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  "body": "\n## Definition\n\nA *pseudorandom number generator* is a tuple $(S, s_0, T, U, G)$ where:\n\n1. $S$ is a finite set, the state space;\n2. $s_0 \\in S$ is the initial state, or seed;\n3. $T \\colon S \\to S$ is a map, the transition function;\n4. $U$ is a set, the output space, usually $[0, 1)$ or a set of integers $\\set{0, \\ldots, m-1}$;\n5. $G \\colon S \\to U$ is a map, the output function.\n\nThe generator produces the states $s_i = T(s_{i-1})$ for $i \\ge 1$ and the output sequence $u_i = G(s_i)$ for $i \\ge 0$.\n\n## Remarks\n\nExample: $S = \\set{0, \\ldots, 7}$, $T(s) = (5s + 1) \\bmod 8$, $U = [0, 1)$, $G(s) = s/8$ and $s_0 = 0$; all five clauses hold, and the outputs are $0, 0.125, 0.75, 0.875, 0.5, 0.625, 0.25, 0.375$, then repeat.\nNon-example: the sequence of decimal digits of $\\pi$ is not eventually periodic, because $\\pi$ is irrational.\nIt is not the output sequence of any such tuple, since a finite state space as in clause 1 forces the outputs to be eventually periodic.\nSince $S$ is finite, the sequence $(s_i)$ is eventually periodic, with period at most $\\abs{S}$; if $T$ is a bijection it is purely periodic.\nThe definition is purely structural; the quality of a generator is judged by its period, by the uniformity of tuples $(u_i, \\ldots, u_{i+k-1})$, and by statistical tests.\nGenerators for cryptography must in addition be unpredictable from past outputs, which is a stronger requirement.\n",
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