Pairing-Friendly Elliptic Curves with Various Discriminants

Woo Sug KANG; Ki Taek KIM

doi:10.1587/transfun.E93.A.1032

Pairing-Friendly Elliptic Curves with Various Discriminants

Woo Sug KANG, Ki Taek KIM

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This paper extends the Brezing-Weng method by parameterizing the discriminant D by a polynomial D(x). To date, the maximum of CM discriminant can be adequately addressed is about 14-digits. Thus the degree of the square free part of D(x) has to be sufficiently small. By making the square free part of D(x) a linear monomial, the degree of the square free part is small and by substituting x to some quadratic monomial, pairing-friendly curves with various discriminants can be constructed. In order that a square free part of D(x) is of the form ax, ax has to be a square element as a polynomial representation in a number field. Two methods are introduced to apply this construction. For k = 5, 8, 9, 15, 16, 20, 24 and 28, the proposed method gives smaller ρ value than those in previous studies.

Publication: IEICE TRANSACTIONS on Fundamentals Vol.E93-A No.6 pp.1032-1038

Publication Date: 2010/06/01

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Online ISSN: 1745-1337

DOI: 10.1587/transfun.E93.A.1032

Type of Manuscript: Special Section PAPER (Special Section on Discrete Mathematics and Its Applications)

Category: Cryptography and Information Security

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Woo Sug KANG, Ki Taek KIM, "Pairing-Friendly Elliptic Curves with Various Discriminants" in IEICE TRANSACTIONS on Fundamentals, vol. E93-A, no. 6, pp. 1032-1038, June 2010, doi: 10.1587/transfun.E93.A.1032.
Abstract: This paper extends the Brezing-Weng method by parameterizing the discriminant D by a polynomial D(x). To date, the maximum of CM discriminant can be adequately addressed is about 14-digits. Thus the degree of the square free part of D(x) has to be sufficiently small. By making the square free part of D(x) a linear monomial, the degree of the square free part is small and by substituting x to some quadratic monomial, pairing-friendly curves with various discriminants can be constructed. In order that a square free part of D(x) is of the form ax, ax has to be a square element as a polynomial representation in a number field. Two methods are introduced to apply this construction. For k = 5, 8, 9, 15, 16, 20, 24 and 28, the proposed method gives smaller ρ value than those in previous studies.
URL: https://global.ieice.org/en_transactions/fundamentals/10.1587/transfun.E93.A.1032/_p

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@ARTICLE{e93-a_6_1032,
author={Woo Sug KANG, Ki Taek KIM, },
journal={IEICE TRANSACTIONS on Fundamentals},
title={Pairing-Friendly Elliptic Curves with Various Discriminants},
year={2010},
volume={E93-A},
number={6},
pages={1032-1038},
abstract={This paper extends the Brezing-Weng method by parameterizing the discriminant D by a polynomial D(x). To date, the maximum of CM discriminant can be adequately addressed is about 14-digits. Thus the degree of the square free part of D(x) has to be sufficiently small. By making the square free part of D(x) a linear monomial, the degree of the square free part is small and by substituting x to some quadratic monomial, pairing-friendly curves with various discriminants can be constructed. In order that a square free part of D(x) is of the form ax, ax has to be a square element as a polynomial representation in a number field. Two methods are introduced to apply this construction. For k = 5, 8, 9, 15, 16, 20, 24 and 28, the proposed method gives smaller ρ value than those in previous studies.},
keywords={},
doi={10.1587/transfun.E93.A.1032},
ISSN={1745-1337},
month={June},}

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TY - JOUR
TI - Pairing-Friendly Elliptic Curves with Various Discriminants
T2 - IEICE TRANSACTIONS on Fundamentals
SP - 1032
EP - 1038
AU - Woo Sug KANG
AU - Ki Taek KIM
PY - 2010
DO - 10.1587/transfun.E93.A.1032
JO - IEICE TRANSACTIONS on Fundamentals
SN - 1745-1337
VL - E93-A
IS - 6
JA - IEICE TRANSACTIONS on Fundamentals
Y1 - June 2010
AB - This paper extends the Brezing-Weng method by parameterizing the discriminant D by a polynomial D(x). To date, the maximum of CM discriminant can be adequately addressed is about 14-digits. Thus the degree of the square free part of D(x) has to be sufficiently small. By making the square free part of D(x) a linear monomial, the degree of the square free part is small and by substituting x to some quadratic monomial, pairing-friendly curves with various discriminants can be constructed. In order that a square free part of D(x) is of the form ax, ax has to be a square element as a polynomial representation in a number field. Two methods are introduced to apply this construction. For k = 5, 8, 9, 15, 16, 20, 24 and 28, the proposed method gives smaller ρ value than those in previous studies.
ER -