Where might pseudorandom number generators come from? There are several general principles useful in constructing deterministic sequences exhibiting quasi-random behavior: expansiveness, nonlinearity, and computational complexity.Įxpansiveness is a concept arising from dynamical systems. If a deterministic function is unpredictable, then it is difficult to prove anything about it in particular, it is difficult to prove that it is unpredictable.Ħ.2 Explicit Constructions of Pseudorandom Bit Generators A major difficulty in settling the existence problem for this theory is summarized in the following heuristic: However, good candidates are known for one-way functions.
The central unsolved question is whether any of these objects exist.
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The basic properties characterizing a secure pseudorandom bit generator are ''randomness-increasing" and "computationally unpredictable." Recently obtained results are that if one of the following objects exist then they all exist:Ī secure (block-type) private-key cryptosystem. The basic object in this theory is the concept of a secure pseudorandom bit generator, which was proposed by Blum and Micali (1982) and Yao (1982). Section 6.3 describes the subject of computational information theory and indicates its connection to cryptography. Most of these generators have underlying group-theoretic or number-theoretic structure. Section 6.2 describes explicitly some pseudorandom bit generators, as well as some general principles for constructing pseudorandom bit generators.
Hence the problem of constructing pseudorandom numbers is in principle reducible to that of constructing pseudorandom bits. 0-1 valued random variables (see Devroye, 1986, and Knuth and Yao, 1976). It is possible to simulate samples of any reasonable distribution using as input a sequence of i.i.d. A third reason arises from cryptography: the existence of secure pseudorandom bit generators is essentially equivalent to the existence of secure private-key cryptosystems.įor Monte Carlo simulations, one often wants pseudorandom numbers, which are numbers simulating either independent draws from a fixed probability distribution on the real line R, or more generally numbers simulating samples from a stationary random process. Second, the deterministic character of pseudorandom bit sequences permits the easy reproducibility of computations.
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