Authentication system and method using elliptic equation

Abstract

An authentication system using an elliptic equation, according to an embodiment of the present application, comprises: a first apparatus that generates ellipse information regarding a predefined elliptic equation; and a second apparatus that is provided to communicate with the first apparatus, receives at least a portion of the ellipse information, transmits, to the first apparatus, credential information generated on the basis of geometric properties of an ellipse defined in the received ellipse information, and is authenticated by the first apparatus, wherein the first apparatus may authenticate the second apparatus by using the credential information received from the second apparatus.

Description

Authentication system and method using elliptic equations

[0001] The present invention relates to an authentication system and method using an elliptic equation.

[0002] In the modern online environment, personal information protection and identity management are emerging as critical issues. Anonymity is a technical and social concept that conceals an individual's identity or prevents identification; by enabling online users to operate without exposing their personal information, it reduces the risk of data leakage or theft. This anonymity plays an important role in enhancing individual privacy while simultaneously promoting free expression and participation online.

[0003] With the increasing importance of anonymity in recent years, there is a growing demand for technologies that go beyond simply concealing identities to verify the accuracy and integrity of information without directly disclosing sensitive data. This trend is particularly pronounced in fields such as online financial transactions, identity authentication, and blockchain, where accurate verification is required without infringing upon personal information.

[0004] To address these demands, Zero-Knowledge Proof (ZKP) technology is gaining attention. ZKP provides a method to verify the authenticity of specific information without disclosing the information itself. Consequently, it is emerging as a powerful technological tool capable of guaranteeing information integrity while maintaining user anonymity.

[0005] The purpose of the present invention is to provide an authentication system and method using elliptic equations that can perform authentication without directly disclosing the user's personal information, and can verify integrity without exposure to third parties by securely encrypting and verifying the user's attribute data.

[0006] In addition, the present invention aims to provide an authentication system and method using elliptic equations that reduces the amount of computation required compared to conventional zero-knowledge proof techniques and can operate even in resource-constrained environments.

[0007] An authentication system using an elliptic equation according to one embodiment of the present invention for solving the above technical problem comprises a first device for generating elliptic information regarding a predefined elliptic equation, and a second device configured to communicate with the first device, receive at least a portion of the elliptic information, transmit proof information generated based on the geometric properties of the ellipse defined in the received elliptic information to the first device, and perform authentication by the first device, wherein the first device can perform authentication for the second device using the proof information received from the second device.

[0008] Additionally, the ellipse includes a first focus, a second focus, and a first point on the ellipse, and the first device transmits information of the first point as part of the ellipse information to the second device, and the second device can calculate the sum of the lengths of the three sides of the focus triangle formed by the first focus, the second focus, and the first point as the proof information.

[0009] Additionally, the second device selects a second point on the ellipse, generates angle information corresponding to the second point as proof information, and transmits it to the first device, and the first device verifies the location of the second point based on the received angle information and performs authentication for the second device based on the verified location.

[0010] In addition, the second device can generate a symmetric key based on the selected second point and transmit encrypted attribute data, which is encrypted using the generated symmetric key, to the first device.

[0011] In addition, the first device can calculate the second point based on the received angle information, restore the symmetric key based on the calculated second point, and decrypt the encrypted attribute data using the restored symmetric key.

[0012] In addition, the above elliptical information can be generated and managed through blockchain-based smart contracts.

[0013] In addition, the first device may be a service provider server, and the second device may be a service user terminal.

[0014] An authentication method using an elliptic equation according to another embodiment of the present invention for solving the above technical problem may include: (a) a first device generating elliptic information regarding a predefined elliptic equation; (b) the first device transmitting at least a portion of the elliptic information to a second device; (c) the second device generating proof information based on the geometric properties of the ellipse defined in the received elliptic information; (d) the second device transmitting the generated proof information to the first device; and (e) the first device performing authentication for the second device using the received proof information.

[0015] According to one embodiment of the present invention described above, an authentication system and method using an elliptic equation can be provided that can perform authentication without directly disclosing the user's personal information, and can verify integrity without being exposed to a third party by securely encrypting and verifying the user's attribute data.

