| Presentation | 2019-11-01 Character on elliptic curves over finite fields and even-oddness of the order of points Masaaki Shirase, |
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| PDF Download Page |  PDF download Page Link |
| Abstract(in Japanese) | (See Japanese page) |
| Abstract(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) |
| Keyword(in English) | Elliptic curve / Finite field / Character / Order / Legendre symbol |
| Paper # | ISEC2019-66,SITE2019-60,LOIS2019-25 |
| Date of Issue | 2019-10-25 (ISEC, SITE, LOIS) |
| Conference Information | |
| Committee | ISEC / SITE / LOIS |
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| Conference Date | 2019/11/1(2days) |
| Place (in Japanese) | (See Japanese page) |
| Place (in English) | Osaka Univ. |
| Topics (in Japanese) | (See Japanese page) |
| Topics (in English) | |
| Chair | Shiho Moriai(NICT) / Tetsuya Morizumi(Kanagawa Univ.) / Tomohiro Yamada(NEL) |
| Vice Chair | Shoichi Hirose(Univ. of Fukui) / Tetsuya Izu(Fujitsu Labs.) / Masaru Ogawa(Kobe Gakuin Univ.) / Takushi Otani(Kibi International Univ.) / Toru Kobayashi(Nagasaki Univ.) |
| Secretary | Shoichi Hirose(NICT) / Tetsuya Izu(Tsukuba Univ.) / Masaru Ogawa(Toyo Eiwa Univ.) / Takushi Otani(KDDI Research) / Toru Kobayashi(Research Organization of Information and Systems) |
| Assistant | Dai Yamamoto(Fujitsu Labs.) / Yuuji Suga(IIJ) / Nobuyuki Yoshinaga(Yamaguchi Pref Univ.) / Daisuke Suzuki(Hokuriku Univ.) / Kenichi Arai(Nagasaki Univ.) |
| Paper Information | |
| Registration To | Technical Committee on Information Security / Technical Committee on Social Implications of Technology and Information Ethics / Technical Committee on Life Intelligence and Office Information Systems |
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| Language | JPN |
| 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 |
| 1st Author's Name | Masaaki Shirase |
| 1st Author's Affiliation | Future University Hakodate(FUN) |
| Date | 2019-11-01 |
| Paper # | ISEC2019-66,SITE2019-60,LOIS2019-25 |
| Volume (vol) | vol.119 |
| Number (no) | ISEC-257,SITE-258,LOIS-259 |
| Page | pp.pp.25-32(ISEC), pp.25-32(SITE), pp.25-32(LOIS), |
| #Pages | 8 |
| Date of Issue | 2019-10-25 (ISEC, SITE, LOIS) |