-1
votes
0answers
23 views

Find point of maximal order $\operatorname{lcm}(a,b)$ on a curve

In Elliptic curve okamoto uchiyama there is a condition for picking base point $G$ such that $G$ belongs to $E_n$ of maximal order $\operatorname{lcm}(|E_{p2}|,|E_q|)$. I am not getting these ...
1
vote
1answer
25 views

Let E be defined over Fq and let n ≥ 1. Show that E(Fq)[n] and E(Fq)/nE(Fq) have the same order.

Let E be an Elliptic Curve defined over $F_q$ and let n ≥ 1. Show that $E(F_q)[n]$ and $E(F_q)$/$nE(F_q)$ have the same order. I feel like this is obvious. The n-th torsion group $E(F_q)[n]$ ...
0
votes
1answer
28 views

Finding the inverse of P on the generalized weierstrass equation

If P = (x, y) = ∞ is on a monic cubic polynomial, then −P is the other finite point of intersection of the curve and the vertical line through P. Show that −P = (x, −$a_{1}x$ − $a_3$ − y). (Hint: This ...
1
vote
1answer
43 views

Prove that the Frobenius map is a homomorphism

I want to prove that the Frobenius map $\phi$ is a homomorphism from the group of points on an elliptic curve $E(F_{2^k})$ to itself (endomorphism). It is trivial to check that if a point $P \in E$ ...
1
vote
1answer
52 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 ...
0
votes
1answer
54 views

Point multiplication in elliptic curve

Suppose $a$ is an integer and $Q$ is a point on an elliptic curve and $(x,y)$ are $x$ and $y$ coordinates of this point. My question is: Whether $a\cdot Q$ is equal to $(ax, ay)$?
0
votes
2answers
133 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 ...
1
vote
1answer
54 views

About the quotient group of degree zero divisors on $C$ by the principal divisors on $C$

Let $C$ be an elliptic curve with distinguished point $O$. My question is about a mathematical desription of this set denoted by $Pic(C)$ which is the quotient group of degree zero divisors on $C$ by ...
1
vote
1answer
75 views

Diffie-Hellman key exchange for three user.

Assume that there are three users that have their own secret key $d_i$ and corresponding public key $Q_i = d_i G$ such that $Q_i$ is a point in an elliptic curve. Now I'm looking for a solution to ...
0
votes
1answer
139 views

Point Division in Elliptic Curve Cryptography?

I want to implement a crypto protocol using Elliptic Curve Cryptography. However, it requires a division which I cannot handle. In multiplicative notation, it requires: Let $\mathbb{G}=\left ...
0
votes
0answers
109 views

conversion between multiplicative and additive group notation

I'm quite new to Groups and I'm using them for cryptology purpose. Currently, I'm learning about Elliptic Curve Cryptography and facing a notation problem. Since Elliptic Curves are abelian the group ...
5
votes
3answers
149 views

Purpose of cusps

In the theory of of modular forms, there is the set of of cusps defined by $\mathbb{P}^1 (\mathbb{Q})= \mathbb{Q} \cup \{\infty\}$. For an subgroup $\Gamma < \text{SL}_2(\mathbb{Z})$ of finite ...
0
votes
1answer
79 views

Addition on elliptic curves

assume $a$, $b$ are two integer numbers, and $G$ is a basepoint in an elliptic curve. Is $(a+b)G$ equal to $aG+bG$ or not?
0
votes
0answers
54 views

Can we extend the map $φ$ to $ℝ^{r}×C(ℚ)^{\text{tors}}→C(ℚ)$ as an isomorphism or not?

The motivation to this question can be found in How I can express $(x,y)∈G$ by using the $r$ independent points $P_1,P_2,\ldots,P_r$ We know that there is an isomorphism ...
8
votes
1answer
350 views

Automorphism group of the elliptic curve $y^2 + y = x^3$

Consider the elliptic curve $E : y^2+y = x^3$ over $\overline{\mathbb{F}_2}$. It has the biggest automorphism group $G$ among all elliptic curves, namely with order $24$. What is the structure of $G$? ...
3
votes
0answers
96 views

Why does Lenstra ECM work?

I came across Lenstra ECM algorithm and I wonder why it works. Please refer for simplicity to Wikipedia section Why does the algorithm work I NOT a math expert but I understood first part well enough ...
1
vote
1answer
236 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 ...
3
votes
1answer
173 views

Ring on an Elliptic Curve

I know that for a given elliptic curve $E$ we can define a group $G$ with the points on this curve. However, can we define a ring on it? That is, can we define a multiplication on the curve, where we ...
2
votes
1answer
114 views

Extracting the value of $y$ from $x$ in an elliptic curve over a finite field

Given an elliptic curve $y^2 = x^3 + ax + b$ over a finite field $\mathbf{F}_p$, how can I retrieve the value of $y$ given the value of $x$? My knowledge in this area is quite limited, so I ...