ec-ecdlp.html

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          <h2 class="title">ECDLP</h2>
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    <p>Since elliptic curves are being used in cryptography there should be a presumable 'hard' problem that lives in the finite group K of an elliptic curve E. And this problem is the elliptic curve discrete logarithm problem. Knowing that DLP in a finite field $\mathbb{F}_p$ is as secure as the largest prime number.
Given $P=(x,y) \in E_K$ and $n$, we compute 
$$\color{blue}{Q}=\color{red}{n}\color{blue}{P}=\underbrace{(x,y)+(x,y)+(x,y)+ \cdots + (x,y)}_{n}$$

The order $l$ of $P$ is a 'large' prime such that $lP=\mathcal{O}_E$. The DLP is parametrized by the largest prime divisors of the order $P$. So we choose $P$ such that should have a large prime order. Then DLP is to find the number $\color{red}{n}$
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<p>*Notes: chosing finite fields of this form: $\mathbb{F}_{p^{n*m}}$, that is $p$ to a composite number is susceptible to $\color{blue}{Weil }$ descent attacks. The interesting thing here is that this is not bad when you do a pairing.} </p>