KR101006358B1 - Elliptic curve cryptography system based on real domain and method thereof - Google Patents

Abstract

Disclosed are a public key generation method, a message encryption / decryption method using the method, and a system thereof. The elliptic curve cryptography method for generating a public key may include selecting at least one point on an elliptic curve defined as a finite body; And generating a public key of a real body group using an elliptic curve equation and a characteristic equation defining the elliptic curve.

Elliptic curve, password, decryption

Description

Real Elliptic Curve Cryptosystem and Its Method {ELLIPTIC CURVE CRYPTOGRAPHY SYSTEM BASED ON REAL DOMAIN AND METHOD THEREOF}

The present invention relates to an elliptic curve cryptography technique, and more particularly, to a method for generating a real object-based public key based on an elliptic curve defined by a finite body, a message encryption / decryption method using the method, and a system thereof. will be.

Elliptic Curve Cryptography (ECC) algorithm is a public key algorithm independently proposed by N. Koblitz and V.Miller in 1985. Is a cryptographic algorithm.

 Recently, Elliptic Curve Method (ECM) has been used to design various algorithms such as factorization problem, decimal decision method and public key cryptography which are the basis of RSA cryptosystem.

On the other hand, RSA (Rivest Shamir Adleman), which is a representative public key cryptosystem, is a public key cryptosystem that is widely known and used.However, for sufficient security, it is necessary to encrypt / decrypt using a large bit of key and takes a lot of computation time. have.

On the other hand, the encryption method using an elliptic curve has a relatively small number of bits in comparison with other encryption methods. That is, according to the encryption method using the elliptic curve, since the resources and time required for the operation are reduced, high performance can be obtained when the encryption method using the elliptic curve is actually applied to the system as software.

1 is a graph for explaining the characteristics of a finite body included on a general elliptic curve. Referring to FIG. 1, since the group (or points) constituting an elliptic curve, which may be defined as y ² = x ³ − x, is an addition group, addition to two points (eg, P and Q) is performed. In this case, the elliptic curve encryption system draws a line connecting point P and point Q.

Subsequently, the elliptic curve encryption system can calculate another point S where the vertical line of the point R where the line connecting the point P and the point Q meets another ellipse meets the elliptic curve.

In this case, if two points (for example, P and Q) are the same, the elliptic curve encryption system draws a tangent to the point on the elliptic curve, and adds a point where the vertical line of the point where the line meets another ellipse meets the elliptic curve. Calculate as a result.

However, since the elliptic curve group on the finite body is composed of a set of finite points, if the elliptic curve group of the finite body is used as it is, the safety of the elliptic curve cryptographic system may be reduced.

Accordingly, a technical problem to be solved by the present invention is to provide a method for generating a high security public key using an elliptic curve based on a real number domain, a message encryption / decryption method using the method, and a system thereof. It is.

The public key generation method for achieving the technical problem comprises the steps of selecting at least one point on the elliptic curve defined by a finite body; And generating a public key of a real body group using an elliptic curve equation and a characteristic equation defining the elliptic curve.

The elliptic curve equation may correspond to y ² = a * x ³ + b * x + c, where a, b, and c are real, wherein a is 1, b is -1, and c may be 1.

The elliptic curve equation corresponds to y ² = x ³ -p * x-q, where p and q are real, and the characteristic equation is: (s = (3x _p ² -p) / (2y _p ) (where x _p is the x coordinate of the selected point and y _p is the y coordinate of the selected point), and generating the real object group comprises at least using a solution of the characteristic equation. generating a single real chegun coordinates, x-coordinate of the at least one real chegun is s ² - ² * x _p corresponds to the (here, s is a solution of the characteristic equation), and the at least one real number The y-coordinate of the body group may correspond to y _p + s * (x _R -x _p ).

Generating the real object group, matching the coordinates of the at least one real body group generated using the s with the coordinates of the finite body on the elliptic curve and replacing the matched coordinates with the coordinates of the real body group It may further include.

