ABSTRACT
We prove that the pseudorandom generator introduced by Impagliazzo et al. (1994) with proper choice of parameters fools group products of a given finite group G. The seed length is O((|G|O(1) + log 1/δ)log n), where n is the length of the word and δ is the allowed error. The result implies that the pseudorandom generator with seed length O((2O(w log w) + log 1/δ)log n) fools read-once permutation branching programs of width w. As an application of the pseudorandom generator one obtains small-bias spaces for products over all finite groups Meka and Zuckerman (2009).
Supplemental Material
- N. Alon, O. Goldreich, J. Håstad, and R. Peralta. Simple construction of almost k-wise independent random variables. Random Structures and Alg., 3(3):289--304, 1992.Google Scholar
Cross Ref
- M. Ajtai, J. Komlós, and E. Szemerédi. Deterministic simulation in logspace. In Proc. of ACM Symp. on Theory of Computing (STOC), pages 132--140, 1987. Google Scholar
Digital Library
- Y. Azar, R. Motwani, and J. Naor. Approximating arbitrary probability distributions using small sample spaces. Combinatorica, 18:151--171, 1998.Google Scholar
Cross Ref
- N. Alon and J. Spencer. The Probabilistic Method. John Wiley, 1992.Google Scholar
- D.A. Barrington. Bounded-width polynomial-size branching programs recognize exactly those languages in NC$^1$. J. of Comp. and System Sci., 38(1):150 -- 164, 1989. Google Scholar
Digital Library
- L. Babai, N. Nisan, and M. Szegedy. Multiparty protocols and logspace-hard pseudorandom sequences (extended abstract). In Proc. of ACM Symp. on Theory of Computing (STOC), pages 1--11, 1989. Google Scholar
Digital Library
- M. Braverman, A. Rao, R. Raz, and A. Yehudayoff. Pseudorandom generators for regular branching programs. In Proc. of Symp. on Foundations of Computer Science (FOCS), pages 40--47, 2010. Google Scholar
Digital Library
- J. Brody and E. Verbin. The coin problem, and pseudorandomness for branching programs. In Proc. of the fifty first annual Symp. on Foundations of Computer Science (FOCS), pages 30--39, 2010. Google Scholar
Digital Library
- O. Gabber and Z. Galil. Explicit constructions of linear-sized superconcentrators. J. of Comp. and System Sci., 22(3):407--420, 1981.Google Scholar
Cross Ref
- R. Impagliazzo, N. Nisan, and A. Wigderson. Pseudorandomness for network algorithms. In Proc. of ACM Symp. on Theory of Computing (STOC), pages 356--364, 1994. Google Scholar
Digital Library
- L. Lovász. Combinatorial Problems and Exercises. Akadémiai Kiadó, Budapest, 1993.Google Scholar
- S. Lovett, O. Reingold, L. Trevisan, and S. P. Vadhan. Pseudorandom bit generators that fool modular sums. In APPROX-RANDOM, pages 615--630, 2009. Google Scholar
Digital Library
- R. Meka and D. Zuckerman. Small-bias spaces for group products. In APPROX-RANDOM, pages 658--672, 2009. Google Scholar
Digital Library
- N. Nisan. Pseudorandom generators for space-bounded computations. Combinatorica, 12(4):449--461, 1992.Google Scholar
Cross Ref
- N. Nisan. RL ⊆ SC. Computational Complexity, 4(1):1--11, 1994. Google Scholar
Digital Library
- J. Naor and M. Naor. Small-bias probability spaces: Efficient constructions and applications. SIAM J. on Computing, 22(4):838--856, 1993. Google Scholar
Digital Library
- N. Nisan and D. Zuckerman. Randomness is linear in space. J. of Comp. and System Sci., 52(1):43--52, 1996. Google Scholar
Digital Library
- R. Raz and O. Reingold. On recycling the randomness of states in space bounded computation. In Proc. of ACM Symp. on Theory of Computing (STOC), pages 159--168, 1999. Google Scholar
Digital Library
- E. Rozenman and S.P. Vadhan. Derandomized squaring of graphs. In APPROX-RANDOM, pages 436--447, 2005. Google Scholar
Digital Library
- O. Reingold, S. Vadhan, and A. Wigderson. Entropy waves, the zig-zag graph product, and new constant-degree expanders and extractors. In Annals of Mathematics, pages 157--187, 2000.Google Scholar
Cross Ref
- W.J. Savitch. Relationships between nondeterministic and deterministic tape complexities. J. of Comp. and System Sci., 4(2):177--192, 1970. Google Scholar
Digital Library
- M.E. Saks and S. Zhou. BP_HSpace(S) ⊆ DSPACE(3/2). J. of Comp. and System Sci., 58(2):376--403, 1999. Google Scholar
Digital Library
- J. S'ıma and S. Zák. A polynomial time construction of a hitting set for read-once branching programs of width 3. Technical Report TR10-088, Electronic Colloquium on Computational Complexity (ECCC), 2010.Google Scholar
Index Terms
Pseudorandom generators for group products: extended abstract
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