期刊名称：Electronic Colloquium on Computational Complexity
印刷版ISSN：1433-8092
出版年度：2003
卷号：2003
出版社：Universität Trier, Lehrstuhl für Theoretische Computer-Forschung
摘要：Cryan and Miltersen recently considered the question of whether there can be a pseudorandom generator in NC0, that is, a pseudorandom generator such that every bit of the output depends on a constant number k of bits of the seed. They show that for k=3 there is always a distinguisher; in fact, they show that it is always 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>3, and conjecture that every NC0 generator can be broken by a statistical test that simply XORs some bits of the input. Equivalently, they conjecture that no NC0 generator can sample an epsilon-biased space with negligible epsilon. We refute the conjecture for k>4, and we give a generator 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 exponentially small in n/c^4. For large values of k, we construct generators that map n bits to n^{Omega(sqrt k})} bits and such that every XOR of outputs has bias exponentially small in n^{1/(2 sqrt k)}. We also present a polynomial-time distinguisher for the case k=4, assuming the number of output bits is at least 24n. Our distinguisher has constant distinguishing probability. For large values of k we show that a linear distinguisher with a constant distinguishing probability exists once the number of output bits is bigger than 2^k n^{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 pseudo random generator for which the XOR of every subset of the outputs has exponentially small bias in n, and which maps n bits to Omega(n^2) bits.
