KR20120028432A - Calculating apparatus and method for elliptic curve cryptography - Google Patents

Abstract

PURPOSE: An operation apparatus and a method thereof for ECC(Elliptic Curve Cryptography) are provided to reduce the number of operations and increase performance speed based on converted scalar multiplication. CONSTITUTION: Scalar multiplication about a point on an elliptic curve is set up through a first coordinate axis(310). The preset scalar multiplication is converted into a square form(320). The scalar multiplication of a square form is transformed into the form of a projective coordinate(330). The coordinate value about the first coordinate axis is calculated by using the scalar multiplication(340).

Description

Calculating apparatus and method for elliptic curve cryptography

The present invention relates to a computing device and method for encryption using an elliptic curve, and more particularly, a computing device for processing a scalar multiplication, which is a geometric operation on points represented on an elliptic curve for an elliptic curve encryption, and a recording medium recording the same. It is about.

Cryptology is the study of linguistic and mathematical methodologies to protect information. It is jointly researched and developed in various disciplines such as computer and communication, mainly mathematics. In a typical cryptographic system, when a sender encrypts and transmits a plaintext that can know the meaning of a message, the receiver decrypts the received ciphertext again and restores the plaintext. Certain rules or methods used in the encryption and decryption process are called encryption keys.

The key may be classified into a symmetric cipher key having the same encryption key used in the encryption process and a decryption key used in the decryption process, and an asymmetric cipher key having a different encryption key and decryption key. The security of the cryptographic system depends on the length of the key and the secure management of the key. In a symmetric cryptographic key system, the cryptographic key and the decryption key are the same. That is, in the symmetric encryption key system, it is preferable that both the creator (sender) and the receiver of the ciphertext belong to a closed user group. However, such a symmetric cryptographic key system is not only difficult to maintain the security of the key, but also has the inconvenience that the number of keys to be managed increases rapidly as the number of users belonging to the system group increases.

A cryptographic system that emerged to overcome this limitation is an asymmetric cryptographic system. In asymmetric cryptography, each user is given two keys. One of the two keys (public key) is public and the other (private key) must be secretly managed by the user. Thus, asymmetric cryptosystems are also referred to as public key cryptosystems. Since the asymmetric encryption system has a different encryption key and decryption key, several arithmetic operations are performed in the encryption and decryption process based on the mathematical characteristics of both. As a result, the asymmetric cryptosystem has a weakness compared to the symmetric cryptographic key system.

The technical problem to be solved by the present invention is to overcome the limitation that the performance slows down due to the need for many operations in the encryption and decryption process in the asymmetric encryption system, and also consumes a lot of system resources for encryption and decryption, so that encryption and decryption It is to solve the problem that the implementation environment of the system is restricted.

In order to solve the above technical problem, the operation method for the elliptic curve encryption according to the present invention is to set the scalar multiplication for the point using only the first coordinate axis of the coordinate space representing the point on the elliptic curve ; Converting the set scalar multiplication into a square form; Converting the scalar multiplication of the square form into a form of projective coordinate; And calculating a coordinate value with respect to the first coordinate axis using scalar multiplication in the form of the projection coordinate.

In order to solve the above technical problem, the calculation method for the elliptic curve encryption according to the present invention further comprises the step of calculating the coordinate value for the second coordinate axis in the coordinate space from the scalar multiplication of the projection coordinate form for the first coordinate axis. .

In order to solve the other technical problem, the operation method for the elliptic curve encryption according to the present invention comprises the steps of receiving a secret key; Setting a scalar multiplication for the point using only a first coordinate axis in a coordinate space representing a point on an elliptic curve; Converts the set scalar multiplication into a square form, converts the converted scalar multiplication into a form of projection coordinates, and calculates a coordinate value with respect to the first coordinate axis using scalar multiplication of the converted projection coordinate form Making; Calculating coordinate values for a second coordinate axis in the coordinate space from a scalar multiplication in the form of projection coordinates for the first coordinate axis; And generating a public key from the coordinate values for the first and second coordinate axes using the received secret key. In addition, the operation method for the elliptic curve encryption includes the steps of: receiving an encrypted cipher text using the generated public key; And decrypting the plain text from the received cipher text using the secret key.

