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Paper Abstract and Keywords
Presentation 2011-09-09 15:15
Determining All Traces of Elliptic Curves $y^{2}=x^{3} \pm 2^{i}3^{j}$ over BN Field
Toshiya Nakajima (QFR Lab.) ISEC2011-30
Abstract (in Japanese) (See Japanese page) 
(in English) A prime $p$ of the form $p=36z^{4}+36z^{3}+24z^{2}+6z+1\ (z \in \mathbb{Z})$ is called a BN(Barreto-Naehrig) prime and a finite prime field with characteristic of BN prime is called a BN field. Shirase[5] pointed out every BN prime can be represented as $p=(6z^{2}+3z+1)^{2}+3z^{2}$, and using the fact he determined group orders of elliptic curves $E:\ y^{2}=x^{3}+2^{i}3^{j}\ (i,j \in \mathbb{Z})$ over BN field for most cases of $i$ and $j$. In special cases of the result those curves are pairing-friendly curves (BN curves) with embedding degree 12. However, in some cases of $i$ and $j$ the orders were stated as conjecture induced by experiments. This paper extends results of [5] and theoretically determines traces (equivalent to determine orders) of elliptic curves $y^{2}=x^{3} \pm 2^{i}3^{j}$ over BN field for all cases of $i$ and $j$ by deriving values of sixth power residue symbols in the ring $\mathbb{Z}[\omega]\ (\omega=(-1+\sqrt{-3})/2)$. As a result, the conjecture of [5] is proven true.
Keyword (in Japanese) (See Japanese page) 
(in English) pairing-friendly curve / BN curve / BN prime / primary / cubic residue / sixth-power residue / trace /  
Reference Info. IEICE Tech. Rep., vol. 111, no. 204, ISEC2011-30, pp. 25-28, Sept. 2011.
Paper # ISEC2011-30 
Date of Issue 2011-09-02 (ISEC) 
ISSN Print edition: ISSN 0913-5685    Online edition: ISSN 2432-6380
Copyright
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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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Conference Information
Committee ISEC  
Conference Date 2011-09-09 - 2011-09-09 
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 2011-09-ISEC 
Language Japanese 
Title (in Japanese) (See Japanese page) 
Sub Title (in Japanese) (See Japanese page) 
Title (in English) Determining All Traces of Elliptic Curves $y^{2}=x^{3} \pm 2^{i}3^{j}$ over BN Field 
Sub Title (in English)  
Keyword(1) pairing-friendly curve  
Keyword(2) BN curve  
Keyword(3) BN prime  
Keyword(4) primary  
Keyword(5) cubic residue  
Keyword(6) sixth-power residue  
Keyword(7) trace  
Keyword(8)  
1st Author's Name Toshiya Nakajima  
1st Author's Affiliation QFR Laboratory (QFR Lab.)
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Speaker Author-1 
Date Time 2011-09-09 15:15:00 
Presentation Time 25 minutes 
Registration for ISEC 
Paper # ISEC2011-30 
Volume (vol) vol.111 
Number (no) no.204 
Page pp.25-28 
#Pages 4 
Date of Issue 2011-09-02 (ISEC) 

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