An elliptic curve is a set of points satisfying y² = x³ + ax + b. Cryptographic implementations operate on these points with a specialized addition operation that keeps results on the same curve. section 3.2.2
Equation 3.2
\[y^2 = x^3 + ax + b \label{equation:elliptic-curves}\]Use the defining equation to check whether coordinates can belong to the curve. section 3.2.2
The operations used by cryptographic code are:
- Add two points P and Q using the curve's special addition operation; the result P + Q remains on the curve and satisfies properties such as P + Q = Q + P. section 3.2.2
- Multiply a point P by an integer n through repeated point addition, written nP and implemented as scalar multiplication. section 3.2.2
- Represent coordinates with arbitrary-precision integers and use Go's crypto/elliptic operations instead of implementing the point arithmetic directly. section 3.2.2
Go's crypto/elliptic package supplies standard curves, including P256, together with Add and ScalarMult operations. This lets cryptographic code work with curve points without relying on ordinary coordinate addition. section 3.2.2
Recovering a missing coordinate
DUAL_EC_DRBG exposes only 30 bytes of a 32-byte x coordinate. Trying the missing two bytes and computing y = √(x³ + ax + b) identifies valid curve points; failed modular square roots reject guesses, while P256 validation confirms candidates. section 3.2.4
