On the Complexity of Non-adaptively Increasing the Stretch of Pseudorandom Generators
We study the complexity of black-box constructions of linear-stretch pseudorandom generators starting from a 1-bit stretch oracle generator G. We show that there is no construction which makes non-adaptive queries to G and then just outputs bits of the answers. The result extends to constructions that both work in the non-uniform setting and are only black-box in the primitive G (not the proof of correctness), in the sense that any such construction implies NP/poly \(\ne\) P/poly. We then argue that not much more can be obtained using our techniques: via a modification of an argument of Reingold, Trevisan, and Vadhan (TCC ’04), we prove in the non-uniform setting that there is a construction which only treats the primitive G as black-box, has polynomial stretch, makes non-adaptive queries to the oracle G, and outputs an affine function (i.e., parity or its complement) of the oracle query answers.
- 2.Bronson, J., Juma, A., Papakonstantinou, P.A.: Limits on the stretch of non-adaptive constructions of pseudo-random generators. In: Ishai, Y. (ed.) TCC 2011. LNCS, vol. 6597, pp. 522–539. Springer, Heidelberg (2011)Google Scholar
- 6.Goldreich, O., Levin, L.: A hard-core predicate for all one-way functions. In: 21st Annual ACM Symposium on Theory of Computing (STOC), pp. 25–32 (1989)Google Scholar
- 7.Haitner, I., Reingold, O., Vadhan, S.P.: Efficiency improvements in constructing pseudorandom generators from one-way functions. In: 42nd ACM Symposium on Theory of Computing (STOC), pp. 437–446 (2010)Google Scholar
- 9.Impagliazzo, R.: Very strong one-way functions and pseudo-random generators exist relative to a random oracle (1996) (manuscript)Google Scholar
- 15.Zimand, M.: Efficient privatization of random bits. In: Randomized Algorithms Satellite Workshop of the 23rd International Symposium on Mathematical Foundations of Computer Science (1998), http://triton.towson.edu/~mzimand/pub/rand-privat.ps