What hard problem does elliptic curve cryptography use?

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.

Elliptic Curve Cryptography (ECC) is a type of public-key cryptosystem that uses the mathematics of an elliptic curve to create public and private keys.

Conditions: It is a type of public-key cryptosystem.; It uses elliptic-curve mathematics to create key pairs.

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).

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.