In order to solve the another technical problem, the operation method for the elliptic curve encryption according to the present invention comprises the steps of receiving a public key generated using a predetermined operation method; And generating a cipher text from the plain text using the received public key, wherein the predetermined operation method sets a scalar multiplication for the point using only a first coordinate axis in a coordinate space representing a point on an elliptic curve. And converting the set scalar multiplication into a square form, converting the converted scalar multiplication into a form of projection coordinates, and using the scalar multiplication of the converted projected coordinate form, a coordinate value for the first coordinate axis. Calculates a coordinate value for a second coordinate axis in the coordinate space from a scalar multiplication in the form of projection coordinates with respect to the first coordinate axis, and uses the received secret key for the first and second coordinate axes. It is preferable to generate a public key from coordinate values.

Further, hereinafter, a computer readable recording medium having recorded thereon a program for executing the calculation method for the elliptic curve cipher described above is provided.

In order to solve the above technical problem, the computing device for the elliptic curve encryption according to the present invention comprises an input unit for receiving a coordinate value representing a point on the elliptic curve; A calculation unit for calculating a scalar multiplication result of the elliptic curve from the input coordinate value using a predetermined calculation method; And an output unit configured to output the calculated scalar multiplication result value, wherein the predetermined calculation method comprises setting a scalar multiplication of the point using only a first coordinate axis in a coordinate space representing a point on the elliptic curve, It is preferable to convert the set scalar multiplication into a square form, convert the square scalar multiplication into a projection coordinate form, and calculate a coordinate value with respect to the first coordinate axis using the scalar multiplication of the projection coordinate form.

According to the present invention, a relatively small number of operations are required for the encryption and decryption process of an asymmetric cryptographic system by calculating coordinate values for points on an elliptic curve using the transformed scalar multiplication, thereby improving the execution speed. In addition, by reducing the system resources required for encryption and decryption, it is possible to implement an effective and reliable encryption and decryption system even in an environment with weaker computing power, such as smart cards.

1 is a view for explaining a process of encrypting a plain text and decrypting it again in a public key cryptosystem.
FIG. 2 is a pseudo code of the Montgomery ladder algorithm illustrated to explain an algorithm of an elliptic curve password.
3 is a flowchart illustrating a calculation method for an elliptic curve encryption according to an embodiment of the present invention.
4 is a diagram for describing a method of restoring coordinate values for a second coordinate axis from scalar multiplication in the form of projection coordinates for a first coordinate axis in an operation method for an elliptic curve cipher according to an embodiment of the present invention.
5 is a pseudo code illustrating an algorithm for explaining a calculation method for an elliptic curve encryption according to an embodiment of the present invention.
FIG. 6 is a diagram for describing a process of a ciphertext sender and a ciphertext receiver, respectively, for an operation method for an elliptic curve cipher according to another embodiment of the present invention.
FIG. 7 is a diagram illustrating an operation device for an elliptic curve encryption according to an embodiment of the present invention.

Before describing the embodiments of the present invention in detail, a public key cryptographic system, which is a representative environment in which embodiments of the present invention can be implemented, will be briefly described with reference to FIG. 1. 1 is a view for explaining a process of encrypting and decrypting plain text again in a public key cryptosystem, in which a public key cryptosystem includes a sender 10 representing an encryption side, a receiver 20 representing a decryption side, and a public key management. System 30.

As briefly introduced above, in a public key cryptosystem, each user is given two keys, a secret key 41 and a public key 42. In the public key cryptosystem, each user only needs to manage his or her private key, thereby reducing the difficulty of key management. In this case, it is assumed that the public key is used as an encryption key when the key is exchanged, and the public key is decrypted using the secret key. Typically, public key cryptosystems generally have the same structure as described above, but they can also be used in reverse when signing electronically. Hereinafter, in the embodiment of FIG. 1, a public key and a private key will be described based on the receiver 20.

