### Abstract

In this paper, we propose a scalar multiplication algorithm on elliptic curves over GF(2^{m}). The proposed algorithm is an extended version of the Montgomery ladder algorithm with the quaternary representation of the scalar. In addition, in order to improve performance, we have developed new composite operation formulas and apply them to the proposed scalar multiplication algorithm. The proposed composite formulas are 2P_{1} + 2P_{2}, 3P_{1} + P_{2}, and 4P_{1}, where P _{1} and P_{2} are points on an elliptic curve. They can be computed using only the x-coordinate of a point P = (x, y) in the affine coordinate system. However, the proposed scalar multiplication algorithm is vulnerable to simple power analysis attacks, because different operations are performed depending on the bits of the scalar unlike the original Montgomery ladder algorithm. Therefore, we combine the concept of the side-channel atomicity with the proposed composite operation formulas to prevent simple power analysis. Furthermore, to optimize the computational cost, we use the Montgomery trick which can reduce the number of finite field inversion operations used in the affine coordinate system. As the result, the proposed scalar multiplication algorithm saves at least 26% of running time with small storage compared to the previous algorithms such as window-based methods and comb-based methods.

Original language | English |
---|---|

Pages (from-to) | 304-312 |

Number of pages | 9 |

Journal | Information Sciences |

Volume | 245 |

DOIs | |

Publication status | Published - 2013 Oct 1 |

### Fingerprint

### Keywords

- Composite formulas
- Elliptic curve
- Montgomery ladder algorithm
- Side-channel atomicity
- Simple power analysis

### ASJC Scopus subject areas

- Artificial Intelligence
- Software
- Control and Systems Engineering
- Theoretical Computer Science
- Computer Science Applications
- Information Systems and Management

### Cite this

*Information Sciences*,

*245*, 304-312. https://doi.org/10.1016/j.ins.2013.05.009

**Extended elliptic curve Montgomery ladder algorithm over binary fields with resistance to simple power analysis.** / Cho, Sung Min; Seo, Seog Chung; Kim, Tae Hyun; Park, Young Ho; Hong, Seokhie.

Research output: Contribution to journal › Article

*Information Sciences*, vol. 245, pp. 304-312. https://doi.org/10.1016/j.ins.2013.05.009

}

TY - JOUR

T1 - Extended elliptic curve Montgomery ladder algorithm over binary fields with resistance to simple power analysis

AU - Cho, Sung Min

AU - Seo, Seog Chung

AU - Kim, Tae Hyun

AU - Park, Young Ho

AU - Hong, Seokhie

PY - 2013/10/1

Y1 - 2013/10/1

N2 - In this paper, we propose a scalar multiplication algorithm on elliptic curves over GF(2m). The proposed algorithm is an extended version of the Montgomery ladder algorithm with the quaternary representation of the scalar. In addition, in order to improve performance, we have developed new composite operation formulas and apply them to the proposed scalar multiplication algorithm. The proposed composite formulas are 2P1 + 2P2, 3P1 + P2, and 4P1, where P 1 and P2 are points on an elliptic curve. They can be computed using only the x-coordinate of a point P = (x, y) in the affine coordinate system. However, the proposed scalar multiplication algorithm is vulnerable to simple power analysis attacks, because different operations are performed depending on the bits of the scalar unlike the original Montgomery ladder algorithm. Therefore, we combine the concept of the side-channel atomicity with the proposed composite operation formulas to prevent simple power analysis. Furthermore, to optimize the computational cost, we use the Montgomery trick which can reduce the number of finite field inversion operations used in the affine coordinate system. As the result, the proposed scalar multiplication algorithm saves at least 26% of running time with small storage compared to the previous algorithms such as window-based methods and comb-based methods.

AB - In this paper, we propose a scalar multiplication algorithm on elliptic curves over GF(2m). The proposed algorithm is an extended version of the Montgomery ladder algorithm with the quaternary representation of the scalar. In addition, in order to improve performance, we have developed new composite operation formulas and apply them to the proposed scalar multiplication algorithm. The proposed composite formulas are 2P1 + 2P2, 3P1 + P2, and 4P1, where P 1 and P2 are points on an elliptic curve. They can be computed using only the x-coordinate of a point P = (x, y) in the affine coordinate system. However, the proposed scalar multiplication algorithm is vulnerable to simple power analysis attacks, because different operations are performed depending on the bits of the scalar unlike the original Montgomery ladder algorithm. Therefore, we combine the concept of the side-channel atomicity with the proposed composite operation formulas to prevent simple power analysis. Furthermore, to optimize the computational cost, we use the Montgomery trick which can reduce the number of finite field inversion operations used in the affine coordinate system. As the result, the proposed scalar multiplication algorithm saves at least 26% of running time with small storage compared to the previous algorithms such as window-based methods and comb-based methods.

KW - Composite formulas

KW - Elliptic curve

KW - Montgomery ladder algorithm

KW - Side-channel atomicity

KW - Simple power analysis

UR - http://www.scopus.com/inward/record.url?scp=84880313577&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84880313577&partnerID=8YFLogxK

U2 - 10.1016/j.ins.2013.05.009

DO - 10.1016/j.ins.2013.05.009

M3 - Article

AN - SCOPUS:84880313577

VL - 245

SP - 304

EP - 312

JO - Information Sciences

JF - Information Sciences

SN - 0020-0255

ER -