral non-exchangeable group ring, so that the long-term security is provided, and the attack of a quantum computer can be resisted. Compared with a previous scheme based on ring SIS, the method provided by the invention has very high efficiency. Meanwhile, many attacks for ideal lattice passwords proposed at present can be avoided by using the construction based on the non-swapping group, so that the method has potential higher security.

Description

Technical Field

[0001] The present invention relates to a signature method, device, system and equipment based on the GR-SIS problem, and belongs to the technical field of information security. Background Art

[0002] Digital signatures, due to their advantages of authentication, integrity, and non-repudiation, have become a key technology for maintaining internet security. Digital signature design frameworks primarily include those based on one-way functions, trapdoor one-way functions, and exchange one-way functions. Digital signatures provide long-term authenticity and integrity, and support data non-repudiation. Digital signatures are widely used in identification and authentication protocols, such as software downloads. Therefore, the security of digital signature algorithms is crucial to maintaining IT security.

[0003] With the advancement of quantum computing technology, relevant operations have theoretically shifted from exponential to polynomial levels, potentially enabling quantum computers to break existing public-key cryptography. To address emerging threats, post-quantum cryptography has emerged, aiming to investigate the security of cryptographic algorithms in a quantum environment. Lattice-based cryptography is one of the most attractive areas within post-quantum cryptography. Due to its advantages, such as worst-case to average-case hardness reduction, high implementation efficiency, and support for a wide range of cryptographic functions, it has been widely researched and applied over the past decade. Post-quantum signatures include lattice-based signatures, hash-based signatures, encoding-based signatures, and multivariate-based signatures. Lattice-based signature schemes can be broadly divided into two categories: those based on hash-and-sign methods, such as the GPV scheme, and those based on the Fiat-Shamir structure, such as the BG, GLP, and BLISS schemes. GPV is a provably secure framework for deriving hash-and-sign lattice-based signature schemes. GGH and NTRUsign are two of the earliest proposed lattice-based signature schemes, but neither has been proven secure. Implementing lattice-based digital signatures with Gaussian sampling is challenging due to the use of Gaussian samplers. In addition, the use of hash functions and rejection sampling techniques makes digital signatures more complex.

[0004] Existing lattice-based signature schemes are primarily based on the Small Integer Solution (SIS) problem, particularly the Ring-SIS problem (Ring-SIS), which leverages the structure of algebraic integer rings to improve computational efficiency. However, in recent years, researchers have developed more efficient algorithms for solving ideal lattices of algebraic integer rings, threatening the security of traditional Ring-SIS schemes. To balance efficiency and security, this paper proposes a signature scheme based on the GR-SIS problem for noncommutative group rings. Summary of the Invention

[0005] In order to solve the above problems, the present invention proposes a signature method, device, system and equipment based on the GR-SIS problem, which can utilize the non-commutative characteristics of the group ring structure to circumvent existing attack algorithms while maintaining the computational efficiency of the ring structure lattice.

[0006] The technical solution adopted by the present invention to solve the technical problem is: In a first aspect, an embodiment of the present invention provides a signature method based on the GR-SIS problem, comprising the following steps: Step 1: Based on the dihedral non-commutative group ring construction, define the group ring structure and sampling rules, and generate a key pair consisting of a public key matrix and a private key vector; Step 2: Map the target message into a group ring element through a hash function, and generate a signature vector by combining the private key and grid sampling; Step 3: Use linear operations between the public key matrix and the signature vector to verify the legitimacy of the message.

[0007] As a possible implementation of this embodiment, step 1 includes: Step 11, define the dihedral group For collection , satisfying the group operation rules and , t is the rotation generator, s is the flip generator, i, j are the index parameters of the group elements, and n is the dihedral group The number of sides of the corresponding regular polygon; Step 12: Construct a polynomial group ring ,in is a finite field modulo q, q is an odd prime number, is a dihedral group Polynomial group rings over ; Step 13: Select the hash function H and define the primitive vector , where each element is A constant polynomial on , k is the primitive vector The number of components; Step 14, from the grid In the example, the parameter pair is generated by Gaussian sampling. , construct the public key A= , the private key is ,in obey The upper deviation is Gaussian distribution , It's a ring Medium elements, is the private key component, is the error term, is the Gaussian parameter, u is the ring The target element in .

