### Abstract

Hyperelliptic curve cryptosystems (HECC) can be good alternatives to elliptic curve cryptosystems, and there is a good possibility to improve the efficiency of HECC due to its flexible algebraic structure. Recently, an efficient scalar multiplication technique for application to genus 2 curves using a degenerate divisor has been proposed. This new technique can be used in the cryptographic protocol using a fixed base point, e.g., HEC-DSA. This paper considers two important issues concerning degenerate divisors. First, we extend the technique for genus 2 curves to genus 3 curves. Jacobian variety for genus 3 curves has two different degenerate divisors: degree 1 and 2. We present explicit formulae of the addition algorithm with degenerate divisors, and then present the timing of scalar multiplication using the proposed formulae. Second, we propose several window methods using the degenerate divisors. It is not obvious how to construct a base point D such that deg(D) = deg(aD) < g for integer a, where g is the genus of the underlying curve and deg(D) is the degree of divisor D. We present an explicit algorithm for generating such divisors. We then develop a window-based scheme that is secure against side-channel attacks.

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

Pages (from-to) | 296-312 |

Number of pages | 17 |

Journal | Lecture Notes in Computer Science |

Volume | 3506 |

Publication status | Published - Sep 27 2005 |

Event | 7th International Conference on Information Security and Cryptology - ICISC 2004 - Seoul, Korea, Republic of Duration: Dec 2 2004 → Dec 3 2004 |

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### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Lecture Notes in Computer Science*,

*3506*, 296-312.

**Some improved algorithms for hyperelliptic curve cryptosystems using degenerate divisors.** / Katagi, Masanobu; Akishita, Tom; Kitamura, Izuru; Takagi, Tsuyoshi.

Research output: Contribution to journal › Conference article

*Lecture Notes in Computer Science*, vol. 3506, pp. 296-312.

}

TY - JOUR

T1 - Some improved algorithms for hyperelliptic curve cryptosystems using degenerate divisors

AU - Katagi, Masanobu

AU - Akishita, Tom

AU - Kitamura, Izuru

AU - Takagi, Tsuyoshi

PY - 2005/9/27

Y1 - 2005/9/27

N2 - Hyperelliptic curve cryptosystems (HECC) can be good alternatives to elliptic curve cryptosystems, and there is a good possibility to improve the efficiency of HECC due to its flexible algebraic structure. Recently, an efficient scalar multiplication technique for application to genus 2 curves using a degenerate divisor has been proposed. This new technique can be used in the cryptographic protocol using a fixed base point, e.g., HEC-DSA. This paper considers two important issues concerning degenerate divisors. First, we extend the technique for genus 2 curves to genus 3 curves. Jacobian variety for genus 3 curves has two different degenerate divisors: degree 1 and 2. We present explicit formulae of the addition algorithm with degenerate divisors, and then present the timing of scalar multiplication using the proposed formulae. Second, we propose several window methods using the degenerate divisors. It is not obvious how to construct a base point D such that deg(D) = deg(aD) < g for integer a, where g is the genus of the underlying curve and deg(D) is the degree of divisor D. We present an explicit algorithm for generating such divisors. We then develop a window-based scheme that is secure against side-channel attacks.

AB - Hyperelliptic curve cryptosystems (HECC) can be good alternatives to elliptic curve cryptosystems, and there is a good possibility to improve the efficiency of HECC due to its flexible algebraic structure. Recently, an efficient scalar multiplication technique for application to genus 2 curves using a degenerate divisor has been proposed. This new technique can be used in the cryptographic protocol using a fixed base point, e.g., HEC-DSA. This paper considers two important issues concerning degenerate divisors. First, we extend the technique for genus 2 curves to genus 3 curves. Jacobian variety for genus 3 curves has two different degenerate divisors: degree 1 and 2. We present explicit formulae of the addition algorithm with degenerate divisors, and then present the timing of scalar multiplication using the proposed formulae. Second, we propose several window methods using the degenerate divisors. It is not obvious how to construct a base point D such that deg(D) = deg(aD) < g for integer a, where g is the genus of the underlying curve and deg(D) is the degree of divisor D. We present an explicit algorithm for generating such divisors. We then develop a window-based scheme that is secure against side-channel attacks.

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UR - http://www.scopus.com/inward/citedby.url?scp=24944586307&partnerID=8YFLogxK

M3 - Conference article

AN - SCOPUS:24944586307

VL - 3506

SP - 296

EP - 312

JO - Lecture Notes in Computer Science

JF - Lecture Notes in Computer Science

SN - 0302-9743

ER -