We revisit the problem of basing pseudorandom generators on regular one-way functions, and present the following constructions:1. For any known-regular one-way" function (on n-bit inputs) that is known to be ε-hard to invert, we give a neat (and tighter) proof for the folklore construction of pseudorandom generator of seed length Θ(n) by making a single call to the underlying one-way function.2. For any unknown-regular one-way function with known ε-hardness, we give a new construction with seed length Θ(n) and O(n/ log (1/ε)) calls. Here the number of calls is also optimal by matching the lower bounds of Holenstein and Sinha (FOCS 2012). Both constructions require the knowledge about s, but the dependency can be removed while keeping nearly the same parameters. In the latter case, we get a construction of pseudo-random generator from any unknown-regular one-way function using seed length O(n) and O(n/log n) calls, where O omits a factor that can be made arbitrarily close to constant (e.g. log log log n or even less). This improves the randomized iterate approach by Haitner, Harnik and Reingold (CRYPTO 2006) which requires seed length O(n·logn) and O(n/ log n) calls.
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