Efficient arithmetic on Hessian curves

Reza R. Farashahi; Marc Joye

doi:10.1007/978-3-642-13013-7_15

Efficient arithmetic on Hessian curves

Reza R. Farashahi, Marc Joye

School of Computing

Research output: Chapter in Book/Report/Conference proceeding › Conference proceeding contribution › peer-review

71 Citations (Scopus)

Abstract

This paper considers a generalized form for Hessian curves. The family of generalized Hessian curves covers more isomorphism classes of elliptic curves. Over a finite field , it is shown to be equivalent to the family of elliptic curves with a torsion subgroup isomorphic to Z/3Z. This paper provides efficient unified addition formulas for generalized Hessian curves. The formulas even feature completeness for suitably chosen parameters. This paper also presents extremely fast addition formulas for generalized binary Hessian curves. The fastest projective addition formulas require 9M∈+∈3S, where M is the cost of a field multiplication and S is the cost of a field squaring. Moreover, very fast differential addition and doubling formulas are provided that need only 5M∈+∈4S when the curve is chosen with small curve parameters.

Original language	English
Title of host publication	Public Key Cryptography, PKC 2010 - 13th International Conference on Practice and Theory in Public Key Cryptography, Proceedings
Editors	Phong Q. Nguyen, David Pointcheval
Pages	243-260
Number of pages	18
Volume	6056 LNCS
DOIs	https://doi.org/10.1007/978-3-642-13013-7_15
Publication status	Published - 2010
Event	13th International Conference on Practice and Theory in Public Key Cryptography, PKC 2010 - Paris, France Duration: 26 May 2010 → 28 May 2010

Publication series

Name	Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume	6056 LNCS
ISSN (Print)	03029743
ISSN (Electronic)	16113349

Other

Other	13th International Conference on Practice and Theory in Public Key Cryptography, PKC 2010
Country/Territory	France
City	Paris
Period	26/05/10 → 28/05/10

Keywords

cryptography
Elliptic curves
Hessian curves

Access to Document

10.1007/978-3-642-13013-7_15

Cite this

Farashahi, R. R., & Joye, M. (2010). Efficient arithmetic on Hessian curves. In P. Q. Nguyen, & D. Pointcheval (Eds.), Public Key Cryptography, PKC 2010 - 13th International Conference on Practice and Theory in Public Key Cryptography, Proceedings (Vol. 6056 LNCS, pp. 243-260). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 6056 LNCS). https://doi.org/10.1007/978-3-642-13013-7_15

Farashahi, Reza R. ; Joye, Marc. / Efficient arithmetic on Hessian curves. Public Key Cryptography, PKC 2010 - 13th International Conference on Practice and Theory in Public Key Cryptography, Proceedings. editor / Phong Q. Nguyen ; David Pointcheval. Vol. 6056 LNCS 2010. pp. 243-260 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).

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Farashahi, RR & Joye, M 2010, Efficient arithmetic on Hessian curves. in PQ Nguyen & D Pointcheval (eds), Public Key Cryptography, PKC 2010 - 13th International Conference on Practice and Theory in Public Key Cryptography, Proceedings. vol. 6056 LNCS, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 6056 LNCS, pp. 243-260, 13th International Conference on Practice and Theory in Public Key Cryptography, PKC 2010, Paris, France, 26/05/10. https://doi.org/10.1007/978-3-642-13013-7_15

Efficient arithmetic on Hessian curves. / Farashahi, Reza R.; Joye, Marc.
Public Key Cryptography, PKC 2010 - 13th International Conference on Practice and Theory in Public Key Cryptography, Proceedings. ed. / Phong Q. Nguyen; David Pointcheval. Vol. 6056 LNCS 2010. p. 243-260 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 6056 LNCS).

Research output: Chapter in Book/Report/Conference proceeding › Conference proceeding contribution › peer-review

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AB - This paper considers a generalized form for Hessian curves. The family of generalized Hessian curves covers more isomorphism classes of elliptic curves. Over a finite field , it is shown to be equivalent to the family of elliptic curves with a torsion subgroup isomorphic to Z/3Z. This paper provides efficient unified addition formulas for generalized Hessian curves. The formulas even feature completeness for suitably chosen parameters. This paper also presents extremely fast addition formulas for generalized binary Hessian curves. The fastest projective addition formulas require 9M∈+∈3S, where M is the cost of a field multiplication and S is the cost of a field squaring. Moreover, very fast differential addition and doubling formulas are provided that need only 5M∈+∈4S when the curve is chosen with small curve parameters.

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KW - Elliptic curves

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Farashahi RR, Joye M. Efficient arithmetic on Hessian curves. In Nguyen PQ, Pointcheval D, editors, Public Key Cryptography, PKC 2010 - 13th International Conference on Practice and Theory in Public Key Cryptography, Proceedings. Vol. 6056 LNCS. 2010. p. 243-260. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). doi: 10.1007/978-3-642-13013-7_15

Efficient arithmetic on Hessian curves

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