KR101524661B1 - Calculating apparatus and method for Triple-Base Chain elliptic curve scalar multiplication by reordering - Google Patents

Abstract

The present invention relates to a calculating apparatus and method of a {2,3,5} triple-base chain-based elliptic curve scalar multiplication by reordering. According to the present invention, the calculating method suggests a method for speeding up the elliptic curve scalar multiplication, by replacing an operation which can utilize an overlapping operation in the scalar multiplication operation with another operation by utilizing a pre-calculated operation, and reordering the operation of a base by using the overlapping operation.

Description

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a triple-base chain-based elliptic curve scalar multiplication by reordering,

The present invention relates to an arithmetic unit and method for an elliptic curve scalar multiplication based on a triple-base chain, and more particularly, to an arithmetic unit and method for scalar multiplication using an elliptic curve scalar multiplication, A computing device and a method in which a curved scalar multiplication expression is modified, and a recording medium on which the program is recorded.

의 부분집합

는 위수

인 순환군이 되고, 이때 생성원

를

의 원시근이라 한다. 즉, 적당한

가 존재하여

이다. 그러므로

의 모든 원소

은 법

에 관하여 적당한

가 존재하여

이다. 이 때,

를 원소

를 밑으로 하는 지수(index) 또는 이산로그(discrete logarithm)라 하고,

또는

로 나타낸다. 이산대수 문제란, 소수

에 대하여

의 한 원시근을

라 할 때,

가 주어졌을 때,

의 지수(이산로그)

를 구하는 문제이다.

와

를 알고

를 계산하는 문제는 상당히 어려운 것으로 알려져 있다. 이 성질을 이용하여 인수분해문제와 함께 공개키 암호 또는 키교환 프로토콜에 이용된다.Elliptic curve cryptosystem is mainly used for digital signature algorithm and key exchange algorithm by using discrete logarithm problem defined in operation between elliptic curve points on finite field. Finite element

Subset of

And

, And at this time,

To

. That is,

Is present

to be. therefore

All elements of

Law

Suitable for

Is present

to be. At this time,

Element

Is referred to as an index or a discrete logarithm,

or

Respectively. The discrete logarithm problem,

about

Of a circle

In other words,

When given,

Exponential (Discrete Log)

.

Wow

Know

Is known to be quite difficult. Using this property, it is used in public key cryptography or key exchange protocols with factoring problems.

{2,3,5} Triple-based chain-based elliptic curve Scalar multiplication is one of the operations between elliptic curve points (Non-Patent Document 1). The scalar multiplication of points on the elliptic curve has a great effect on the execution time of the entire encryption system because it takes a lot of time and resources. Therefore, it is a method to improve the stability and efficiency of the elliptic curve cryptosystem, Improvement is needed.

 P. K. Mishra, V. S. Dimitrov. Efficient Quintuple Formulas for Elliptic Curves and Efficient Scalar Multiplication Using Multibase Number Representation. Springer-Verlag, volume 4779, pages 390-406, 2007.

SUMMARY OF THE INVENTION It is an object of the present invention to improve the performance and efficiency of an encryption system by reducing the amount of operation of an elliptic curve scalar multiplication used for encryption considering the influence of the operation amount of the elliptic curve scalar multiplication on the execution speed of the elliptic curve encryption system There is. In particular, we propose a scheme to reduce the computation amount in scalar multiplication of a triple-base chain with {2,3,5} as a base, thereby speeding up the overall encryption performance.

를 표현하는 3차원 좌표값

및

(

는 양의 정수)를 입력받는 단계; 연산 모듈이 상기

를 {2,3,5} 밑수(Base)로 재 표현하는 단계; 상기 연산 모듈이 상기 타원곡선 스칼라 곱셈 연산식 내에 존재하는 중복 연산 활용이 가능한 연산을 추출하고, 상기 추출된 중복 연산 활용이 가능한 연산을 이용하여 상기 타원곡선 스칼라 곱셈 연산식을 변형하는 단계; 상기 연산 모듈이 상기 점

에 대하여 상기 변형된 타원곡선 스칼라 곱셈 연산식을 사용하여 타원곡선 스칼라 곱셈

를 수행하는 단계; 및 출력 모듈이 상기 수행된 타원곡선 스칼라 곱셈 결과값을 출력하는 단계;를 포함한다. According to an aspect of the present invention, there is provided an operation method for an elliptic curve scalar multiplication according to an embodiment of the present invention, including a {2,3,5} triple-base chain-based elliptic curve scalar multiplication ), The input module is a point on the elliptic curve

A three-dimensional coordinate value

And

(

Receiving a positive integer; When the operation module

To {2,3,5} base (Base); Extracting an operation that can be performed by the operation module existing in the elliptic curve scalar multiplication operation expression and transforming the elliptic curve scalar multiplication operation expression using an operation capable of utilizing the redundant operation; The operation module

Using the modified elliptic curve scalar multiplication equation for the elliptic curve scalar multiplication < RTI ID = 0.0 >

; And outputting the result of performing the elliptic curve scalar multiplication performed by the output module.