[0016] In addition, the present invention can provide an authentication system and method using elliptic equations that reduces the amount of computation required compared to conventional zero-knowledge proof techniques and can operate even in resource-constrained environments.

[0017] FIG. 1 is a diagram illustrating the configuration of an authentication system using an elliptic equation according to one embodiment of the present invention.

[0018] FIG. 2 is a diagram illustrating an example of an elliptic equation and a focal triangle through an authentication system using an elliptic equation according to one embodiment of the present invention.

[0019] FIG. 3 is a diagram illustrating an example of a notation method of an authentication system using an elliptic equation according to one embodiment of the present invention.

[0020] FIG. 4 is a diagram illustrating an algorithm representing a simple authentication process of an authentication system using an elliptic equation according to one embodiment of the present invention.

[0021] FIG. 5 is a diagram illustrating an algorithm representing the attribute authentication process of an authentication system using an elliptic equation according to one embodiment of the present invention.

[0022] FIG. 6 is a diagram illustrating an overview of an online public opinion survey scenario that performs authentication through an authentication system using an elliptic equation according to one embodiment of the present invention.

[0023] FIG. 7 is a diagram illustrating an overview of a data transaction scenario that performs authentication through an authentication system using an elliptic equation according to one embodiment of the present invention.

[0024] FIG. 8 is a diagram illustrating the flow of an authentication method using an elliptic equation according to one embodiment of the present invention.

[0025] The objectives and effects of the present invention, and the technical configurations for achieving them, will become clear by referring to the embodiments described in detail below in conjunction with the accompanying drawings. In describing the present invention, if it is determined that a detailed description of known functions or configurations may unnecessarily obscure the essence of the invention, such detailed description will be omitted.

[0026] However, this is not intended to limit the invention to specific embodiments, and it should be understood that it includes all modifications, equivalents, and substitutions that fall within the spirit and scope of the invention.

[0027] Furthermore, the terms described below are defined considering their functions in the present invention, and these may vary depending on the intentions or practices of the user or operator.

[0028] However, the present invention is not limited to the embodiments disclosed below but may be implemented in various different forms. These embodiments are provided merely to ensure that the disclosure of the present invention is complete and to fully inform those skilled in the art of the scope of the invention, and the present invention is defined only by the scope of the claims. Therefore, such definition should be based on the content throughout this specification.

[0029] The terms used in this application are used merely to describe specific embodiments and are not intended to limit the invention. The singular expression includes the plural expression unless the context clearly indicates otherwise. In this application, terms such as "comprising" or "having" are intended to specify the presence of the features, numbers, steps, actions, components, parts, or combinations thereof described in the specification, and should be understood as not precluding the existence or addition of one or more other features, numbers, steps, actions, components, parts, or combinations thereof.

[0030] Unless otherwise defined, all terms used herein, including technical or scientific terms, have the same meaning as generally understood by those skilled in the art to which the present invention pertains. Terms such as those defined in commonly used dictionaries should be interpreted as having a meaning consistent with their meaning in the context of the relevant technology, and should not be interpreted in an ideal or overly formal sense unless explicitly defined in this application.

[0031] Hereinafter, preferred embodiments of the present invention will be described in more detail with reference to the attached drawings. In order to facilitate an overall understanding of the present invention, the same reference numerals are used for identical components in the drawings, and redundant descriptions of identical components are omitted.

[0032] FIG. 1 is a diagram illustrating the configuration of an authentication system using an elliptic equation according to one embodiment of the present invention.

[0033] Referring to FIG. 1, an authentication system (100) using an elliptic equation may include a first device (110) and a second device (120).

[0034] An authentication system (100) using an elliptic equation according to one embodiment of the present invention relates to a Zero-Knowledge Proof (ZKP)-based user authentication technology that can verify the integrity of information while ensuring personal information protection and anonymity in an online environment, and in particular, can lighten authentication by utilizing the geometric properties of the elliptic equation (invariance of the sum of distances of the focal triangle, angle representation of points on the ellipse).