The public key management system 30 is also called a public key directory, and a user using the public key cryptosystem can freely access the management system 30 to read other users' public keys. On the other hand, the physical location of the public key management system 30 may be located on the sender 10 or the receiver 20 side for convenience of implementation, or may be located in a separate space as shown in FIG. . In FIG. 1, a plurality of public keys may be stored in the public key management system 30, and a public key 42 corresponding to the secret key 41 of the receiver 20 may also be stored.

First, the sender 10 accesses the public key management system 30 to generate a cipher text by encrypting the plain text using the public key 42 of the receiver 20. The cipher text thus generated is transmitted to the receiver 20. Next, the receiver 20 decrypts the received cipher text using its secret key 41. That is, the receiver 20 decrypts the cipher text and restores the plain text.

Although the public key management system 30 stores public keys of various users in the structure of the cipher system above, anyone can freely generate a cipher text using the public key management system. However, the cipher text can be decrypted as plain text. Only the user who owns the private key corresponding to the public key used in the process is allowed. For example, in the example of FIG. 1, even if a third party other than the receiver 20 intercepts the cipher text generated by the sender 10 in the middle, the cipher text cannot be decrypted unless there is a secret key unique to the receiver 20.

As described above, since the public key and the private key granted to each user in the public key cryptosystem are mathematically related, encryption and decryption are possible. In other words, public key cryptography is based on mathematical problems that are very difficult to solve without a specific kind of information that only the designer knows. The creator of the key exposes the problem (public key) to the public and hides certain information (secret key) so that only he or she can know it. Then, no matter whoever uses the problem to encrypt the message, only the creator of the key can solve the problem and decrypt the original message.

Embodiments of the present invention are based on such a public key cryptographic system, but employ a method called elliptic curve cryptography, particularly for this mathematical problem. Elliptic Curve Cryptography (ECC) is a type of public key cryptography based on elliptic curve theory that solves the Elliptic Curve Discrete Logarithm Problem (ECDLP), which is known to be difficult to solve in finite fields. The password is based on. The elliptic curve discrete log problem means that when Q = kP is multiplied by an arbitrary point P on an elliptic curve, it is difficult to calculate the integer k even if the points Q and P are known. Therefore, the key operation of the elliptic curve cryptosystem is to find the scalar multiplication, that is, Q = kP, and this scalar multiplication has the greatest impact on the safety and efficiency of the elliptic curve cryptosystem.

The elliptic curve in space is composed of points of (x, y) and has the form of an elliptic curve represented by the relationship between the x and y coordinates. The scalar multiplication operation is performed by iteratively using a doubling operation that sums two identical points and an addition operation that sums two different points. However, the elliptic curve doubling (ECDBL) operation and the elliptic curve addition (ECADD) operation used in the elliptic curve scalar multiplication have a difference in the operation performed according to the key bit. Many analyzes were performed by SPA).

As simple power analysis of elliptic curve scalar multiplication algorithm proceeds, many countermeasures have been proposed to defend against this. Among these countermeasures, the Montgomery algorithm is the first algorithm proposed by Montgomery, and it is a method that can speed up operations by performing scalar multiplication operations on Montgomery forms using only x coordinates. Since then, Lopez and Dahab and Izu and Takagi have proposed equations for the Montgomery ladder algorithm using only x-coordinates in the general Weierstrass form. The Montgomery ladder algorithm is known to be able to perform fast and efficient scalar multiplication on elliptic curves by using only x coordinates, and is known to be very efficient in terms of point compression.

FIG. 2 is a pseudo code of the Montgomery ladder algorithm exemplified for explaining an algorithm of an elliptic curve password, which is performed in the following steps.

