1
vote
1answer
65 views

Structure of $C(F_5)$ from Rational Points on Elliptic Curves

In the book Rational Points on Elliptic Curves by Silverman/Tate one examines the elliptic curve $y^2 = x^3 + x + 1$ over $F_5$. One can then easily determine the group $$ C(F_5) = \lbrace ...
4
votes
1answer
128 views

Group structure of an elliptic curve

Let $E$ be an elliptic curve over field $\mathbb{Z}/p\mathbb{Z}$. The curve group $E(\mathbb{Z}/p\mathbb{Z})$ is always a) cyclic or b) direct product of two cyclic groups. First question: How do I ...
0
votes
2answers
160 views

Why must the order of basepoint of elliptic curve be prime?

Let $E$ be an elliptic curve defined over a finite field $F(q)$. Let $G\in E(F(q))$ be a point of order $n$, where $n$ is a prime number and $n>2^{160}$. The elliptic curve discrete logarithm ...
3
votes
1answer
60 views

Explicitly computing finite subgroups on elliptic curves

I have a simple cubic curve, say $y^2 = x^3 - x.$ Is there a simple way to find a small finite subgroup of points lying on this curve? (with respect to the elliptic curve group law.) Otherwise, does ...
2
votes
1answer
127 views

Inverse Scalar Multiplication of a point over elliptic curve

I was implementing point arithmetic operation, and was exploring the properties of point arithmetic, and I am unable to conclude whether $$ k^{-1}(kP) = P $$ where P is a point over elliptic curve $ ...
0
votes
1answer
121 views

Prove that there are $p+1$ points on the elliptic curve $y^2 = x^3 + 1$ over $\mathbb{F}_p$, where $p > 3$ is a prime such that $p \equiv 2 \pmod 3$.

Let $p > 3$ be a prime such that $p \equiv 2 \pmod 3$. Define the elliptic curve $E$ over $\mathbb{F}_p$ by $y^2 = x^3 + 1$. Prove that $E(\mathbb{F}_p)$ consists of $p+1$ points. Using Fermat's ...
1
vote
3answers
1k views

How to find the order of elliptic curve over finite field extension

I want to find the order of elliptic curve over the finite field $\mathbb{F}_{5^2}$, where $E(\mathbb{F}_{5^2}):y^2=x^3+10x+17$. I am using the method illustrated by John J. McGee in his thesis ...
1
vote
1answer
53 views

The group $E(\mathbb{F}_q)/NE(\mathbb{F}_q)$

Let $E$ be an elliptic curve defined over the finite field $\mathbb{F}_q$ and let $N\geq 1$. We also make the following assumptions: $T \in E(\mathbb{F}_q)[N]$ is a point of exact order $N$, ...
1
vote
1answer
245 views

Fencing the Group size,and its implication to Finiteness of Tate-Shafarevich Group

This question is an interesting one,not like my previous one. Can we judge the size of a Quotient Group by seeing the size of its constituents ? To add something ,Suppose consider a group ...