### Abstract

In most algorithms involving elliptic curves, the most expensive part consists in computing multiples of points. This paper investigates how to extend the τ-adic expansion from Koblitz curves to a larger class of curves defined over a prime field having an efficiently-computable endomorphism φ in order to perform an efficient point multiplication with efficiency similar to Solinas' approach presented at CRYPTO '97. Furthermore, many elliptic curve cryptosystems require the computation of k_{0}P + k_{1}Q. Following the work of Solinas on the Joint Sparse Form, we introduce the notion of φ-Joint Sparse Form which combines the advantages of a φ-expansion with the additional speedup of the Joint Sparse Form. We also present an efficient algorithm to obtain the φ-Joint Sparse Form. Then, the double exponentiation can be done using the φ endomorphism instead of doubling, resulting in an average of l applications of φ and l/2 additions, where l is the size of the k_{i}'s. This results in an important speed-up when the computation of φ is particularly effective, as in the case of Koblitz curves.

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

Pages (from-to) | 388-400 |

Number of pages | 13 |

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

Volume | 2656 |

Publication status | Published - 2003 |

Externally published | Yes |

### Fingerprint

### Keywords

- Elliptic curves
- Fast endomorphisms
- Joint Sparse Form

### 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)*,

*2656*, 388-400.

**Improved algorithms for efficient arithmetic on elliptic curves using fast endomorphisms.** / Ciet, Mathieu; Lange, Tanja; Sica, Francesco; 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. 2656, pp. 388-400.

}

TY - JOUR

T1 - Improved algorithms for efficient arithmetic on elliptic curves using fast endomorphisms

AU - Ciet, Mathieu

AU - Lange, Tanja

AU - Sica, Francesco

AU - Quisquater, Jean Jacques

PY - 2003

Y1 - 2003

N2 - In most algorithms involving elliptic curves, the most expensive part consists in computing multiples of points. This paper investigates how to extend the τ-adic expansion from Koblitz curves to a larger class of curves defined over a prime field having an efficiently-computable endomorphism φ in order to perform an efficient point multiplication with efficiency similar to Solinas' approach presented at CRYPTO '97. Furthermore, many elliptic curve cryptosystems require the computation of k0P + k1Q. Following the work of Solinas on the Joint Sparse Form, we introduce the notion of φ-Joint Sparse Form which combines the advantages of a φ-expansion with the additional speedup of the Joint Sparse Form. We also present an efficient algorithm to obtain the φ-Joint Sparse Form. Then, the double exponentiation can be done using the φ endomorphism instead of doubling, resulting in an average of l applications of φ and l/2 additions, where l is the size of the ki's. This results in an important speed-up when the computation of φ is particularly effective, as in the case of Koblitz curves.

AB - In most algorithms involving elliptic curves, the most expensive part consists in computing multiples of points. This paper investigates how to extend the τ-adic expansion from Koblitz curves to a larger class of curves defined over a prime field having an efficiently-computable endomorphism φ in order to perform an efficient point multiplication with efficiency similar to Solinas' approach presented at CRYPTO '97. Furthermore, many elliptic curve cryptosystems require the computation of k0P + k1Q. Following the work of Solinas on the Joint Sparse Form, we introduce the notion of φ-Joint Sparse Form which combines the advantages of a φ-expansion with the additional speedup of the Joint Sparse Form. We also present an efficient algorithm to obtain the φ-Joint Sparse Form. Then, the double exponentiation can be done using the φ endomorphism instead of doubling, resulting in an average of l applications of φ and l/2 additions, where l is the size of the ki's. This results in an important speed-up when the computation of φ is particularly effective, as in the case of Koblitz curves.

KW - Elliptic curves

KW - Fast endomorphisms

KW - Joint Sparse Form

UR - http://www.scopus.com/inward/record.url?scp=35248868300&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=35248868300&partnerID=8YFLogxK

M3 - Article

VL - 2656

SP - 388

EP - 400

JO - Lecture Notes in Computer Science

JF - Lecture Notes in Computer Science

SN - 0302-9743

ER -