First, input a point P on the elliptic curve and a constant d (scalar). In the Montgomery ladder algorithm of FIG. 2, an elliptic curve doubling (ECDBL) operation and an elliptic curve addition (ECADD) operation are repeatedly performed in each loop. Therefore, the Montgomery ladder algorithm performs the same operation every loop of scalar multiplication and thus may be safe for simple power analysis. The Montgomery ladder algorithm is characterized in that the x coordinate of P + Q can be obtained using only the X coordinates of three points P, Q and P-Q in the Bayerstrass form.

However, the various methods proposed above for the elliptic curve scalar multiplication also require a lot of calculations, resulting in slow performance and excessive system resources. Accordingly, embodiments of the present invention propose a method capable of reducing the number of operations required for a scalar operation and the resources required for each operation. Hereinafter, various embodiments of the present invention will be described in detail with reference to the accompanying drawings.

3 is a flowchart illustrating a calculation method for an elliptic curve encryption according to an embodiment of the present invention. For convenience of description, the following embodiments of the present invention illustrate an elliptic curve of Jacobi quartic form as an elliptic curve function. The elliptic curve of the Jacobi fourth order can be expressed as Equation 1 below, where the degree of x is the fourth order, and a person having ordinary knowledge in the art to which the present invention pertains has various equivalents in addition to the Jacobi fourth elliptic curve. It can be seen that the elliptic curve can be used to implement the technical idea of the present invention.

이고 소수 p를 선택하였다고 가정하면, 유한체 F_p 상의 자코비 4차 타원곡선은 상기된 수학식 1과 같이 정의될 수 있다. 이 타원곡선 위의 점

,

및

과 두 점의 차가 다음의 수학식 2와 같이 유지되면 이하에서 소개될 수학식 3 내지 수학식 5를 만족한다.In other words,

Assuming a select small number p, the finite field F _p The Jacobian fourth-order elliptic curve of the image may be defined as in Equation 1 above. A point on this elliptic curve

,

And

If the difference between and is maintained as in Equation 2 below, Equations 3 to 5 described below are satisfied.

In particular, Equation 2 indicates that the algorithm of FIG. 2 regarding the Montgomery ladder algorithm is also satisfied, and the difference between two points in each loop maintains a constant value P.

Referring to FIG. 3 in order based on the above contents, it is as follows.

In step 310, a scalar multiplication is set for the point using only the first coordinate axis in the coordinate space representing the point on the elliptic curve. In the embodiment of the present invention, in applying the Montgomery ladder algorithm in the Jacobi fourth order elliptic curve, a scalar multiplication operation is performed using only x ² coordinates. That is, only the coordinate value of the x coordinate axis which is one coordinate axis in the two-dimensional coordinate space of (x, y) is used.

More specifically, the addition operation and the doubling operation used in scalar multiplication are defined as in Equation 3 below.

In step 320, the scalar multiplication set in step 310 is converted into a square form. That is, Equation 3 can be summarized as Equation 4 below with respect to the square of the x coordinate.

Through this transformation, it is possible to express addition and doubling operations, which are major operations of scalar multiplication operations, using x ² coordinates. As such, when the expression method using the x ² coordinate is used, the square operation is reduced compared to the scalar multiplication using the x coordinate. In order to calculate the sum of two points in the scalar multiplication process in an elliptic curve, multiplication, division and addition operations are required as shown in Equations 3 to 4 above. Among these, the operation that consumes the most time and resources corresponds to the division operation. This is because these division operations include inverse operations. Therefore, in order to improve the performance of scalar multiplication in the elliptic curve cryptography method, it is necessary to exclude such inverse arithmetic operation as much as possible for the efficiency of the computation.

의 투영 좌표(projective coordinate)의 형태로 변환한다. 이러한 투영 좌표 변환은 나눗셈 연산(역원 연산)을 소거시키는 역할을 한다. 따라서, 상기 수학식 4는 투영 좌표계

에 의해 다음의 수학식 5와 같이 표현될 수 있다.In step 330, the scalar multiplication of the square form transformed in step 320 is performed.

