Elliptic Curve (EC)

Explore the fascinating world of Elliptic Curve cryptography through the following resources:

1. Exploring Elliptic Curve Pairings by Vitalik Buterin
2. Cloudflare’s blog An easy to understand primer on elliptic curve cryptography
3. Cryptography Algorithms A comprehensive book on cryptography algorithms

• Elliptic Curve (EC)
• What Is An Elliptic Curve?
• Terminology
  • Point of Infinity
  • Order Of A Curve
  • Generator Point
• Calculations On An Elliptic Curve
• Elliptic Curve Pairings
• Applications

What Is An Elliptic Curve?

The equation defining an elliptic curve is \(y^{2} = x^{3} + ax + b\).

Warning

An elliptic curve is not an ellipse.

You should also read about elliptic curve group.

Terminology

Point of Infinity

\(O\): a point that goes to infinity if we intersect two points in the curve symmetrically with respect to the y-axis. You can imagine the third point as an infinite point lying at the infinite end of the y-axis.

\(P + O = P\)

Order Of A Curve

A curve has an “order”; there exists a number \(n\) such that \(P * n = O\) for any \(P\).

\(=> P∗(n+1)=P\)

\(=> P∗(7∗n+5)=P∗5\)

Generator Point

There is also a commonly agreed-upon “generator point” \(g\) (or \(G\) interchangeably), which is understood to, in some sense, represent the number “1”.

Denote: \(\langle g \rangle = \mathbb{G}\)

Theoretically, any point on a curve (except \(O\)) can be \(g\); all that matters is that \(g\) is standardized.

Calculations On An Elliptic Curve

Given 2 points \(A\) and \(B\) on the elliptic curve, there is no simple/quick way to calculate \(k\) so that: \(A*k = B\) How much energy needed to break EC

\(P + Q = R\)

\(P + Q = O\) if and only if \(x_{p} = x_{q}\), and \(P + P = O\) if and only if \(y_{p} = 0\).

\(P * n = P + P + … + P\)

Elliptic Curve Pairings

Applications

• Encryption
• Signatures
• Deterministic threshold signatures
• Quadratic arithmetic programs

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