Pairwise Independent Hash Functions and Hash Mixing Lemma

A randomized algorithm A can be thought of as a function A : {0, 1} n × {0, 1} R(n) → {Accept, Reject}, that is, function A is a deterministic algorithm that takes two input strings x and y, where x is the " real " input to the randomized algorithm and y is the random string used during the computation. Definition 1 A (deterministic) function G : {0, 1} m → {0, 1} R(n) is a pseudorandom generator (PRG) for algorithm A with parameter ǫ if for all x, Pr y [A(x, y) accepts] − Pr z [A(x, G(z)) accepts] ≤ ǫ. We are going to study the following PRG construction. Theorem 2 (Nisan 1990) For any algorithm A that runs in S(n) = Ω(log n) space and uses R(n) random bits, there is a pseudorandom generator for A with parameter 1 10 that uses O(S(n) log R(n)) random bits and runs in O(S(n) log R(n)) space. The following claim easily follows. Corollary 3 (Nisan 1990) If a randomized algorithm A runs in S(n) = Ω(log n) space and uses R(n) random bits, then A can be converted into a randomized algorithm A ′ that runs in O(S(n) log R(n)) space and uses O(S(n) log R(n)) random bits. Consider the model of Turing machine computation when space complexity is our main concern. The TM has two tapes, one is read-only input tape of size n and the other is a work tape of size S(n). The space complexity of the TM is S(n) (that is, the read-only input tape will not be counted). Such a TM has at most n · 2 O(s(n)) states, and if s(n) = Ω(log n), this can be bounded by 2 O(s(n)) states.