According to an aspect of the present invention, there is provided an operation method for an elliptic curve scalar multiplication according to an embodiment of the present invention, wherein the step of modifying an elliptic curve scalar multiplication operation expression includes: And replacing the operation with another operation.

According to an aspect of the present invention, there is provided a method for computing an elliptic curve scalar multiplication, the method comprising: rearranging a base operation sequence to 2 -> 3 -> 5.

The present invention also provides a computer-readable recording medium having recorded thereon a program for causing a computer to execute an arithmetic method for the above-described elliptic curve scalar multiplication.

를 표현하는 3차원 좌표값

및

(

는 양의 정수)를 입력받는 입력부; 상기

를 {2,3,5} 밑수(Base)로 재 표현하고, {2,3,5} 트리플-베이스 체인(Triple-Base Chain) 기반 타원곡선 스칼라 곱셈 연산식 내에 존재하는 중복 연산 활용이 가능한 연산을 추출하고, 상기 추출된 중복 연산 활용이 가능한 연산을 이용하여 상기 타원곡선 스칼라 곱셈 연산식을 변형하고, 상기 점

에 대하여 상기 변형된 타원곡선 스칼라 곱셈 연산식을 사용하여 타원곡선 스칼라 곱셈

를 수행하는 연산부; 및 상기 수행된 타원곡선 스칼라 곱셈 결과값을 출력하는 출력부를 포함한다. According to an aspect of the present invention, there is provided an arithmetic and logic unit for an elliptic curve scalar multiplication,

A three-dimensional coordinate value

And

(

A positive integer); remind

Is re-represented as {2,3,5} base and {2,3,5} triple-base chain based elliptic curve scalar multiplication operation And transforms the elliptic curve scalar multiplication operation expression using the extracted operation that can utilize the redundant operation,

Using the modified elliptic curve scalar multiplication equation for the elliptic curve scalar multiplication < RTI ID = 0.0 >

; And an output unit for outputting the result of performing the elliptic curve scalar multiplication.

The present invention speeds up the operation speed of the elliptic curve scalar multiplication by using the redundant operation and rearranges the base operation order of the {2,3,5} triple-base chain-based elliptic curve scalar multiplication operation expression to reduce the total operation amount. Therefore, it is possible to implement an efficient and reliable encryption / decryption system by improving the efficiency and performance in the encryption system using the elliptic curve calculation.

를 {2,3,5} 밑수로 나타내는 알고리즘의 의사코드이다.
도 2는 Pradeep Kumar Mishra 등이 제안한 {2,3,5} 트리플-베이스 체인에 기반한 타원곡선 스칼라 곱셈

를 구하는 알고리즘의 의사코드이다.
도 3은 본 발명의 일 실시예에 따른 트리플-베이스 체인 기반 타원곡선 스칼라 곱셈을 위한 연산 방법을 도시한 흐름도이다.
도 4는 Pradeep Kumar Mishra 등이 제안한

연산을 위한 중간 과정을 도시한 도면이다.
도 5는 본 발명의 일 실시예에 따른 중복 연산을 이용하여 변형한

연산을 위한 중간 과정을 도시한 도면이다.
도 6은 {2,3,5} 트리플-베이스 체인의 밑수 연산 순서를 종래 것과 본 발명의 일 실시예에 따른 것을 도시한 그림이다.
도 7은

의 연산 중간 값을 도시한 도면이다.
도 8은

의 연산 중간 값을 도시한 도면이다.
도 9는 본 발명의 일 실시예에 따른 중복 연산을 이용하여 변형한 {2,3,5} 트리플-베이스 체인 방법의 밑수 연산 순서를 재배치한 타원곡선 스칼라 곱셈 알고리즘을 예시한 의사코드이다.
도 10은 본 발명의 일 실시예에 따른 타원곡선 스칼라 곱셈 연산을 위한 장치를 도시한 도면이다. Figure 1 shows an algorithm

Is the pseudo code of the algorithm represented by {2,3,5} base.
FIG. 2 shows an example of an elliptic curve scalar multiplication based on {2,3,5} triple-base chains proposed by Pradeep Kumar Mishra et al.

