Requested material on Bilinear Pairing

Question

Bilinear map/pairing is widely used in Pairing based Cryptography. I am new to this area. Can anyone suggest me some good reference on Bilinear pairings? I need at least an example of Bilinear map which I have failed to find.

Answer 1

The dot product is a bilinear pairing. More specifically, for some fixed positive integer $n$, a bilinear pairing on $\Bbb{R}^n$ is a map $$ B: \Bbb{R}^n \times \Bbb{R}^n \to \Bbb{R} $$ such that for any $x, x', y, y' \in \Bbb{R}^n$, $$ \left. \begin{align} B(x + x', y) &= B(x, y) + B(x', y) \\ B(x, y + y') &= B(x, y) + B(x, y') \end{align} \right\} \quad \text{linearity of addition in each factor individually} $$ and for any scalar $c \in \Bbb{R}$, $$ B(cx, y) = cB(x, y) + B(x, cy) \quad \text{linearity of scalar multiplication in each factor individually} $$ The ordinary Euclidean dot product is given by the formula $$ B \left( (x_1, x_2, \ldots, x_n), (y_1, y_2, \ldots, y_n) \right) = x_1 y_1 + x_2 y_2 + \cdots x_n y_n, $$ and you can check that it satisfies the formulas above.

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It's worth noting that in the special case of $n = 1$, the dot product reduces to ordinary multiplication, and the bilinearity condition amounts to the distributive law of multiplication over addition: $$ (x + x')y = xy + x'y \\ x(y + y') = xy + xy' $$ as well as the associative and commutative laws for multiplication: $$ (cx)y = c(xy) = x(cy), $$ where all symbols are real numbers.

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Also, note that we could substitute any commutative ring to take the place of $\Bbb{R}$ as the scalars.

Answer 2

There are bilinear pairings known as Weil pairingsand Tate pairings which allows tripartite Diffie-Hellman to take place. A good place to learn them would be from Silverman's "The Arithmetic of Elliptic Curves".