[0008] As a possible implementation of this embodiment, step 2 includes: Step 21: Target message Calculating the hash value ; Step 22, in the grid Perform pre-image sampling in and obtain vector , making ; Step 23: Construct signature vector .

[0009] As a possible implementation of this embodiment, step 3 includes: Calculate the dot product of the transpose of the public key matrix and the signature vector , verify whether it satisfies , if the verification formula is established, the signature is valid.

[0010] As a possible implementation of this embodiment, verify The process of establishment is: .

[0011] As a possible implementation of this embodiment, the dihedral group The non-commutative property of ,Depend on It can be deduced that , that is, the order of multiplication of group elements affects the operation results.

[0012] As a possible implementation of this embodiment, the grid The dimension is k, and its basis matrix is ​​composed of primitive vectors Null space generation, sampling solution The norm satisfies ,in A preset small constant.

[0013] In a second aspect, an embodiment of the present invention provides a signature device based on the GR-SIS problem, including: The key generation module is used to define the group ring structure and sampling rules based on the dihedral non-commutative group ring construction, and generate a key pair consisting of a public key matrix and a private key vector; The signature generation module is used to map the target message into a group ring element through a hash function, and generate a signature vector by combining the private key and lattice sampling; The signature verification module is used to verify the legitimacy of the message by using linear operations between the public key matrix and the signature vector.

[0014] In a third aspect, an embodiment of the present invention provides a signature system based on the GR-SIS problem, including a sender and a receiver. The recipient is responsible for generating the public key and private key , where the public key Contains primitive vectors , random polynomial and trapdoor parameters ; The sender uses the public key A to perform hash calculation and grid sampling on the message m to generate a signature z; The recipient uses the public key A and signature z to verify the Is it true to verify the message Legitimacy, among which For the preset modulus parameter, determine the finite field and group ring The value range of the elements.

[0015] In a fourth aspect, an embodiment of the present invention provides an electronic device, comprising a processor, a memory and a bus, wherein the memory stores machine-readable instructions executable by the processor. When the electronic device is running, the processor communicates with the memory through the bus, and the processor executes the machine-readable instructions to perform any steps of the above-mentioned signature method based on the GR-SIS problem.

[0016] In a fifth aspect, an embodiment of the present invention provides a storage medium on which a computer program is stored. When the computer program is run by a processor, the steps of any of the above-mentioned signing methods based on the GR-SIS problem are executed.

[0017] The technical solution of the embodiment of the present invention can have the following beneficial effects: This paper extends the signature scheme to dihedral noncommutative group rings, proposing a signature method based on GR-SIS that not only provides long-term security but is also resistant to attacks by quantum computers. Compared to previous schemes based on ring SIS, this method is highly efficient. Furthermore, the use of a construction based on noncommutative groups circumvents many currently proposed attacks against ideal lattice cryptography, potentially offering even higher security. BRIEF DESCRIPTION OF THE DRAWINGS

[0018] Figure 1 is a flowchart of a signature method based on a GR-SIS problem according to an exemplary embodiment; Figure 2 is a schematic diagram showing a signature device based on the GR-SIS problem according to an exemplary embodiment; Figure 3Flowchart of signing using the GR-SIS problem-based signature system of the present invention. DETAILED DESCRIPTION

[0019] The present invention will be further described below with reference to the accompanying drawings and embodiments: In order to clearly illustrate the technical features of this solution, the present invention is described in detail below through specific implementation methods and in conjunction with the accompanying drawings. The disclosure below provides many different embodiments or examples for realizing different structures of the present invention. In order to simplify the disclosure of the present invention, the components and settings of specific examples are described below. In addition, the present invention may repeat reference numbers and / or letters in different examples. This repetition is for the purpose of simplicity and clarity and does not itself indicate the relationship between the various embodiments and / or settings discussed. It should be noted that the components illustrated in the accompanying drawings are not necessarily drawn to scale. The present invention omits descriptions of well-known components and processing technologies and processes to avoid unnecessary limitations on the present invention.

