Affine short weierstrass curves are general ways to describe elliptic curves

More specialized representations for faster algorithms for group law or scalar multiplication.

Montgomery Curves

Allow constant time algos for elliptic curve scalar multiplication

Let F be a prime field of order p>3.

$$ By^2 = x^3+Ax^2+x $$

This is montgomery curve M(F) over field F, all pairs of elements that satisfy the Montgomery cubic equation.

Every Montgomery curve is a short weierstrass curve

Conditions for Montgomery:

• number of points must be divisible by 4
• There must be at least one root
• $3z^2+a$ is a quadratic residue. Meaning its a y such that sqrt(y) = x is whole number.

TJJ_13 is Montgomery Curve

1. order is 20 which is divisible by 4