### 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 |
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Pages (from-to) | 304-312 |

Number of pages | 9 |

Journal | Information Sciences |

Volume | 245 |

DOIs | |

Publication status | Published - 2013 Oct 1 |

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### Keywords

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

### ASJC Scopus subject areas

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

### Cite this

*Information Sciences*,

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