Paper Abstract and Keywords |
Presentation |
2011-05-13 15:50
Order of Elliptic Curve $y^2=x^3+2^i3^j$ Over Barreto-Naehrig Field Masaaki Shirase (FUN) ISEC2011-6 |
Abstract |
(in Japanese) |
(See Japanese page) |
(in English) |
Barreto-Naehrig (BN) curve is an elliptic curve over $\FP$ whose order is $36z^4+36z^3+18z^2+6z+1$ and the embedding degree of which is 12, where $p$ is a BN prime given by $p=p(z)=36z^4+36z^3+24z^2+6z+1$ with some integer $z$, and is a pairing-friendly curve. BN curve has the form $E_b:y^2=x^3+b,\ b \in \FP^{\,*}$. If $b$ is randomly selected, $E_b$ becomes a BN curve with 1/6 possibility. Any BN prome has a property that it is easily to apply Euler's conjecture which describes cubic residues of $2$ and $3$ modulo a prime to any BN prime $p$ because any BN prime can be represented as $p=U^2+3V^2,\ U=6z^2+3z+1,\ V=z$. The purpose of this paper is to classify the order of $E_b:y^2=x^3+b$ over $\FP$ with BN prime $p$ by $z \bmod{36}$ using this property, Gauss' theorem, and properties of twists for $b=2^i3^j$. Although most parts of results of this paper are theoretical, some parts of those are experimental. |
Keyword |
(in Japanese) |
(See Japanese page) |
(in English) |
BN curve / Gauss' theorem / Euler's conjecture / twist / pairing-friendly elliptic curve / / / |
Reference Info. |
IEICE Tech. Rep., vol. 111, no. 34, ISEC2011-6, pp. 37-44, May 2011. |
Paper # |
ISEC2011-6 |
Date of Issue |
2011-05-06 (ISEC) |
ISSN |
Print edition: ISSN 0913-5685 Online edition: ISSN 2432-6380 |
Copyright and reproduction |
All rights are reserved and no part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Notwithstanding, instructors are permitted to photocopy isolated articles for noncommercial classroom use without fee. (License No.: 10GA0019/12GB0052/13GB0056/17GB0034/18GB0034) |
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ISEC2011-6 |
Conference Information |
Committee |
ISEC |
Conference Date |
2011-05-13 - 2011-05-13 |
Place (in Japanese) |
(See Japanese page) |
Place (in English) |
Kikai-Shinko-Kaikan Bldg. |
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(See Japanese page) |
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Paper Information |
Registration To |
ISEC |
Conference Code |
2011-05-ISEC |
Language |
Japanese |
Title (in Japanese) |
(See Japanese page) |
Sub Title (in Japanese) |
(See Japanese page) |
Title (in English) |
Order of Elliptic Curve $y^2=x^3+2^i3^j$ Over Barreto-Naehrig Field |
Sub Title (in English) |
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BN curve |
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Gauss' theorem |
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Euler's conjecture |
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twist |
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pairing-friendly elliptic curve |
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1st Author's Name |
Masaaki Shirase |
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Future University Hakodate (FUN) |
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Speaker |
Author-1 |
Date Time |
2011-05-13 15:50:00 |
Presentation Time |
25 minutes |
Registration for |
ISEC |
Paper # |
ISEC2011-6 |
Volume (vol) |
vol.111 |
Number (no) |
no.34 |
Page |
pp.37-44 |
#Pages |
8 |
Date of Issue |
2011-05-06 (ISEC) |
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