| Presentation | 2011-05-13 Order of Elliptic Curve y^2=x^3+2^i3^j Over Barreto-Naehrig Field Masaaki SHIRASE, |
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| Abstract(in Japanese) | (See Japanese page) |
| Abstract(in English) | Barreto-Naehrig (BN) curve is an elliptic curve over F_p 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∈F_p. 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=z^3+b over F_p with BN prime p by z mod 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) |
| Keyword(in English) | BN curve / Gauss' theorem / Euler's conjecture / twist / Pairing-friendly elliptic curve |
| Paper # | ISEC2011-6 |
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| Conference Information | |
| Committee | ISEC |
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| Conference Date | 2011/5/6(1days) |
| Place (in Japanese) | (See Japanese page) |
| Place (in English) | |
| Topics (in Japanese) | (See Japanese page) |
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| Paper Information | |
| Registration To | Information Security (ISEC) |
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| Language | JPN |
| 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 |
| 1st Author's Name | Masaaki SHIRASE |
| 1st Author's Affiliation | Future University Hakodate() |
| Date | 2011-05-13 |
| Paper # | ISEC2011-6 |
| Volume (vol) | vol.111 |
| Number (no) | 34 |
| Page | pp.pp.- |
| #Pages | 8 |
| Date of Issue | |