On ε-biased generators in NC⁰

Cry an and Miltersen (Proceedings of the 26th Mathematical Foundations of Computer Science, 2001, pp. 272-284) recently considered the question of whether there can be a pseudorandom generator in NC⁰, that is, a pseudorandom generator that maps n-bit strings to m-bit strings such that every bit of the output depends on a constant number k of bits of the seed. They show that for k = 3, if m ≥ 4n + 1, there is a distinguisher; in fact, they show that in this case it is possible to break the generator with a linear test, that is, there is a subset of bits of the output whose XOR has a noticeable bias. They leave the question open for k ≥ 4. In fact, they ask whether every NC⁰ generator can be broken by a statistical test that simply XORs some bits of the input. Equivalently, is it the case that no NC⁰ generator can sample an ε-biased space with negligible ε? We give a generator for k = 5 that maps n bits into cn bits, so that every bit of the output depends on 5 bits of the seed, and the XOR of every subset of the bits of the output has bias 2^-Ω(n/c4). For large values of k, we construct generators that map n bits to n^Ω(√k) bits such that every XOR of outputs has bias 2^{-n 1/2√k}. We also present a polynomial-time distinguisher for k = 4, m ≥ 24n having constant distinguishing probability. For large values of k we show that a linear distinguisher with a constant distinguishing probability exists once m ≥ Ω(2^kn^[k/2]). Finally, we consider a variant of the problem where each of the output bits is a degree k polynomial in the inputs. We show there exists a degree k = 2 pseudorandom generator for which the XOR of every subset of the outputs has bias 2^-Ω(n) and which maps n bits to Ω(n²) bits.