### Abstract

In this work we analyse the GLV method of Gallant, Lambert and Vanstone (CRYPTO 2001) which uses a fast endomorphism Φ with minimal polynomial X^{2} + rX + s to compute any multiple kP of a point P of order n lying on an elliptic curve. First we fill in a gap in the proof of the bound of the kernel K vectors of the reduction map f: (i,j) → i + λj (mod n). In particular, we prove the GLV decomposition with explicit constant kP = k_{1}P + k_{2}Φ(P), with max{|k_{1}|,|k_{2}|} ≤ √1 + |r| + s√n . Next we improve on this bound and give the best constant in the given examples for the quantity sup_{k,n} max{|k_{1}|,|k_{2}|}/√n. Independently Park, Jeong, Kim, and Lim (PKC 2002) have given similar but slightly weaker bounds. Finally we provide the first explicit bounds for the GLV method generalised to hyperelliptic curves as described in Park, Jeong and Lim (EUROCRYPT 2002).

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

Pages (from-to) | 21-36 |

Number of pages | 16 |

Journal | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |

Volume | 2595 |

Publication status | Published - 2003 |

Externally published | Yes |

### Fingerprint

### Keywords

- Algebraic number fields
- Efficiently-computable endomorphisms
- Elliptic curve cryptography
- Fast performance

### ASJC Scopus subject areas

- Computer Science(all)
- Biochemistry, Genetics and Molecular Biology(all)
- Theoretical Computer Science

### Cite this

*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)*,

*2595*, 21-36.

**Analysis of the gallant-lambert-vanstone method based on efficient endomorphisms : Elliptic and hyperelliptic curves.** / Sica, Francesco; Ciet, Mathieu; Quisquater, Jean Jacques.

Research output: Contribution to journal › Article

*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)*, vol. 2595, pp. 21-36.

}

TY - JOUR

T1 - Analysis of the gallant-lambert-vanstone method based on efficient endomorphisms

T2 - Elliptic and hyperelliptic curves

AU - Sica, Francesco

AU - Ciet, Mathieu

AU - Quisquater, Jean Jacques

PY - 2003

Y1 - 2003

N2 - In this work we analyse the GLV method of Gallant, Lambert and Vanstone (CRYPTO 2001) which uses a fast endomorphism Φ with minimal polynomial X2 + rX + s to compute any multiple kP of a point P of order n lying on an elliptic curve. First we fill in a gap in the proof of the bound of the kernel K vectors of the reduction map f: (i,j) → i + λj (mod n). In particular, we prove the GLV decomposition with explicit constant kP = k1P + k2Φ(P), with max{|k1|,|k2|} ≤ √1 + |r| + s√n . Next we improve on this bound and give the best constant in the given examples for the quantity supk,n max{|k1|,|k2|}/√n. Independently Park, Jeong, Kim, and Lim (PKC 2002) have given similar but slightly weaker bounds. Finally we provide the first explicit bounds for the GLV method generalised to hyperelliptic curves as described in Park, Jeong and Lim (EUROCRYPT 2002).

AB - In this work we analyse the GLV method of Gallant, Lambert and Vanstone (CRYPTO 2001) which uses a fast endomorphism Φ with minimal polynomial X2 + rX + s to compute any multiple kP of a point P of order n lying on an elliptic curve. First we fill in a gap in the proof of the bound of the kernel K vectors of the reduction map f: (i,j) → i + λj (mod n). In particular, we prove the GLV decomposition with explicit constant kP = k1P + k2Φ(P), with max{|k1|,|k2|} ≤ √1 + |r| + s√n . Next we improve on this bound and give the best constant in the given examples for the quantity supk,n max{|k1|,|k2|}/√n. Independently Park, Jeong, Kim, and Lim (PKC 2002) have given similar but slightly weaker bounds. Finally we provide the first explicit bounds for the GLV method generalised to hyperelliptic curves as described in Park, Jeong and Lim (EUROCRYPT 2002).

KW - Algebraic number fields

KW - Efficiently-computable endomorphisms

KW - Elliptic curve cryptography

KW - Fast performance

UR - http://www.scopus.com/inward/record.url?scp=35248862660&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=35248862660&partnerID=8YFLogxK

M3 - Article

VL - 2595

SP - 21

EP - 36

JO - Lecture Notes in Computer Science

JF - Lecture Notes in Computer Science

SN - 0302-9743

ER -