Twisted Curve

• twists of an elliptic curve have:
  • similar properties and are related to each other
  • may have different cryptographic properties and performance characteristics

Types of Twist

Quadratic Twist

• Given an elliptic curve in Weierstrass form, $y^2=x^3+ax+b$, the quadratic twist of the curve is obtained by replacing the equation with its quadratic twist equation.
  • $y^2=x^3+(a\ast d^2)\ast x+(b\ast d^3)$
    • where d is a non-square element in the underlying field of the curve
  • called quadratic twist because it involves a square ($d^2$) in the coefficients of the curve equation.

Sextic Twist

• $y^2=x^3+(a\ast d^4)\ast x+(b\ast d^6)$
  • where d is a non-sixth power element in the underlying field of the curve
• called sextic twist because it involves a sixth power ($d^6$) in the coefficients of the curve equation