Modern cryptography runs on a strange kind of arithmetic where you add points on a curve. See the elegant geometric rule that secures Bitcoin and TLS.
Elliptic Curve AdditionLive
the geometry behind ECC
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On an elliptic curve y² = x³ + ax + b, you "add" two points with a geometric rule: draw the line through P and Q, find where it meets the curve again, and reflect that point over the x-axis. Repeating this addition is easy, but reversing it — the elliptic-curve discrete log — is brutally hard, which is what makes ECC secure with far smaller keys than RSA.
Reading this result: The secant through P and Q meets the curve at a third point; reflecting it across the x-axis gives P+Q. Repeating this is easy, but undoing it (the discrete log) is what secures ECC.
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How it works
On an elliptic curve y² = x³ + ax + b, adding two points means drawing the line through them, finding the third intersection with the curve, and reflecting it over the x-axis. Repeating this scalar multiplication is fast, but undoing it — the elliptic-curve discrete logarithm — is infeasible. That asymmetry gives ECC RSA-grade security with far smaller keys, ideal for phones and embedded devices.
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