For questions regarding elliptic curves. Questions on ellipses should be tagged [conic-sections] instead.

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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 ...
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86 views

How is this an isomorphism?

$\newcommand\O{\mathcal O}$I was reading Silverman and Tate's Rational Points on Elliptical Curves. In page 21 of the same book it was written We also want to mention that there is nothing ...
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68 views

What is the point $\{∞\}$?

The set of all rational points in an elliptic curve $C$ over $ℚ$ is denoted by $C(ℚ)$ and called the Mordell-Weil group, i.e., $C(ℚ)=\{\text{points on } $C$ \text{ with coordinates in } ℚ\}∪\{∞\}$. ...
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35 views

Integer points belonging to two distinct elliptic curves.

Two different circles can have an integer point in common (for example, $P=(1,1)$ belongs to both $x^2+y^2-2=0$ and $x^2+y^2-4(x+y)+6=0$) but any pair of distinct elliptic curves on the class defined ...
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70 views

Points on an elliptic curve over $\mathbb F_p$

Let $E$ be an elliptic curve over $\mathbb F_p$ (the finite field of $p$ elements) defined by $y^2=x(x-n)(x-m)$ where $p\nmid nm(n-m)$. Let $N$ be the number of $\mathbb F_p$-valued points of $E$. ...
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57 views

Projective coordinates for point at infinity on elliptic curve

What is the unique characteristic of the projective coordinates of a point at infinity? I am specifically looking for a characteristic on (short) weierstrass curves. I know that the point at infinity ...
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67 views

Elliptic curves find points with rational coordinates

The elliptic curve $y^2=x^3+3x+4$ has points O,(-1,0) and (0,2). Find five more points with rational coordinates. The answer to this example gives: (0,-2) (5,-12) (5,12) (71/25,744/125) and ...
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34 views

Multiply point by scalar in elliptic curve group

I'm trying to understand how to multiply a point by a scalar to get a point in elliptic curve cryptography. Here's an example from my textbook. The group is E257(0, -4). That's shorthand for y2 = x3 ...
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96 views

defining the group law on elliptic curves in general

Let $k$ be an arbitrary field and $C \subset \mathbb{P}^2(k)$ an elliptic curve. In order to define the group law on $C$ we need to establish some geometric facts first, e.g. Any line intersects $C$ ...
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46 views

Derivative of Integral of (g) with g in the limit

I would like to evaluate the following: $$\frac{\partial }{\partial \beta }\int _0^{\cos ^{-1}(\beta )}\text{dx} \sqrt{\beta +\cos (x)}$$ given that $0\leq\beta\leq1$ basically I'd like to find ...
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79 views

an Example of Elliptic Curve over finite field has no CM

I have known this property (from Silverman's The Arithmetic of Elliptic Curves): Let $\operatorname{char}(K)=p>0,$ and let $E/K$ be an elliptic curve with $j(E)~ \overline{\in}~ \overline{ ...
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121 views

Elliptic Curves and “roots”

Given elliptic curve $\omega$ in $\mathbb{R}^2$ such that $y^2 = x^3 + ax + b$, how can you find how many solutions (and what they are) of $x^3+ax+b$ have a $y$ value of $0$; or as they call it, a ...
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125 views

Elliptic Curve: Deduce the formula for doubling a point

Given an elliptic curve $E=\{ (x,y) \in \mathbb{F}_q^2 | y^2=x^3+ax+b \}$. Now deduce the general equation for doubling a point $P:=(x,y) \in E$. --- Firstly I constructed the function f ...
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Solving an equation in charcateristic 2 in sage OR finding 3-torsion points of an elliptic curve over field with char 2

Problem: show that an elliptic curve over a field of char 2 has nontrivial 3-torsion points Method: I used SAGE to unwind the duplication formula for an elliptic curve given in short Weierstrass form ...
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38 views

Why is the answer set limited here?

