Random Function

In probability theory and its applications, such as statistics and cryptography, a random function is a function chosen randomly from a family of possible functions. Each realisation of a random function would result in a different function. Thus the concept of a random function is one example of a random element and hence is a generalization of the simpler idea of a random variable.

In probability and statistics, one important type of random function is studied under the name of stochastic processes, for which there are a variety of models describing systems where an observation is a random function of time or space. However, there are other applications where there is a need to describe the uncertainty with which a function is known and where the state of knowledge about the true function can be expressed by saying that it is an unknown realisation of a random function, for example in the Dirichlet process.

A special case of a random function is a random permutation, where a realisation can be interpreted as being in the form of a function on the set of integers describing the original location of an item, where the value of the function provides the new (permuted) location of the item that was in a given location.

In cryptography, a random function can be a useful building block in enabling cryptographic protocols.