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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
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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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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. 
Topics (in Japanese) (See Japanese page) 
Topics (in English)  
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)  
Keyword(1) BN curve  
Keyword(2) Gauss' theorem  
Keyword(3) Euler's conjecture  
Keyword(4) twist  
Keyword(5) pairing-friendly elliptic curve  
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1st Author's Name Masaaki Shirase  
1st Author's Affiliation 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
Date of Issue 2011-05-06 (ISEC) 


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