A random point is an elliptic-curve point obtained by randomizing a curve value rather than choosing a fixed point. A common construction selects a random x value and embeds it on the curve; over polynomial extensions, a random polynomial is embedded instead. Because each embedded value can produce two points, one additional random bit selects the result. section 3.6 section 13.2.4
A polynomial-extension implementation follows this flow:
- Inspect the lowest coefficient's least-significant bit to choose between the two embedded points. section 13.2.4
Pairing experiments select random points from precomputed groups by choosing an index with rand() % nmpnt and filtering it by group type. The selected point can then serve as P, Q, or T in Tate-pairing tests. The selector must have eligible points available; otherwise its retry loop never exits.
Tiny Tate pairing test
For an 11×165 test, a point S of order 5 is selected in G₁, P is selected from the order-11 group, and Q and T from the order-165 group. The computed values satisfy τ(P,Q)τ(P,T) = τ(P,T+Q) in the displayed runs.
