Hessian form of an elliptic curve: Difference between revisions
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The '''Hessian curve''' is a curve similar to [[Folium of Descartes]]. It is named after the German mathematician [[Otto Hesse]]. |
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To define the Hessian curve over a field $K&=&Fq$ where q≡2 (mod3). |
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Let E denote an elliptic curve over &K& which has a $K$-rational point of order 3 (but we also can define the curve for any $K$ with characteristic p>3). |
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It is interesting to analyze this curves, because we can get a 33% performance improvement as compared to the best reported methods and requires much less memory. For this, we only need to use the same procedure to compute the addition, doubling or subtraction of points. |
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Since we assume that E has a point of order 3, now we define the elliptic curve so that this point is moved to the origin: |
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Y<sup>2</sup>+a<sub>1</sub>XY+a<sub>3</sub>Y=X<sup>3</sup> |
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This curve has discriminant &\deltha=(a<sub>3</sub>(a<sub>1</sub><sup>3</sup>-27a<sub>3</sub>))=a<sub>3</sub><sup>3</sup>&\deltha |
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and the points of order 3 are given by (0,0) and (0,-a<sub>3</sub>) |
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So, to obtain our Hessian curve, we are going to do the following transformation: |
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First let &\mu denote a root of the polynomial T<sup>3</sup>- &\delthaT<sup>2</sup>+ &\deltha2T/3+a<sub>3</sub> &\deltha2=0 |
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Since q≡2 (mod3) every element in &K& has a unique cube root and we can determine &\mu from the formula: |
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==References== |
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* Otto Hesse (1844), ''Uber die Elimination der Variabeln aus drei algebraischen Gleichungen vom zweiten Grade mit zwei Variabeln'', Journal für die reine und angewandte Mathematik, 10, pp. 68-96 |
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[[Category:Curves]] |
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{{geometry-stub}} |
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