How to construct random functions

A constructive theory of randomness for functions, based on computational complexity, is developed, and a pseudorandom function generator is presented. This generator is a deterministic polynomial-time algorithm that transforms pairs (<italic>g</italic>, <italic>r</italic>), where <italic>g</italic> is <italic>any</italic> one-way function and <italic>r</italic> is a random <italic>k</italic>-bit string, to polynomial-time computable functions ƒ<italic><subscrpt>r</subscrpt></italic>: {1, … , 2<italic><supscrpt>k</supscrpt></italic>} → {1, … , 2<italic><supscrpt>k</supscrpt></italic>}. These ƒ<italic><subscrpt>r</subscrpt></italic>'s cannot be distinguished from <italic>random</italic> functions by any probabilistic polynomial-time algorithm that asks and receives the value of a function at arguments of its choice. The result has applications in cryptography, random constructions, and complexity theory.

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