Index set

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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)j∈J.

In 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]

[edit] Examples

  • An enumeration of a set S gives an index set J \sub \mathbb{N}, where f:J \rarr \mathbb{N} is the particular enumeration of S.
\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}.

[edit] References

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


[edit] See also