[0020] To maintain the efficiency of the ring structure while resisting attacks on the difficult problem of ideal lattices of algebraic integer rings, this paper considers using group rings to construct the group ring small integer solution problem, particularly when the underlying basic group is a non-commutative group. This paper proposes a signature scheme based on the group ring small integer solution problem, which maintains the efficiency of previous lattice schemes while resisting currently proposed attack algorithms on ideal lattices of algebraic integer rings, providing higher security.

[0021] This paper proposes a group ring small integer solution (GR-SIS) problem. In particular, when the fundamental group is a cyclic group, the construction on the cyclotomic integer ring can be considered a special case. When the underlying structure uses a non-commutative group, a new instance of the difficult problem is obtained. Furthermore, based on a dihedral non-commutative group ring, this paper proposes a signature scheme based on GR-SIS. The invention involves three algorithms: key generation, signature, and verification; and involves two participants: a sender and a receiver.

[0022] First, we give the definitions of group rings and dihedral groups. First, we give the definitions of group rings and dihedral groups. Group rings: Given a finite group and commutative rings (n is the order of the group,) group ring The elements in are the form and . Dihedral group: one order dihedral group, using Represents a set ,satisfy Then there is , Right now , which has the property of being non-commutative. Define the polynomial group ring .

[0023] Constructing a group ring with small integer solutions: Given a dihedral group , construct group ring ,in, is the modulus, It's a model Finite field. Given and ,in is a real number, find the non-zero vector satisfy ,in The present invention uses a dihedral group ring to specifically give an example of the GR-SIS problem and constructs a signature scheme based on GR-SIS.

[0024] like Figure 1 As shown, an embodiment of the present invention provides a signature method based on the GR-SIS problem, including the following steps: Step 1: Based on the dihedral non-commutative group ring construction, define the group ring structure and sampling rules, and generate a key pair consisting of a public key matrix and a private key vector; Step 2: Map the target message into a group ring element through a hash function, and generate a signature vector by combining the private key and grid sampling; Step 3: Use linear operations between the public key matrix and the signature vector to verify the legitimacy of the message.

[0025] This paper reduces the security of traditional lattice signatures to the problem of solving small integers in a group ring by constructing a difficult problem instance on a noncommutative group ring. The key generation phase utilizes the noncommutative properties of the dihedral group to construct a special algebraic structure. Preimage sampling is used during signature generation to achieve efficient signature generation, and verification is performed through matrix-vector operations, providing a theoretical basis for resisting quantum attacks.

[0026] As a possible implementation of this embodiment, step 1 includes: Step 11, define the dihedral group For collection , satisfying the group operation rules and , t is the rotation generator, s is the flip generator, i, j are the index parameters of the group elements, and n is the dihedral group The number of sides of the corresponding regular polygon; Step 12: Construct a polynomial group ring ,in is a finite field modulo q, q is an odd prime number, is a dihedral group Polynomial group rings over ; Step 13: Select the hash function H and define the primitive vector , where each element is A constant polynomial on , k is the primitive vector The number of components; Step 14, from the grid In the example, the parameter pair is generated by Gaussian sampling. , construct the public key A= , the private key is ,in obey The upper deviation is Gaussian distribution , It's a ring Medium elements, is the private key component, is the error term, is the Gaussian parameter, u is the ring The target element in .

[0027] As a possible implementation of this embodiment, the dihedral group The non-commutative property of ,Depend on It can be deduced that , that is, the order of multiplication of group elements affects the operation results.

[0028] The non-commutativity of the dihedral group (st≠ts) destroys the symmetry of the traditional algebraic integer ring ideal lattice, making it impossible for algebraic attacks on the ideal lattice (such as the SVP solution algorithm) to use the ring structure to quickly decompose difficult problems, thereby improving the anti-attack capability of the signature scheme; in the group ring construction, the quotient ring operation is used to Constraining the degree of polynomials improves computational efficiency; Gaussian sampling ensures the randomness and small norm of private key parameters, providing a foundation for signature security; the construction of the public key matrix incorporates random elements , primitive vector With private key parameters ,The private key information is hidden by linear combination, and the matrix structure design enables the message hash value u to be quickly restored during signature verification by performing the dot product operation of the matrix transpose and the signature vector, thus achieving efficient verification.

[0029] As a possible implementation of this embodiment, step 2 includes: Step 21: Target message Calculating the hash value ; Step 22, in the grid Perform pre-image sampling in and obtain vector , making ; Step 23: Construct signature vector .