The elliptic curve cryptographic public key generation method may further include changing at least one real object group corresponding to the matched coordinates and setting the changed real object group as the public key.

The encryption method for achieving the technical problem can encrypt the message using the public key generated by the public key generation method of the elliptic curve encryption method.

Decryption method for achieving the technical problem It is possible to decrypt the cipher text using the public key generated by the public key generation method of the elliptic curve encryption method.

Message encryption method for achieving the above technical problem, the step of selecting a predetermined finite group on the elliptic curve is defined as a finite body; And generating a public key using the predetermined finite group, and generating a cipher text for a message using a sender private key and a receiver public key existing on the generated public key.

The message encryption method may further include encrypting information about a predetermined finite group, and may further include transmitting information on the encrypted finite field and a ciphertext to a receiver.

The public key generator for achieving the technical problem, the selection unit for selecting at least one point on the elliptic curve defined by a finite body; And a public key generator including generating a public key using an elliptic curve equation and a characteristic equation defining the elliptic curve.

The elliptic curve equation corresponds to y ² = x ³ -p * x-q (where a, b, and c are real numbers), and the characteristic equation is: (s = (3x _p ² -p) / (2y _p ), where x _p is the x coordinate of the selected point and y _p is the y coordinate of the selected point), and the public key generator is at least one using a solution of the characteristic equation. for generating a coordinate of the real chegun is, x coordinate (x _R) of the at least one real chegun is s ² - ² * x _p corresponds to the (here, s is a solution of the characteristic equation), and the The y coordinate (y _R ) of the at least one real body group may correspond to y _p + s * (x _R − x _p ).

The public key generator may match the coordinates of at least one real body group generated using the s with the coordinates of the finite body on the elliptic curve, and replace the matched coordinates with the coordinates of the real body group.

Message encryption system for achieving the technical problem may include the public key generator.

The public key generation method in the elliptic curve cryptography method, the message encryption / decryption method using the method, and the system construct an elliptic curve cryptographic system based on a real object, thereby establishing an existing finite body based elliptic curve cryptography. There is an effect that can provide more reliable encryption performance than the system.

The public key generation method in the elliptic curve cryptography method, the message encryption / decryption method using the method, and the system change the points on the elliptic curve based on the finite field and generate the public key using the changed points. Performing encryption can provide the effect of creating a new encryption algorithm.

In order to fully understand the present invention, the operational advantages of the present invention, and the objects achieved by the practice of the present invention, reference should be made to the accompanying drawings which illustrate preferred embodiments of the present invention and the contents described in the accompanying drawings. Hereinafter, exemplary embodiments of the present invention will be described in detail with reference to the accompanying drawings. Like reference numerals in the drawings denote like elements.

2 is a block diagram of an encryption / decryption system according to an embodiment of the present invention, FIG. 3 is a graph showing a real object group generated according to an embodiment of the present invention, and FIG. 4 is created according to an embodiment of the present invention. It is a graph for explaining the process of matching the real body group to the finite body.

5 is a graph showing a result of matching a real body group generated according to an embodiment of the present invention to a finite body. 2 to 5, the encryption / decryption system 10 may include a public key generator 11, an encoder 12, and a decoder 18.

In this case, the encryptor 12 and the decryptor 18 may be implemented independently of the encryption / decryption system 10, and the encryption / decryption system 10, the encryptor 12, and / or the decryptor. 18 may be implemented as a portable application.

The portable application includes a portable computer, a digital camera, a personal digital assistance (PDA), a cellular telephone, an MP3 player, a portable multimedia player, and a vehicle navigation system. system, memory card, system card, game machine, electronic dictionary, or solid state disk.