Convert to the form of projective coordinates. This projection coordinate transformation serves to cancel the division operation (inverse operation). Therefore, Equation 4 is a projection coordinate system

It can be expressed by the following equation (5).

의 연산량은 '4M+2S'로 표현될 수 있으며, 타원곡선 더블링 연산

의 연산량은 '2M+2S+1C'로 표현될 수 있다. 여기서, M은 곱셈(multiplication)을 의미하고, S는 제곱(square)을 의미하며, C는 상수(constant)를 의미한다.At this time, elliptic curve addition calculation

Can be expressed as '4M + 2S' and elliptic curve doubling operation

May be expressed as '2M + 2S + 1C'. Here, M means multiplication, S means square, and C means constant.

In operation 340, a coordinate value of the first coordinate axis is calculated using scalar multiplication of the converted projection coordinates in operation 330. That is, step 340 calculates the square value of the x coordinate of kP.

Steps 310 to 340 described above correspond to a process of converting the set scalar multiplication to be performed through fewer operations and calculating a calculation result according to the conversion formula, so that information and mathematical expressions in electronic form can be processed. It can be implemented through a processor and a memory required for such an operation, and a controller capable of appropriately controlling data processing between the processor and the storage space may be utilized as necessary. . Such a processor, a storage space, and a controller may be appropriately selected by those skilled in the art in consideration of the utilization environment or operating environment of the technical field to which the present invention belongs. Furthermore, additional software code for controlling the hardware exemplified above may be utilized in the process of setting and converting the scalar multiplication.

Meanwhile, step 350 describes a process of restoring the y coordinate after the above-described process, and step 350 may be additionally performed. Through the above steps 310 to 340, the embodiment of the present invention can obtain the square value of the x coordinate of kP through the proposed scalar multiplication. However, in the elliptic curve cryptography system, since not only the x coordinate but also the y coordinate value is needed, it is necessary to restore it.

Therefore, in operation 350, a coordinate value for the second coordinate axis in the coordinate space is calculated from a scalar multiplication in the form of projection coordinates for the first coordinate axis. In other words, a coordinate value for the y axis that is the second coordinate axis is calculated from the scalar multiplication for the x axis that is the first coordinate axis. More specifically, the following equation (6) is used to restore the y coordinate value.

의 연산량은 '4M+4S'이다.Equation 6 is introduced to reduce the amount of computation required to restore the y coordinate,

The computation amount of is '4M + 4S'.

A method of restoring y coordinates is now presented with reference to FIG. 4. 4 is a diagram for describing a method of restoring coordinate values for a second coordinate axis from scalar multiplication in the form of projection coordinates for a first coordinate axis in an operation method for an elliptic curve cipher according to an embodiment of the present invention.

,

및

과 두 점의 차

는 일정한 값으로 유지된다. 즉, 타원곡선상에 위치한 2개의 점들의 차이가 P로서 유지되는 특징을 갖는다.Figure 4 also illustrates the Jacobian fourth elliptic curve, the point on the elliptic curve

,

And

And the difference between two points

Is kept constant. That is, the difference between the two points located on the elliptic curve is maintained as P.

Referring to FIG. 4, first, after the end of the loop when i = 1 in the Montgomery ladder algorithm, Q [0] = ((kk ₀ ) / 2) P, and Q [1] = ((kk ₀ ) / 2) You can get P + P.

를 이용하여 Q[0], Q[1] 및 -P=(-x,y)로부터 Q[0]-P를 산출한다. 이어서,

를 이용하여 Q[0]-P, Q[1] 및 2P로부터 2Q[0]=(k-k₀)P를 산출하고, Q[0], Q[1] 및 P로부터 2Q[0]+P=(k-k₀)P+P를 각각 산출한다.after that,

Calculate Q [0] -P from Q [0], Q [1] and -P = (-x, y) using. next,

Calculate 2Q [0] = (kk ₀ ) P from Q [0] -P, Q [1] and 2P using 2Q [0] + P = from Q [0], Q [1] and P Compute (kk ₀ ) P + P respectively.

