Requested material on Bilinear Pairing 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.
 A: 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.

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.

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