Generators from Communication Complexity

where x ∈ {0, 1}ks, y ∈ {0, 1}s, and Γk : {0, 1}ks × {0, 1}s → {0, 1}ks is some function that can “recycle the randomness in x”. In the INW-generator, Γ is the neighbor function of a suitable expander graph or it is a suitable extractor. Nisan’s generator can also be cast in the framework of (1), but the definition is a bit subtle. Here the string x is of length (2k − 1) · s and of the form x = σ, h1, . . . , hk−1 and y is of the form y = hk. The string σ is of length s and corresponds to Nisan’s “inner seed” as discussed in Lecture 15. The “outer seed” consists of the k strings h1, . . . , hk of length 2s each, and they determine k pair-wise uniform hash functions from {0, 1}s → {0, 1}s. Then we obtain Nisan’s generator Gk when Γk is of the form