Pairings or Bilinear Maps¶
Read the full article on pairings or bilinear maps here.
Definition¶
A function \(e\) is considered a bilinear function or map if it satisfies the following conditions:
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Let \(\mathbb{G}_1\), \(\mathbb{G}_2\), and \(\mathbb{G}_T\) be cyclic groups of order \(n\). The bilinear pairing is represented as \(e : \mathbb{G}_1 \times \mathbb{G}_2 \rightarrow \mathbb{G}_T\).
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Bilinearity: \(e(aP, bQ) = e(P, Q)^{ab}\).
- Non-degeneracy: \(e(P, Q) \neq 1\).
- Computability: \(e(P, Q)\) can be computed efficiently.
Properties¶
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\(e(aP, bQ) = e(P, Q)^{ab}\)
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\(e(P, {a}Q) = e(P, Q)^{a}\)
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\(e(P, Q + R) = e(P, Q) \cdot e(P, R)\)
Example¶
Consider simple numbers \(P\), \(Q\), \(R\), and \(S\). Creating a pairing can be done using \(e(x, y) = 2^{x \cdot y}\). Then, observe:
\(e(6, 4) \cdot e(6, 5) = 2^{{6} \cdot {4}} \cdot 2^{{6} \cdot {5}} = 2^{{6} \cdot ( {4} + {5})} = 2^{{6} \cdot {9}}\)
\(e(6, 4 + 5) = 2^{{6} \cdot {9}}\)
\(e(6 \cdot 2, 4 \cdot 3) = 2^{{6} \cdot {2} \cdot {4} \cdot {3}}\)
\(e(6, 4)^{{2} \cdot {3}} = 2^{{6} \cdot {4} \cdot {2} \cdot {3}}\)
What Can It Do?¶
The expression \(e(P, Q) \cdot e(G, G \cdot 5) = 1\) is equivalent to \(p \cdot q + 5 = 0\).