We shall use the following notation. For our purposes, the pairing is a bilinear map from two cyclic groups, G1 and G2 to a third group GT, where each group has prime order r.
pbc/pbc and type:
g := rnd(G1); g;
The first line generates a random element g of the group G1,
while the second prints out the value of g. (The syntax was
bc, an arbitrary
precision calculator.) Next, enter:
h := rnd(G2); h;
This assigns h to a random element of the group G2.
Actually, the default pairing
uses is symmetric so G1 and G2 are in fact the same group, but
in general they are distinct. To compute the pairing applied to
g and h, type:
The order of both g and h is r. Let’s generate two random numbers between 1 and r:
a := rnd(Zr); b := rnd(Zr);
By bilinearity, the resulting output of both of these lines should be identical:
This program has other features but the commands shown here should be enough to quickly and interactively experiment with many pairing-based cryptosystems using real numbers.