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5.1. Further background on elliptic curves 1372. Non-degeneracy: if K is finite and contains the m-th roots of unity, then the Tate pairingis non-degenerate.3. Galois invariance: 〈σP, σQ〉 m = σ〈P, Q〉 m .The bilinearity is a consequence of Weil’s reciprocity. A proof of the non-degeneracy is givenin [104]. A more elementary and specific one has been devised by Heß [130].We now define the Weil pairing. With the same notation as before, let P ∈ E[m] be anm-torsion point. There exists a function f ∈ K(E) such that:div(f) = m(P ) − m(O E ) .Multiplication by m is a surjective map on E(K), so we can choose P ′ such that [m]P ′ = P .There exists a function g ∈ K(E) such thatdiv(g) = [m] ∗ ((P ) − (O E )) =∑(P ′ + R) − (R) ,because [m 2 ]P ′ = O E . Furthermoreso that (g ◦ τ Q /g) ∈ K. FinallyR∈E[m]div(g ◦ τ Q ) = div(τ ∗ Qg) = τ ∗ Q div(g) = div(g) ,div(f ◦ [m]) = div([m] ∗ f) = [m] ∗ div(f) = m div(g) = div(g m )and so f ◦ [m] = g m up to a non-zero multiplicative constant. If Q ∈ E[m] and S ∈ E(K), theng((S + Q)) m = f([m](S + Q)) = f([m](S)) = g((S)) m , so g(S+Q)g(S)is an m-th root of unity.Definition 5.1.24 (Weil pairing). Let K be a perfect field and E an elliptic curve defined over K.Let m ≥ 2 be an integer co-prime to the characteristic of K. Let P, Q ∈ E(K)[m] be m-torsionpoints. We define the Weil pairing as above:e m (P, Q) =g(S + Q)g(S).Theorem 5.1.25 ([19, Theorem IX.10], [244, Proposition III.8.1]). Let K be a perfect field andE an elliptic curve defined over K. Let m ≥ 2 be an integer co-prime to the characteristic of K.Let P, P 1 , P 2 , Q, Q 1 , Q 2 ∈ E(K)[m] and σ ∈ Gal(K/K). The Weil pairing satisfies the followingproperties:1. Bilinearity:2. Alternance: e m (P, Q) = e m (Q, P ) −1 .e m (P 1 + P 2 , Q) = e m (P 1 , Q)e m (P 2 , Q) ,and e m (P, Q 1 + Q 2 ) = e m (P, Q 1 )e m (P, Q 2 ) .3. Non-degeneracy: if E(K) contains E[m], then e m (P, Q) ≠ 1.4. Galois invariance: e m (σP, σQ) = σe m (P, Q).The Weil pairing and the Tate pairing are closely related. In fact, the following proposition isoften used to define the Weil pairing.