[0030] The original image sampling uses the trapdoor property of the lattice to ensure the existence of a small norm solution x; the signature vector is generated by a linear combination of the private key parameters and the sampling vector, and the first two components are and The last k components are the sampling vectors themselves, forming a compact structure containing message binding information and private key authentication information.

[0031] As a possible implementation of this embodiment, step 3 includes: Calculate the dot product of the transpose of the public key matrix and the signature vector , verify whether it satisfies , if the verification formula is established, the signature is valid.

[0032] As a possible implementation of this embodiment, verify The process of establishment is: .

[0033] This equation uses the property of algebraic elimination to retain only the terms related to the primitive vector, ensuring the correctness and simplicity of the verification.

[0034] As a possible implementation of this embodiment, the grid The dimension is k, and its basis matrix is ​​composed of primitive vectors Null space generation, sampling solution The norm satisfies ,in A preset small constant.

[0035] The lattice dimension k determines the security parameter of the signature, and is usually set to k ≥ 256 to resist quantum attacks. Ensure the compactness of the signature vector to avoid verification failure due to excessive norm, while ensuring the worst-case to average-case reduction of difficult problems.

[0036] like Figure 2 As shown, an embodiment of the present invention provides a signature device based on the GR-SIS problem, including: The key generation module is used to define the group ring structure and sampling rules based on the dihedral non-commutative group ring construction, and generate a key pair consisting of a public key matrix and a private key vector; The signature generation module is used to map the target message into a group ring element through a hash function, and generate a signature vector by combining the private key and lattice sampling; The signature verification module is used to verify the legitimacy of the message by using linear operations between the public key matrix and the signature vector.

[0037] It should be noted that the above three modules correspond to three steps in a signature method based on the GR-SIS problem, and the examples and application scenarios implemented by the above modules and the corresponding steps are the same.

[0038] like Figure 3 As shown, an embodiment of the present invention provides a signature system based on the GR-SIS problem, including a sender and a receiver, The recipient is responsible for generating the public key and private key , where the public key Contains primitive vectors , random polynomial and trapdoor parameters ; The sender uses the public key A to perform hash calculation and grid sampling on the message m to generate a signature z; The recipient uses the public key A and signature z to verify the Is it true to verify the message Legitimacy, among which For the preset modulus parameter, determine the finite field and group ring The value range of the elements.

[0039] like Figure 3 As shown, the specific implementation process of the GR-SIS signature based on the present invention is as follows.

[0040] Step 1: Key generation.

[0041] Randomly select a hash function ,make is the primitive vector, and each term is considered as a constant polynomial. Define a sampling function , from the grid mid-sampling , making .for Instances ,in , , express The upper deviation is The output public key is , the private key is .

[0042] Step 2: Sign.

[0043] Target message ,make In order to respond to the news Sign, Perform original image sampling to obtain , the signature is .

[0044] Step 3: Verify.

[0045] Verification formula Is it established? If the signature is correct, then: .

[0046] The present invention not only avoids the existing attack algorithm by utilizing the non-commutative property of the group ring structure, but also maintains the computational efficiency of the ring structure lattice.

[0047] An embodiment of the present invention provides an electronic device, including a processor, a memory and a bus, wherein the memory stores machine-readable instructions executable by the processor. When the device is running, the processor and the memory communicate through the bus, and the processor executes the machine-readable instructions to perform any steps of the above-mentioned signature method based on the GR-SIS problem.

[0048] Specifically, the above-mentioned memory and processor can be general-purpose memory and processor, which are not specifically limited here. When the processor runs the computer program stored in the memory, the above-mentioned signature method based on the GR-SIS problem can be executed.

[0049] Those skilled in the art will understand that the structure of the electronic device does not constitute a limitation of the electronic device, and may include more or fewer components than shown in the figure, or combine certain components, or separate certain components, or arrange the components differently.