In addition, the encryption / decryption system 10, the public key generator 11, the encryptor 12, and / or the decryptor 18 may be implemented in a communication system (not shown), wherein the communication system ( Not shown) may include any wireless system, such as a portable terminal, PDA, portable computer, wireless telephone, satellite system, pager, RFID reader, or RFID system.

In addition, the communication system may be a wireless local area network (WLAN) system or a wireless personal area network (WPAN) system.

The public key generator 11, which may be implemented inside the encryptor 12 and the decryptor 18, is defined as a finite body (e.g., G ₁ to G _{17 in} FIG. 5, hereinafter referred to as 'G'). Real body group E (G) by selecting at least one point (eg, P in FIG. 3) on the elliptic curve ECC1 to be obtained, and using an elliptic curve equation and a characteristic equation defining the elliptic curve ECC1. For example, EG ₁ , EG ₂ , ..., EG _n )) may be generated.

In this case, the elliptic curve equation of the elliptic curve ECC1 may correspond to Equation 1 below.

y ² = a * x ³ + b * x + c, where a, b, and c are real

In this case, a may be 1, b may be -1, and c may be 1, but is not limited thereto. Meanwhile, when Equation 1 is defined as Equation 2 below,

y ² = x ³ -p * x-q, where p and q are real

The public key generator 11 may generate coordinates of at least one real body group (E (G), for example, EG ₁ , EG ₂ , ..., EG _n ) using a solution of a characteristic equation. The characteristic equation may correspond to Equation 3 below.

s = (3x _p ² -p) / (2y _p ), where s is the solution of the characteristic equation, x _p is the x coordinate of the selected point, and y _p is the y coordinate of the selected point

In addition, the public key generator 11 may calculate the x-coordinate of the at least one real object group E (G) by the following Equation 4 based on the calculation result of Equation 3 above.

x _R = s ² - (x-coordinate of where, _R x is at least one real chegun (E (G))) 2 * x p

In addition, the public key generator 11 may calculate the y-coordinate of at least one real object group E (G) by the following Equation 5 based on the calculation result of Equation 3 above.

y _R = y _p + s * (x _R -x _p ), where y _R is the y-coordinate of at least one real family (E (G))

In this case, the public key generator 11 may calculate the plurality of real object groups E (G) by repeatedly performing (or calculating) Equations 3 to 5.

For example, the public key generator 11 inputs a first real object coordinate (x _R1 , y _R1 ) corresponding to the operation result of the first equation (5) instead of (x _p , y _p ) of the equation (3), By substituting the calculation result s of the value into equations (4) and (5), the second real object coordinates (x _R2 , y _R2 ) can be calculated.

In addition, the public key generator 11 inputs the second real object coordinates (x _R2 , y _R2 ) instead of (x _p , y _p ) in Equation 3, and inputs the calculation result (s) of the input value. By substituting into 4 and (5), the third real object coordinates (x _R3 , y _R3 ) can be calculated.

In the same manner as above, the public key generator 11 repeats Equations 3 to 5 n times (n is a natural number), whereby n real object groups (E (G), for example, EG ₁ (x _R1 , y _R1) ), EG ₂ (x _R2 y _R2 ), ..., EG _n (x _Rn , y _Rn )).

For example, the public key generator 11 may use the real coordinates EG ₁ (x _R1 , y _R1 ) and EG _{2 of} each of the at least one real object group E (G) through Equations 2 to 5 as shown in FIG. 3. (x _R2 y _R2 ), ..., EG _n (x _Rn , y _Rn )) can be calculated.

At this time, each of the real coordinates (EG ₁ (x _R1 , y _R1 ), EG ₂ (x _R2 y _R2 ), ... of each of the at least one real body group E (G) calculated through the equations (2) to (5). , EG _n (x _Rn , y _Rn ) may not be located on the elliptic curve ECC1 as shown in FIG. 3.

That is, according to the embodiment of the present invention, the public key generator 11 does not select only the coordinates on the elliptic curve ECC1 defined by the finite body G as the public key when the public key is generated. Therefore, the public key generated by the public key generator 11 has higher stability. There is an effect that can increase the reliability of encryption and decryption.

