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Questions tagged [elliptic-curves]

For questions about elliptic curves.

An elliptic curve is a smooth, nonsingular projective curve of genus 1 with a specified point $\mathcal{O}$, defined over any field $K$. They form abelian groups under point addition. They are much studied in number theory, for example in cryptography and integer factorization.

An elliptic curve can be defined by an equation of the form: $$E:y^2=x^3+ax+b$$ with the discriminant $\triangle_E=-16(4a^3+27b^2)\ne 0$ so the curve is nonsingular, i.e. its graph has no cusps or intersections.

The elliptic curves with $a=0$ are .

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Counting point on twists of elliptic curves

Is there an "easy" way to count points on a twists of elliptic curves? Say we consider the curve $\newcommand{\F}{\mathbb F} E(\F_p):\ y^2=x^3+x$ and her twist $E'(\F_{p^2})=x^3+2^{1/4}x$. (Some may see, this is a curve from the KSS-16 family.) Our…
Shalec
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understanding the resulted lambda for 2 points on the elliptic curve

I'm following on a tutorial and the resulted lambda for the 2 points p & q is: $\lambda=\frac{-1}{2} = 11 mod 23$ but I'm not sure how the author got 11?
adhg
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Difference between affine and projective elliptic curve

where is the difference between an elliptic curve in affin and in projective representation? I know that an projective elliptic curve can create an abelian group. Does the affine elliptic curve does the same? Thank you very much!
matzzzz
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Elliptic curves - Weierstrass normal form

Are there any reasons why we transform elliptic curves with the equation $Y^2Z+a_1XYZ+a_3YZ^2=X^3+a_2X^2Z+a_4XZ²+a_6Z³$ over $\mathbb{R}$ or a finite field $F_q$ ($q$ is prime) to the shorter Weierstrass forms? Are the shorter forms more often…
matzzzz
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Number of points of two elliptic curves

Please help or hints me to solve this question: Suppose that $E_1: y^2= x^3+2x+5$ and $E_2: y^2= x^3+3x+5$ are two elliptic curve on $\Bbb F_{361}$. Show that #$E_1(\Bbb F_{361})$=#$E_2(\Bbb F_{361})$.
Masoud
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How do I prove that for an elliptic curve the only isomorphism is the following way?

I habe an elliptic curve over a field $K$ given by: $$E: y^2=x^3+Ax+B$$ Now I want to show that all changes of variables preserving this form are given by $x=u^2x$ and $y=u^3y$ with $u \in \bar{K}^*$... Any idea how to show it?
user299124
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Lines Intersecting with Elliptic Curves Help

I have an elliptic curve $E$ over $\mathbb{F}_{7}$ defined by $y^2=x^3+2$ with the point at infinity $\mathcal{O}$ I am given the point $(3,6)$ and need to find the line which intersects with $E$ at only this point I am told that this line is…
lioness99a
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Elliptic Curve and Divisor Example help (Step 1)

I have an elliptic curve $E$ over $\mathbb{F}_{11}$ defined by $y^2=x^3+4x$ with the point at infinity $\mathcal{O}$ I have a divisor of $E$, defined by…
lioness99a
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Ordinary or Supersingular?

Can someone verify that the elliptic curve $y^2 = x^3 + x +1$ over $\mathbb{F}_{3^2}$ is indeed supersingular? $\textbf{My solution:}$ The characteristic of the field is 3 so we are looking for the coefficient of $x^2$ in $(x^3+x+1)^1$, obviously…
Junsworth
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Torsion points of $y^2=x^3+1$.

Im trying to find the torsion points of $y^2=x^3+1$. If $y=0$, then $x=-1$. If $y^2 \mid 27$, then $y= \pm 1$, $\pm 3$. We get $y= \pm 3$, $x=2$ and $y= \pm 1$, $x=0$. Thus, $E(Q)_{\text{tors}}= \{O,(-1,0), (2, \pm 3), (0, \pm1) \}$. Here is the…
usere5225321
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Elliptic curve point multiplication with division polynomials

Given an elliptic curve $E: y^2 = x^3 + ax + b$ over a finite field $\mathbb{F}_q$ and a point $P$ with coordinates $(x,y)$, we have from the point addition formula that \begin{align*} 2P =&(\lambda^2-2x,\: \lambda(x-(\lambda^2-2x))-y) \\ =&…
user406579
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Prove that an algorithm that solves Basis Problem for $E[m]$ can be used to solve the ECDLP (Elliptic Curve Discrete Logarithm Problem)

Q: Let $\{P_1, P_2\}$ be a basis for $E[m]$. The Basis Problem for $\{P_1, P_2\}$ is to express an arbitrary point $P \in E[m]$ as a linear combination of the basis vectors, i.e., to find $n_1$ and $n_2$ so that $P = n_1 P_1 + n_2 P_2$. Prove that…
clay
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Possible cardinalities of an elliptic curve over $\mathbb F_7$

Possible cardinalities of an elliptic curve over $\mathbb F_7$ Where the equation is given by $y^2=x^3+ax+b$. Hasse Bound gives something between $3$ and $13$ but my question is, can all these values be attained ? Playing around here a bit, I got…
user257
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finding y-coordinate of a 4-torsion point on elliptic curve

I found the x-coordinate of my 4-torsion point on E: $y^2=x^3-3267x+45630$ given by $x=15\pm 36B$ with $B^2=-2$. My question here is how am I to find the y-coordinate of this 4-torsion point?
shahrina ismail
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Schoof's algorithm: when we find a factor of $\psi_\ell$, why do its roots correspond to the kernel of some endomorphism?

I'm busy self-studying Schoof's algorithm from Andrew Sutherland's notes. In section 9.6, he states that when we happen to find some factor $g$ of the division polynomial $\psi_\ell$, then the roots of $g$ must be the $x$-coordinates of points in…
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