[0035] Conventional zero-knowledge proof techniques are useful in that they can verify the truthfulness of specific information without directly disclosing personal information, but they have high computational costs and scalability issues due to multiple interactions. These limitations cause serious performance degradation, particularly when applied in resource-constrained Internet of Things (IoT) environments or large-scale network environments. To overcome these limitations, the authentication system (100) utilizes the geometric properties of the elliptic equation and the focal triangle derived therefrom.

[0036] An ellipse has two foci, and possesses the property that the sum of the distances from any point to the two foci is constant. This property has long been known in geometry and has been applied in various fields, such as optics, satellite orbits, and cryptographic structures, due to the fact that the sum of the distances is constant. In particular, this paper utilizes the mathematical invariance of the focal triangle and angle representation of an ellipse. The focal triangle consists of two foci and a point on the ellipse, and possesses the characteristic that the sum of the lengths of its three sides is constant. Additionally, a point on the ellipse can be converted into a polar coordinate representation and expressed as an angle, which can then be substituted for an angle θ using the arctangent function for each quadrant.

[0037] To understand this topic more clearly, we first examine the elliptic equation. The elliptic equation is an equation distinct from the Elliptic Curve Cryptography (ECC) and Elliptic Curve Digital Signature (EDCSA) algorithms.

[0038] FIG. 2 is a diagram illustrating an example of an elliptic equation and a focal triangle through an authentication system using an elliptic equation according to one embodiment of the present invention.

[0039] Referring to Fig. 2, an ellipse is a type of conic section and is generally defined by the equation of mathematical formula 1 below.

[0040] [Mathematical Formula 1]

[0041]

[0042] In the elliptic equation of Equation 1, x and y represent the coordinates of points on the ellipse, h and k represent the coordinates of the ellipse's center, and a and b represent the lengths of the major and minor axes, respectively. If w is greater than z, w can be considered the major axis and z the minor axis. Additionally, the focus is located along the major axis, and according to the relationship between the major and minor axes, the coordinates of the focus ∝ are ∝ 2 =a 2 -b 2 It can be calculated as.

[0043] In the elliptic equation of mathematical formula 1, the two foci (P a , P b ) and the point on the ellipse (P c The triangle formed by ) is called the focus triangle (F), and the focus triangle F is F = △P a P b P c It can be expressed as follows. The sum of the lengths of the sides of the triangle is point P on the ellipse. c It can remain constant regardless of the position of . In particular, the equation α = d(P a P b )+ (P a P c )+ (P b P c ) is P c It always produces the same α value even if varies.

[0044] For example, the elliptic equation is (x-0) 2 / 985 2 + (y-0) 2 / 975 2 In the case where =1 is defined, the points P a =(-140,0), P b =(140,0), P c Set to =(0, 975). Distance d(P a P b ) is 280, and the distance d(P a P c ) and d(P b Pc ) are all 985. Additionally, P c Calculating the value of α using yields α=2250, and P' c Using , α'=2250 is produced.

[0045] FIG. 3 is a diagram illustrating an example of a notation method of an authentication system using an elliptic equation according to one embodiment of the present invention.

[0046] An authentication system (100) using an elliptic equation according to one embodiment of the present invention can provide an authentication method that integrates simple authentication, which performs user authentication simply without requiring attribute values, and attribute authentication, which authenticates by encrypting attribute values. The notation used in the following description is as shown in FIG. 3.

[0047] In the following description, the first device (110) represents a service provider server (sp), and the second device represents a service user terminal (su).

[0048] First, we examine simple authentication. The first device (110) can generate elliptical information regarding a predefined elliptical equation. In this case, the elliptical information can be generated and managed through a blockchain-based smart contract. Additionally, the ellipse may include a first focus, a second focus, and a first point on the ellipse. The first device (110) can generate a set of parameters corresponding to the elliptical equation of Equation 1. The parameters belonging to the elliptical information are the center coordinates (h, k), the major and minor radii (w, z), and the coordinates (P) of the first focus and the second focus. a , P b It may include ).

