Pairings

Let G1G_1, G2G_2, and GtG_t be three cyclic groups of the same order. A map e:G1×G2Gte : G_1×G_2 → G_t is said to be bilinear if u,vG;a,b{0,...,G1}:e(ua,vb)=e(u,v)ab\forall u, v ∈ G;\forall a, b ∈ \{0, . . . , |G|−1\}: e(u^a, v^b) = e(u, v)^{ab}. If a bilinear map ee is also non-degenerate (meaning, it does not map all pairs in G1×G2G_1 × G_2 to the identity element 1Gt1_{G_t}) and ee is efficiently computable, then ee is called a pairing. This terminology refers to the fact that ee associates each pair of elements in G1×G2G_1 × G_2 to an element of GtG_t .

In the general case that G1G2G_1 \neq G_2, the pairing is said to be asymmetric, while the case that G1=G2G_1 = G_2 is called symmetric. Asymmetric pairings are much more efficient in practice than symmetric pairings.

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