Tagged Questions
1
vote
2answers
68 views
How do you determine if an elliptic curve over a finite field is cyclic?
I know the group order and the points of the elliptic curve $y^2 = x^3 + Ax + B$, but I am confused on how to determine if they from a cyclic group
The curve $y^2 = x^3 + 2x +2$ in $\Bbb F_{11}$ ...
2
votes
2answers
82 views
order of elliptic curve $y^2 = x^3 - x$ defined over $F_p$, where $p \equiv 3 \mod{4}$
It is said that the elliptic curve $y^2 = x^3 - x$ defined over a prime field $\mathbb{F}_p$, where $p \equiv 3 \mod{4}$ has an order $p + 1$.
When I tried to get the elements of $E = \{(x,y) \in ...
0
votes
1answer
49 views
Find lift($E_{p^2}$) of an elliptic curve $E_p$ defined in field $F_p$ where $p$ is a prime
How to find $E_{p^2}$ of an elliptic curve $E_p$ defined over finite field $F_p$ where $p$ is a prime number?
3
votes
2answers
257 views
Elliptic curves over a finite field $\mathbb{F}_p$ where $p$ is prime.
Let $Y^2=f(X)$ be an Elliptic curve over a finite field $\mathbb{F}_p$ where $f(X)=X^3+aX+b$
In an undergraduate coursebook on an Applied Algebra course it states that "It is plausible to suggest ...
0
votes
0answers
103 views
Showing Elliptic Curves are irreducible
Suppose an elliptic curve defined over a field $K$ has form $y^2 + h(x)y = f(x)$ where $h(x), f(x) \in K[x]$. $h(x)$ has degree 0 or 1 and $f(x)$ has degree 3 or 4. Moreover, this equation is ...
3
votes
1answer
296 views
Intersection of a line with an Elliptic Curve
I am trying to show that if a line given by $y = mx + b$ intersects an Elliptic Curve given by $E(\mathbb{K}): y^2 = x^3 + Ax + B$ in three points then the line is not tangent to the curve.
Given ...
3
votes
1answer
133 views
Groups where discrete logarithm is hard
What are examples of groups, where DLP (discrete logarithm problem) is hard?
Two obvious ones are: integers modulo $p$ ($p$ being prime) and elliptic curves over finite fields. What are the others?
1
vote
1answer
144 views
How to nicely extend finite field?
I'm working on an implementation of Miller's algorithm that computes the Weil pairing (elliptic curves, cryptography). In order to do that, I have to implement finite fields.
So far I have managed to ...
23
votes
3answers
605 views
What is an elliptic curve, and how are they used in cryptography?
I hear a lot about Elliptic Curve Cryptography these days, but I'm still not quite sure what they are or how they relate to crypto...