Is the pseudo-code of the algorithm for obtaining the.
3 is a flowchart illustrating a computation method for triple-based chain-based elliptic curve scalar multiplication according to an embodiment of the present invention.
Fig. 4 is a schematic diagram

Lt; RTI ID = 0.0 > operation. ≪ / RTI >
FIG. 5 is a block diagram of a modified embodiment of the present invention,

Lt; RTI ID = 0.0 > operation. ≪ / RTI >
FIG. 6 is a diagram illustrating a base operation order of a {2,3,5} triple-base chain according to an embodiment of the present invention.
Figure 7

Of FIG.
Figure 8

Of FIG.
FIG. 9 is a pseudo code illustrating an elliptic curve scalar multiplication algorithm that rearranges the base operation order of the {2,3,5} triple-based chain method modified using the redundant operation according to an embodiment of the present invention.
10 is a diagram illustrating an apparatus for an elliptic curve scalar multiplication operation according to an embodiment of the present invention.

Before describing embodiments of the present invention in detail, an elliptic curve and an elliptic curve scalar multiplication used in embodiments of the present invention will be defined.

에 대하여,

위에서 정의된 정칙 타원곡선(nonsingular elliptic curve)을

라고 하자.

를 포함한

의 집합은 수학식 1에 표현된 바이어슈트라스 식(Weierstrass equation)을 만족한다. Finite element

about,

The nonsingular elliptic curve defined above

Let's say.

Including

Satisfies the Weierstrass equation expressed in Equation (1).

의 값에 따라 간결한 형태로 표현될 수 있다. 만약

이라면, 수학식 1은 아래 수학식 2와 같이 표현 된다. 이 때의 점들의 집합을 아핀 좌표계(Affine Coordinates)라 한다. Equation (1)

Can be expressed in a concise form according to the value of < RTI ID = 0.0 > if

, Equation (1) is expressed as Equation (2) below. The set of points at this time is called Affine Coordinates.

이다. 만약

이면, 수학식 1은 수학식 3과 같이 표현된다. At this time,

to be. if

, Equation (1) is expressed as Equation (3).

이다.

인 타원곡선에 대해서만 고려한다고 했을 때, 타원곡선 상에 존재하는 임의의 두 점을 각각

,

라 하면,

연산을 포인트 애디션(point addition, ADD)이라 하고,

연산은 포인트 더블링(point doubling, DBL)이라 한다. 아핀 좌표계에서 ADD, DBL 연산식은 수학식 4, 수학식 5와 같다. Equation 3 is non-supersingular,

to be.

When considering only the elliptic curve, two arbitrary points on the elliptic curve are referred to as

,

In other words,

The operation is called point addition (ADD)

The operation is called point doubling (DBL). In the affine coordinate system, the ADD and DBL arithmetic expressions are expressed by Equations (4) and (5).

,

을 포함한다. 역원 연산은 유한체 곱셈 연산의 30배가 소요될 정도로 비효율적이다. 역원 연산을 제외하고 효율적인 타원곡선 스칼라 곱셈을 수행할 방안으로 자코비안(Jacobian) 좌표계가 제안되었다. 자코비안 좌표계에서 ADD, DBL 연산식은 수학식 6, 수학식 7과 같다. Equations (4) and (5)

,

. Inverse operation is as inefficient as it takes 30 times of finite field multiplication operation. A Jacobian coordinate system has been proposed to perform efficient elliptic curve scalar multiplication except for inverse operations. In the Jacobian coordinate system, the ADD and DBL operations are expressed by Equations (6) and (7).

When the ADD and DBL operations are performed using the Jacobian coordinate system as shown in Equations (6) and (7), there is no inverse operation. Therefore, the present invention uses a Jacobian coordinate system.

를 양의 정수라 하고,

를 타원곡선 위의 한 점으로 정의하면, 타원곡선 스칼라 곱셈은

를

번 더하는 연산이다. You can define an elliptic curve scalar multiplication using ADD and DBL in an elliptic curve.

Is a positive integer,

Is defined as a point on the elliptic curve, the elliptic curve scalar multiplication

To

It is an operation to add.

