CurveEnthusiast
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11 answers
48 votes
4k views
Why study finite-dimensional vector spaces in the abstract if they are all isomorphic to $R^n$?
8 votes

A very concrete example of why we should not stop studying something once we have isomorphisms, although going away from vector spaces. A lot of modern (asymmetric) cryptography is based on discrete ...

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2 answers
8 votes
2k views
Galois theory and cryptography
4 votes

I think Jyrki's answer is great, and I completely agree with it. It focuses on public key cryptography, which is probably most interesting from a mathematical point of view. Let me try to give what I ...

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1 answers
1 votes
1k views
Determinant of Hessian matrix and inflection point on elliptic curve
4 votes

Short answer: the point at infinity is an easy case. For any affine point, notice that $\det H=8\psi_3$, where $\psi_3$ is a division polynomial. From standard properties of division polynomials, the ...

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1 answers
3 votes
556 views
Automorphism Elliptic Curve - nth Root of Unity
Accepted answer
3 votes

Let $E/\mathbb{F}_p$ be the elliptic curve which you describe. I'm going to assume that $E(\mathbb{F}_p)=\langle P\rangle$, because you mention that $P$ is a generator. Therefore $\#E(\mathbb{F}_p)=n$,...

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1 answers
1 votes
441 views
What is the kernel polynomial of a degree $l$ isogeny?
Accepted answer
3 votes

I'm not sure what you mean by a standard way, here's an answer from someone working with elliptic curves related to cryptography. Steven Galbraith has a nice chapter on Elliptic Curves in his book ...

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2 answers
4 votes
444 views
Non-commutative formal groups- proof of non-commutativity
2 votes

To add to my comment: the problem is that in $\mathbb{F}_p$, $p$ prime, we'll have $x^p=x$ for all $x\in\mathbb{F}_p$. Therefore your group law is almost trivially commutative. Instead you can ...

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1 answers
3 votes
116 views
Show that ${\phi^2}_p =-p$
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2 votes

Let $E/\mathbb{F}_2:y^2+y=x^3+x$ be an elliptic curve. It is easily checked that it is supersingular, and that $\#E(\mathbb{F}_2)=5.$ That means that $a=p+1-\#E(\mathbb{F}_2)=-2$. Suppose that $\...

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1 answers
3 votes
234 views
Building blocks to understanding Elliptic Curves
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2 votes

Disclaimer: I don't consider myself a mathematician, but I have some background in it. I am currently working on topics related to elliptic-curve cryptography, so I hope my perspective might be useful....

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1 answers
-2 votes
135 views
Twin prime conjecture and finite groups. Help with proof of basic equivalence.
Accepted answer
2 votes

Let $n=14$. Then $(n-2)n=12\cdot 14=168$, and $n^2-1=14^2-1=195$. Take $X=2$. Then $$2^{168}\equiv1\bmod{195}.$$ But $n-1=13$, while $n+1=15$, and 15 is clearly not prime. Your statement seems to be ...

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1 answers
1 votes
139 views
Order of an Elliptic curve
Accepted answer
2 votes

By Lagrange $\#E(\Bbb F_p)=kp$ for some integer $k$. By Hasse's theorem $$kp<p+1+2\sqrt{p}<2p\quad (\text{since }1+2\sqrt{p}<p\text{ for }p>7)$$ and $$kp>p+1-2\sqrt{p}>0\quad (\text{...

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1 answers
1 votes
154 views
Alternative to Pohlig-Hellman with given generator order
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2 votes

The hint seems to just point you to using Pohlig-Hellman. We can use this approach whenever the order of our base element is smooth, in this case $g$ which has smooth order $2\cdot11\cdot19\cdot281$. ...

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1 answers
1 votes
67 views
Cryptographic Reduction: Last Bit of RSA Suffices to Recover the Plaintext
Accepted answer
2 votes

It seems that we are browsing the same SE pages :-). I think the idea of the result has not sunk in yet, so let me try to explain what we're actually trying to do. We are assuming a $k$-bit RSA-...