On the other hand, the public key generator 11 has coordinates EG ₁ (x _R1 , y _R1 ), EG ₂ (x _R2 y _R2 ), ..., EG _n (x) of at least one real object group E (G). _{R n} , y _{R n} )) to the coordinates of the finite body on the elliptic curve (eg, G ₁ , G ₂ , ..., G _{n in} FIG. 4) and matched coordinates G ₁ , G ₂ , ... , G _n ) can be selected as the public key.

In more detail, the public key generator 11 includes the real coordinates EG ₁ (x _R1 , y _R1 ) of each of the at least one real object group E (G) calculated through Equations 2 to 5; EG ₂ (x _R2 y _R2 ), ..., EG _n (x _Rn , y _Rn ) coordinates (G ₁ , G ₂ , ..) of the finite body (Q) located on the elliptic curve (ECC1). ., G _n ).

In this case, the coordinate value of the finite body Q matching each of the at least one real body group E (G) is the elliptic curve ECC1 at the closest distance to the at least one real body group E (G). The public key generator 11 may have approximate points G ₁ , G ₂ , and the like on the elliptic curve ECC1 matching the at least one real object group E (G), as shown in FIG. 5. ..., G _n ) can be chosen as the public key.

In addition, the encoder 12 may change the approximations G ₁ , G ₂ ,..., G _n on the elliptic curve ECC1 and select the changed approximations as the public key. For example, the encoder 12 erases at least one approximation point among the approximations G ₁ , G ₂ ,..., G _n on the elliptic curve ECC1 and selects the remaining approximations as the public key. can do.

In this case, the encryptor 12 may encrypt information about the approximate points remaining in the erased result and transmit the encrypted information to the decryptor 18.

The encryptor 12 uses Diffie-Hellman's key exchange scheme, ElGamal, and Schnorr and NIST signature schemes based on the public keys of the real object group E (G) generated by the public key generator 11. Messages (or plain text, m) can be encrypted.

For example, the encryptor 12 converts the public key of the real object group E (G) generated by the public key generator 11 into an ElGamal public key cryptography based on a discrete logarithm to encrypt the message m. Can be.

Encryptor 12 using the sender's private key (j) and the sender's public key (c ₁₎ the generation and the generated the sender's private key (j) and the sender's public key, (c ₁₎ of the message (m) It is possible to generate a cipher text (c ₂ ).

At this time, at least one of the sender private key (j), the sender public key (c ₁ ), the cipher text (c ₂ ) is the coordinates of the real object group (eg, EG ₁ (x _R1 , y _R1 ), EG ₂ (x _R2 y _R2 ), ..., EG _n (x _Rn , y _Rn )), and the sender public key c ₁ is selected on the sender private key j and on an elliptic curve ECC1. It may be an operation result (eg, a multiplication) of coordinates for at least one point.

Finally, the encryptor 12 may transmit the generated sender public key c ₁ and the cipher text c ₂ for the message m to the decryptor 18.

Meanwhile, the encryptor 12 may include a sender key generator 14 and a message encryptor 16. The sender key generation unit 14 may generate a sender private key j and a sender public key c ₁ .

In this case, the sender public key c ₁ may be a calculation result of coordinates of the sender private key j and at least one point (for example, P of FIG. 3).

The message encryptor 16 may generate a ciphertext c ₂ for the message m using the sender private key j and the sender public key c ₁ generated by the sender key generator 14. .

For example, the message encryptor 16 performs an operation (e.g., a multiplication operation) on the message m with the sender private key j and the receiver public key j to generate the ciphertext c ₂ for the message m. Can be generated.