에 대하여

와

은 동치 관계이고 무한원점은

으로 표현된다. 바이어슈트라스 식은 다음 수학식 8과 같다.P1363 Using the Jacobian coordinate system proposed in IEEE Standard, fast scalar multiplication is possible. The representation of a point in a Jacobian coordinate system is not unique. In other words,

about

Wow

And the infinite origin

. The Bayer-Strasse equation is shown in Equation (8).

,

,

이라 하면, 덧셈 연산과 두 배 연산은 수학식 9와 같다. In the Jacobian coordinate system

,

,

, The addition operation and the double operation are expressed by Equation (9).

를 밑수 {2,3,5}로 나타내어 스칼라 곱셈

를 수행하는 것이다. 예를 들어

이면 다음 수학식 10과 같이 표현할 수 있다.In the present invention, {2,3,5} triple-based chain-based elliptic curve scalar multiplication is used, which will be described first. In 2007, Pradeep Kumar Mishra and others proposed a triple-base chain with {2,3,5} as the base. Triple - a base chain is a positive integer

Is represented as a base < RTI ID = 0.0 > {2,3,5}

. E.g

The following equation (10) can be obtained.

를 {2,3,5} 밑수로 나타낼 때 각 밑수의 지수는 다음 항의 밑수의 지수보다 항상 같거나 커야 한다. 예를 들면 밑수 2의 지수는 5, 3, 1로 감소하는 형태이다. essence

Is {2,3,5}, the exponent of each base must always be greater than or equal to the exponent of the base of the next term. For example, the index of base 2 is reduced to 5, 3, 1.

를 {2,3,5} 밑수로 재표현하는 알고리즘의 의사 코드(pseudo code)이다. 본 발명이 속하는 기술분야에서 통상의 지식을 가진 자에게 이러한 정수를 밑수 {2,3,5}의 표현으로 변환하는 도 1의 방법은 하나의 예시일 뿐, 그 외의 다양한 알고리즘을 이용할 수 있음이 자명하다. 1 is a method proposed by Pradeep Kumar Mishra et al.

Is the pseudo code of the algorithm re-expressing {2,3,5} as the base. It should be understood that the method of FIG. 1, which converts such integers to the representation of base 2, 3, 5, is of course only one example to those of ordinary skill in the art to which the present invention pertains, It is obvious.

를 이용하여 타원곡선 스칼라 곱셈

는 수학식 11과 같이 수행된다.Re-expressed integer

Using elliptic curve scalar multiplication

Is performed as shown in Equation (11).

를 구하는 알고리즘의 의사코드이다. 도 2을 보면 {2,3,5} 트리플-베이스는 5, 3, 2 순서로 연산이 진행된다. 그리고 마지막에

(Triple and Add, TA)와

(Double and Add, DA)를 한다. 도 2에서는

(Quintuple and add, QA) 연산은 사용하지 않았다. FIG. 2 is a schematic diagram of a conventional {2,3,5} triple-base chain based elliptic curve scalar multiplication

Is the pseudo-code of the algorithm for obtaining the. Referring to FIG. 2, the {2,3,5} triple-base is operated in the order of 5, 3, and 2. And finally

(Triple and Add, TA) and

(Double and Add, DA). 2,

(Quintuple and add, QA) operation is not used.

FIG. 3 is a flowchart illustrating a calculation method for a triple-base chain-based elliptic curve scalar multiplication according to an embodiment of the present invention, including the following steps.

를 표현하는 3차원 좌표값

및

(

는 양의 정수)를 입력받고, S320 단계에서 연산 모듈이 상기

를 {2,3,5} 밑수(Base)로 재표현하고, S330 단계에서 상기 연산 모듈이 상기 타원곡선 스칼라 곱셈 연산식 내에 존재하는 중복 연산 활용이 가능한 연산을 추출하고, S340 단계에서 상기 추출된 중복 연산 활용이 가능한 연산을 이용하여 상기 타원곡선 스칼라 곱셈 연산식을 변형하며 S350 단계에서 상기 연산 모듈이 상기 점 P에 대하여 상기 변형된 타원곡선 스칼라 곱셈 연산식을 사용하여 타원곡선 스칼라 곱셈

를 수행하여 S360 단계는 출력 모듈이 상기 수행된 타원곡선 스칼라 곱셈 결과값을 출력하는 타원곡선 스칼라 곱셈을 위한 연산 방법이다. 보다 구체적으로 본 발명은 타원곡선 스칼라 곱셈 과정에서 발생하는 중복 연산을 이용하는 것이다. 첫 번째 제안하는 방법은 중복 연산을 이용하여 5P 연산 식을 변형하는 것이다. If it is determined in step S310 that the input module is a point on the elliptic curve