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2 answers
0 votes
225 views
Math Olympiad Question- Proof
Accepted answer
2 votes

Since we can pay $A$ cents with $B$ coins, there are non-negative integers $b_1,b_2,b_5,b_{10},b_{20},b_{50}$ and $b_{100}$ such that \begin{align*} A &= b_1+2b_2+\ldots+100b_{100}, \\ B &= ...

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1 answers
3 votes
246 views
Silverman AEC exercise V.5.10(e)
1 votes

I've managed to solve it, posting it here for whoever is interested. Let $E/\mathbb{F}_q$ be an elliptic curve, and assume that $\operatorname{tr}(\phi)=0\pmod{p}$. Then by (a), we know that $E$ is ...

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1 answers
1 votes
1k views
How do you compute order of points in elliptic curve?
1 votes

Hints: a) A point $P$ has order 2 if $P+P=\mathcal{O}$, and therefore $P=-P$. What is $-P$? Then equate the $y$-coordinates. b) A point $Q$ has order 3 if $Q+Q+Q=\mathcal{O}$, and therefore $Q+Q=-Q$....

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1 answers
1 votes
94 views
Holomorphic differentials on $\mathbb{P^1}$?
1 votes

Any $\omega\in\Omega_C$ is holomorphic if $\operatorname{ord}_P(g)\geq0$ for all $P\in C$, where $\omega=g\,dt$ and $t$ is a uniformizer at $P$. Note that $g\in\bar{K}(C)$. Explanation: On page 31 ...

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1 answers
0 votes
246 views
Shanks Algorithm for composite orders
Accepted answer
1 votes

Shank's algorithm can be used for any group, it does not use any specific properties. The same is true for the Pohlig-Hellman algorithm. Suppose we have a group of order $$r=\prod_ip_i^{e_i},$$ then ...

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1 answers
0 votes
68 views
Find $log_{P}R$ on this Elliptic curve
Accepted answer
1 votes

Multiplying the second equation by 2 we have the system \begin{align*} 6P+2Q+R=\infty, \\ 4P+2Q+4R=\infty. \end{align*} Subtracting the second from the first gives $2P-3R=\infty$, i.e. $2P=3R$. Since $...

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1 answers
0 votes
74 views
Prove that an algorithm that solves Basis Problem for $E[m]$ can be used to solve the ECDLP (Elliptic Curve Discrete Logarithm Problem)
1 votes

The points $P_1,P_2$ together form a basis of $E[m]$. Hence for any $\alpha,\beta\in\mathbb{Z}/m\mathbb{Z}$ such that $\alpha P_1+\beta P_2=\mathcal{O}$ we can conclude that $\alpha=\beta=0$. As ...

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1 answers
2 votes
187 views
Avoiding the principal square root issue in Discrete Logarithm?
1 votes

Not sure if I understand correctly, but here's my viewpoint on where you go wrong. In the first step to obtain $z_0$, you check whether $g^z$ is a quadratic residue modulo $p$. You then reduce to $...

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1 answers
1 votes
344 views
Silverman Arithmetic of Elliptic Curves Exercise 3.5(a)(ii)
1 votes

As far as I know, answering my own question is considered okay. Here is my very computational approach: there is likely still a much more elegant solution. Suppose $(x,y)\in E_{\text{ns}}(L)$ such ...

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1 answers
3 votes
921 views
Why is the abelian group of points on an elliptic curve over a finite field isomorphic to the product of at most two cyclic groups?
0 votes

Let $n=\#E(\mathbb{F}_q)$, which is finite because $\mathbb{F}_q$ is. Then $E(\mathbb{F}_q)\subset E[n]\cong \mathbb{Z}/n\mathbb{Z}\times \mathbb{Z}/n\mathbb{Z}$.

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1 answers
2 votes
583 views
Generator of group, Computation of discrete logarithm
0 votes

Let's write $y\equiv g^x\bmod{p}$, so that the goal is to find $x$. By exponentiating both sides we have the equalities \begin{align*} y^{33}\equiv (g^{33})^{x}\bmod{p}, \\ y^{22}\equiv (g^{22})^{x}\...

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