Further, the encryptor 12 is an ellipse irrespective of the coordinates of the real object group described above (for example, (x _R1 , y _R1 ), (x _R2 y _R2 ), ..., (x _Rn , y _Rn )). Select a predetermined finite group on the curve, generate a public key using the selected finite group, and use the sender private key (j) and the recipient public key (h) on the generated public key It is also possible to generate a ciphertext c ₂ for m).

In this case, the encryptor 12 may encrypt information about the predetermined finite body group, and may transmit information about the encrypted finite body group and the ciphertext c ₂ to the decryptor 18.

In addition, the encoder 12 has an approximation point in which the coordinates of at least one real body group E (G) are matched with the coordinates of the finite body on the elliptic curve (for example, G ₁ , G ₂ , ..., G _n ). Only coordinates arbitrarily selected from among the coordinates of the above may be selected as the public key and encryption of the message m may be performed based on the selection result.

In this case, the encryptor 12 may encrypt information on arbitrarily selected coordinate values, and transmit the encrypted information and the ciphertext c ₂ to the decryptor 18.

The decoder 18 uses Diffie-Hellman's key exchange exchange scheme, ElGamal, and Schnorr and NIST signature schemes based on the public keys of the real object group E (G) generated by the public key generator 11. The ciphertext c ₂ can be decrypted.

For example, the decoder 18 decrypts the ciphertext c ₂ by converting the public key of the real object group E (G) generated by the public key generator 11 into an ElGamal public key cryptography based on a discrete logarithm. can do.

The decryptor 18 receives the sender public key c ₁ and the cipher text c ₂ for the message m from the encryptor 12, the receiver private key k, the receiver public key h, and The ciphertext c ₂ for the message m can be decrypted based on at least one of the sender public key c ₁ .

The decryptor 18 may include a receiver key storage unit 20 and a message decryption unit 22. The receiver key storage unit 20 may receive a sender public key (c ₁ ) and a cipher text (c ₂ ) for the message (m) from the encryptor 12, the message decryption unit 22 is the receiver private key The cipher text c ₂ for the message m can be decrypted based on at least one of (k), the recipient public key h, and the sender public key c ₁ .

In this case, the receiver public key h is a result of calculation of coordinates for at least one point (for example, P of FIG. 3) selected on the receiver private key k and the elliptic curve ECC1. Can be.

The message decryption unit 22 may decrypt the cipher text c ₂ using Equation 6 below.

(여기서, 는 실수체에서 정의되는 연산으로, 은 를 연산후(즉, 타원 곡선(ECC1) 상의 점(예컨대, 도 3의 P)에 대하여 상기 수학식 2 내지 수학식 5를 k번 수행하고, k번 수행된 연산결과에 대하여 상기 수학식 2 내지 수학식 5를 c₁회 더 수행함을 의미한다.)

(Wherein, is an operation defined in the real body, and after performing the operation (that is, performing the above Equations 2 to 5 k times on the point on the elliptic curve (ECC1) (for example, P in Fig. 3), Equations 2 to 5 are performed c ₁ more times for the result of the calculation performed k times.)

Equation 6 may correspond to Equation 7 below.

Equation 7 may correspond to Equation 8 below.

과

은 동일하다. 이에 대해서 보다 상세히 설명하면, 도 6은 본 발명의 실시 예에 따라 생성된 실수체 군들의 특성을 설명하기 위한 그래프로서, 타원 곡선(ECC1) 상에서 제1 점(2P)와 제2 점(3P)의 연산시(즉, 2P

3P), 5P가 연산결과로서 출력되고, 제2 점(3P)과 제1 점(2P)의 연산시(즉, 3P

2P), 5P가 연산결과로서 출력된다.In Equation 8

and

Is the same. In more detail, FIG. 6 is a graph illustrating characteristics of the real object groups generated according to an exemplary embodiment of the present invention, and includes a first point 2P and a second point 3P on an elliptic curve ECC1. In operation of (i.e. 2P

3P), 5P is outputted as a calculation result, and when computing the 2nd point 3P and the 1st point 2P (that is, 3P)

2P) and 5P are output as calculation results.

