# Timofey Yaluhin - Elliptic Curves: Cheat Sheet (Highlights)
## Metadata
**Review**:: [readwise.io](https://readwise.io/bookreview/39016329)
**Source**:: #from/readwise #from/reader
**Zettel**:: #zettel/fleeting
**Status**:: #x
**Authors**:: [[Timofey Yaluhin]]
**Full Title**:: Elliptic Curves: Cheat Sheet
**Category**:: #articles #readwise/articles
**Category Icon**:: 📰
**URL**:: [hackmd.io](https://hackmd.io/@timofey/rJ8HP8Yaj)
**Host**:: [[hackmd.io]]
**Highlighted**:: [[2024-03-27]]
**Created**:: [[2024-03-27]]
## Highlights
- Small ρ is desirable to speed up arithmetic on the elliptic curve. ([View Highlight](https://read.readwise.io/read/01hsvzj3wbmw4n6xzbg5qxb3cj)) ^698047068
Larger scalar field and smaller base field.
- **Cofactor** h=n/r measures curve's order relative to its largest prime subgroup. Cofactor of prime-order curves is always equal to 1. ([View Highlight](https://read.readwise.io/read/01hsvzn6t7zb8gaf66ydqzdemj)) ^698048422
- The **embedding degree** is the smallest integer k that lets you transform an instance of the ECDLP over an elliptic $E(F_q)$ into an instance of the DLP over the field $F_{q^k}$. ([View Highlight](https://read.readwise.io/read/01hsvzrdn70rspkh8ah3h3310v)) ^698049164
- **Weierstrass curves** are defined as $y^2=x^3+Ax+B$. This is arguably the most common form for elliptic curves. ([View Highlight](https://read.readwise.io/read/01hsvzsp3azy96ysgnwbysy362)) ^698049335
- **Order** n is the maximum number of points on the curve and is sometimes called *cardinality*. ([View Highlight](https://read.readwise.io/read/01hsv0jt2xwt4n4krs0qb1dq98)) ^697755642
- **Base field** $F_q$ of an elliptic curve is the field over which the curve is defined. The **base field size** q thereby defines the number of elements of the finite field $F_q$. ([View Highlight](https://read.readwise.io/read/01hsv0jxm2r3g990ygw2xddy9m)) ^697755648
- **Scalar field** $F_r$ is the field of scalars used in the operations performed on the curve, such as point addition, scalar multiplication, and pairing. The **scalar field size** r is also the size of the largest subgroup of prime order. Interestingly, the Elliptic Curve DLP (ECDLP) of an elliptic curve is only as hard as that curve's largest prime order subgroup, not its order. However, when curve's order is prime, its largest prime subgroup is the group itself, so r=n. ([View Highlight](https://read.readwise.io/read/01hsv0k1bf2vv4hsdvzc5vce78)) ^697755654