Finite field integers

Question

can some one explain the following terms

$Z_n^N$, $F_q^N$ and $F_q^*$

and

why the bi-linear pairing is used in cryptography.

Answer 1

${\bf Z}_n^N$ is the set of all ordered $N$-tuples of elements of ${\bf Z}_n$; ${\bf Z}_n$ can be thought of as the set $\{{\,0,1,2,\dots,n-1\,\}}$ with addition and multiplication carried out modulo $n$.

${\bf F}_q^N$ is the set of all ordered $N$-tuples of elements of ${\bf F}_q$; ${\bf F}_q$ is the field of $q$ elements. There is a field of $q$ elements if, and only if, $q$ is a positive power of a prime number. Moreover, given such a $q$, there is (up to isomorphism) a unique such field.

${\bf F}_q^*$ is the nonzero elements of ${\bf F}_q$; these form a group under multiplication.

Sorry, I have no idea about the bilinear pairing. Maybe you could give us a little more information --- what bilinear pairing? what is being paired with what? how is it being used? where did you come across the concept? does anything good happen if you type $$\rm bilinear\ pairing\ cryptography$$ into a search engine?