As a result, as the exchange law of operation is established, the message decoder 22 may restore (or output) the message m by Equation (8).

7 is a flowchart of a method for generating an elliptic curve cryptography public key according to an embodiment of the present invention. 2, 3, and 7, the selector 11-1 of the public key generator 11 selects at least one point P on an elliptic curve ECC1 defined by a finite body ( S10).

The public key generator 11-3 may generate a public key of a real object group using an elliptic curve equation and a characteristic equation (for example, Equation 3) defining an elliptic curve (S12).

8 is a flowchart illustrating a message encryption method according to an embodiment of the present invention. 2, 3, and 8, the selector 11-1 of the public key generator 11 selects at least one point P on an elliptic curve ECC1 defined by a finite body ( S20).

The public key generator 11-3 may generate a public key of a real object group using an elliptic curve equation and a characteristic equation (for example, Equation 3) defining an elliptic curve (S22).

The encryptor 12 generates the ciphertext c ₂ for the message m based on at least one of the public key, the sender private key j, and the recipient public key k generated by the public key S22. Can be generated (S24).

9 is a flowchart illustrating a message encryption method according to another embodiment of the present invention. 2, 3, and 9, the sender key generation unit 14 of the encryptor 12 selects a predetermined finite group on an elliptic curve ECC1 defined as a finite body (S30).

The message encryptor 16 generates a public key using the finite group selected by S30, and uses the sender private key j and the recipient public key h on the generated public key to send the message m. Ciphertext c _{2 for} ) can be generated (S32).

10 is a flowchart illustrating a message decoding method according to an embodiment of the present invention. 2, 3, and 10, the selector 11-1 of the public key generator 11 selects at least one point P on an elliptic curve ECC1 defined by a finite body ( S40).

The public key generator 11-3 may generate a public key of a real object group using an elliptic curve equation and a characteristic equation (for example, Equation 3) defining an elliptic curve (S42).

The decryptor 18 is configured to at least one of the public key generated by step S42, the receiver private key k received from the encryptor 12, the receiver public key h, and the sender public key c ₁ . Based on the ciphertext (c ₂ ) for the message (m) can be decrypted (S44).

The present invention can be embodied as computer readable codes on a computer readable recording medium. The computer-readable recording medium includes all kinds of recording devices in which data that can be read by a computer system is stored. Examples of computer-readable recording media include ROM, RAM, CD-ROM, magnetic tape, optical data storage, and the like. The computer readable recording medium can also be distributed over network coupled computer systems so that the computer readable code is stored and executed in a distributed fashion. And functional programs, codes and code segments for implementing the present invention can be easily inferred by programmers in the art to which the present invention belongs.

Although the present invention has been described with reference to one embodiment shown in the drawings, this is merely exemplary, and those skilled in the art will understand that various modifications and equivalent other embodiments are possible therefrom. Therefore, the true technical protection scope of the present invention will be defined by the technical spirit of the appended claims.

BRIEF DESCRIPTION OF THE DRAWINGS In order to better understand the drawings cited in the detailed description of the invention, a brief description of each drawing is provided.

1 is a graph for explaining the characteristics of a finite body included on a general elliptic curve.

2 is a block diagram of an encryption / decryption system according to an embodiment of the present invention.

3 is a graph showing a group of real bodies generated according to an embodiment of the present invention.

4 is a graph illustrating a process of matching a real body group generated according to an embodiment of the present invention to a finite body.

5 is a graph showing a result of matching a real body group generated according to an embodiment of the present invention to a finite body.

6 is a graph illustrating characteristics of the real object groups generated according to an embodiment of the present invention.

7 is a flowchart of a method for generating an elliptic curve cryptography public key according to an embodiment of the present invention.

8 is a flowchart illustrating a message encryption method according to an embodiment of the present invention.

