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Constructing Elliptic Curves for the GLV Method with Low-cost Decomposition.

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Title: Constructing Elliptic Curves for the GLV Method with Low-cost Decomposition.
Authors: Wroński, Michał1 michal.wronski@wat.edu.pl, Dryło, Robert2 rdrylo@sgh.waw.pl, Kijko, Tomasz1 tomasz.kijko@wat.edu.pl, Bora, Piotr1 piotr.bora@wat.edu.pl
Source: Fundamenta Informaticae. 2017, Vol. 153 Issue 4, p399-413. 15p.
Subjects: Elliptic curves, Complex multiplication, Computable functions, Endomorphisms, Mathematical decomposition, Integers
Abstract: The GLV method allows to improve scalar multiplication on an elliptic curve E/Fq with an efficiently computable endomorphism Φ : E → E over Fq. For points in a subgroup of large prime order r this requires decomposition of scalar k = k0 + k1λ mod r, where Φ acts on the subgroup of order r as multiplication by λ ∊ Fr and k0, k1 are integers O(√r) . In this note we consider the case when λ is of the form λ = 2s + a, where a is a small integer and λ=O(√r), which allows very easy and fast decomposition of k especially in hardware implementations. We give a method to construct such elliptic curves based on the complex multiplication method, and give examples of elliptic curves for λ ∊ {2s, 2s - 1} and various security levels. [ABSTRACT FROM AUTHOR]
Copyright of Fundamenta Informaticae is the property of Polskie Towarzystwo Matematyczne and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: <searchLink fieldCode="DE" term="%22Elliptic+curves%22">Elliptic curves</searchLink><br /><searchLink fieldCode="DE" term="%22Complex+multiplication%22">Complex multiplication</searchLink><br /><searchLink fieldCode="DE" term="%22Computable+functions%22">Computable functions</searchLink><br /><searchLink fieldCode="DE" term="%22Endomorphisms%22">Endomorphisms</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematical+decomposition%22">Mathematical decomposition</searchLink><br /><searchLink fieldCode="DE" term="%22Integers%22">Integers</searchLink>
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  Data: The GLV method allows to improve scalar multiplication on an elliptic curve E/Fq with an efficiently computable endomorphism Φ : E → E over Fq. For points in a subgroup of large prime order r this requires decomposition of scalar k = k0 + k1λ mod r, where Φ acts on the subgroup of order r as multiplication by λ ∊ Fr and k0, k1 are integers O(√r) . In this note we consider the case when λ is of the form λ = 2s + a, where a is a small integer and λ=O(√r), which allows very easy and fast decomposition of k especially in hardware implementations. We give a method to construct such elliptic curves based on the complex multiplication method, and give examples of elliptic curves for λ ∊ {2s, 2s - 1} and various security levels. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
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  Data: <i>Copyright of Fundamenta Informaticae is the property of Polskie Towarzystwo Matematyczne and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.3233/FI-2017-1546
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      – Code: eng
        Text: English
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        PageCount: 15
        StartPage: 399
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      – SubjectFull: Elliptic curves
        Type: general
      – SubjectFull: Complex multiplication
        Type: general
      – SubjectFull: Computable functions
        Type: general
      – SubjectFull: Endomorphisms
        Type: general
      – SubjectFull: Mathematical decomposition
        Type: general
      – SubjectFull: Integers
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              Text: 2017
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