🧩 Point Addition on Elliptic Curves
Learn how elliptic curve math creates cryptographic magic
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Introduction
📐 How Elliptic Curves Work
Elliptic curves have unique mathematical properties that make them perfect for cryptography. Let's explore the magic!
📊 The Elliptic Curve Equation
Standard Weierstrass Form:
y² = x³ + ax + b
Where 4a³ + 27b² ≠ 0 (ensures the curve has no singularities)
Bitcoin's Curve (secp256k1):
y² = x³ + 7
Simplified: a = 0, b = 7. Chosen for its computational efficiency.
Why This Shape?
- • Symmetric across x-axis
- • Continuous and smooth
- • Infinite number of points
- • Group structure for math
🎮 Interactive: Curve Operations
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Point Addition: P + Q = R
Adding two different points on the curve
Visual Process:
- 1.Draw a straight line through points P and Q
- 2.This line intersects the curve at a third point
- 3.Reflect that point across the x-axis to get R
Mathematical Formula:
slope = (yQ - yP) / (xQ - xP)
xR = slope² - xP - xQ
yR = slope(xP - xR) - yP
xR = slope² - xP - xQ
yR = slope(xP - xR) - yP
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Why this matters: This operation is fast (just algebra) but hard to reverse. If you know R = P + Q, finding P or Q from R is extremely difficult!
🔐 The Discrete Logarithm Problem
ECC security relies on the Elliptic Curve Discrete Logarithm Problem (ECDLP):
Given:
Q = kP
Where Q and P are points on the curve
✓ Easy Direction:
Computing Q from k and P takes ~256 operations (milliseconds)
✗ Hard Direction:
Finding k from Q and P requires ~2¹²⁸ operations (universe's lifetime)
Why It's So Hard:
- •No shortcuts: Unlike factoring (RSA), no known algorithm breaks ECDLP efficiently
- •Exponential growth: Each bit doubles the security (256-bit ≈ 2¹²⁸ operations)
- •Best attack: Pollard's rho algorithm still needs sqrt(n) operations
- •Quantum threat: Shor's algorithm could solve it, but quantum computers don't exist yet
🎯 Key Concepts Summary
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Curve Equation
y² = x³ + ax + b defines the shape and properties of the curve
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Point Operations
Addition, doubling, and scalar multiplication form the mathematical foundation
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Trapdoor Security
Easy to compute forward (kP → Q), impossible to reverse (Q → k)