[0049] The first device (110) can transmit information of the first point as part of the ellipse information to the second device (120). The second device (120) is configured to communicate with the first device and can receive at least part of the ellipse information. Additionally, the second device (120) transmits proof information generated based on the geometric properties of the ellipse defined in the received ellipse information to the first device (110), and authentication can be performed by the first device (110).

[0050] The first device (110) can perform authentication for the second device (120) using authentication information received from the second device (120). That is, the second device (120) can receive at least a portion of the elliptical information, generate authentication information, and then transmit it back to the first device (110), and the first device (110) can perform authentication for the second device (120) by verifying the authentication information.

[0051] FIG. 4 is a diagram illustrating an algorithm representing a simple authentication process of an authentication system using an elliptic equation according to one embodiment of the present invention.

[0052] Referring to FIG. 4, the ellipse used herein includes a first focus, a second focus, and a first point on the ellipse, and the geometric properties of the focus triangle formed by these are utilized in the authentication process. F = △P a P b P c In the focus triangle represented as, the first point P c It exists on an ellipse generated by an elliptic equation G. This can be represented as P∈Ellipse(G). Specifically, the first device (110) can generate an elliptic equation G, select a specific point P on the ellipse, and transmit that information to the second device (120). For the authentication of the second device (120), the first device (110) uses the information of the first point, G sp It can generate. In this case, G sp is (P) = W2 -Z 2 class The conditions must be satisfied. In addition, the generated G must be unique and must not have been previously generated by another sp.

[0053] The second device (120) can calculate the sum of the lengths of the three sides of a focal triangle formed by the first focal point, the second focal point, and the first point as the proof information. That is, the second device (120) receives the G sp and P c Using this, you can construct the focal triangle F and calculate the distance d for each point. This distance is d(P a P b ), d(P b P c ), d(P a P c It can be expressed as ). The second device (120) calculates the lengths of the three sides of the focus triangle based on the received focus and point information, and the sum thereof is proof information α su It can be generated and transmitted to the first device (110).

[0054] Here, the sum of the lengths of the three sides α su It can be expressed by the following mathematical formula 2.

[0055] [Mathematical Formula 2]

[0056]

[0057] The sum of the lengths of the three sides α su has mathematical properties that are maintained regardless of the position of P. Therefore, the proof information α calculated and transmitted by the second device (120) su If the value matches the reference value held by the first device (110), the first device (110) can authenticate that the second device (120) is the correct user.

[0058] As examined above, simple authentication has the advantage of performing authentication without directly exposing users' attribute data or personal information. Furthermore, because it utilizes the equation of an ellipse and the geometric invariance of the focal triangle, it is difficult to forge or arbitrarily guess the authentication value. Accordingly, it can provide efficient and secure authentication even in resource-constrained environments and can be applied in various fields, such as online services, financial transactions, and blockchain-based services.

[0059] Next, we will examine attribute authentication. The authentication system (100) of the present invention can perform attribute authentication not only to simply authenticate the existence of a user, but also to securely transmit and verify the user's attribute data.

[0060] FIG. 5 is a diagram illustrating an algorithm representing the attribute authentication process of an authentication system using an elliptic equation according to one embodiment of the present invention.

[0061] Referring to FIG. 5, first, when the second device (120) requests attribute authentication from the first device (110), the first device (110) [requests] G to the second device (120). sp It can provide. The second device (120) can select a second point on the ellipse and generate angle information corresponding to the second point as proof information and transmit it to the first device (110).

[0062] The first device (110) can determine the location of the second point based on received angle information and perform authentication for the second device (120) based on the determined location. Specifically, the second device (120) is a specific point of the ellipse, i.e., the second point P Ω You can select and generate angle information θ corresponding to the point as proof information and transmit it to the first device (110). The first device uses the received θ to P Ω The location of can be checked, and based on this, authentication of the second device (120) can be performed.

