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Elliptic Curve Point Addition Calculator

Enter the curve coefficient a and the two points P and Q to compute P+Q.

How it is calculated

The curve is y² = x³ + ax + b over the real numbers. You enter a and the two points; the tool derives b from P as b = y₁² − x₁³ − a·x₁ and then checks that Q lies on the same curve. If it does not, there is no result. For two points with different x, the chord slope is m = (y₂ − y₁) ÷ (x₂ − x₁). If P and Q are the same point, the tangent slope m = (3x₁² + a) ÷ (2y₁) is used instead (doubling). Then x₃ = m² − x₁ − x₂ and y₃ = m(x₁ − x₃) − y₁, which is the third intersection reflected in the x-axis. Results are shown to four decimals. If P and Q share an x but have opposite y (P + (−P)), or the tangent is vertical (y = 0), the sum is the point at infinity and the tool shows nothing.

Worked example

Take the curve y² = x³ − x + 1 (a = −1, so b = 1² − 0 − 0 = 1). P = (0, 1) defines it, and Q = (3, 5) lies on it since 27 − 3 + 1 = 25 = 5². The slope is m = (5 − 1) ÷ (3 − 0) = 1.3333. Then x₃ = 1.3333² − 0 − 3 = 1.7778 − 3 = −1.2222, and y₃ = 1.3333 × (0 − (−1.2222)) − 1 = 1.6296 − 1 = 0.6296. The tool shows R = P + Q as (-1.2222, 0.6296). Check: (−1.2222)³ + 1.2222 + 1 = 0.3964, and 0.6296² = 0.3964, so R is on the curve.

Try this example →

Common mistakes

  • Entering a Q that is not on the curve defined by a and P. The tool rejects it and shows nothing; for P = (0, 1) and a = −1, Q = (3, 9) fails because 81 is not 25. Substitute Q into y² = x³ + ax + b first.
  • Entering a different a than the one the points were taken from. The value of b is calculated from P and a, so a wrong a moves the curve and Q will usually fail the on-curve test.
  • Entering P and Q with the same x but opposite y (P and −P), expecting a number. Their sum is the point at infinity, which has no (x, y) coordinates, so the tool returns nothing. The same happens when P = Q and y = 0.
  • Thinking that equal points are an error. Entering the same point twice is allowed and gives the doubled point 2P using the tangent slope.
  • Treating the output as an ECC computation. Real cryptography does the same operations modulo a prime on exact integers, whereas these are decimal values over the reals, rounded to four places.

About this calculator

An elliptic curve defined by y² = x³ + ax + b, together with a geometric rule for "adding" points on it, forms an algebraic group — the mathematical structure underneath modern public-key cryptography. To add two distinct points P and Q, you draw the line through them, find where it intersects the curve a third time, and reflect that point across the x-axis to get R = P + Q; this calculator carries out that construction and reports the resulting point along with the chord's slope. It works out b from P, shows no result when Q is not on the same curve, and when P = Q it uses the tangent slope (3x² + a)/2y instead.

This point-addition rule (and the related point-doubling rule for adding a point to itself) is the foundation of elliptic curve cryptography (ECC), used in ECDSA digital signatures, TLS/HTTPS key exchange, and Bitcoin and other cryptocurrency wallets — where the same group law is applied over a finite field rather than the real numbers, and the difficulty of reversing repeated point addition (the elliptic curve discrete logarithm problem) is what makes the cryptography secure. Working through it over the reals, as this calculator does, is exactly how cryptography and coding-theory courses introduce the geometry before moving to finite fields.

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Comments (1)

  • Editorial team

    Try swapping P and Q: the sum should come out identical, because adding points on a curve is commutative, and that is a quick way to catch a typing slip. Then enter the same point twice to see the tangent rule double it. These are real-number points; cryptography does the same steps modulo a prime. Would a version over a prime field be useful? Tell us the prime sizes you would want.

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