[0050] In some embodiments, the electronic device may also include a touch screen that can be used to display a graphical user interface (e.g., an application startup interface) and receive user operations on the graphical user interface (e.g., application startup operations). Specifically, the touch screen may include a display panel and a touch panel. The display panel may be configured in the form of an LCD (Liquid Crystal Display), an OLED (Organic Light-Emitting Diode), or other devices. The touch panel can detect contact or non-contact operations performed by the user on or near it and generate pre-set operation instructions. For example, the user can use a finger, stylus, or any other suitable object or accessory to perform operations on or near the touch panel. Furthermore, the touch panel may include two parts: a touch detection device and a touch controller. The touch detection device detects the user's touch position and posture, detects signals generated by the touch operation, and transmits the signals to the touch controller. The touch controller receives the touch information from the touch detection device, converts it into information that can be processed by the processor, and then transmits it to the processor. The touch controller can also receive and execute commands from the processor. In addition, the touch panel can be implemented using various types such as resistive, capacitive, infrared, and surface acoustic wave, and any technology developed in the future can also be used to implement the touch panel. Furthermore, the touch panel can cover the display panel, and the user can operate on or near the touch panel covered on the display panel according to the graphical user interface displayed on the display panel. After the touch panel detects the operation on or near it, it transmits it to the processor to determine the user input, and then the processor provides a corresponding visual output on the display panel in response to the user input. In addition, the touch panel and the display panel can be implemented as two independent components or integrated.

[0051] Corresponding to the above-mentioned application startup method, an embodiment of the present invention further provides a storage medium on which a computer program is stored. When the computer program is run by a processor, the steps of any of the above-mentioned signing methods based on the GR-SIS problem are executed.

[0052] The startup device of the application provided in the embodiment of the present application can be specific hardware on the device or software or firmware installed on the device. The implementation principle and technical effects of the device provided in the embodiment of the present application are the same as those of the aforementioned method embodiment. For the sake of brief description, for any part not mentioned in the device embodiment, reference can be made to the corresponding content in the aforementioned method embodiment. Those skilled in the art can clearly understand that for the convenience and brevity of description, the specific working processes of the systems, devices and units described above can all refer to the corresponding processes in the aforementioned method embodiment, and will not be repeated here.

[0053] Those skilled in the art will appreciate that the embodiments of the present application may be provided as methods, systems, or computer program products. Therefore, the present application may take the form of an entirely hardware embodiment, an entirely software embodiment, or an embodiment combining software and hardware. Furthermore, the present application may take the form of a computer program product implemented on one or more computer-usable storage media (including but not limited to magnetic disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.

[0054] In the embodiments provided in this application, it should be understood that the disclosed devices and methods can be implemented in other ways. The device embodiments described above are merely schematic. For example, the division of modules is only a logical function division. There may be other division methods in actual implementation. For example, multiple modules or components can be combined or integrated into another system, or some features can be ignored or not executed. Another point is that the mutual coupling or direct coupling or communication connection shown or discussed can be through some communication interface, the indirect coupling or communication connection of the device or module can be electrical, mechanical or other forms.

[0055] Modules described as separate components may or may not be physically separate, and components shown as modules may or may not be physical modules, that is, they may be located in one place or distributed across multiple network modules. Some or all of these modules may be selected to achieve the purpose of this embodiment based on actual needs.

[0056] In addition, each functional module in the embodiments provided in the present application may be integrated into one processing module, or each module may exist physically separately, or two or more modules may be integrated into one module.

[0057] The present application is described with reference to the flowcharts and / or block diagrams of the methods, devices (systems), and computer program products according to the embodiments of the present application. It should be understood that each process and / or block in the flowchart and / or block diagram, as well as the combination of processes and / or blocks in the flowchart and / or block diagram, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, a special-purpose computer, an embedded processor, or other programmable data processing device to produce a machine, so that the instructions executed by the processor of the computer or other programmable data processing device generate instructions for implementing the processes in the flowchart and / or block diagram. Figure 1 a process or multiple processes and / or boxes Figure 1 A device that provides the functions specified in a block or multiple blocks.

[0058] These computer program instructions may also be stored in a computer readable memory that can direct a computer or other programmable data processing device to work in a specific manner, so that the instructions stored in the computer readable memory produce an article of manufacture comprising an instruction device, which implements the process Figure 1 a process or multiple processes and / or boxes Figure 1 The function specified in one or more boxes.

