Scalar multiplication multiplies an elliptic-curve point by an integer using the curve’s special addition operation. Writing nP means adding P to itself n times; the result remains a point on the same curve. section 3.2.2
In Go’s elliptic-curve implementation:
- curve.ScalarMult(...) performs the operation without requiring the caller to implement curve-point addition. section 3.2.2
- Coordinates use arbitrary-precision integers from math/big, supporting cryptographic point values. section 3.2.2
- DUAL_EC_DRBG multiplies its internal state by points P and Q, then extracts the x-coordinate; the Q result is truncated before output. section 3.2.3
Using scalar multiplication in DUAL_EC_DRBG
For internal state n, the generator advances with gP(n) = X(nP) and produces output with gQ(n) = t(X(nQ)). The Go implementation calls curve.ScalarMult for P and Q, discards each y-coordinate, and returns the final 30 bytes. section 3.2.3
Equation 3.3
\[\begin{aligned} \begin{aligned} g_P(n) &= X(nP) \\ g_Q(n) &= t(X(nQ)) \end{aligned} \label{equation:dualec-p-and-q} \end{aligned}\]Equation 3.3 makes the two scalar multiplications explicit: one updates the state, while the other transforms it into output. section 3.2.3