A three-dimensional coordinate value

And

(

Is a positive integer), and in step S320,

Is expressed as {2,3,5} base (Base), and in step S330, the operation module extracts an operation that is available in the elliptic curve scalar multiplication operation expression and that can utilize the redundant operation. In step S340, The operation module modifies the elliptic curve scalar multiplication operation expression by using an operation capable of utilizing redundant operation, and in step S350, the operation module performs an elliptic curve scalar multiplication operation using the modified elliptic curve scalar multiplication expression for the point P

And step S360 is an operation method for an elliptic curve scalar multiplication in which the output module outputs the performed elliptic curve scalar multiplication result. More specifically, the present invention utilizes a redundant operation that occurs in an elliptic curve scalar multiplication process. The first proposed method is to modify the 5P equation using redundant operations.

연산식을 소개한다.

을 타원곡선 상의 한 점이라 할 때,

는 수학식 12와 같다.First, Pradeep Kumar Mishra et al.

The equation is introduced.

Is a point on an elliptic curve,

Is expressed by Equation (12).

연산을 위한 중간 과정을 정리한 것으로, 중복 연산 활용이 가능한 연산을 추출하면

,

,

의 3개의 연산이 추출된다.

은 선 계산된

를 이용하여 수학식 13와 같이 연산 가능하다.FIG. 4 is a graph showing the relationship

If you extract the operations that can use redundant operation

,

,

Are extracted.

The line is calculated

(13). ≪ / RTI >

의 연산식을 수학식 13으로 대체하면 중복 연산을 이용하게 되어

의 연산량은 1S-1M의 연산 효율성을 갖는다. 마찬가지로, 중복 연산 활용이 가능한 연산

,

도 각각 다른 연산식으로 변형할 수 있으며, 도 5는 상기 식을 변형한 연산식을 표시하였다. 즉, 도 5는 본 발명의 실시예에 따른 중복 연산이 활용 가능한 연산을 이용하여 타원곡선 스칼라 곱셈 연산식, 구체적으로

연산식을 변형한 연산식을 정리한 것이다. 요약하건대, 타원곡선 스칼라 곱셈 연산식을 변형하는 단계는 선 계산된 연산을 이용하여 중복 연산 활용이 가능한 연산을 다른 연산식으로 대체하는 것을 특징으로 한다. therefore,

The equation (13) is replaced with the equation

Has an operation efficiency of 1S-1M. Similarly, operations capable of utilizing redundant operations

,

Can also be transformed into different arithmetic expressions, and FIG. 5 shows arithmetic expressions obtained by modifying the above expressions. That is, FIG. 5 illustrates an elliptic curve scalar multiplication operation expression using an operation that can utilize redundant operation according to an embodiment of the present invention,

It is summarized the operation formula which modified the operation formula. In summary, the step of modifying the elliptic curve scalar multiplication operation expression is characterized by replacing operations that can utilize redundant operation by using a pre-computed operation with another operation expression.

연산식의 연산량은 15M + 10S 이고, 본 발명의 실시예에 따른 도 5의 연산량은 12M + 13S 이다. 1M = 0.8S 라 가정하면, 본 발명의 실시예에 따른

연산은 기존 방법보다 0.6M 감소하는 효과가 있다. 그러므로

연산에서는

만큼 연산이 감소하는 효과가 발생한다. 4

The calculation amount of the calculation formula is 15M + 10S, and the calculation amount of FIG. 5 according to the embodiment of the present invention is 12M + 13S. Assuming 1M = 0.8S,

The operation is 0.6M less than the conventional method. therefore

In the operation

There is an effect that the number of operations is reduced.

연산을 수행하고, 그 다음

연산을 수행한다. 그리고 마지막으로

연산 후 DA를 수행한다. 본 발명의 일 실시예로 밑수 연산 순서를 2->3->5로 재배치(Reordering)하는 방법을 제안한다. 재배치를 하면 2->3, 3->5, 2->5로 연산이 진행될 때 효율성이 나타난다.

을 타원곡선 한 점이라 하자.

