Nondegeneracy is the requirement that a pairing not collapse every admissible pair of inputs to the target identity. For an elliptic-curve pairing, the source consists of points of order m, while outputs lie among mth roots of unity. The identity is the point at infinity for point addition and 1 for field multiplication. section 15.1
This condition matters because a pairing must preserve meaningful information when it moves from point arithmetic to field arithmetic. Point multiplication is expressed as addition, whereas operations on pairing outputs are expressed as powers and multiplication. Nondegeneracy, together with bilinearity, is presented as fundamental to pairing operations. section 15.1
A practical check begins by establishing that the inputs and output live in the intended order-m structures:
- Find the order of each candidate point and reject cases where a known factor does not produce the point-at-infinity identity. section 16.3
- Evaluate the pairing on points of order 11 and verify that its output is an 11th root of unity. section 16.3
- Treat a nonidentity pairing result as evidence that the pairing has not collapsed all tested inputs; a G1 × G1 test is identified as showing nondegeneracy, even though that pairing combination is otherwise not useful. section 16.3
Tiny order-11 pairing output
For extension-field points, pairing values can be represented as polynomials such as e_m(P,Q) = 34x + 19. Combining outputs follows field multiplication: e_m(P,Q)e_m(P,T) = 32x + 34, matching e_m(P,Q+T). Raising that result to the 11th power gives 1, confirming that the output is an 11th root of unity. section 16.3
Equation 16.8
\[\label{equ168} \begin{gathered} e_m(P, Q) = 34x + 19\\ e_m(P, T) = 11x + 2\\ e_m(P, Q) e_m(P, T) = 32x + 34\\ e_m(P, Q+T) = 32x + 34\\ e_m(P, Q+T)^{11} = 1 \end{gathered}\]This calculation traces pairing values through multiplication and confirms that the final field element has order dividing 11 by raising it to the 11th power. section 16.3