As a result, kP is calculated as follows. If k ₀ = 0 (which means the last bit is 0), then restore the y coordinate as 2Q [0] = kP-k ₀ P; if k ₀ = 1 (which means the last bit is 1), 2Q. [0] = kP-k ₀ Restore the y coordinate as P + P.

The method of calculating the y coordinate through this restoration procedure is expressed by Equation 7 below.

The calculation amount of Equation 7 may be expressed as '1I + 4M'. Where I stands for inverse.

,

및

를 사용하여 y 좌표를 복원하는 본 발명의 실시예를 전체적으로 도시하고 있다.FIG. 5 is a pseudo code illustrating an algorithm for explaining an operation method for an elliptic curve cipher according to an embodiment of the present invention, wherein the scalar multiplication transformed into a square form and a projection coordinate form is applied to a point on an elliptic curve. Compute the x coordinate value for

,

And

The embodiment of the present invention for restoring the y coordinate using is illustrated as a whole.

)과 타원곡선 애디션 연산(

)을 수행한다. 여기서, 타원곡선 애디션 연산과 타원곡선 더블링 연산은 각각 다른 저장공간을 활용하여 각각 다른 점들에 대한 연산을 수행하므로 병렬적으로 수행될 수 있다.In FIG. 5, the exemplary algorithm of the present invention receives a point P and a scalar d on an elliptic curve. The algorithm now uses an elliptic curve doubling operation on the square of the x coordinate,

) And elliptic curve addition operations (

). Here, the elliptic curve addition operation and the elliptic curve doubling operation may be performed in parallel because each operation is performed on different points using different storage spaces.

Subsequently, the algorithm of FIG. 5 has a computation amount of '4M' because the ECADD operation yields a value of (x, z) through lines 8 and 9. Finally, we compute the y coordinates using the ECRE ^{(x, y)} operation.

According to the embodiments of the present invention described above, by using a scalar multiplication transformed into a square form and a projection coordinate form, a coordinate value for a point on an elliptic curve is calculated so that a relatively small number of encryption and decryption processes of an asymmetric encryption system are performed. Operation is required, which speeds up its execution. In addition, by reducing the system resources required for encryption and decryption, it is possible to implement an effective and reliable encryption and decryption system even in an environment with low computing power.

Comparing the operation method for the elliptic curve encryption proposed in the improved embodiments of the present invention with the known operation method as follows.

)을 수행하기 위해 필요공간(데이터 저장을 위한 레지스터를 의미한다.)이 2로 감소하였고, 연산량도 감소함으로 인해 연산 속도가 향상될 수 있음을 알 수 있다.Elliptic curve addition calculation proposed in the embodiment of the present invention in Table 1

The required space (meaning a register for storing data) is reduced to 2, and the computation speed can be improved by reducing the amount of computation.

)을 수행하기 위해 필요공간(데이터 저장을 위한 레지스터를 의미한다.)이 2로 감소하였고, 연산량도 감소함으로 인해 연산 속도가 향상될 수 있음을 알 수 있다.Subsequently, in Table 2, an elliptic curve doubling operation proposed in the embodiment of the present invention (

The required space (meaning a register for storing data) is reduced to 2, and the computation speed can be improved by reducing the amount of computation.

This reduction in required space can be more clearly seen with reference to the algorithm illustrated in FIG. 5 described above. That is, in FIG. 4, the elliptic curve addition operation uses registers for storing Q [0] and Q [1], respectively, and the elliptic curve doubling operation uses Q [0] and Q [2], respectively. We will use register to save.

Now, an embodiment of applying to a practical public key cryptosystem by using the operation method for elliptic curve cryptography proposed above is proposed.

FIG. 6 is a diagram for describing a process of a ciphertext sender and a ciphertext receiver, respectively, for an operation method for an elliptic curve cipher according to another embodiment of the present invention. It illustrates how the scalar multiplication operation is utilized.