연산을 먼저 수행하고,

연산을 수행하면

연산 중간 값을 이용하여

의 연산량이 2S 감소한다. 또한,

연산을 먼저 수행하고

를 연산을 하면, 중간 값을 이용한 연산 감소가 발생한다.To illustrate another embodiment according to the present invention, Fig. 6 shows a flowchart of a method for calculating the number of operations of the present invention in accordance with an existing base number 5- >3-> 2 operation sequence in the {2,3,5} triple-based chain-based elliptic curve scalar multiplication, 2 -> 3 -> 5 according to the base order. The triple-base chain based on {2,3,5} is shown in FIG. 2

Operation, and then

. And finally

Perform DA after the operation. According to an embodiment of the present invention, a method of reordering the base calculation order to 2->3->> 5 is proposed. When relocation is performed, efficiency is shown when the operation is performed in 2-> 3, 3-> 5, 2-> 5.

Let 's assume an elliptic curve.

Operation is performed first,

When you perform an operation

Using the intermediate value of the operation

The amount of computation of 2S decreases. Also,

The operation is performed first

The operation is reduced using the intermediate value.

연산 중간 값을 나타내고, 도 8은

연산 중간 값을 나타낸다. 도 4의

연산 중에

연산이 수행된다. 도 7의

연산에는

연산이 포함된다. 따라서

은 선 계산된 연산을 이용하여 수학식 14와 같이 나타낼 수 있다. Figure 7

Represents the intermediate value of the operation, and Fig. 8

Represents the intermediate value of the operation. 4

During operation

An operation is performed. 7

The operation

Operation. therefore

Can be expressed by Equation (14) using a linear computation.

의 연산량은 기존

의 연산량에 비해 1M - 2S 만큼 연산 효율성이 있다. 마지막으로

연산 후

연산이 수행될 때의 연산 효율성도 위와 같은 방법으로 보일 수 있다. 도 8에

연산에는

포함되고, 선 연산된 상기 값을 이용하여 도 4의

연산 과정 중의

연산은 수학식 15와 같이 나타낼 수 있다. As a result,

The amount of computation

The computation efficiency is as much as 1M - 2S. Finally

After calculation

The computational efficiency when the computation is performed can also be seen in the same manner. 8

The operation

4, using the above-mentioned pre-

During the calculation process

The computation can be expressed as: < EMI ID = 15.0 >

연산에 의해 2S 만큼 연산량이 감소한다. Replaced

The amount of computation is reduced by 2S by the operation.

Table 1 shows the increase / decrease of the computation amount according to the rearranged base 2-> 3-> 5 computation method compared with the existing base 5 -> 3 -> 2 computation method.

division Change in calculation amount

2S reduction

2S reduction

2S reduction 1M increase

에 대하여 스칼라 곱셈을 하기 위해 {2,3,5} 트리플-베이스 체인 방법의 밑수 연산순서를 재배치한 타원곡선 스칼라 곱셈 알고리즘의 의사 코드이다. 이에 따라 실험을 통해 정수

의 크기가 160bit 또는 256bit인 일 때, 타원곡선 스칼라 곱셈 연산량을 각각 비교하였다. 실험에 사용한 정수

는 랜덤 생성기를 사용하여 160bit, 256bit 랜덤 값을 각각 1000개씩 생성하였다. 모든 실험에는 미리 생성된 1000개의 랜덤 값을 사용하였다. 표 2는

의 크기가 160bit에 대해서, 표 3은 256bit에 대해서 도 2와 도 9를 구현한 후, 타원 곡선 스칼라 곱셈

를 {2,3,5} 트리플-베이스 체인 스칼라 곱셈을 수행할 때 나오는 항의 개수와 연산량을 각각 정리한 것이다. FIG. 9 is a graph of a point on an elliptic curve according to an embodiment of the present invention.

Is a pseudo-code of an elliptic curve scalar multiplication algorithm that rearranges the order of the bases of the {2,3,5} triple-based chain method for scalar multiplication. As a result,

Is 160 bits or 256 bits, the elliptic curve scalar multiplication operation amounts are compared with each other. The constant used in the experiment

Using the random generator, 1000 random values of 160 bits and 256 bits were generated, respectively. For all experiments, 1000 randomly generated values were used. Table 2

Table 2 shows the results of the Eb curve scalar multiplication < RTI ID = 0.0 >

And {2, 3, 5} triple-base-chain scalar multiplication.

