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| Presentation | 2019-11-01 13:10 Character on elliptic curves over finite fields and even-oddness of the order of points Masaaki Shirase (FUN)  ISEC2019-66 SITE2019-60 LOIS2019-25 |
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| Abstract | (in Japanese) | (See Japanese page) |
| (in English) | Let $p$ be a prime $ge 5$, $q$ be a power of $p$, and $Fq$ be a finite field with $q$ elements. Let $E/Fq$ be an elliptic curve, $#E(Fq)$ is an even number, and a point $P_0$ having order 2 is given, where $#$ means the number of elements. In a previous study, it is shown that a homomorphism map (or character) $chi_E: E(Fq) rightarrow {1, -1 }$ is defined using the coordinate of $P_0$, the Legendre symbol, and the norm. The main result of the previous study is that when $#E(Fq)=2 times$ odd, if $chi_E^{-1}(-1)ne emptyset$ then $Pin E(Fq)mbox{ has odd order}Leftrightarrow chi_E(P)=1$ holds. Also another proposition in the previous study needs the condition, lqlq if $chi_E^{-1}(-1)ne emptyset$.'' This report shows that if (a) $#E(Fp)=2times$ odd, or (b) $#E(Fq)[2]=4$ and $#E(Fq)=4times$ odd, then $chi_E^{-1}(-1)ne emptyset$. Also, this report shows if (b) holds then three characters are defined, and it holds that $P in E(Fq) mbox{ has odd order }Leftrightarrow chi_{E0}(P) =chi_{E1}(P)=1$, where $chi_{E0}$ and $chi_{E1}$ are two characters of them. | |
| Keyword | (in Japanese) | (See Japanese page) |
| (in English) | Elliptic curve / Finite field / Character / Order / Legendre symbol / / / | |
| Reference Info. | IEICE Tech. Rep., vol. 119, no. 257, ISEC2019-66, pp. 25-32, Nov. 2019. | |
| Paper # | ISEC2019-66 | |
| Date of Issue | 2019-10-25 (ISEC, SITE, LOIS) | |
| 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) | |
| Download PDF | ISEC2019-66 SITE2019-60 LOIS2019-25 |
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| Conference Information | |
| Committee | ISEC SITE LOIS |
| Conference Date | 2019-11-01 - 2019-11-02 |
| Place (in Japanese) | (See Japanese page) |
| Place (in English) | Osaka Univ. |
| Topics (in Japanese) | (See Japanese page) |
| Topics (in English) | |
| Paper Information | |
| Registration To | ISEC |
| Conference Code | 2019-11-ISEC-SITE-LOIS |
| Language | Japanese |
| Title (in Japanese) | (See Japanese page) |
| Sub Title (in Japanese) | (See Japanese page) |
| Title (in English) | Character on elliptic curves over finite fields and even-oddness of the order of points |
| Sub Title (in English) | |
| Keyword(1) | Elliptic curve |
| Keyword(2) | Finite field |
| Keyword(3) | Character |
| Keyword(4) | Order |
| Keyword(5) | Legendre symbol |
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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 | 2019-11-01 13:10:00 |
| Presentation Time | 25 minutes |
| Registration for | ISEC |
| Paper # | ISEC2019-66, SITE2019-60, LOIS2019-25 |
| Volume (vol) | vol.119 |
| Number (no) | no.257(ISEC), no.258(SITE), no.259(LOIS) |
| Page | pp.25-32 |
| #Pages | 8 |
| Date of Issue | 2019-10-25 (ISEC, SITE, LOIS) |
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