The utilization process in terms of the receiver 20 is as follows. First, in step 21, the receiver 20 receives a secret key. This input may be directly input by the receiver 20, or may be automatically generated by the encryption system of the receiver 20 as necessary.

Next, in step 22, coordinate values for the first and second coordinate axes of the points on the elliptic curve are calculated using the transformed scalar multiplication. Herein, the transformed scalar multiplication means the scalar multiplication transformed into the above-described square form and projection coordinates. That is, step 22 calculates x and y coordinate values by using a scalar multiplication proposed through embodiments of the present invention.

In step 23, the receiver 20 generates a public key from coordinate values for the first and second coordinate axes calculated in step 22 using the secret key input in step 21. The generated public key may be registered in the public key management system described above with reference to FIG. 1, and then may be viewed by the sender 10 who wants to create a cipher text.

Now, when it comes to the side of the sender 10, the sender 10 obtains the public key generated by the receiver 20 in step 11 (meaning public key reading). Next, in step 12, the sender 10 generates a cipher text from the plain text using this public key. The generated cipher text can be restored only by the secret key of the receiver 20, and even if obtained by a third party, decryption is impossible.

Returning to the side of the receiver 20 again, the receiver 20 receives the encrypted text encrypted by the sender 10 in step 24. Finally, the receiver 20 decrypts the plain text from the cipher text using its private key in step 25.

According to another embodiment of the present invention described above, a relatively small number of operations are required for the encryption and decryption process of the public key cryptographic system, thereby improving its execution speed and implementing an elliptic curve encryption system with only a small amount of system resources. It is possible to do

FIG. 7 illustrates an operation device 700 for elliptic curve encryption according to an embodiment of the present invention, and includes an input unit 721, an operation unit 722, and an output unit 723. In addition, the storage unit 724 may be optionally included for an addition operation and a doubling operation for substantially implementing scalar multiplication.

The input unit 721 receives a coordinate value representing a point on an elliptic curve. The input unit 721 may be implemented as an input device capable of receiving data in an electronic form, and may include a physical interface as necessary.

The calculation unit 722 calculates a scalar multiplication result of the elliptic curve from the coordinate value received through the input unit 721 using the transformed scalar multiplication calculation method proposed by the embodiments of the present invention. The operation unit 722 corresponds to steps 310 to 350 described above with reference to FIG. 3 and a detailed description thereof will be omitted.

However, in implementing the arithmetic unit of the present embodiment in hardware, the arithmetic unit 722 may be implemented as a combination of a modular multiplier, a modular adder, and a subtractor for performing the enumerated operations. . The design of such hardware can be implemented by those skilled in the art of circuit design in the technical field to which the present invention belongs.

The output unit 723 outputs a scalar multiplication result value calculated through the operation unit 722. The output unit 723 may be implemented as an input device capable of outputting data processed in an electronic form, and may include a physical interface as necessary.

On the other hand, since the scalar multiplication set by the operation unit 722 includes an elliptic curve addition operation and an elliptic curve doubling operation, each of the at least two storage units 724 is used to perform the elliptic curve addition operation and the elliptic curve doubling operation. Can be. As described above in Tables 1 and 2, the two operations of scalar multiplication require two registers each. Furthermore, the computing device of this embodiment may include a controller for controlling the hardware devices described above and a data bus for transmitting input / output data.

According to the above-described embodiments of the present invention, a relatively small number of operations are required for the scalar multiplication operation process, thereby improving its execution speed, and it is possible to implement an elliptic curve encryption system with only a few system resources.

Meanwhile, 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, floppy disks, optical data storage devices, and the like, which may also be implemented in the form of carrier waves (for example, transmission over the Internet). Include. 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. In addition, functional programs, codes, and code segments for implementing the present invention can be easily deduced by programmers skilled in the art to which the present invention belongs.