max2 max3 max5 Number 2, 9, 160 103 69 30.84 1655 [m] 1582 [m] 100 85 45 32.56 1704 [m] 1623 [m] 90 75 35 32.90 1701 [m] 1622 [m] 85 60 25 32.78 1686 [m] 1616 [m] 85 38 18 31.61 1636 [m] 1571 [m]

max2 max3 max5 Number 2, 9, 256 161 110 47.68 2658 [m] 2539 [m] 160 103 69 52.66 2764 [m] 2618 [m] 120 50 25 50.05 2648 [m] 2539 [m]

를 {2,3,5} 트리플-베이스로 나타내는 시간을 고려한다면 max2 = 120, max3 = 50, max5 = 25가 더 효율적이다. 그러므로, 본 발명에서 제안하는 방법으로 트리플-베이스 체인 스칼라 곱셈 연산을 수행하면 기존의 트리플-베이스 체인 스칼라 곱셈보다 4∼6% 효율성이 증가한다. Table 2 shows that triple-base-chain scalar multiplication is most efficient when max2 = 85, max3 = 38, max5 = 18. The most efficient operations in Table 3 are max2 = 120, max3 = 50, max5 = 25. Comparing only the computation amounts, it is equal to max2 = 256, max3 = 161 and max5 = 110. However,

Max2 = 120, max3 = 50, max5 = 25 are more efficient considering the time represented by {2,3,5} triple-base. Thus, performing the triple-base-chain scalar multiplication operation in accordance with the method proposed by the present invention increases the efficiency by 4 to 6% compared with the existing triple-base-chain scalar multiplication.

연산을 수행할 때, 중복 연산을 이용하여 밑수의 연산 순서를 재배치하여 구현할 수 있다.

연산을 비교하기 위해서 먼저

,

,

연산에 대해 설명한다.

연산은

연산 후

연산을 하는 것으로 중복 연산이 발생한다. 도 8을 보면

이다.

,

는

연산을 수행할 때 계산된다. 따라서

연산에는

연산이 필요하지 않다. 그래서

의 총 연산량은

이다. In another embodiment according to the present invention

When performing an operation, it is possible to implement by rearranging the operation order of the base by using redundant operation.

To compare operations,

,

,

The operation will be described.

The operation

After calculation

Duplicate operation occurs by performing an operation. 8,

to be.

,

The

Calculated when performing an operation. therefore

The operation

No computation is required. so

The total amount of computation

to be.

,

,

연산량을 정리한 것이다.

,

,

연산량도 중복 연산을 고려하여 정리하면 표 5와 같다. Table 4

,

,

The computation is summarized.

,

,

Table 5 shows the computational complexity considering the redundant operation.

division Duplicate operation Total operation amount

-

division

Operation amount

연산량을 정리하면, 표 6은

연산량, 표 7은

연산량, 표 8은

연산량에 관한 것이다. {2,3,5} triple-based chain change in base chain

Table 6 summarizes the computational complexity.

The amount of computation, Table 7,

The amount of computation, Table 8

.

division Operation amount

division Operation amount

division Operation amount

,

,

연산량의 합을 비교해 보면,

연산이 가장 효율성이 높다. 따라서

연산도 밑수 순서를 2->3->5 로 재배치함으로서 연산 효율성이 높아지는 것을 확인할 수 있다. Assuming 1M = 0.8S,

,

,

Comparing the sum of operations,

Operation is the most efficient. therefore

It can be seen that the computational efficiency is improved by rearranging the base order of the operations to 2->3->> 5.

10 illustrates an arithmetic unit 10 for elliptic curve cryptosystem according to an embodiment of the present invention. The arithmetic unit 10 includes an input unit 11, an arithmetic unit 12, and an output unit 13.

를 표현하는 3차원 좌표값

및

(

는 양의 정수)를 입력받는다. 이러한 입력부(11)는 전자적인 형태의 데이터를 수신할 수 있는 입력 장치로서 구현될 수 있으며, 필요에 따라 물리적인 인터페이스를 포함할 수 있다.The input unit 11 receives points on the elliptic curve

A three-dimensional coordinate value

And

(

Is a positive integer). The input unit 11 may be implemented as an input device capable of receiving electronic data, and may include a physical interface as needed.

