← Mathematical Foundations of CryptographyElliptic curves

Adding points on an elliptic curve

The chord-and-tangent rule over the real numbers, then the same formulas over 𝔽ₚ, where the line wraps around the grid.

PQ
P = (-1.00, 1.00)Q = (1.40, -1.53)
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The curve y² = x³ − x + 1. Move P and Q with the sliders, or flip them between the upper and lower branch.

What to notice

  • Over 𝔽ₚ the line through P and Q still exists: it is the set of points y = λx + ν (mod p), shown as faint dots wrapping around the grid.
  • The line meets the curve in exactly one more point. Reflecting it across the middle row gives P + Q.
  • Scalar multiplication walks G, 2G, 3G, … and comes back to O after #E steps. Going forward is easy; finding k from kG is the discrete log problem.