9 is a flowchart illustrating a message encryption method according to another embodiment of the present invention.

10 is a flowchart illustrating a message decoding method according to an embodiment of the present invention.

Claims (15)

Selecting at least one point on an elliptic curve defined by a finite body; Calculating at least one real body group using an elliptic curve equation and a characteristic equation defining the elliptic curve; Matching the at least one real body group to finite body coordinates on the elliptic curve at a distance closest to the at least one real body group; And And generating a public key coordinates on the matched elliptic curve. The method of claim 1, wherein the elliptic curve equation, y ² = a * x ³ + b * x + c (where a, b, and c are real numbers) corresponding to elliptic curve cryptography. 3. The method of claim 2, wherein a is 1, b is −1, and c is 1. 4. The method of claim 1, wherein the elliptic curve equation, y ² = x ³ − p * x − q where p and q are real numbers, and the characteristic equation corresponds to Equation 9, Computing the real object group, The x coordinate (x _R) for generating a least one of the real chegun coordinates by using the solutions of the characteristic equation, and the at least one real chegun is s ² - ² * x _p (where, s is the characteristic And y coordinate (y _R ) of the at least one real object group corresponds to y _p + s * (x _R -x _p ). s = (3x _p ² -p) / (2y _p ), where x _p is the x coordinate of the selected point and y _p is the y coordinate of the selected point The method of claim 4, wherein the matching comprises: Using the s, matching the at least one real body group to finite field coordinates on the elliptic curve at a distance closest to the at least one real body group, and matching the matched coordinates on the elliptic curve to the at least one. Elliptic curve cryptography public key generation method comprising the step of replacing the coordinates of the real object group of. The method of claim 5, wherein the generating with the public key comprises: And generating a public key other than at least one of the matched coordinates on the elliptic curve as a public key. An encryption method for encrypting a message based on a public key generated by the elliptic curve cryptography public key generation method of claim 1. A decryption method for decrypting a cipher text based on a public key generated by the public key generation method of the elliptic curve cryptography method of claim 1. Selecting at least one point on an elliptic curve defined by a finite body; Calculating at least one real body group using an elliptic curve equation and a characteristic equation defining the elliptic curve; Matching the at least one real body group to finite body coordinates on the elliptic curve at a distance closest to the at least one real body group; Generating coordinates on the matched elliptic curve with a public key; And Generating a cipher text for the message using a sender private key and a receiver public key existing on the generated public key. delete A computer-readable recording medium having recorded thereon a program for performing the method of claim 1. A selection unit for selecting at least one point on an elliptic curve defined by a finite body; And Calculating at least one real body group using an elliptic curve equation and a characteristic equation defining the elliptic curve, Matching the at least one real body group with finite field coordinates on the elliptic curve at a distance closest to the at least one real body group, An elliptic curve cryptography public key generator comprising a public key generator for generating coordinates on the matched elliptic curve as a public key. The method of claim 12, wherein the elliptic curve equation, y ² = x ³ -p * x-q, where a, b, and c are real, and the characteristic equation corresponds to Equation 10, The public key generation unit, The x coordinate (x _R) of generating at least one coordinate of the real chegun of using the solutions of the characteristic equation, and the at least one real chegun is s ² - ² * x _p (Here, s is a solution of the characteristic equation And y-coordinate (y _R ) of the at least one real object group corresponds to y _p + s * (x _R -x _p ). s = (3x _p ² -p) / (2y _p ), where x _p is the x coordinate of the selected point and y _p is the y coordinate of the selected point The method of claim 12, wherein the public key generation unit, Using the s, matching the at least one real body group to finite field coordinates on the elliptic curve at a distance closest to the at least one real body group, and matching the matched coordinates on the elliptic curve to the at least one. Elliptic curve cryptographic public key generator that substitutes the coordinates of the real family of. A message encryption system comprising the public key generator of claim 12.

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