Explicit Formulas for Real Hyperelliptic Curves of Genus 2 in Affine Representation

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Author

Stefan Erickson and Michael J. Jacobson and Ning Shang and Shuo Shen and Andreas Stein

Entry type

inproceedings

Abstract

In this paper, we present for the first time efficient explicit formulas for arithmetic in the degree 0 divisor class group of a real hyperelliptic curve. Hereby, we consider real hyperelliptic curves of genus 2 given in affine coordinates for which the underlying finite field has characteristic > 3. These formulas are much faster than the optimized generic algorithms for real hyperelliptic curves and the cryptographic protocols in the real setting perform almost as well as those in the imaginary case. We provide the idea for the improvements and the correctness together with a comprehensive analysis of the number of field operations. Finally, we perform a direct comparison of cryptographic protocols using explicit formulas for real hyperelliptic curves with the corresponding protocols presented in the imaginary model.

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Date

2007

Booktitle

WAIFI

Key alpha

DBLP:conf/waifi/EricksonJSSS07

Pages

202-218

Publisher

Springer Berlin / Heidelberg

Series

LNCS

Volume

4547/2007

Publication Date

2007-01-01

Isbn

978-3-540-73073-6

Issn

0302-9743 (Print) 1611-3349 (Online)

Keywords

hyperelliptic curve, reduced divisor, infrastructure and dis- tance, Cantorâ€™s algorithm, explicit formulas, efficient implementation, cryptographic key exchange

Subject

Application of low genus curves in cryptography.

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@Inproceedings{ DBLP:conf/waifi/EricksonJSSS07,
	title = "Explicit Formulas for Real Hyperelliptic Curves of Genus
 2 in Affine Representation",
	author = "Stefan Erickson and Michael J. Jacobson and Ning Shang and Shuo Shen and Andreas Stein",
	year = "2007",
	booktitle = "WAIFI",
	pages = "202-218",
	publisher = "Springer Berlin / Heidelberg",
	series = "LNCS",
	volume = "4547/2007",
	abstract = "In this paper, we present for the first time efficient explicit formulas for arithmetic in the degree 0 divisor class group of a real hyperelliptic curve. Hereby, we consider real hyperelliptic curves of genus 2 given in affine coordinates for which the underlying finite field has characteristic > 3. These formulas are much faster than the optimized generic
algorithms for real hyperelliptic curves and the cryptographic protocols
in the real setting perform almost as well as those in the imaginary case.
We provide the idea for the improvements and the correctness together with a comprehensive analysis of the number of field operations. Finally, we perform a direct comparison of cryptographic protocols using explicit formulas for real hyperelliptic curves with the corresponding protocols presented in the imaginary model.
",
	isbn = "978-3-540-73073-6",
	issn = "0302-9743 (Print) 1611-3349 (Online)",
	keywords = "hyperelliptic curve, reduced divisor, infrastructure and dis-
tance, Cantorâ€™s algorithm, explicit formulas, efficient implementation,
 cryptographic key exchange",
	subject = "Application of low genus curves in cryptography.",
}