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## 학위논문 상세정보

### Study of Efficient Scalar Multiplication in Elliptic Curve

• 저자

곽승규

• 학위수여기관

고려대학교 정보보호대학원

• 학위구분

국내석사

• 학과

정보보호학과

• 지도교수

홍석희

• 발행년도

2014

• 총페이지

66

• 키워드

Scalar Multiplication;

• 언어

eng

• 원문 URL
• #### 초록

Elliptic curve cryptography (ECC), was first proposed by V.Miller and N.Koblitz \cite{1}. In elliptic curve cryptography, finite abelian groups to implement public key cryptography primitives uses to elliptic curves over finite fields. And elliptic curve cryptography is a cryptography based on the discrete logarithm problem which seems to be even more difficult than other problems defined in other groups. Although elliptic curve cryptography uses smaller key compared with other public key cryptography such as RSA and elgamal, it provides security similar to existing public key cryptography. For example, ECC using 160-bit key is considered to provide equivalent level of security of RSA and DSA using 1024-bit key size. Since elliptic curve cryptography uses short length key, it is suitable for mobile phones and PDA with low power consumption and limited memory space to implement cryptography. Scalar multiplication of elliptic curve is an operation that adds a point P on elliptic curve k times. Q = kP = P+ P ...+P, k times where k is a positive integer, P is a point on elliptic curve. Scalar multiplication of elliptic curve is a major part of elliptic curve cryptography. Thus, there are many researches have been proposed to speed up scalar multiplication of elliptic curve. Jacobian coordinate had been proposed to eliminate the costly inversion operation in affine coordinate \cite{2}. A mixed addition, which is a combination of projective and affine coordinates, makes the operation more efficiently \cite{3}. Dmitrov, et al. showed a method reusing intermediate values while point tripling and point doubling \cite{4}. P. Longa, et al. replaced multiplication by squaring during scalar multiplication kP operation \cite{5}. On 2006, the double-base chain which represents integer k using two base {2,3} had been introduced by C. Doche, et al \cite{6}. The triple-base chain using three base {2,3,5} is an advanced method of the double-base chain \cite{7}. A method to reordering the operation order of base {2,3} in double-base chain had been studied \cite{8}. In this paper, we propose a method to improve the scalar multiplication algorithm based on triple-base chain. Proposed method is to optimize 5P operation and to reorder the operation order of base {2,3,5}. We also proposed an algorithm using the reordering method. Our experimental results on curves defined over prime field in Jacobian coordinates showed reductions of cost from 4 to 6 percent using general scalar multiplication method of elliptic curve based on {2,3,5} triple-base chain. This paper is organized as follows: section 2 announces public key cryptography. section 3 introduces basic knowledge of elliptic curve cryptography and scalar multiplication base on {2,3,5} triple chain. Section 4 proposes our method to make scalar multiplication efficient. Our experimental results are described in section 5 and show that our method is more efficient than existing method. Finally, section 6 addresses our conclusion.

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