[0063] In other words, the second device (120) is at the second point P Ω Angle information θ must be provided to the first device (110) so that the position of can be verified. θ is the major axis and P Ω , P a , P b Center P between the points Φ It is defined as the angle between the straight lines passing through. It is desirable to accurately specify the angle information θ value to 10 decimal places using the following mathematical formulas 3 to 5.

[0064] [Mathematical Formula 3]

[0065]

[0066] [Mathematical Formula 4]

[0067]

[0068] [Mathematical Formula 5]

[0069]

[0070] Upon closer examination, the second device (120) is at the selected second point P Ω Symmetric key K based on sym Generate and the generated symmetric key K sym The encrypted attribute data, encrypted using [the method], can be transmitted to the first device (110). Specifically, the symmetric key K sym It is used to encrypt attribute data Attri, and the encrypted attribute data Enc(Attri) can be transmitted to the first device (110).

[0071] The first device (110) can calculate a second point based on received angle information, restore a symmetric key based on the calculated second point, and decrypt encrypted attribute data using the restored symmetric key. Specifically, the first device (110) can determine the second point P through the previously received angle information θ. Ω Calculate , and in the same way, symmetric key Ksym It can be restored.

[0072] The first device (110) is based on the following mathematical formula 6, P Ω The position of can be calculated.

[0073] [Mathematical Formula 6]

[0074]

[0075] Here, w and z are the elliptic equations It is a parameter of, and π represents the radius of the ellipse. Subsequently, the first device (110) is the restored K sym By using the decryption of the encrypted attribute data Enc(Attri), the attribute provided by the second device (120) can be verified.

[0076] Through this procedure, the first device (110) can authenticate that the second device (120) is the correct user and simultaneously confirm that the attribute data has been safely transmitted without being tampered with. In particular, the attributes provided by the second device (120) are encrypted using a symmetric key, so they are not exposed to third parties. Furthermore, since the angle information based on the elliptic equation and point selection has randomness, it is virtually impossible for an attacker to forge and transmit the correct attribute information.

[0077] Below, we describe the process in which an authentication system (100) performs authentication by integrating simple authentication and attribute authentication based on two scenarios: online public opinion surveys and data trading.

[0078] First, I will explain the online public opinion survey scenario.

[0079] FIG. 6 is a diagram illustrating an overview of an online public opinion survey scenario that performs authentication through an authentication system using an elliptic equation according to one embodiment of the present invention.

[0080] The system model used in this scenario is as follows.

[0081] Requesting Organization (hereinafter, co): The requesting organization is an organization that requests voting through a survey agency. This organization is primarily interested in determining the total number of participants who took part in the vote.

[0082] Survey agency (hereinafter, ca): A survey agency is an organization responsible for conducting voting at the request of a survey-commissioning agency. This agency focuses solely on collecting and analyzing the responses of participants.

[0083] Participants (hereinafter, par): Participants aim to notify the survey commissioning organization of their participation in the vote without disclosing their identity. Additionally, they want the survey organization to receive and process their voting response results.

[0084] Here, ca represents the first device (110) and par represents the second device (120).

[0085] co requests ca (the first device) to proceed with the vote. Accordingly, ca (the first device) [represents] the unique elliptic equation G ca Generates, and G ca is a verified smart contract (SC G Duplication is checked by ). Smart Contract (SC G The above G by ) ca If verified, ca (first device) smart contract SC F Distribute and point P on the ellipse a , P b , P c It stores . Subsequently, ca ca (first device) is the above G ca and the above SC F Provide the address of to co and par (second device).

[0086] par (the second device) sends the encrypted response Enc(re) and θ to ca (the first device), and ca (the first device) transmits the voting result through this. In addition, par (the second device) uses α to confirm its participation. parTransmits to co. co receives α from par (the second device). par Review the value to verify the total number of participants in the requested vote.

[0087] ca (the first device) verifies that the vote was cast by a legitimate voter by verifying the received authentication value and encrypted attribute data. In addition, the entire voting process is managed through a blockchain-based smart contract, thereby making it impossible to falsify or alter the voting record and ensuring the transparency of the vote. Thus, the present invention enables online voting that is impossible to forge while guaranteeing anonymity.

