A Tate pairing is a bilinear, nondegenerate elliptic-curve pairing that maps suitable point groups into the multiplicative group of an extension-field root-of-unity group. Unlike the Weil pairing, it is not alternating. section 17.0 section 17.5
Equation 17.1
\[\tau: E[m] \times \dfrac{E}{mE} \longrightarrow \mu_m \label{tq1}\]The pairing has type τ: E[m] × E/mE → μm. The first point has order m; the second needs m as a factor of its order, rather than having exactly order m. section 17.1
For points P, Q and an unrelated reference point S, compute τ(P,Q) by evaluating two Miller functions, dividing them, and raising the result to (p^k−1)/m: τ(P,Q) = (fP(Q+S)/fP(S))^((p^k−1)/m). The exponent is integral because m divides p^k−1. section 17.1 section 17.2
An implementation follows this execution flow:
- Choose S unrelated to either input point; an x-coordinate of zero is not universally safe. section 17.2
Tiny-field bilinearity test
A finite-field extension example uses group order 11, curve coefficients a4=23 and a6=42, and brute-force point-order classification. For selected P, Q, and T, the computed values satisfy τ(P,T+Q)=τ(P,T)τ(P,Q), demonstrating bilinearity with printable polynomial values. section 17.3
