ECC Summary

Cryptography · Axiom Academy

Let's review key concepts from Elliptic Curve Cryptography. Smaller Keys: 256-bit ECC offers same security as 3072-bit RSA Faster Computation: Efficient point operations mean quicker encryption and signing Less Bandwidth: Smaller keys and signatures reduce network overhead Mobile-Friendly: Lower computational requirements ideal for constrained devices Equation: y² = x³ + ax + b defines the elliptic curve Discriminant: Δ = -16(4a³ + 27b²) ≠ 0 ensures no singularities Points: Pairs (x, y) satisfying the equation plus point at infinity O Group Structure: Points form an abelian group under point addition Geometric Method: Draw line through P and Q, find third intersection, reflect over x-axis Slope Formula: λ = (y₂ - y₁) / (x₂ - x₁) for distinct points Result: x₃ = λ² - x₁ - x₂, y₃ = λ(x₁ - x₃) - y₁ Identity: P + O = P for point at infinity O Tangent Method: Draw tangent line at P, find second intersection, reflect Slope Formula: λ = (3x₁² + a) / (2y₁) for doubling point P Efficiency: Special case of addition optimized for same-point operation Critical Role: Foundation of scalar multiplication algorithms Scalar Multiplication & Efficiency Definition: nP = P + P + ... + P (n times) Double-and-Add: O(log n) algorithm using binary representation of n Example: 25P = 16P + 8P + P via repeated doubling Trapdoor Function: Easy to compute nP, hard to find n from P and Q = nP Performance: Makes ECC practical for real-world cryptography

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