This question is based on pp $67$ - $68$ of Ash and Gross's "Elliptic Tales". Here the authors discuss points on a curve in the projective plane. We have an equation $f(x,y) = x^2+y^2$ We can ...
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32 views

On the rank of $y^2=x^3+a^2x^2-a^4x$

How can I prove that the rank of $y^2=x^3+a^2x^2-a^4x$ is zero where $a$ is rational and positive?
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72 views

Order of subgroup on elliptic curve over $Z_p$

I should determine the order of subgroup on elliptic curve over $\mathbb{Z}_p$ where $p$ is prime, and point $X$ is generator of some subgroup. While generating the subgroup by points addition I found ...
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86 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?
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47 views

Elliptic curve and restriction

Let $E$ be an elliptic curve. Let $\xi$ be a class of $H^{1}(\mathbb{Q}, E[m])$ unramified at the prime $\ell$. Then $\xi$ restricted to $H^{1}(I_{\ell}, E[m])$ where $I_{\ell}\subset ...
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140 views

Are elliptic curves also Galois covers of degree 3

Let E be an elliptic curve with equation $y^2=x^3+Ax+B$. The projection onto the $x$-coordinate is a Galois morphism of degree $2$. But what about the projection onto the $y$-coordinate? Is it ...
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question about j-invariant

In Hartshorne, there's a formula for the j-invariant in terms of $\lambda$. It says that $$ j = 2^8 \frac{(\lambda^2-\lambda+1)^3}{\lambda^2(\lambda-1)^2}.$$ Can one reverse this formula? That is, can ...
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How to calculate an elliptic curve

I need to find an elliptic curve in $F_{19}$ that has $|E(F_{19})|=18$. I am really stuck here. Can anyone help?
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Subgroups of points of order 2 in an elliptic curve

Depending on the roots of $y^2 - x(x^2+ax+b) = 0$ being real or not, we can have 2 subgroups of points of order 2 for a given elliptic curve- Kelin-4 group or a cyclic group of order 2. How does one ...
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Conditions of $f=a+bx+cz+dx^2+exz+fz^2+…$ such that its tangent line is $z=0$ and inflection point is at the origin.

Let $x,z$ be coordinates on $k^2$ and $f\in k[x,z]$; write $f$ as $$f=a+bx+cz+dx^2+exz+fz^2+...$$ Write down the conditions in terms of $a,b,c,...$ such that (a) $P=(0,0)\in C: (f=0)$; (b) the ...
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22 views

How to show rational points of finite order on an elliptic cure are closed under addition

I would like to show that rational points of finite order on an elliptic curve are closed under addition. If $P_1$ and $P_2$ are rational (actually integral) points of finite order, say $nP_1= O$ and ...
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71 views

Abelian group of rational points of an elliptic curve

I want to find the abelian group of rational points $E(\mathbb{Q})_{\text{torsion}}$ of the elliptic curve $y^2=x^3-2$. $$E(\mathbb{Q})_{\text{torsion}}=\{P \in E(\mathbb{Q}) | P \text{ of finite ...
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48 views

Elliptic curve-point at infinity

In my lecture notes we have the following: $$P \oplus Q \oplus R =O \Leftrightarrow P, Q, R \text{ are collinear }$$ So $$P \oplus Q \oplus O =O \Leftrightarrow Q=-P$$ that means that $Q=-P$ ...
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71 views

graduate level introduction to elliptic curve cryptography

I am looking for a good modern book / lecture-notes about elliptic curve cryptography. Does anyone have good recommendations?
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87 views

Compute principal divisor for a rational function on a curve

During the lecture we defined the principal divisor of a rational function on a smooth curve as it follows: Consider the smooth curve $C\subseteq\mathbb{P}^2$. Take $g\in{K(C)^*}$. Then the principal ...
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Tangent line at $x_1$ to polynomial curve $p(x)$ of degree at least $2$ implies $x_1$ is a double root of $p(x) - p^{'}(x_1)(x-x_1)$?

Tangent line at $x_1$ to polynomial curve $p(x)$ of degree at least $2$ implies $x_1$ is a double root of $p(x) - p^{'}(x_1)(x-x_1)$ ?. Suppose I have a polynomial function $p(x): \mathbb R ...
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87 views

Third point of elliptic curve $E: y^2 = x^3 + Ax + B$ given points $P_1=(x_1,y_1), P_2=(x_2, y_2)$ on $E$ (Weierstrass equation).

