What is Elliptic Curve Cryptography (ECC) and what mathematical structure does it use to generate keys?

Scalar multiplication on an elliptic curve is the repeated application of point addition. Starting from a point G and adding it to itself k times produces another point P, written as $P = [k]G$.

Conditions: G is a point on the elliptic curve.; k is an integer scalar counting the number of repeated point additions.

ECC offers the same level of security as RSA but with much smaller keys. For example, a 256-bit ECC key is roughly as secure as a 3072-bit RSA key.

Conditions: The keys are generated using standard parameters for their respective cryptosystems.

The standard Weierstrass equation for an elliptic curve is y² = x³ + ax + b, and the coefficients must satisfy the non-singularity condition 4a³ + 27b² ≠ 0.

Conditions: (x, y) are coordinates of points on the curve.; a and b are coefficients.; 4a³ + 27b² ≠ 0 (non-singularity condition).

Elliptic curve cryptography uses the discrete logarithm problem, which involves finding a scalar x such that X = xP given group elements X and P.

Conditions: X and P are elements of the group 𝔾.; x is the unknown scalar to be recovered.; The problem is treated as computationally infeasible when only X and P are known.

Point addition on an elliptic curve is defined geometrically by drawing a line through two points A and B, finding its third intersection with the curve, and reflecting that intersection point across the x-axis to obtain the sum C.

Conditions: A and B are points on the elliptic curve.; The line through A and B intersects the curve at a third point.; The curve is symmetric about the x-axis.