를 {2,3,5} 밑수로 재표현하고, {2,3,5} 트리플-베이스 체인 기반 타원곡선 스칼라 곱셈 연산식 내에 존재하는 중복 연산 활용이 가능한 연산을 추출하고, 상기 추출된 중복 연산 활용이 가능한 연산을 이용하여 상기 타원곡선 스칼라 곱셈 연산식을 변형하고, 상기 점

에 대하여 상기 변형된 타원곡선 스칼라 곱셈 연산식을 사용하여 타원곡선 스칼라 곱셈

를 수행하여 결과값을 산출한다. The arithmetic operation unit 12 is an integer that is proposed by embodiments of the present invention

Is expressed as a base of {2,3,5}, an operation that can utilize the redundant operation existing in the {2,3,5} triple-base chain-based elliptic curve scalar multiplication expression is extracted, and the extracted redundant operation Modifies the elliptic curve scalar multiplication operation expression by using an operable operation,

Using the modified elliptic curve scalar multiplication equation for the elliptic curve scalar multiplication < RTI ID = 0.0 >

To calculate the resultant value.

In implementing the arithmetic unit of the present embodiment in hardware, the arithmetic unit 12 may be implemented by a combination of a modular multiplier, a modular adder, and a subtractor for performing the arithmetic operations. Such a hardware design can be implemented by a person having ordinary skill in circuit design in the technical field to which the present invention belongs.

The output unit 13 outputs the scalar multiplication result value calculated through the arithmetic unit 12. The output unit 13 may be implemented as an output device capable of outputting data processed in an electronic form, and may include a physical interface as needed.

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

The present invention can be embodied in computer readable code on a computer readable recording medium. The computer readable recording medium may be any type of recording apparatus for storing data that can be read by a computer system, .

Examples of the computer-readable recording medium include a ROM, a RAM, a CD-ROM, a magnetic tape, a floppy disk, an optical data storage device and the like, and also a carrier wave (for example, transmission via the Internet) . In addition, the computer-readable recording medium may be distributed over network-connected computer systems so that computer readable codes can be stored and executed in a distributed manner. 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. It will be understood by those skilled in the art that various changes in form and details may be made therein without departing from the spirit and scope of the invention as defined by the appended claims. Therefore, the disclosed embodiments should be considered in an illustrative rather than a restrictive sense. The scope of the present invention is defined by the appended claims rather than by the foregoing description, and all differences within the scope of equivalents thereof should be construed as being included in the present invention.

10: Elliptic Curve Scalar Multiplier
11: Input unit
12:
13: Output section

Claims (9)

{2,3,5} Triple-Base Chain Based Elliptic Curve Scalar multiplication,
If the input module is a point on an elliptic curve

A three-dimensional coordinate value

And

(

Receiving a positive integer;
When the operation module

To {2,3,5} base (Base);
Extracting an operation that can be performed by the operation module existing in the elliptic curve scalar multiplication operation expression and transforming the elliptic curve scalar multiplication operation expression using an operation capable of utilizing the redundant operation;
The operation module

Using the modified elliptic curve scalar multiplication equation for the elliptic curve scalar multiplication < RTI ID = 0.0 >

; And
And outputting the result of the elliptic curve scalar multiplication performed by the output module. The method according to claim 1,
Wherein transforming the elliptic curve scalar multiplication expression is performed by replacing an operation that can utilize the redundant operation with another operation using a pre-calculated operation. The method according to claim 1,
Further comprising the step of reordering a base operation sequence of the operation module. The method of claim 3,
Wherein the rearranged base arithmetic operation sequence is 2 > 3- > 5. 2. The arithmetic operation method for an elliptic curve scalar multiplication according to claim 1, A computer-readable recording medium storing a program for causing a computer to execute the method according to any one of claims 1 to 4. Point on an elliptic curve

A three-dimensional coordinate value

And

(

A positive integer);
We re-express the k as {2,3,5} base and use the redundant operations in the {2,3,5} triple-base chain based elliptic curve scalar multiplication expression Extracts possible arithmetic operations, modifies the elliptic curve scalar multiplication arithmetic expression using an operation capable of utilizing the extracted redundant arithmetic,

Using the modified elliptic curve scalar multiplication equation for the elliptic curve scalar multiplication < RTI ID = 0.0 >

; And
And an output unit for outputting the result of performing the elliptic curve scalar multiplication. The method according to claim 6,
Wherein the arithmetic unit replaces the arithmetic operation that can utilize the redundant arithmetic operation with another arithmetic operation using the arithmetic operation, thereby modifying the elliptic curve scalar multiplication arithmetic expression. The method according to claim 6,
Wherein the arithmetic unit further comprises reordering a base arithmetic operation sequence for an elliptic curve scalar multiplication. 9. The method of claim 8,
Wherein the rearranged base arithmetic operation sequence is 2 > 3- > 5. 2. The arithmetic apparatus for an elliptic curve scalar multiplication according to claim 1,

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