[0088] Next, we examine data trading scenarios.

[0089] FIG. 7 is a diagram illustrating an overview of a data transaction scenario that performs authentication through an authentication system using an elliptic equation according to one embodiment of the present invention.

[0090] Referring to FIG. 7, the data buyer (db), which is the first device (110), conducts a transaction with the data seller (ds) through the data market (dm). In the initial stage of the transaction, the dm establishes a blockchain-based smart contract (SC) to manage unique elliptic equation G and point information.

[0091] ds is the secret value P Ω Using symmetric key K sym Generates, encrypts the data, and sends it to the DB. The DB is the same P Ω The data is decrypted using [the method]. Subsequently, both parties submit their respective authentication value point information P(db) and α to dm, and dm verifies whether these values ​​match. If a match is confirmed, dm declares that the transaction has been completed as agreed and pays ds. In this way, the data transaction achieves the effects of privacy protection, integrity guarantee, and enhanced transparency.

[0092] As examined above, an authentication system utilizing elliptic equations can satisfy the essential attributes of ZKP, namely integrity, soundness, and zero-knowledge. The probability of an attacker randomly forging authentication values ​​or angles is extremely low, and user attribute data is transmitted in an encrypted state, ensuring it remains secure.

[0093] FIG. 8 is a diagram illustrating the flow of an authentication method using an elliptic equation according to one embodiment of the present invention.

[0094] The authentication method using an elliptic equation illustrated in FIG. 8 can be performed by the authentication system (100) using an elliptic equation described through FIG. 1 to 7. Therefore, even if the content is omitted below, the content described regarding the authentication system (100) using an elliptic equation through FIG. 1 to 7 can be applied in the same way to FIG. 8.

[0095] Referring to FIG. 8, in step S110, the first device (110) can generate elliptical information regarding a predefined elliptical equation. At this time, the elliptical information can be generated and managed through a blockchain-based smart contract. Additionally, the ellipse may include a first focus, a second focus, and a first point on the ellipse.

[0096] In step S120, the first device (110) may transmit information of the first point as part of the elliptical information to the second device (120). The second device (120) is configured to communicate with the first device and may receive at least part of the elliptical information.

[0097] In step S130, the second device (1220) can generate proof information based on the geometric properties of the ellipse defined in the received ellipse information. At this time, the second device (120) can calculate the sum of the lengths of the three sides of the focal triangle formed by the first focal point, the second focal point, and the first point as the proof information. Additionally, the second device (120) can select a second point on the ellipse and generate angle information corresponding to the selected second point as proof information.

[0098] Specifically, the second device (120) can generate a symmetric key based on the selected second point and encrypt attribute data using the generated symmetric key to generate encrypted attribute data. Additionally, the second device (120) can transmit the encrypted attribute data to the first device (110).

[0099] In step S140, the second device (120) can transmit the generated proof information to the first device (110).

[0100] In step S150, the first device (110) can perform authentication for the second device (120) using the received proof information. At this time, the first device (110) can calculate a second point based on the received angle information and restore a symmetric key based on the calculated second point. The first device (110) can complete the verification for the second device (120) by decrypting the encrypted attribute data using the restored symmetric key and verifying the decrypted attribute data.

[0101] An authentication method using an elliptic equation according to one embodiment of the present invention may be implemented in the form of program instructions that can be executed through various computer means and recorded on a computer-readable medium. The computer-readable medium may include program instructions, data files, data structures, etc., either individually or in combination. The program instructions recorded on the medium may be those specifically designed and configured for the present invention, or may be those known and available to those skilled in the art of computer software. Examples of computer-readable recording media include magnetic media such as hard disks, floppy disks, and magnetic tapes; optical recording media such as CD-ROMs and DVDs; magneto-optical media such as floptical disks; and hardware devices specifically configured to store and execute program instructions, such as ROM, RAM, and flash memory. Examples of program instructions include machine code, such as that generated by a compiler, as well as high-level language code that can be executed by a computer using an interpreter, etc. The above-described hardware device may be configured to operate as one or more software modules to perform the operation of the present invention, and vice versa.

