Elliptic Curve Playground

Curve: y2 = x3 + x +    (mod )

Arithmetic Operations

 Find Point:  (,
 Add Points:  (, ) + (, ) = (, )
 Multiply Points:  * (, ) = (, )

Group Operations

 Generate Cyclic Group:
 (, )

 Generated Points: (0 found)

 Discovered Points: 1


  1. Before anything can be done, choose a random curve and a modulus. Elliptic curves also have prominent roles in factoring, so if the modulus isn't a prime, things might break horribly on occasion.
  2. Check random x-coordinates until you find a point on the curve. (With the "Find Point" helper.)
  3. Generate a cyclic group or add/multiply points to find more points. The addition helper is animated, to demonstrate the chord-tangent law of composition (a line through any two points necessarily intersects a third point) and the group law (the chord-tangent law + an x-axis flip). The addition helper also doesn't verify that points are on the curve, so you can play with (as they're called in crypto) invalid-curve attacks.
  4. If the modulus is q, then the number of points on the curve N is bounded by: (Hasse's Theorem)
  5. The order of a generated cyclic group is a divisor of N. (Lagrange's Theorem)
  6. Most elliptic curves create modules (Abelian ring acting on a set of vectors). If you find a subgroup of prime order, that subgroup/subspace is a vector space (field acting on a set of vectors)
  7. The curve E: y2 = x3 (mod p) is called a singular curve and is always isomorphic to Zp, disregarding the singularity (0, 0). There's an efficiently computable group isomorphism from E(Zp) to Zp defined by (x, y) → x/y, ∞ → 0.
  8. Given a curve E: y2 = x3 + Ax + B (mod p) and a quadratic non-residue d, the twist of E is defined as Ed: y2 = x3 + Ad2x + Bd3 (mod p)
    1. E and Ed share 2, 1, or 0 points. The x-coordinate of these points is (Bd3 - B) / (A - Ad2)
    2. If E, mod p, has p + 1 - a points, then Ed has p + 1 + a points. As mentioned above, knowledge of the order of all curves Ed allows us to classify the structures of all of the quadratic twists of E.

Interesting Curves