Abstract
Small-bias, or ε-biased, spaces have found many applications in complexity theory, coding theory, and derandomization. We generalize the notion of small-bias spaces to the setting of group products. Besides being natural, our extension captures some of the difficulties in constructing pseudorandom generators for constant-width branching programs - a longstanding open problem. We provide an efficient deterministic construction of small-bias spaces for solvable groups. Our construction exploits the fact that solvable groups have nontrivial normal subgroups that are abelian and builds on the construction of Azar et al. [AMN98] for abelian groups. For arbitrary finite groups, we give an efficient deterministic construction achieving constant bias. We also construct pseudorandom generators fooling linear functions mod p for primes p.
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Alon, N., Goldreich, O., Håstad, J., Peralta, R.: Simple construction of almost k-wise independent random variables. Random Struct. Algorithms 3(3), 289–304 (1992)
Ajtai, M., Komlós, J., Szemerédi, E.: Deterministic simulation in logspace. In: STOC, pp. 132–140 (1987)
Azar, Y., Motwani, R., Naor, J.: Approximating probability distributions using small sample spaces. Combinatorica 18(2), 151–171 (1998)
Barrington, D.A.M.: Bounded-width polynomial-size branching programs recognize exactly those languages in NC 1. In: STOC, pp. 1–5 (1986)
Bogdanov, A., Dvir, Z., Verbin, E., Yehudayoff, A.: Pseudorandomness for width 2 branching programs (manuscript, 2008)
Blum, M., Evans, W.S., Gemmell, P., Kannan, S., Naor, M.: Checking the correctness of memories. Algorithmica 12(2/3), 225–244 (1994)
Bogdanov, A., Viola, E.: Pseudorandom bits for polynomials. In: FOCS, pp. 41–51 (2007)
Herstein, I.: Topics in algebra. Wiley, Chichester (1975)
Impagliazzo, R., Nisan, N., Wigderson, A.: Pseudorandomness for network algorithms. In: STOC, pp. 356–364 (1994)
Lovett, S.: Unconditional pseudorandom generators for low degree polynomials. In: STOC, pp. 557–562 (2008)
Lovett, S., Reingold, O., Trevisan, L., Vadhan, S.: Pseudorandom bit generators that fool modular sums. In: RANDOM (2009)
Lund, C., Yannakakis, M.: On the hardness of approximating minimization problems. J. ACM 41(5), 960–981 (1994)
Nisan, N.: Pseudorandom generators for space-bounded computation. Combinatorica 12(4), 449–461 (1992)
Naor, J., Naor, M.: Small-bias probability spaces: Efficient constructions and applications. SIAM J. Comput. 22(4), 838–856 (1993)
Nisan, N., Zuckerman, D.: Randomness is linear in space. J. Comput. Syst. Sci. 52(1), 43–52 (1996)
Reingold, O.: Undirected connectivity in log-space. J. ACM 55(4) (2008)
Raz, R., Reingold, O.: On recycling the randomness of states in space bounded computation. In: STOC, pp. 159–168 (1999)
Reingold, O., Vadhan, S.: Personal communication (2006)
Saks, M., Zuckerman, D. (unpublished manuscript)
Viola, E.: The sum of d small-bias generators fools polynomials of degree d. In: IEEE Conference on Computational Complexity, pp. 124–127 (2008)
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Meka, R., Zuckerman, D. (2009). Small-Bias Spaces for Group Products. In: Dinur, I., Jansen, K., Naor, J., Rolim, J. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2009 2009. Lecture Notes in Computer Science, vol 5687. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03685-9_49
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DOI: https://doi.org/10.1007/978-3-642-03685-9_49
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