[0102] The features, structures, effects, etc. described in the above-described embodiments are included in at least one embodiment of the present invention and are not necessarily limited to only one embodiment. Furthermore, the features, structures, effects, etc. exemplified in each embodiment may be combined or modified and implemented in other embodiments by a person skilled in the art to which the embodiments belong.

[0103] Therefore, details regarding such combinations and variations should be interpreted as being included within the scope of the present invention. Furthermore, although the above description has focused on embodiments, this is merely illustrative and does not limit the present invention. Those skilled in the art will understand that various modifications and applications not exemplified above are possible within the scope of the essential characteristics of the embodiments. For example, each component specifically shown in the embodiments may be modified and implemented. Differences related to such modifications and applications should be interpreted as being included within the scope of the present invention as defined in the appended claims.

Claims

1. In an authentication system using elliptic equations, A first device for generating elliptical information regarding a predefined elliptical equation; and A second device configured to communicate with the first device, receiving at least a portion of the ellipse information, transmitting proof information generated based on the geometric properties of the ellipse defined in the received ellipse information to the first device, and performing authentication by the first device; An authentication system in which the first device performs authentication for the second device using the certification information received from the second device.

2. In Paragraph 1, The above ellipse includes a first focus, a second focus, and a first point on the ellipse, and The first device transmits information of the first point to the second device as part of the elliptical information, and The second device is an authentication system that calculates the sum of the lengths of the three sides of a focal triangle formed by the first focal point, the second focal point, and the first point as the proof information.

3. In Paragraph 1, The second device selects a second point on the ellipse, generates angle information corresponding to the second point as proof information, and transmits it to the first device. An authentication system in which the first device determines the location of the second point based on the received angle information and performs authentication for the second device based on the determined location.

4. In Paragraph 3, An authentication system in which the second device generates a symmetric key based on the selected second point and transmits encrypted attribute data, encrypted using the generated symmetric key, to the first device.

5. In Paragraph 4, An authentication system wherein the first device calculates the second point based on the received angle information, restores the symmetric key based on the calculated second point, and decrypts the encrypted attribute data using the restored symmetric key.

6. In Paragraph 1, An authentication system in which the above elliptical information is generated and managed through a blockchain-based smart contract.

7. In Paragraph 1, An authentication system in which the first device is a service provider server and the second device is a service user terminal.

8. In an authentication method using an elliptic equation, (a) A step in which the first device generates elliptical information regarding a predefined elliptical equation; (b) a step in which the first device transmits at least a portion of the elliptical information to the second device; (c) A step in which the second device generates proof information based on the geometric properties of the ellipse defined in the received ellipse information; (d) the second device transmitting the generated proof information to the first device; and (e) a step in which the first device performs authentication for the second device using the received authentication information; comprising an authentication method.

9. In Paragraph 8, The above ellipse includes a first focus, a second focus, and a first point on the ellipse, and The above (c) step is, An authentication method comprising the step of the second device calculating the sum of the lengths of the three sides of a focal triangle formed by the first focal point, the second focal point, and the first point as the proof information.

10. In Paragraph 8, The above (c) step is, An authentication method comprising the step of the second device selecting a second point on the ellipse and generating angle information corresponding to the selected second point as proof information.

11. In Paragraph 10, After step (c) above, (c-1) A step in which the second device generates a symmetric key based on the selected second point; (c-2) A step in which the second device encrypts attribute data using the generated symmetric key to generate encrypted attribute data; and (c-3) A step of the second device transmitting the encrypted attribute data to the first device; further comprising an authentication method.

12. In Paragraph 11, The above step (e) is, (e-1) A step in which the first device calculates the second point based on the received angle information; (e-2) A step in which the first device restores the symmetric key based on the calculated second point; and (e-3) A step in which the first device decrypts the encrypted attribute data using the restored symmetric key and verifies the decrypted attribute data; an authentication method comprising.

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