Compute slopes, sums, doubles, and repeated point operations. Switch between real and finite fields effortlessly. See every step, table, graph, and export clearly today.
| Setting | Example Value | Why it matters |
|---|---|---|
| Field type | Finite field | Switches the calculator from continuous geometry to modular arithmetic. |
| Curve | y² = x³ + 2x + 2 over F17 | This is a non-singular sample curve used often for practice. |
| Point P | (5, 1) | P lies on the sample curve and supports addition tests. |
| Point Q | (6, 3) | Q is also valid, so P + Q can be evaluated. |
| Scalar k | 5 | Shows repeated addition through binary scalar multiplication. |
| Expected highlights | P + Q = (10, 6), 2P = (6, 3), 5P = (9, 16) | These outputs confirm the finite-field formulas are working. |
Curve equation: y² = x³ + ax + b. The curve is valid only when the discriminant is non-zero.
Discriminant: Δ = -16(4a³ + 27b²). If Δ = 0, the curve has a singular point and group operations break down.
j-invariant: j = 1728(4a³) / (4a³ + 27b²). It helps classify elliptic curves up to isomorphism over algebraically closed fields.
Point membership test: A point (x, y) belongs to the curve when y² equals x³ + ax + b, either exactly or modulo p.
Point addition: For P ≠ Q, λ = (y₂ - y₁) / (x₂ - x₁). Then x₃ = λ² - x₁ - x₂ and y₃ = λ(x₁ - x₃) - y₁.
Point doubling: For P = Q, λ = (3x₁² + a) / (2y₁). Then the same x₃ and y₃ formulas apply.
Finite fields: Division becomes multiplication by a modular inverse. The calculator uses λ = numerator × denominator⁻¹ mod p.
Scalar multiplication: kP is computed by repeated doubling and conditional addition. This is the standard double-and-add method used in elliptic curve arithmetic.
It checks curve validity, tests point membership, computes P + Q, computes 2P, evaluates kP, displays discriminant and j-invariant, and plots the curve or modular points.
A zero discriminant means the curve is singular. Singular curves contain cusps or self-intersections, so the usual elliptic curve group law no longer works correctly.
Real mode treats the curve as a continuous shape and plots branches. Finite-field mode works with modular arithmetic, so only discrete coordinate pairs can appear.
Elliptic curves over finite fields typically use Fp, where p is prime. Prime moduli guarantee that non-zero values have inverses, which addition formulas require.
That occurs when Q is the inverse of P, or when a vertical tangent appears during doubling. The point at infinity acts as the identity element.
Warnings appear when one or both points fail the membership test. The calculator still shows curve metrics, but point operations require valid points on the chosen curve.
Yes. kP means adding P to itself k times. The calculator speeds this up with the double-and-add algorithm instead of a slow direct loop.
Yes. It is useful for learning elliptic curve arithmetic, slopes, inverses, and repeated multiplication. It is educational code, not a hardened cryptographic library.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.