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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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How exactly is a specific message sent in ECC

I tried to think about it and came up with no answer. I read how the Elliptic curve cryptography works, I understand that Bob and Alice have their own key which they multiply the generator of the elliptic curve points (which is a group), and then do…
Jungle
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Doubt about Lang's "Elliptic Curves Diophantine Analysis"

I am trying to understand Theorem VII.3.4 of AEC without using formal groups. I am reading then Chapter III.1 in Elliptic Curves Diophantine Analysis by Lang. In the proof of theorem 1.3 he says that $t(P)\mid t(p^{m-1}P)$ and I don't understand…
cartesio
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Proposition III.3.1 in Silverman's The Arithmetic of Elliptic Curves

I'm reading Silverman's The Arithmetic of Elliptic Curves, and I have a question concerning his proof of proposition III.3.1a, which states that any elliptic curve $E$ over $K$ is isomorphic to a plane curve given by a Weierstrass equation. Using…
Sha Vuklia
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Understanding Elliptic Divisibility Sequence (EDS)

I am sorry if this is a basic question but I am pretty new to elliptic curves. More precisely I am trying to understand elliptic divisibility sequence. While I was searching online I came across to: ''Given an elliptic curve $E$ in short…
Kate Jns
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When is a $j$-invariant supersingular?

Assume $j \in \mathbb{F}_p$ is the $j$-invariant of a curve $E/\mathbb{F}_p$. The modular polynomial $\Phi_l(T,S)$ is such that the roots of $\Phi_l(T,j)$ are the $j$-invariants of curves $l$-isogenous to $E$, but apparently there are cases where…
José
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Determining the kernel of an isogeny

Let $$g:(x,y)\rightarrow \Big(\frac{x^2+x+3}{x+1},\frac{x^2y+2xy+15y}{x^2+2x+1}\Big)$$ be a map $E_1\rightarrow E_2$ where $$E_1/\mathbb{F}_{17}:y^2=x^3+1$$ $$E_2/\mathbb{F}_{17}:y^2=x^3-10$$ I want to determine the kernel of $g$. Apparently the…
Henkie
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Notation $\overline{K}(C)^*$ in Silverman's "The Arithmetic of Elliptic Curves"

In Joe Silverman's "The Arithmetic of Ecliptic Curves" he talks a lot about the integral domain $\overline{K}(C)$. On page $27$ he suddenly decides to chose an element $f$ from some set $\overline{K}(C)^*$. What does this notation mean? After page…
Milo Moses
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Prime Numbers and Elliptic Equations

I came across one elliptic equation of the form $y^2 = x^3 + p^2$ being $p$ prime, and taking $p \neq 3$, I want to have more understanding why there is no rational point $x$, for $y = 3p$ or $y = 3p^2$, such that: $$y^2 = x^3 + p^2$$ I want to know…
João Víctor Melo
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Equivalent definition of supersingular elliptic curves

Using the definitions that an elliptic curve $E$ over a finite field $K$ of characteristics $p$ is ordinary if $E[p]\cong \mathbb{Z}/p\mathbb{Z}$ and supersingular if $E[p]\cong 0$, how can I show that $E$ is supersingular if and only if…
Christina
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Elliptic curves under field automorphism

I was reading Silverman's Advanced Topics in the Arithmetic of Elliptic Curves book and I saw a notation which it wasn't defined before (or I missed): Let $E$ be an elliptic curve over $\mathbf{C}$ and $\sigma: \mathbf{C} \longrightarrow \mathbf{C}$…
user794509
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Multiples of a point in a non-elliptic curve

Let $E:y^2-xy+y-x^3=0$ over a field $K$, $P=(0,0)$. If $\text{char}(K)\neq2$, $E$ is an elliptic curve for which I can easily get $n*P$. Now, something goes wrong with $\text{char}(K)=2$: Basically, I draw the tangent line in $P$ (a horizontal line)…
mikemike
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Why is a solution colinear to two rational solutions of a Weierstrass-form elliptic curve also rational?

I learned this fact and it blew my mind: given an equation $y^{2}=x^{3}+ax+b$ and two rational solutions: $(x_1, y_1), (x_2, y_2)$ with $x_1, y_1, x_2, y_2 \in \mathbb{Q}$, then any other solution colinear with the first two solutions is also…
Gabi
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Elliptic Curve and Consecutive X Coordinates

My question is brief, is there an algorithm or method to calculate the consecutive X coordinates on an elliptic curve to find out if the distance between is constant? The reason for the question is as follows: 1. Base Generator Point is given. …
kmart875
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Abscissa of a point of infinite order in an elliptic curve

Any hints on this? I am fine with having a proof only for the case $a_1 = a_2 = a_3 = 0$ (if necessary, assume also $a_4 = 0$). I know how to work with the group law in general, but have no ideas apart from tackling an expression for $x([m]P)$…
DesmondMiles
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Result that determine the values of the integers $a,b$ such that $r=0$

Let $C$ be an elliptic curve over $ℚ$ that has the form: $$y²=x³+ax+b...........................(1)$$ where $a,b$ are integers. The group $C(ℚ)$ is a finitely generated Abelian group and we have $C(ℚ)≃ℤ^{r}⊕C(ℚ)^\mathrm{tors}$, where…
Safwane
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