The present invention has been described above with reference to various embodiments thereof. Those skilled in the art will understand that the present invention can be implemented in a modified form without departing from the essential features of the present invention. Therefore, the disclosed embodiments should be considered in an illustrative rather than a restrictive sense. The scope of the present invention is shown in the claims rather than the foregoing description, and all differences within the scope will be construed as being included in the present invention.

10: sender (encryption side) 20: receiver (decoding side)
30: public key management system (public key directory)
41: secret key 42: public key
700: elliptic curve encryption device
721 input unit 722 operation unit
723: output unit 724: storage unit

Claims (12)

Setting a scalar multiplication for the point using only a first coordinate axis in a coordinate space representing a point on an elliptic curve;
Converting the set scalar multiplication into a square form;
Converting the scalar multiplication of the square form into a form of projective coordinate; And
Calculating a coordinate value with respect to the first coordinate axis by using a scalar multiplication in the form of the projection coordinate. The method of claim 1,
Calculating coordinate values for a second coordinate axis in the coordinate space from scalar multiplication in the form of projection coordinates for the first coordinate axis. The method of claim 2,
The set scalar multiplication method of claim 2, wherein the difference between the two points located on the elliptic curve is maintained at a predetermined value. The method of claim 1,
The set scalar multiplication includes an elliptic curve addition operation and an elliptic curve doubling operation,
The elliptic curve addition operation and the elliptic curve doubling operation are performed in parallel. Receiving a secret key;
Setting a scalar multiplication for the point using only a first coordinate axis in a coordinate space representing a point on an elliptic curve;
Converts the set scalar multiplication into a square form, converts the converted scalar multiplication into a form of projection coordinates, and calculates a coordinate value with respect to the first coordinate axis using scalar multiplication of the converted projection coordinate form Making;
Calculating coordinate values for a second coordinate axis in the coordinate space from a scalar multiplication in the form of projection coordinates for the first coordinate axis; And
And generating a public key from the coordinate values of the first and second coordinate axes using the received secret key. 6. The method of claim 5,
Receiving an encrypted cipher text using the generated public key; And
And decrypting the plain text from the received cipher text using the secret key. Receiving a public key generated using a predetermined calculation method; And
Generating a cipher text from the plain text using the received public key,
The predetermined calculation method,
Sets a scalar multiplication for the point using only the first coordinate axis of the coordinate space representing the point on the elliptic curve,
Converts the set scalar multiplication into a square form, converts the converted scalar multiplication into a projection coordinate form, and calculates a coordinate value with respect to the first coordinate axis by using scalar multiplication of the converted projection coordinate form and,
Calculating a coordinate value for a second coordinate axis in the coordinate space from a scalar multiplication in the form of projection coordinates for the first coordinate axis,
And a public key is generated from the coordinate values of the first and second coordinate axes using the received secret key. A computer-readable recording medium storing a program for causing a computer to execute the method according to any one of claims 1 to 7. An input unit for receiving a coordinate value representing a point on an elliptic curve;
A calculation unit for calculating a scalar multiplication result of the elliptic curve from the input coordinate value using a predetermined calculation method; And
An output unit for outputting the calculated scalar multiplication result value,
The predetermined calculation method,
Sets a scalar multiplication for the point using only a first coordinate axis in a coordinate space representing a point on the elliptic curve,
Convert the set scalar multiplication into a square form,
Converts the scalar multiplication of the square form into the form of projection coordinates,
And calculating a coordinate value with respect to the first coordinate axis by using scalar multiplication of the projection coordinate type. The method of claim 9,
And the calculating unit further calculates a coordinate value for a second coordinate axis in the coordinate space from a scalar multiplication in the form of projection coordinates for the first coordinate axis. The method of claim 10,
The set scalar multiplication apparatus of claim 11, wherein the difference between the two points on the elliptic curve is kept at a predetermined value. The method of claim 9,
The set scalar multiplication includes an elliptic curve addition operation and an elliptic curve doubling operation,
And at least two storage units for performing the elliptic curve addition operation and the elliptic curve doubling operation, respectively.

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