[0059] These computer program instructions can also be loaded onto a computer or other programmable data processing device so that a series of operational steps are executed on the computer or other programmable device to produce a computer-implemented process, thereby providing the instructions executed on the computer or other programmable device for implementing the process. Figure 1 a process or multiple processes and / or boxes Figure 1 A step that specifies a function in one or more boxes.

[0060] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and not to limit it. Although the present invention has been described in detail with reference to the above embodiments, ordinary technicians in the field should understand that the specific implementation methods of the present invention can still be modified or replaced by equivalents. Any modification or equivalent replacement that does not depart from the spirit and scope of the present invention should be covered by the scope of protection of the claims of the present invention.

Claims

1. A signature method based on the GR-SIS problem, characterized in that: The following steps are involved: Step 1: Based on the dihedral non-commutative group ring construction, define the group ring structure and sampling rules, and generate a key pair consisting of a public key matrix and a private key vector; Step 2: Map the target message into a group ring element through a hash function, and generate a signature vector by combining the private key and grid sampling; Step 3: Use linear operations between the public key matrix and the signature vector to verify the legitimacy of the message.

2. The GR-SIS problem-based signature method according to claim 1, characterized in that: The step 1 comprises: Step 11, define the dihedral group For collection , satisfying the group operation rules and , t is the rotation generator, s is the flip generator, i, j are the index parameters of the group elements, and n is the dihedral group The number of sides of the corresponding regular polygon; Step 12: Construct a polynomial group ring ,in is a finite field modulo q, q is an odd prime number, is a dihedral group Polynomial group rings over ; Step 13: Select the hash function H and define the primitive vector , where each element is A constant polynomial on , k is the primitive vector The number of components; Step 14, from the grid In the example, the parameter pair is generated by Gaussian sampling. , construct the public key A= , the private key is ,in obey The upper deviation is Gaussian distribution , It's a ring Medium elements, is the private key component, is the error term, is the Gaussian parameter, u is the ring The target element in .

3. The GR-SIS problem-based signature method according to claim 2, characterized in that: The step 2 comprises: Step 21: Target message Calculating the hash value ; Step 22, in the grid Perform pre-image sampling in and obtain vector , making ; Step 23: Construct signature vector .

4. The GR-SIS problem-based signature method according to claim 3, characterized in that: The step 3 comprises: Calculate the dot product of the transpose of the public key matrix and the signature vector , verify whether it satisfies , if the verification formula is established, the signature is valid.

5. The GR-SIS problem-based signature method according to any one of claims 2 to 4, characterized in that: The dihedral group The non-commutative property of ,Depend on It can be deduced that , that is, the order of multiplication of group elements affects the operation results.

6. The GR-SIS problem-based signature method according to any one of claims 2 to 4, characterized in that: The grid The dimension is k, and its basis matrix is ​​composed of primitive vectors Null space generation, sampling solution The norm satisfies ,in A preset small constant.

7. A signature device based on the GR-SIS problem, characterized in that: include: The key generation module is used to define the group ring structure and sampling rules based on the dihedral non-commutative group ring construction, and generate a key pair consisting of a public key matrix and a private key vector; The signature generation module is used to map the target message into a group ring element through a hash function, and generate a signature vector by combining the private key and lattice sampling; The signature verification module is used to verify the legitimacy of the message by using linear operations between the public key matrix and the signature vector.

8. A signature system based on the GR-SIS problem, characterized in that: Including sender and receiver, The recipient is responsible for generating the public key and private key , where the public key Contains primitive vectors , random polynomial and trapdoor parameters ; The sender uses the public key A to perform hash calculation and grid sampling on the message m to generate a signature z; The recipient uses the public key A and signature z to verify the Is it true to verify the message Legitimacy, among which For the preset modulus parameter, determine the finite field and group ring The value range of the elements.

9. An electronic device, characterized in that: The electronic device includes a processor, a memory, and a bus, wherein the memory stores machine-readable instructions executable by the processor. When the electronic device is running, the processor and the memory communicate with each other through the bus, and the processor executes the machine-readable instructions to perform the steps of the signature method based on the GR-SIS problem as described in any one of claims 1 to 6.

10. A storage medium, characterized in that: The storage medium stores a computer program, which, when executed by a processor, executes the steps of the GR-SIS problem-based signature method according to any one of claims 1 to 6.

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