1
vote
1answer
31 views

Find coordinate $y$ of an elliptic curve point

If I have an elliptic curve over a finite filed $F_p$ ($p$ is prime) defined as $$ y^2 \equiv x^3 + ax + b\pmod p,$$ such that $4a^2 + 27b^2 \neq 0$ and suppose I have only given the coordinate $x$, ...
0
votes
1answer
58 views

Public Key Scheme decryption. [closed]

You have been sent a message based on the following Public Key Scheme. 1) Bob chooses two large primes $\ p,q $ with $ p \equiv q \equiv 2 \pmod 3$ and computes $ n=pq. $ 2) Bob chooses integers $ e,d ...
0
votes
1answer
25 views

Addition on an Elliptic Curve and Modular Arithmetic involving fractions

I'm having a bit of an issue with addition on elliptic curves. For example, I've been given the curve $Y^2 = X^3 + 2X + 1$, working modulo 3. Now, say I want to add the point $(1,2)$ with itself. To ...
0
votes
1answer
22 views

Explanation and validation of point adding/doubling on elliptic curves

I'd like to implement point multiplication on elliptic curves over prime fields. My problem is that I've found different definition how to do it. At adding: the second parameter of the result is not ...
2
votes
1answer
85 views

Number of points on $Y^2 = X^3 + A$ over $\mathbb{F}_p$

Let $p\equiv 2\pmod{3}$ be prime and let $A\in\mathbb{F}^{∗}_p$ . Show that the number of points (including the point at infinity) on the curve $Y^2 = X^ 3 + A$ over $\mathbb{F}_ p$ is exactly $p + 1$ ...
3
votes
1answer
109 views

Finding number of solutions to an equation in $\mathbb F_p$

$p=3 \pmod 4$ is a prime. Let $b\in \mathbb F_p^*$. Show that the equation $v^2=u^4-4b$ has $p-1$ solutions $(u,v)$ with $u, v \in \mathbb F_p$. If we write the given equation as $v+u^2=x$ and ...
2
votes
1answer
135 views

Nef divisors on the compactified modular curve level $N$

Consider the compactified modular curve with full level structure $X=\overline{\Gamma(N)\setminus \mathcal{H}}$. We know the Hodge bundle (the extension of the hodge bundle to the compactification) ...
0
votes
1answer
118 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 ...
2
votes
1answer
173 views

Congruence subgroups and modular curves of type (M,N)

I would like to study the "modular curve" $Y(M,N)$, parametrizing an elliptic curve $E$ together with $p \in E[M]$ and $q \in E[N]$ (here and in the following $M$ divides $N$). Let $\Gamma(M,N)$ be ...
8
votes
2answers
804 views

The modular curve X(N)

I have a question about the modular curve X(N), which classifies elliptic curves with full level N structure. (A level N structure of an elliptic curve E is an isomorphism from $Z/NZ \times Z/NZ$ to ...
4
votes
2answers
921 views

What is a primitive point on an elliptic curve?

While working with elliptic curves for cryptography reasons, I found the notion of a primitive point, but no definition. For example, $P(0,6)$ is a primitive point on the elliptic curve $y^2\equiv ...