# Index set

In mathematics, the elements of a set A may be indexed or labeled by means of a set J that is on that account called an index set. The indexing consists of a surjective function from J onto A and the indexed collection is typically called an (indexed) family, often written as (Aj)jJ.

In computational complexity theory and cryptography, an index set is a set for which there exists an algorithm I that can sample the set efficiently; i.e., on input 1n, I can efficiently select a poly(n)-bit long element from the set. [1]

## Examples

• An enumeration of a set S gives an index set $J \sub \mathbb{N}$, where f : JS is the particular enumeration of S.
• Any countably infinite set can be indexed by $\mathbb{N}$.
• For $r \in \mathbb{R}$, the indicator function on r is the function $\mathbf{1}_r\colon \mathbb{R} \rarr \{0,1\}$ given by
$\mathbf{1}_r (x) := \begin{cases} 0, & \mbox{if } x \ne r \\ 1, & \mbox{if } x = r. \end{cases}$

The set of all the $\mathbf{1}_r$ functions is an uncountable set indexed by $\mathbb{R}$.

## References

1. ^ Goldreich, Oded (2001). Foundations of Cryptography: Volume 1, Basic Tools. Cambridge University Press. ISBN 0-521-79172-3.