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\section{Introduction}  Elliptic curves are a useful tool within cryptography. An Elliptic Curve is a group, and some Elliptic Curves have this useful property: given a group member (point) $G$ and an integer $n$, the point $H = nG$ can be computed in $\log n$ time; however given two points $G, H$, computing the integer $n$ such that $H = nG$ takes $\sqrt{n}$ time.  The most common Elliptic Curves used in practice is defined over a prime field; field $GF(p)$;  what this means is that a majority of the time taken computing $nG$ is done computing the modular multiplication $a \times b \pmod p$, for a large (perhaps 256 bit) prime $p$ that we have picked. - What are elliptic curves  - How we do math in them  - Sparce primes  - Side channel attacks