Third point of elliptic curve $E: y^2 = x^3 + Ax + B$ given points $P_1=(x_1,y_1), P_2=(x_2, y_2)$ on $E$ (Weierstrass equation). Assume $x_1 \neq x_2$. I create the straight line $y = m(x-x_1) ...
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55 views

Existence of certain homogenous forms

Let $D(X,Y), E(X,Y)\in\mathbb{Z}[X,Y]$ forms of the same degree $n$ and suppose that the resultant $R=Res(D,E)$ of $D$ and $E$ is not $0$. Show that there are homogenous forms $L_0(X,Y),M_0(X,Y), ...
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43 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 ...
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38 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 ...
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82 views

Trivial torsion subgroup

I am just wondering, suppose we have a curve $y^2 = x^3+ax + b$ defined over $\mathbb{Q}$ and suppose for simplicity $a,b \in \mathbb{Z}$. Can we say something about the torsion subgroup with the only ...
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63 views

Weierstrass normal form

How can I show that the Weierstrass normal form $u^3 + v^3 = \alpha$,with $x=12\alpha/(u+v)$ and $y=36\alpha (u-v)/(u+v)$, satisfy $y^2=x^3-432α^2$ ?
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Points on elliptic curves

I am learning elliptic curves theorem and I have read in more papers that for two distinct points $P$ and $Q$ there is always point $R$ such that $P+Q+R = \infty$. I know that this point should be ...
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1answer
39 views

isogenies between tori

Let Hom$(\mathbb{C}/\Lambda_1,\mathbb{C}/\Lambda_2)$ be the set of isogenies between $\mathbb{C}/\Lambda_1$ and $\mathbb{C}/\Lambda_2$, where $\Lambda_1,\Lambda_2$ are lattices. I am asked to prove ...
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55 views

Find the bound for [K(E[p]):K]

Let E be an elliptic curve over a field K of characteristic p > 0, we know that E[p] has order 1 or p, how to bound [K(E[p]):K]?
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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 ...
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74 views

This proof is completely unclear for me

Let $$L(C,s)=\sum_{n=1}^\infty \frac{a_n}{n^s}$$ be the Dirichlet series of the Hasse--Weil L-function of an elliptic curve $C$ over $ℚ$. I find the following proof in an old Russian book: I want to ...
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351 views

Subtraction two points on elliptic curve.

Suppose Q, T and S are three points on an elliptic curve, such that Q+T = S. With knowing Q and S, can we compute T? In other word whether exists subtraction operation on elliptic curve, or not?
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419 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 ...
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70 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?
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58 views

Equation of elliptic curve

Let $E/\mathbb{Q}$ be an elliptic curve with a 3-torsion point $T$. One can write a Weierstrass equation for $E$. If I define $C := E/\langle T \rangle$, what is the Weierstrass equation for $C$? Is ...
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106 views

Solutions of $a^2 = b^d -3^c$

The solutions of $a^2 = b^d -3^c$ are in the form $(a, b, c, d) = ((46)27^t, (13)9^t, 6t+4, 3)$. This is done by using calculator. As per my calculator, I have checked some terms, which are satisfied ...
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114 views

Maximal small lattice points of an elliptic curve

The elliptic curve $-4 x^3 + 4 x^2 y + 16 x - y^3 + 9 y$ goes through $21$ integer points in the range $-9$ to $9$. Is that the maximum?
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Representations of algebraic group ($S_{\mathfrak{m}}$)

I'm studying Serre's book "Abelian $\ell$-adic Representations and Elliptic Curves" and in chapter II $\S$2.4 we have this proposition: Consider $v$ a finite place of $K$ and $F_v \in Gal(K^{ab}/K)$ ...
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Framing a lattice problem from information available on multiple runs of GLV decomposition

I have posted a similar question here. The GLV method [ref] is used to speed up ECDSA signature generation. In this method, an input scalar $k$ is decomposed into two scalars, $k_1$ and $k_2$. Then ...
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44 views

For which values of $k$ does: $ y^2 = x^3+(2^{2^k}+1)x$ have solutions in integers?

let $E_D$ be elliptic curve and $k$ is integer number $$E_D: y^2 = x^3+px. $$ When $p = 2^{2^k}+1$ is prime fermat . my question is :For which values of $k$ does:E.d $$ y^2 = x^3+px. $$ have ...