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If I have the following Elliptic Curve:

$$E: y^2 \equiv x^3 + 2x + 2 \pmod{17}$$

How can I calculate the number of points on this elliptic curve $E$ ?

and how can I invest the following law which calculates the Number of Points on an Elliptic curve NPE:

$$NPE=1+p+\sum_{x=1}^{p-1} \frac{f(x)}{p}$$

Clayton
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jemmy.w
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  • Just do a search: for $0\le x\le16$, ask first whether $x^3+2x+2$ is zero modulo $17$, and if not, check by Quadratic Reciprocity or any other good method to see whether this number is a square modulo $17$. – Lubin May 10 '18 at 19:40

1 Answers1

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Let $\;f(x) := x^3+2x+2\;$ be a cubic. An Elliptic curve $\;E: y^2 \equiv f(x) \pmod{p}\;$ is defined as the set of points $\;(x,y)\;$ which satisfy the congruence and the point at infinity $O$, on which an addition operation is defined which makes it a group with the additive identity being the point at infinity $O$.

Thus the number of points $\;NPE=1+|\{(x,y):y^2=f(x)\}|\;$ where we work modulo $p$. The number of solutions of the quadratic $\;y^2=f(x)\;$ is given by the Legendre symbol as $\;1\!+\!\left({f(x) \over p}\right)\!.$ Thus
$$\;NPE=1+\sum_{x=0}^{p-1}1+\left({f(x) \over p}\right) = 1 + p +\sum_{x=0}^{p-1}\left({f(x) \over p}\right).\;$$

Somos
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