uplinks:: Algebra Geometry
tags:: #lang/en #type/term #note/tidy

Elliptic Curve

• An elliptic curve $E$ is a plane algebraic curve defined as the set of points to the Weierstrass equation $y^2 = x^3 + ax + b$ and an extra point at infinity $O$ , where the constants $a$ and $b$ must satisfy $4a^3 + 27b^2 \neq 0$
  • The equation was named after the mathematician Karl Weierstrass, who studied elliptic curves extensively during the 19th century

There are other representations of elliptic curves, but technically an elliptic curve is the set points satisfying an equation in two variables with degree two in one of the variables and three in the other. — Cloudflare

Properties

• Non-singular
  • no cusps or self-intersections
• Horizontal symmetric.
  • Any point on the curve can be reflected over the x axis and remain the same curve.
• Any non-vertical line will intersect the curve in at most three places.

Examples

$E_1 : y^2 = x^3 − x$
$E_2: y^2 = x^3 + \frac{1}{4}x + \frac{5}{4}$

#todo/picture

See Also

1. Common Elliptiic Curves in Blockchains
2. Elliptic Curve Cryptography
3. Elliptic Curve Arithmetic

References

1. Peking University Introduction to Information Security Class
2. https://crypto.stackexchange.com/questions/26329/what-are-the-differences-between-the-elliptic-curve-equations #todo/read
3. https://safecurves.cr.yp.to/equation.html #todo/read
4. https://blog.cloudflare.com/a-relatively-easy-to-understand-primer-on-elliptic-curve-cryptography/