Elliptic curve representations:

  • Short Weierstrass:
  • Montgomery:
  • Edward:
  • Twisted Edwards:

Elliptic curve

  • Affine point:
  • Projective point:
  • Jacobian point:


Inverse of a point by inverting on x-axis, including the case when :


Addition: where . Using algebraic formula,

Using chord method,


as human species, we have arrived to the unanimous conclusion that we hate divisions, and will do anything to avoid them.


Case where,

Using tangent method, slope of the curve comes out to be:


Scalar Multiplication

scalar multiplication for elliptic curve refers to multiplication of a scalar value with a point . Three different kinds of scalar mult exists:

  • Fixed base: where the point is fixed
  • Variable base: where the point can be different
  • Double base: multiple points with multiple scalars

Montgomery curve formulae

montgomery curves

List of things and resources to learn

  • all different point representations and their advantages/disadvantages.
    • affine
    • projective
    • Jacobian
    • extended jacobian
  • different curve representations
    • short weierstrass
    • montgomery
    • edwards
    • twisted edwards
  • hash to curve
  • constant time operations
  • pairings